The Theory of Tonal Movement

A Unified Foundation for a Pedagogy That Produces Composers

The harmonic series is the acoustic ground. Movement — dissonance seeking consonance — is the central phenomenon. This volume sets out the theory; Volume 2 draws it through to a pedagogy. The pedagogy exists to produce composers.

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Preface

The theory's debts are explicit. To Hermann Helmholtz for the psychoacoustic foundation; to Paul Hindemith for reasserting, in the twentieth century, the relevance of the overtone series to a working theory of music; to Molly Gustin for the rigorous mathematical defense of the unified-ground position in Tonality (1969); and to Heinrich Schenker for the doctrine of hierarchical unfolding. Against: Moritz Hauptmann, Arthur von Oettingen, and Hugo Riemann, whose dualist program this theory reckons with in §2 and shows to be unnecessary. Pedagogical debts — to Johann Joseph Fux, Nadia Boulanger, and Padre Martini (and the Mozart-Attwood lessons as historical model) — are acknowledged in Volume 2, where they do their proper work.

A note on how the two volumes were built. This volume's theoretical apparatus was not constructed from first principles and then handed down to the pedagogy. The pedagogy came first; the theory is the description of what the pedagogy is already doing when a composer's training succeeds. A reader who wants the methodological frame for the whole project — why the Gradus method approaches aesthetics from the ground up rather than from a philosophical framework, why its convergence with Aquinas's three conditions of beauty was a discovery rather than a commitment, and what the project does and does not claim — should read Volume 2 §0.2 before or alongside this volume. The theoretical claims of Volume 1 can be evaluated on their own merits, but the project as a whole becomes intelligible only against that methodological frame.

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§1 · Introduction — The Thesis

§1.1 · One Paragraph

The harmonic dimension of tonal music is the audible structure of pulls and resolutions. Every event in the harmonic surface of a tonal composition — a passing tone, a chordal seventh, a suspension, a secondary dominant, a modulation — is an instance of one phenomenon: dissonance seeking consonance, instability seeking stability, distance from a perceived tonal center seeking return to it. The phenomenon is not stylistic convention. It is a consequence of two facts: that pitched sound contains its overtone series, which fixes a fundamental and ranks the partials around it; and that human listeners are unavoidably conditioned to small-integer ratios between low partials, which they hear in any approximation as the "true" interval. From these two facts follow the major triad's primacy, the status of the diminished triad as functionally dependent, the cross-cultural emergence of the pentatonic scale as the shared skeleton of pitched music, the historical convergence of Western tonal music on major and minor as its two primary scales, the universality of the V–I cadence, and the late-Romantic dissolution of tonality as a deliberate setting-aside of the hierarchy. From these facts also follow the theory's pedagogical consequence — the discipline of writing music that lives on the principle rather than describing it — developed in Volume 2.

§1.2 · Three Moves

The treatise will make three moves in order.

Move 1 — Critical Context (§2). The Leipzig dualist tradition (Hauptmann, Oettingen, Riemann) is reconstructed historically and shown to be unnecessary as a theoretical foundation: the undertone series Riemann hypothesized is superfluous when the harmonic series alone suffices, listeners hear the perceived tonic of a minor triad at its bass and not at its top, and the cross-cultural emergence of the pentatonic scale is direct evidence for one acoustic ground rather than two opposed ones. The unified-ground alternative — major as acoustic baseline, minor as modal modification of the parent diatonic collection — is proposed and defended.

Move 2 — Acoustic Ground and Foundational Claims (§§3–4). The harmonic series is established as the physical foundation; the near-universal pentatonic scale is presented as the cross-cultural empirical evidence; the stability hierarchy is derived from the lower partials and shown to be what listener perception tracks. The ten foundational claims are then stated formally, each with its empirical or perceptual ground.

Move 3 — The Theory of Movement (§5). Movement is established as the central phenomenon of the harmonic dimension, given an operational definition (graduated, falsifiable, anchored in listener perception), and developed through the catalog of local pulls that constitute the surface vocabulary of tonal music. The remainder of the volume (§§6–8) treats disambiguation vocabulary (the three lenses on movement function), modal and atonal practices as positions taken within and against the harmonic hierarchy, and a brief coda.

§1.3 · What This Theory Is Not

This theory is not theory for theory's sake. It is the theoretical foundation of the pedagogy set out in Volume 2, which is the means by which a student becomes a composer (see §0.4). Where a freestanding music theory would extend further, this theory does not.

The theory is not a theory of all music. Its scope is the harmonic dimension of music: pitch, interval, chord, scale, mode, tonal center, voice leading, and the listener's perception of pulls and resolutions in pitch-space. The theory has direct implications for orchestration, since every instrument has a particular relationship to the harmonic series that informs the tonal contribution it makes in a given musical context. The theory does not address rhythm or meter, which constitute their own theoretical domain (the literature on rhythmic structure, metric organization, and temporal perception is substantial and has its own apparatus). The theory does not address form or musical architecture (the literature on formal types — sonata, rondo, theme-and-variations, and the rest — has its own apparatus, see Caplin 1998 and Hepokoski–Darcy 2006 for contemporary treatments). The theory does not commit itself to any particular aesthetic position. Aesthetic principles — balance, proportion, contrast, unity, variety, and the rest — are real and important, and the present author holds substantive positions about them, but those positions do not belong inside this theory. The pedagogical application of aesthetic principles is treated separately in Volume 2. A reader who disagrees with the present author on rhythm, on form, or on aesthetics can still evaluate the harmonic claims of this theory on their own merits. That is the point of restricting the scope.

Within its scope the theory is also not a universal account of all harmonic practice. Its primary subject is Western tonal music as a coherent practice, broadly the repertoire running from late-medieval polyphony through the late Romantics, with extensions into impressionism, modal harmony, and the deliberate setting-aside of tonality in the twentieth century. Other musical traditions — Indian classical, Javanese gamelan, Arabic maqam, the many folk and ritual traditions of the world — are equally grounded in pitched sound and in the same harmonic series this theory takes as its physical foundation, but they have built different harmonic systems on that ground. The near-universal pentatonic scale (§3.3) is shared evidence that the harmonic series is the common ground; what gets built on it is culturally and historically variable. Other harmonic lenses for those traditions remain valid and welcome.

The theory is not a competitor to Schenkerian analysis at the level of analytic practice. It is a foundational layer beneath that practice. A Schenkerian could adopt this theory's acoustic and historical claims without changing how she draws a Bassbrechung. The theory is also not a prescription for what tonal music should sound like; it is a description of how the harmonic dimension of tonal music works. It is also not a refutation of atonal music. Atonality, on this account, is the deliberate setting-aside of the tonal hierarchy as one positive compositional choice among several — see §7.3.

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§2 · Critical Context

§2.1 · The Leipzig Dualist Tradition

Three figures, each building on the last, give the dualist program its nineteenth-century shape. The sequence is genuinely developmental: each of the three takes a step the previous one had not, and the cumulative result is the mature dualist system that came to dominate German-language theory pedagogy. The three figures' theoretical ambitions were not identical, and the contemporary critique benefits from keeping the steps distinct.

Moritz Hauptmann (1792–1868) was chamber-music director of the Leipzig Gewandhaus from 1842 and professor of theory and composition at the newly-founded Leipzig Conservatory from its opening in 1843 — the institution Mendelssohn had established the same year — until his death in 1868. Die Natur der Harmonik und der Metrik (Leipzig: Breitkopf und Härtel, 1853; English trans. W. E. Heathcote, The Nature of Harmony and Metre, London: Swan Sonnenschein, 1888) is the institutional statement of his theory and a direct importation of Hegel's dialectical method into music-theoretical writing. Hauptmann's three primary intervals — the octave, the fifth, and the major third — receive a linguistic gloss meant to capture their dialectical role: the octave is identity ("I am"), the fifth is otherness or separation ("you are"), the major third is return-in-otherness ("he is"). The major triad (Klang) is the concrete synthesis of these three relations: a single sonority that exhibits, simultaneously, identity, separation, and return. The minor triad is the same intervals reflected downward from the top tone, the dialectical antithesis. Tonality proceeds across phrases by the same pattern — thesis, counterstatement, synthesis — and Hauptmann reads sonata-form expositions as the large-scale realization of the same dialectical shape.

A clarification worth making at the outset: Hauptmann's downward reflection of the major triad to produce the minor triad is, in his own treatise, a conceptual operation, not a claim about a physical undertone series. He treats the inversion as a dialectical move required by the metaphysical role minor must play in the system, and he leaves the acoustic question to one side. The physicalization of dualism — the claim that an inverse spectrum actually exists in sounding pitched tones, mirror to the overtone series — was made later, by Oettingen, and the present treatise's rebuttal at §2.2.1 applies in full strength to the Oettingen-Riemann development of the program rather than to Hauptmann's own writings. Hauptmann's bias is metaphysical (dialectic imported as foundation); the empirical error about undertones is downstream.

Arthur Joachim von Oettingen (1836–1920) was a Baltic-German physicist at the University of Dorpat (now Tartu, Estonia), where he held the chair of physics from 1865 to 1893. The institutional setting matters: Oettingen brought a physicist's preference for mathematical symmetry to a music-theoretical problem, and his Harmoniesystem in dualer Entwicklung: Studien zur Theorie der Musik (Dorpat and Leipzig: W. Gläser, 1866) is the result. The treatise gives Hauptmann's dialectical dualism what Hauptmann himself had not attempted — a physical ground. Every major chord is phonisch bestimmt ("phonically determined"), with its Phonus (phonic root) at the bottom and the major triad arising as the first three distinct partials of the lowest tone's overtone series. Every minor chord is tonisch bestimmt ("tonically determined"), with its Tonus (tonic root) at the top and the minor triad arising as the inverted figure 1/4 : 1/5 : 1/6 of a hypothesized undertone series descending from the top tone.

The empirical support Oettingen offered for the undertone series was the phenomenon of combination tones (Tartini tones, difference tones) — known to physicists since Sorge's Anweisung zur Stimmung der Orgelwerke (1744) and re-described systematically by Helmholtz (1863). Two simultaneously sounding tones produce additional tones at frequencies equal to the sum and difference of the originals; the difference tone is audible under controlled conditions and sits below both sounding tones. Oettingen interpreted the difference-tone phenomenon as physical evidence that the auditory system "generates" tones below sounding tones, and from there generalized to a full undertone series mirror to the overtone series. The terminological innovation — phonisch vs tonisch — propagated forward to Riemann unchanged and is the source of the lowercase-vs-uppercase function notation (t, s, d vs T, S, D) still used in German theory pedagogy.

Hugo Riemann (1849–1919) systematized the dualism into a complete theory of harmonic function across a sequence of works whose relations are worth keeping in view. The technical foundation appears in Musikalische Logik (Leipzig: C. F. Kahnt, 1873) and is developed across Musikalische Syntaxis (Leipzig: Breitkopf & Härtel, 1877), Skizze einer neuen Methode der Harmonielehre (Breitkopf & Härtel, 1880), and Handbuch der Harmonielehre (Breitkopf & Härtel, 1887; multiple subsequent editions). Vereinfachte Harmonielehre, oder die Lehre von den tonalen Funktionen der Akkorde (London: Augener, 1893; English trans. H. Bewerunge, Harmony Simplified, Augener, 1895) is the pedagogical statement: a textbook synthesis aimed at conservatory teaching, condensing two decades of technical writing into a working classroom system. Anglophone readers most often encounter Riemann through this text.

The functions Tonic (T), Subdominant (S), and Dominant (D) hold mirror-image positions in major and minor; in the dualist notation inherited from Oettingen, minor functions are lowercase (t, s, d) and read downward. An A-minor triad is the "tonic" of the A-minor system in Oettingen's sense — its functional root sits on E (the top tone), with the chord generated as the figure 1/4 : 1/5 : 1/6 read downward from E. The C-major plagal cadence (IV → I) is, on this account, the mirror of the A-minor authentic cadence (v → t or, in the modern reading, V → i) read upside down: the same harmonic motion in opposite directions across the dualist axis. Riemann's Schritt and Wechsel operations (steps and exchanges) are likewise defined in mirror pairs.

The dualist program was the dominant theoretical paradigm in German-language conservatory teaching from the 1860s through the 1920s, and its institutional inertia outlasted its empirical plausibility. By 1900 the lack of any spectroscopic evidence for an undertone series mirror to the overtone series was widely acknowledged. Riemann's own late writings reflect the strain. In Ideen zu einer Lehre von den Tonvorstellungen (in the Jahrbuch der Musikbibliothek Peters for 1914–15) Riemann recast the dualist framework in psychological rather than acoustic terms: the Tonvorstellungen ("tone representations") of the listener, not the spectrum of sounding tones, are the proper subject of harmonic theory. The recasting acknowledged what Riemann's earlier work had denied — that listeners do hear the perceived tonic of A minor at the bass, not at the top — and attempted to preserve the dualist structure as a property of mental representation while conceding the acoustic ground. The result was an awkward halfway position the mature Riemann did not fully resolve before his death in 1919; the twentieth-century reception of Riemann is largely a reception of the 1893 Vereinfachte Harmonielehre rather than the 1914 Tonvorstellungen paper, and the dualist apparatus the conservatory tradition inherited is the earlier, acoustically committed version.

The influence on twentieth-century theory is substantial. It shaped the Schenker–Riemann polemic (§2.4 below); it shaped the way American theorists handle "parallel" and "relative" major-minor relations; it shaped the function-symbols (T, S, D, t, s, d) still used in German theory today; and through the function-symbols it shaped the analytic vocabulary of the Neo-Riemannian tradition (§7.2), which inherits Riemann's transformational apparatus while quietly setting aside the dualist metaphysics that originally motivated it. The contemporary working musician who writes a German Roman-numeral analysis with capital and lowercase function letters is, whether or not she knows it, working within the notational residue of a philosophical program that was already in retreat in its own author's lifetime.

§2.2 · Why Dualism Fails

Three independent grounds.

§2.2.1 · The undertone series is unnecessary when the overtone series suffices. The dualist appeal to a hypothesized undertone series — sinusoidal components at reciprocal integer multiples of a fundamental, mirror to the harmonic (overtone) series — was meant to provide a physical ground for treating minor as the dialectical inverse of major. The proposal originates with Oettingen (1866), is inherited by Riemann in Vereinfachte Harmonielehre (1893) and presented as the acoustic basis of his function theory, and is acknowledged as empirically problematic in Riemann's late Tonvorstellungen paper (1914–15). The argument below applies in full to the Oettingen-Riemann development of dualism; Hauptmann's earlier treatise treated the major-minor inversion as a dialectical rather than an acoustic operation and so did not himself rest the program on the undertone-series claim (§2.1). The harmonic series itself is a fact of physics: any periodic wave decomposes by Fourier analysis into sinusoidal components at integer multiples of the fundamental, and these components are present in the spectrum of every pitched sound. The undertone series, in the mirror-image form Riemann required, does not appear in any such spectrum. Psychoacoustic phenomena that bear formal resemblance to shadow-undertones (Tartini's combination tones, Terhardt's virtual pitch, the missing-fundamental effect) are real but weak, indirect, and require considerable theoretical interpretation to be heard at all.

The unified-ground position does not need to deny that undertone-like psychoacoustic phenomena exist somewhere in the perceptual landscape. It observes only that they are not necessary to explain tonal music. The overtone series alone — direct, robust, present in every pitched sound — explains the consonance hierarchy, the priority of the major triad, the universality of the perfect fifth, the structure of the dominant seventh, and the cross-cultural emergence of the pentatonic scale (§2.2.3). The dualist apparatus introduces additional structure (a hypothesized inverse spectrum, plus the mechanism by which the ear is supposed to "produce" it) that does not buy additional explanatory power over what the simpler account already delivers. By the standard scientific principle of preferring the simpler explanation that suffices for the data, the dualist apparatus is superfluous — and superfluous structure in a theory is a defect, not a virtue.

§2.2.2 · In the most-encountered voicings, listeners perceive the tonic of a minor triad at its bass, not at its top. Riemann's claim that the root of the A-minor triad sits on E (the top tone) is contradicted by the typical fact of listener perception: an A-minor triad in root position, with A in the bass, is heard as having A as its tonic. Ear-training and analysis pedagogy are organized around this perception; folk and popular practice take it for granted; performers who must choose between rooting the chord on its bass or its top note choose the bass.

The empirical record on minor-triad rooting is more textured than this simple statement suggests. Parncutt's virtual-pitch work (Harmony: A Psychoacoustical Approach, Springer, 1989, and subsequent papers) shows that minor triads in some inversions, registers, and timbres can root ambiguously or unstably, depending on voicing and context. The unified-position critique of dualism does not require that listener perception agree with the unified-position prediction in every voicing of every minor chord. It requires only that the prediction agree with what listeners report for the most-encountered voicings — the root- position triads, the close-position spacings, the registers in which the dualist tradition itself analyzed minor harmony — and on this, the record is unambiguous. The dualist apparatus is being asked to override the listener's perception in the canonical case, not in a marginal voicing. That override is what the unified-position critique rejects.

The simpler account is that the perceived tonic is the tonic, and any theory of minor must respect that perception rather than override it. The harmonic series is the physical foundation on which perception is built, but it does not get to overrule perception by formal derivation. Theories that attempt to are theories that have substituted procedure for phenomenon.

§2.2.3 · The pentatonic universality argues for one acoustic ground, not two. The five-tone pentatonic scale appears across radically different cultures with extraordinary consistency — Chinese ya yueh, Scottish and Irish folk traditions, sub-Saharan African vocal practice, Native American flute repertoires, Indonesian slendro (a five-tone inflection of the broader gamelan system, with intervals close to but not identical to the Western pentatonic), Andean huayno. The scale is derivable by stacking the second-strongest interval of the harmonic series, the perfect fifth (3:2, partial 3 over the fundamental): C–G–D–A–E octave-reduced gives the pentatonic. The fact that isolated cultures, working from different musical premises, independently converged on the same five-tone structure is direct empirical evidence that the harmonic series is the shared physical foundation of pitched music — one foundation, not a dialectical pair of opposed orderings.

The unified-position claim is specifically the claim about the foundation, not the claim that the West's particular harmonic system is universal. Different cultures build different systems on the shared foundation. The harmonic series gives them one set of strongest intervals; the scales, modes, ornaments, and harmonic practices each culture has developed are variations on what gets built on that ground. Indonesian slendro is pentatonic-but- different from the Western pentatonic. Indian classical music uses heptatonic rāga systems with characteristic micro-intervals. Arabic maqām deploys quarter-tones the Western tradition does not. None of these is "incorrect" by the unified-position account; they are different harmonic systems built on the same physical foundation. What the cross-cultural evidence shows is that the foundation is shared, not that the systems built on it are.

This is the unified-position claim in its honest form: one acoustic ground, many cultural realizations. The Leipzig dualist tradition required two acoustic grounds (overtone and undertone) to account for the difference between major and minor; the present theory rejects the second ground as unnecessary and explains the major-minor relation through modal modification of a single foundation (Claim 5). The unification being claimed in this section is between major and minor within the Western system, not across the world's harmonic traditions. The universality claim is about the physical foundation, not about any specific harmonic system built on it.

§2.3 · The Unified-Ground Alternative

The position taken in this treatise: there is one acoustic ground, the harmonic series; major is the directly derivable scale (its tonic triad is the first three distinct partials); minor is a modal modification of the parent diatonic collection (Aeolian with cadential adjustments, of which the raised seventh is the historically and acoustically dominant one); the remaining modes (Dorian, Phrygian, Lydian, Mixolydian, Locrian) are deliberate quietings of major's strongest pulls (Stage VI of the curriculum); atonality is the deliberate setting-aside of the hierarchy as a whole (Stages IX–X). All of these are positions taken by composers, not imposed by physics. But the physics fixes which positions exist to be taken.

The unified-ground position is in the lineage of Schenker (against Riemann), Hindemith (against the Schoenberg circle), and now Gustin (against the informal residue of dualism in mid-century theory pedagogy). It is the position of the Gradus method. The Schenker placement requires qualification — Schenker's anti-Riemann posture is unmistakable but the apparatus he built has features that are not straightforwardly unified-ground in the present treatise's sense — and the qualifications are the subject of §2.4 immediately below. The Hindemith placement is straightforward and is the subject of Volume 1's bibliography (§I); the Gustin placement is straightforward and is the subject of §3.4 and §4.5 above. Schenker is the harder case.

§2.4 · Schenker — Unified-Ground With Qualifications

Schenker's place in the unified-ground lineage is real but it is not the simple matter the previous paragraph might suggest. Schenker explicitly rejected the Riemannian Funktionstheorie and built his analytic apparatus on different foundations; that much is clear and the rejection puts him on the unified-ground side of the nineteenth-century debate. But the apparatus Schenker built has features that are not derived from the harmonic series in the way the present treatise's apparatus is, and his treatment of certain specific questions diverges from the position taken here in ways worth marking. This section sketches the relation in four parts: the anti-Riemann polemic, the Sechter lineage Schenker actually inherits from, the Naturhorn argument as a point of partial alignment, and the qualifications the present treatise enters against Schenker's mature apparatus.

§2.4.1 · The anti-Riemann polemic

Schenker's hostility to Riemann's function theory is documented across the mature work. Der Tonwille (Vienna: A. Gutmann, 1921–24, ten issues; English trans. ed. William Drabkin, Der Tonwille: Pamphlets in Witness of the Immutable Laws of Music, Oxford UP, 2004–05, 2 vols.) attacks the dualist functional reductions in passing comments scattered across the analytic essays. Das Meisterwerk in der Musik (Munich: Drei Masken, 1925–30, three volumes; English trans. ed. William Drabkin, The Masterwork in Music, Cambridge UP, 1994–97, 3 vols.) extends the polemic with more developed analytic counter-demonstrations. Der freie Satz (Vienna: Universal Edition, 1935; English trans. Ernst Oster, Free Composition, Longman, 1979) makes the rejection methodological: function theory operates at the wrong analytic level and misreads contrapuntal-prolongation phenomena as harmonic-functional phenomena. On Schenker's view, what a Riemannian theorist labels as a T → S → D → T progression is, in real tonal music, almost always a contrapuntal elaboration of a single underlying Stufe (scale-step chord), and the function-theoretic label is a kind of analytic nominalism that hides the structural motion the analyst should be trying to recover. The Schenker-Riemann debate was one of the most consequential disagreements in early-twentieth-century music theory and its terms shape Anglophone music theory pedagogy to this day, generally to Schenker's advantage in conservatories with Schenkerian inheritance and to Riemann's advantage in those with German inheritance.

The hostility to Riemann translates into hostility to the dualist apparatus that supports Riemann's theory. Schenker did not write a sustained refutation of the undertone series — he treated it as beneath refutation — but the apparatus he built has no use for it and no place in it. Major and minor in Schenkerian analysis are not mirror duals across a hypothesized inverse spectrum; they are distinct positions on the spectrum of structural possibilities, with different relations to the Naturklang (the natural harmonic-series sound) that grounds Schenker's theory of consonance. This is the sense in which Schenker is correctly placed in the unified-ground lineage: he rejects the dualist program at the foundation, not merely at the surface.

§2.4.2 · The Sechter lineage

Where does Schenker's apparatus come from, if not from Riemann? The answer is the Viennese conservatory tradition, specifically the Stufenlehre of Simon Sechter as transmitted via Anton Bruckner to the conservatory teaching that shaped Schenker's training. Sechter (1788–1867) was the dominant Vienna theorist of the mid-nineteenth century; his Die Grundsätze der musikalischen Komposition (Leipzig: Breitkopf & Härtel, 1853–54, three volumes) is the technical statement of the Stufenlehre, the theory in which the seven diatonic scale-degrees serve as the fundamental harmonic units of tonal motion. Sechter taught at the Vienna Conservatory and the Imperial Court Chapel until his death in 1867; Bruckner succeeded Sechter as Conservatory professor of harmony and counterpoint and continued teaching Sechter's method until Bruckner's own retirement in 1891. Schenker studied at the Vienna Conservatory from 1887 to 1890 — overlapping with Bruckner's tenure — and inherited the Sechter-Bruckner Stufenlehre as the working apparatus of his own Harmonielehre (Vienna: Universal Edition, 1906; English trans. Elisabeth Mann Borgese, ed. Oswald Jonas, Harmony, Chicago: University of Chicago Press, 1954).

The Stufenlehre is, in its origin, a non-dualist apparatus. Sechter predates Oettingen and is not affiliated with the Leipzig program; the scale-step theory inherits from the eighteenth-century fundamental-bass tradition (Rameau → Kirnberger → Albrechtsberger) by way of the early-nineteenth-century Viennese teaching of Beethoven's own theory teachers. The scale-steps are not mirror duals between major and minor; they are scale-degree positions that have particular harmonic functions in any given key, and the functions are described directly rather than as inversions of one another. Schenker's harmonic vocabulary inherits this directness. When Schenker writes about the V → I cadence in C major and the V → i cadence in C minor, he treats them as the same harmonic motion in two different tonal modes, not as mirror images across a dualist axis. The similarity to the present treatise's position is genuine: both treat the major-minor relation as a modal variation rather than a dialectical opposition. The difference, taken up in §2.4.4 below, is in how each accounts for why the modal variation has the acoustic-perceptual properties it has. (Wason 1985 traces the Vienna lineage in detail; the chapter on Sechter is the standard reference.)

§2.4.3 · The Naturhorn argument

Schenker's clearest acoustic claim appears in Harmonielehre's opening chapter, where he derives the priority of the major triad from the harmonic series. The argument is essentially Rameau's corps sonore in twentieth-century dress. Any pitched tone contains its overtones; the lower partials form the major triad; the major triad is therefore the natural sound (the Naturklang) of pitched sound itself; tonal music's privileged status of the major triad is acoustically grounded; major is "what Nature gives." On this much the unified-ground position and Schenker agree, and the agreement is not superficial — both derive major's priority from a common acoustic fact, both decline to derive a mirror dualism from a hypothesized inverse spectrum, and both treat the harmonic series as the foundation of consonance rather than as a contributing factor to be weighed against others.

The agreement extends further. Schenker treats the perfect fifth (partial 3) and the major third (partial 5) as the "natural" intervals the listener tracks; the present treatise's LCM gradient produces the same ranking. Schenker treats the dominant-tonic relation as the deep acoustic relation governing tonal motion; the gradient analysis produces the same conclusion from §3.4's LCM measure. Schenker treats the V–I cadence as the prototype of resolution; Claim 10 of the present treatise treats the PAC as the multi-layer collapse of several stability gradients onto a single arrival, of which the V–I relation is the dominant component. The Schenker-Naturhorn argument and the gradient analysis converge on a wide swath of harmonic practice, and the convergence is what makes the placement of Schenker in the unified-ground lineage substantive rather than nominal.

Schenker's account of minor, however, is where the agreement becomes complicated. In Harmonielehre §13 (the chapter on the minor mode), Schenker writes that minor is not given by Nature — there is no minor triad in the lower partials of any pitched tone, and the minor third (5:6) is a higher-LCM interval than the major third (4:5). Schenker concludes that minor is an "artistic" invention, a creative elaboration of major-key practice that has secured a working place in tonal music through historical convention rather than acoustic necessity. This is closer to the present treatise's modal-modification account (Claim 5) than to dualist mirroring, and Schenker is correct to refuse to treat minor as a parallel system. But Schenker's "artistic invention" account leaves the acoustic status of minor genuinely indeterminate in a way the present treatise does not. The modal-modification account, with Gustin's supporting result that Aeolian + raised 7̂ is the only mode-alteration that produces a second single-rooted set (§4.5), gives minor a positive acoustic grounding Schenker's account does not supply.

§2.4.4 · The Ursatz and the present treatise's three qualifications

The mature Schenkerian apparatus is set out in Der freie Satz (1935). At its center is the Ursatz — the fundamental structure of every tonal masterwork: a Bassbrechung of I → V → I in the bass voice supporting an Urlinie of stepwise descent (from 3̂ to 1̂, from 5̂ to 1̂, or in rare cases from 8̂ to 1̂) in the structural upper voice. The Ursatz is the deepest layer of the multi-level structural hierarchy Schenker calls Schichten (layers); the surface of any tonal masterwork is a prolongation — an elaborated traversal — of the Ursatz across the duration of the piece. Schenkerian analysis is the procedure for recovering the Ursatz from the surface and documenting the prolongations that mediate between them.

The analytic apparatus is genuinely productive. Schenker's analyses of Bach, Beethoven, Brahms, and Mozart yield real structural insights that Riemannian function-labeling, on its own, does not. The present treatise's §1.3 explicitly grants this — the treatise is not a competitor to Schenkerian analysis at the analytic-practice level, and a Schenkerian could adopt the present treatise's foundational claims without changing how she draws a Bassbrechung. The qualifications below concern the foundation of the Ursatz, not its analytic productivity.

Qualification 1: the Ursatz is presented as an analytic axiom, not derived, and the present treatise does not undertake to ground it perceptually. Schenker offers the Ursatz as the fundamental structure of tonal music but does not derive it from the harmonic series or from any other independent ground. His justification is functional: the Ursatz works analytically across the masterwork repertoire, the analyses it produces reveal structural facts the alternative apparatuses miss, and the Ursatz is therefore the underlying form of tonal music. The reasoning is circular at its base — the Ursatz is the underlying form because it produces good analyses; it produces good analyses because it is the underlying form — and Schenker's critics from Cone (1960) and Beach (1969) onward have pressed the circularity.

The natural temptation is to relieve the circularity by grounding the Ursatz in the gradient analysis — to read the Bassbrechung I → V → I as the tonal-gradient traversal at the scale of a whole movement and the Urlinie descent 3̂ → 2̂ → 1̂ as the half-step-and-cadential gradient operating at the same scale. The present treatise declines this temptation, and the reason is empirical rather than diplomatic.

The gradient analysis of §3.4 is grounded in listener perception of the harmonic-series ratios that the ear has been habituated to identify (§3.2). The mechanism is Tonvorstellung: the listener hears a sounding pitch as a position on the gradient relative to a perceived tonal center that the recent context has established. The mechanism is robust at the timescales the auditory short-term memory operates on — pitch-to-pitch, beat-to-beat, phrase-to-phrase. It is not robust at the timescale of whole movements. The empirical record on this is consistent across forty years of cognitive music-psychology research: Krumhansl and Kessler's (1982) probe-tone ratings show tonal-fit decaying rapidly when the establishing context recedes in time; Bigand and Pineau ("Global context effects on musical expectancy," Perception & Psychophysics 59, 1997) and follow-up studies show listeners often fail to identify the global tonic at the end of long pieces, and fail to notice when a section of a Bach prelude is played transposed to a different key from the opening; Bharucha's "tonal anchoring" model (Cognitive Psychology 16, 1984; Music Perception 5, 1987) frames perceived tonality as tracking the most-recent strong tonic rather than a persistent architectural one; Lerdahl's tonal-pitch-space distance (Tonal Pitch Space, Oxford UP, 2001) grows with elapsed time as the establishing context decays from working memory. The aggregate evidence is that perceived tonality shifts at the scale of phrases and short sections, and that what the analytic apparatus represents as a "prolonged structural dominant" the listener typically hears, in real time, as a new local home that the music has briefly moved to.

The treatise's commitment is to ground its claims in what listeners actually track. At the timescales the ear tracks, the gradient analysis is operative and the present account stands. At architectural timescales — the scale at which Schenker's Ursatz operates — the gradient analysis loses its perceptual anchor and the present treatise declines to extend it. The Ursatz is engaged as an analytical construct, useful for organizing the score on the page and useful as a composer's planning notation, but the treatise does not undertake to derive it from listener perception of the gradient because the listener does not perceive the gradient at that scale. Cone's circularity critique therefore stands as the treatise leaves it: the Ursatz's foundation remains a problem for Schenkerians to address on the analytical-construction side, and the present treatise does not offer to resolve it from the perceptual side.

The position has a consequence the next subsection makes explicit. Modulation in the present account is the local home shifting, not an excursion within a still-present larger home. Secondary dominants and tonicizations are brief local-home shifts. Sonata-form key-area motion is a sequence of local homes the listener moves through, with the recapitulation perceived as the establishment of a new (or returned) local home rather than as the collapse of an architectural dominant. These are positions the gradient analysis can speak to, because each operates at the scale at which tonal perception is robust. What the treatise declines to say is anything about what happens across those local-home transitions at movement-length scale, because that is the scale at which the empirical record does not support a perceptual claim.

Qualification 2: the account of minor differs. As noted in §2.4.3, Schenker's "artistic invention" account of minor is closer to the present treatise than to dualism but is less specific than the modal-modification account of Claim 5. The present treatise treats minor as Aeolian + raised 7̂, derives the cadential urgency of harmonic minor from the leading-tone partial-15 pull (§3.4), and supports the privilege of major and harmonic minor as the two operative two-key system with Gustin's single-rooted-set result (§4.5). Schenker's apparatus accommodates the same musical material but leaves the acoustic question for minor unresolved.

Qualification 3: the metaphysical and nationalist framing is set aside. Der freie Satz's Foreword and Introduction contain claims about the German musical genius, the divine election of the tonal masterworks (Bach through Brahms, with the German line privileged), and the Ursatz as the expression of a cosmic order that the Genie of the master composer alone can perceive. These claims are not load-bearing for Schenker's analytic apparatus — a Schenkerian analysis of a Beethoven sonata does not require believing that Beethoven was divinely elected or that the German musical tradition holds a metaphysical privilege over others — and the present treatise sets them aside. The treatise's scope is the harmonic dimension of music (§1.3) and not its metaphysical interpretation. (The treatise's own positions about beauty, design, and meaning are stated in Volume 2, where they belong, and stated as the author's positions rather than as derivable from the harmonic claims of Volume 1.) The qualification is not a dismissal of Schenker as a thinker — Schenker was a serious and original musical mind, and the analytic contribution is independent of the metaphysical framing — but a disciplined restriction of what the present treatise inherits and what it leaves to one side.

§2.4.5 · The convergence in Gustin's reading

A useful bridge between the Schenkerian hierarchical apparatus and the present treatise's gradient-grounded apparatus appears in Gustin's own approving gloss on Schenker, at the conclusion of her chapter on aesthetics (Gustin 1969, p. 82):

"The concept of unity which is achieved by means of the common measure is similar to Heinrich Schenker's idea of tonal unfolding. The tone which is unfolded is the tone whose frequency is the greatest common measure of the music which unfolds it."

Gustin's reading is not strict equivalence — Schenker's tonal unfolding (the Auskomponierung of a structural tone across the prolongational layers) is not identical to Gustin's common measure (the greatest-common-divisor frequency relation among the partials of a set's combined waveform) — but the kinship at the local level is real. Both describe, at the scale of phrases and short sections, the way a single tone organizes the harmonic material that surrounds it. Schenker arrives at the feature through analytic intuition validated by repertoire study; Gustin arrives at it through the mathematics of the LCM measure validated by listener perception. At the architectural scale the two part company more sharply than Gustin's irenic gloss admits: Schenker's apparatus claims an unfolding across the duration of a piece, while Gustin's common-measure apparatus is silent at that scale because the listener-perception ground on which her measure rests is itself silent at that scale. The present treatise takes Gustin's framing as the more secure of the two within its scope — at the timescales the ear tracks, the mathematical apparatus is independently checkable; the analytic intuition is harder to test from outside — and reads Schenker's local hierarchical analyses (the surface and middleground prolongations, voice-leading reductions within a single tonal area) as illuminating what the gradient describes. The two predecessors are kindred at the local level on which the present treatise rests its weight, and the analytic predictions they produce on phrase-length and short-section passages coincide substantially.

The summary of §2.4 is therefore the following. Schenker is engaged as a unified-ground predecessor with three explicit qualifications: the Ursatz is engaged as an analytical and notational construct, not adopted as a foundational claim about what the listener perceives (the gradient analysis does not extend to architectural scale, because perceived tonality decays at that scale); Schenker's account of minor is recognized as non-dualist but is not the modal-modification account this treatise adopts; and the metaphysical and nationalist framing of Der freie Satz is set aside. With those qualifications in place, Schenker is a predecessor whose analytic insights are acknowledged, whose anti-Riemannian critique the present treatise extends and grounds at the local scale where the gradient analysis operates, and whose architectural-scale claims the present treatise declines to inherit on grounds of what listener perception actually tracks.

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§3 · The Acoustic Ground

§3.1 · The Harmonic Series as Physical Fact

Any pitched sound is a periodic vibration, and any periodic vibration decomposes by Fourier analysis into a sum of sinusoidal components whose frequencies stand in integer ratios to the lowest, called the fundamental. This decomposition is not an artifact of how the ear hears the sound; it is a property of the sound wave itself, present whether or not anyone is listening. The component frequencies are called partials. The series of partials over a fundamental of frequency f is f, 2f, 3f, 4f, 5f, 6f, 7f, 8f, ... — the harmonic series.

The relationship between integer ratios and musical consonance is far older than the modern observation of the harmonic series, and it is worth keeping the two discoveries distinct.

The Pythagoreans, in the sixth century BCE, established that pairs of strings whose lengths stood in simple integer ratios produced consonant intervals — the octave at 2:1, the fifth at 3:2, the fourth at 4:3. This is the foundational empirical observation of the integer-ratio basis of consonance, and it underwrote two and a half millennia of subsequent music theory. What the Pythagoreans did not discover, because the apparatus to do so did not yet exist, is that a single pitched string also contains, in its own vibration, a coordinated set of higher pitches at those same integer ratios. The Pythagorean tradition worked with ratios between separate strings; the modern understanding of partials concerns ratios within a single sounding tone. The two are physically related but were observed separately and centuries apart.

The within-a-single-tone observation was first made in something close to its modern form by Joseph Sauveur in 1701, in his Système général des intervalles des sons presented to the Académie Royale des Sciences. Sauveur noticed that long, low-pitched string vibrations were accompanied by faint additional tones at higher pitches and concluded — correctly — that the string was vibrating in halves, thirds, quarters, and so on simultaneously with its fundamental motion. He named the higher tones sons harmoniques, the term that became the modern English "harmonics" and "harmonic series." The discovery completed what the Pythagoreans had begun: the integer ratios of consonance are not merely a relation imposed on separate sounding bodies; they are a relation present inside every single pitched tone.

The argument that this internal harmonic structure is the natural foundation of harmony — that the major triad is natural because it consists of the first three distinct partials of any pitched sound — was set out a generation after Sauveur by Jean-Philippe Rameau in Traité de l'harmonie réduite à ses principes naturels (Paris, 1722). Rameau named the harmonic series the corps sonore and made it the basis of his entire theory of harmony. The argument the present treatise inherits is essentially Rameau's, with refinements drawn from the subsequent literature. The psychoacoustic foundation — the account of how listeners actually hear pitched sounds as decomposing into partials — was set out by Hermann von Helmholtz in Die Lehre von den Tonempfindungen (1863, English translation as On the Sensations of Tone, 1875). The apparatus has been refined but not displaced by subsequent psychoacoustic research.

For a fundamental on C two octaves below middle C, the first sixteen partials produce the following pitch sequence (with cents deviation from equal temperament noted in parentheses for the partials whose ratios are not octaves of the fundamental):

Partial	Pitch	Ratio to fundamental	Cents deviation
1	C₂	1:1	(fundamental)
2	C₃	1:2	0
3	G₃	1:3	+2
4	C₄	1:4	0
5	E₄	1:5	−14
6	G₄	1:6	+2
7	B♭₄	1:7	−31
8	C₅	1:8	0
9	D₅	1:9	+4
10	E₅	1:10	−14
11	F♯₅	1:11	−49
12	G₅	1:12	+2
13	A♭₅	1:13	+41
14	B♭₅	1:14	−31
15	B₅	1:15	−12
16	C₆	1:16	0

The first three distinct pitch classes — C, G, and E — appear at partials 4, 5, and 6, and they form the major triad. This is the headline acoustic fact of the harmonic series: the major triad is not a cultural construction but the first three distinct pitch classes of any pitched sound's natural spectrum. A reader with a stringed instrument can verify this directly by playing harmonics; a reader with a piano can verify it indirectly by listening to which pitches a low fundamental seems to "contain."

One additional partial deserves immediate attention. Partial 7, in our example a B♭ over the C fundamental, is a pitch outside the major triad. Its ratio to the octave-displaced root is 7:4, roughly 31 cents flatter than the equal-tempered minor seventh1. The natural seventh is a real and prominent partial — often more prominent in the spectrum of brass and reed instruments than partials 9, 11, or 13 — and it introduces the next pitch class beyond the major triad. That pitch class is the chordal seventh of V⁷. The dominant-seventh chord, in other words, is what one obtains by reading partials 4, 5, 6, and 7 together: the major triad plus the natural seventh. The cross-cultural prominence of the dominant-seventh sonority — present everywhere from blues to ragtime to common-practice tonality — is partly accounted for by this acoustic fact, which §3.5 will develop further.

Partials 9 and 11 are the next boundary cases for chromatic extensions of tonal vocabulary. Partial 9 is the major second above the fundamental (ratio 9:8 to the octave above), the same interval that appears between G and A in C major. Partial 11 is sharper than the equal-tempered tritone — sometimes called the "natural eleventh" — and sits 49 cents flat of F♯₅ in our example. These partials are weaker than 4, 5, 6, and 7 but still present in the spectrum, and they contribute to the tonal vocabulary that emerges in the chromatic and impressionist repertoires. Beyond partial 16, the pitches become progressively more dissonant and the gaps between them shrink below what the ear can reliably distinguish. The lower partials — 1 through perhaps 8 or 10 — do most of the work of determining what the listener hears as the consonance hierarchy, because those are the partials the ear has been conditioned by experience to identify and round to. §3.2 takes up that conditioning.

Harmonic and inharmonic spectra

One clarification is in order before §3.4 develops the consonance gradient from the table above. The harmonic series as set out here — partials at integer multiples of a fundamental — is the spectrum produced by harmonic sound emitters: bowed and plucked strings, blown pipes and reeds, and the human voice. These instruments produce, to a useful approximation, sound waves whose partial frequencies stand in the integer ratios the table records. The consonance gradient of §3.4 is calibrated against this spectrum, and the cross-cultural pentatonic evidence of §3.3 is drawn from traditions whose primary instruments are in this family.

Other sound-producing bodies produce inharmonic partials — partial frequencies that do not stand in integer ratios to any common fundamental. Bells, tuned gongs, the bronze instruments of the Javanese gamelan, and many electronically-synthesized timbres fall in this category. William Sethares has worked out the generalization the inharmonic case requires: in "Local consonance and the relationship between timbre and scale" (Journal of the Acoustical Society of America 94, 1993) and Tuning, Timbre, Spectrum, Scale (Springer, 1997; 2nd ed. 2005), Sethares shows that the consonance curve must be computed from the partials of the sound producing it. For harmonic timbres his apparatus reduces to the integer-ratio gradient; for inharmonic timbres it predicts different local minima, and the scales those traditions have built — Javanese pélog and sléndro with their distinctive non-just intervals among them — are explicable on his account in a way the integer-ratio gradient alone is not.

The present treatise takes the harmonic-spectrum case as its privileged domain. Strings, pipes, and the voice are harmonic emitters by physics; the stretched string and the open pipe are the simplest pitched bodies a human culture can construct, and both produce harmonic spectra to a high degree of accuracy. The dominance of harmonic-spectrum instruments in pitched-music traditions worldwide is itself a consequence of the physics of pitched sound: human cultures preferentially build instruments that the voice can imitate and that imitate one another, and the voice is harmonic. Inharmonic-timbre traditions remain valid and welcome; they require Sethares' generalization, which the present theory does not pursue. The scope restriction is honest rather than apologetic: the theory describes what it describes, and the harmonic-spectrum family is wide enough to include the entire Western tonal tradition and the closely-related tonal traditions of Indian classical, Arabic maqām, and the near-universal vocal traditions of the world's folk music.

§3.2 · Habit and the Lower Partials

The harmonic series is a fact of physics, but the harmonic series alone does not produce the perception of tonal music. A second factor must be added: the listener's habit of rounding perceived intervals to the small-integer ratios formed by the lower partials. This habit is not innate to the human ear in any strong sense. It is the product of cumulative listening experience with pitched sound — for humans across virtually all cultures, that experience consists of musical and vocal sound whose lower partials follow the same series.

Gustin (1969, pp. 12–13) makes the argument as follows. The intervals between the lower partials of the harmonic series are the intervals that have appeared, in one form or another, in nearly every scale system humans have constructed. Any listener with normal hearing has been exposed to thousands of hours of pitched sound whose acoustic structure is governed by these intervals. The result is a perceptual conditioning: the listener does not consciously decide to round perceived intervals to small-integer ratios; the ear performs the rounding automatically, as a consequence of long exposure. The assumption requires, in Gustin's phrase, "no more dubious law of human nature than 'people are creatures of habit.'"

The strength of the rounding effect is striking. Zuckerkandl, in Sound and Symbol (1956, p. 79, cited in Gustin 1969 p. 9), reports the case of pitched sound emitted by performers on instruments without fixed pitch — solo string and wind playing, vocal performance — measured against idealized equal-tempered tuning. Deviations from the tempered ideal were sometimes as large as 80 cents, nearly a semitone. The audience, including experienced musicians, had not noticed. The ear had silently corrected the input toward the nearest paradigm interval. Whatever the oscilloscope recorded, the ear heard the music as in tune.

This has a direct consequence for the theory of harmony: the true ratio of a perceived interval is the ratio the ear hears, not the ratio physically present in the sound. And which ratio the ear hears depends on context. A minor seventh sounding as the chordal seventh of a V⁷ chord is heard as 7:8 — partial 7 over partial 8 — because the V⁷ chord is itself a recognizable gestalt mapping onto the lower partials of the harmonic series, and the ear corrects toward that mapping. The same minor seventh appearing as a stepwise descent in a diatonic melody (G–F in C major, say) is heard as 8:9 — the major second between partials 8 and 9, octave-displaced — because the diatonic context maps onto a different region of the spectrum. The intervals are physically the same; the ear hears them differently. As Gustin observes (1969, p. 18), "the dominant seventh chord is a gestalt which is in itself a representation of the overtone series; the seventh of the chord will be likely to be heard as the seventh partial of the overtone series. On the other hand, a major second occurring in a diatonic melody is more likely to be perceived as 8:9, its value in the diatonic scale."

This context-dependence of perceived ratio explains why the same physical intervallic content can read as resolution-demanding in one harmonic context and resolution-complete in another. It also explains why the small-integer ratios of the lower partials carry so much of the theoretical weight. The higher partials — 11, 13, 17, 19 — exist physically, but the ear has had less exposure to them in their pure form, and their integer ratios are correspondingly less stable as perceptual paradigms. The rounding effect drops off as one moves up the spectrum, and with it the ear's ability to identify higher-partial ratios as "the" interval being heard.

The harmonic series is therefore the acoustic ground of tonal music, but the listener's habit of mapping perceived intervals to lower-partial ratios is what makes the ground audible. Both facts together produce the tonal experience the rest of this volume describes.

§3.3 · The Pentatonic Universality

The cross-cultural appearance of the pentatonic scale is the single strongest piece of empirical evidence that the harmonic series is the shared physical ground of pitched music — not a Western convention masquerading as universal physics, and not an artifact of any particular pedagogical tradition.

The pentatonic scale appears in extraordinarily consistent form across cultures that had no historical contact with one another at the relevant period of musical development. Chinese ritual music since the Western Zhou dynasty (1046–771 BCE) used a five-tone collection equivalent to the pitches Western theory writes as C, D, E, G, A; the five tones were called gōng, shāng, jué, zhǐ, yǔ and were given cosmological associations, but the intervallic content of the collection itself is identical to the Western pentatonic. Scottish and Irish folk traditions are pervaded by the same five-tone collection, preserved in tunes whose surviving notation predates significant contact with continental theoretical practice. Sub-Saharan African vocal traditions — including the polyphonic singing of the Aka and Mbuti peoples of central Africa, documented by ethnomusicologists from the 1960s onward — frequently use pentatonic collections built on the same intervals. Native American flute repertoires across the continent favor pentatonic scales, often realized on five-hole flutes whose construction enforces the collection mechanically. Indonesian slendro is a five-tone inflection of the broader gamelan system, with intervals close to but not identical to the Western pentatonic — a cultural variation on the same structural pattern. Andean huayno music is overwhelmingly pentatonic. The pattern is not exhaustive; many traditions use scales that are non-pentatonic or that supplement the pentatonic with additional tones. But the recurrence of the pentatonic in approximately the same form across radically separated cultures is the empirical fact that requires explanation.

The pentatonic is derivable from the harmonic series by stacking the second-strongest interval, the perfect fifth, five times. The perfect fifth corresponds to partial 3 over partial 2 of the harmonic series, ratio 3:2 — the strongest interval after the octave. Five superimposed perfect fifths starting from C produce the sequence C–G–D–A–E. Octave-reduced into a single span, this is the pentatonic scale: C, D, E, G, A. The construction works because the perfect fifth is the most consonant non-octave interval available, and stacking five fifths is the smallest stack that produces a pitch-class collection short of the diatonic seven-tone set without leaving disjoined gaps in the octave.

What this evidence supports is not the claim that the pentatonic is the only "natural" scale. Many cultures use scales that are not pentatonic — heptatonic Indian raga systems, Arabic maqam with their characteristic micro-intervals, the seven-tone modes of medieval and Renaissance European practice. The cross-cultural prevalence of the pentatonic does not mean other cultures got the scale wrong; it means that when cultures begin from pitched sound and arrive at a five-tone collection, they tend to arrive at the same one. What it supports is the more limited and more defensible claim that the harmonic series is the shared physical foundation on which different cultures have built different scale systems. The pentatonic is the structurally simplest realization of that foundation; the divergent scales are different positions taken on the same physical ground.

Aside: Why the fifth-stacking principle stops where it does

The stacking-of-fifths construction of pentatonic and diatonic invites an evolutionary projection: if five fifths give the pentatonic and six give the diatonic, can the procedure be extended further to generate richer scales? Joseph Yasser, in A Theory of Evolving Tonality (American Library of Musicology, 1932), proposed exactly this. Yasser argued that the historical progression from pentatonic (five tones) through diatonic (seven) to chromatic (twelve) was an organic developmental sequence and that its natural next stage was a nineteen-tone supra-diatonic scale, which composers would eventually adopt as the chromatic vocabulary had been adopted in the common-practice era.

The proposal does not survive the principle that generates it. Gustin (1969, pp. 47–49) gives the diagnostic in three steps. First, the pentatonic and diatonic emerge from the Pythagorean process of stacking 2 : 3 fifths until a fifth-derived tone comes within a perceptible distance of an octave-derived tone, at which point the two are declared identical and the cycle closes. The cycle closes at twelve tones with the Pythagorean comma (23.46 cents) — small enough that the ear, given its habit of rounding intervals to small-integer paradigms (§3.2), tolerates the discrepancy. The procedure has a natural stopping point because the perception threshold has a natural stopping point. The nineteen-tone set Yasser proposes is not generated by this principle: its closure point falls farther from the octave than tones already produced earlier in the cycle, which means the new "smallest interval" is effectively a negative quantity relative to what the diatonic structure has already established. The construction is internally inconsistent with the procedure that produced the scales it claims to extend.

Second, the nineteen-tone equal-tempered scale does not improve the intervals of the diatonic vocabulary. Its perfect fifth is five cents farther from the just 2:3 ratio than the twelve-tone equal-tempered fifth; it improves the major third only at the cost of degrading the more acoustically primary fifth. Third, like any equal-tempered set the nineteen-tone scale has no root in the operational sense (the ratios between its tones are irrational and so admit no finite-integer representation), which means it has no acoustically privileged center the listener can take as home. Gustin's verdict: the nineteen-tone set is "an arbitrary division of the octave into nineteen equal parts bearing no more organic relationship to the diatonic and pentatonic sets than would the division of the octave into twenty, twenty-one, twenty-two, etc., equal parts." (Gustin 1969, p. 49.)

The diagnosis matters for the present treatise's argument because it explains why the construction stops where it does. The pentatonic and the diatonic are not the first two members of an indefinitely extensible series; they are the points at which the stacking of fifths converges with the octave to within the ear's tolerance for ratio-rounding. Beyond the diatonic the construction can continue into chromatic vocabulary by adding tones one at a time — and the late-Romantic vocabulary developed in exactly this way (§§4.6, 4.7) — but it cannot continue as an organic extension of the same generative principle, because the generative principle has already done its work at twelve. The chromatic vocabulary that emerges beyond the diatonic is an ornament-and-alteration vocabulary, not a new scale; and beyond twelve, the ear's discrimination threshold falls below the resolution the construction would require.

The deeper point: the harmonic series is finite in its perceptually relevant portion, the ear's rounding habits are finite in their tolerance, and these two finitenesses together fix the cardinality of the pitch vocabulary at twelve. Yasser's evolutionary projection mistook a structural endpoint for a developmental midpoint.

This matters for the argument of the present treatise in two ways. First, it grounds the unified-position claim of §2: there is one acoustic ground, which different cultures realize differently, not multiple acoustic grounds in dialectical opposition. The harmonic series produces consistent results across human cultures because it is a fact of physics, not because Western culture has imposed itself. Second, it provides the empirical floor for the harmonic-dimension scope of the theory (§1.3): the harmonic series is universal in its physical presence; what gets built on it is variable. The Theory of Tonal Movement describes one such build — Western tonal music — and does not claim to describe the others. But the foundation is shared.

§3.4 · The Stability Hierarchy

From the harmonic series and the listener's habituation to its lower partials there emerges a stability hierarchy: a perceptual ranking of pitches, intervals, chords, and tonal regions according to how stable they sound. The hierarchy is not a Western theoretical artifact or a pedagogical convention. It is what listeners actually track when they hear pitched sound organized around a center, in any tradition whose harmonic vocabulary is grounded in the lower partials.

A note on terminology

Before the apparatus is introduced, a clarification about the word consonance is in order. James Tenney's A History of Consonance and Dissonance (Excelsior, 1988) distinguishes five historically distinct concepts that the literature has often conflated: a pre-polyphonic melodic concept (consonance as the quality of a good melodic sequence); an early-polyphonic concept (consonance as the vertically permitted interval in organum-era practice); a contrapuntal concept (consonance as the property bound up with voice-leading treatment in figured-bass-period music); a functional concept (consonance and dissonance as properties of chords against a tonal context, the sense in which Rameau and his successors use the term); and a psychoacoustic concept (consonance as the relative absence of roughness, the sense Helmholtz inaugurates). The LCM measure developed in this section quantifies the functional and psychoacoustic concepts, which the unified-ground argument treats as two aspects of the same harmonic-series phenomenon. The contrapuntal concept — Claim 9's dual nature of dissonance, the dissonance-as-motion sense — is developed in §5.4 and §5.5 and is logically derived from the gradient. The earlier melodic and early-polyphonic concepts are historical positions the present treatise does not engage further. The reader encountering older literature in which these senses are entangled should be reassured that the present treatise has them disentangled.

Three families of consonance theory

The chronological narrative the next subsection presents flattens what is actually three theoretically distinct families competing in the consonance-theory literature. A reader trained in twentieth-century psychoacoustics will already have the taxonomy in hand; placing the present treatise within it makes the position the treatise occupies explicit rather than implicit.

Family 1 — Integer-ratio / periodicity. Pythagoras through Zarlino, Rameau, Euler, Hindemith, Gustin, and Stolzenburg. Consonance reduces to the simplicity of integer relations among the partials of the sounding tones, operationalized variously as the LCM of the just-intonation ratio, the prime-factor weight of that LCM, or the periodicity of the combined waveform. The Family-1 view treats the integer ratios as the primary phenomenon; psychoacoustic manifestations are downstream.

Family 2 — Sensory roughness. Helmholtz, Plomp and Levelt, and Sethares. Perceived dissonance reduces to roughness produced when partials beat against one another within the ear's critical band of frequency discrimination. The Family-2 view treats integer-ratio minima as a derivative effect of partial coincidence in harmonic timbres rather than as a primary phenomenon; Sethares' strong form generalizes the apparatus to inharmonic timbres, where integer-ratio minima dissolve and Family 1 fails to predict the consonance gradient the inharmonic case actually exhibits.

Family 3 — Harmonicity / template-matching. Terhardt ("Pitch, consonance, and harmony," Journal of the Acoustical Society of America 55, 1974), Parncutt (Harmony: A Psychoacoustical Approach, Springer, 1989), and Harrison and Pearce (2020). The ear searches the spectrum for a virtual fundamental whose harmonic series best explains the heard partials; consonance is harmonicity — the degree to which a chord's combined spectrum matches a single harmonic series. Integer-ratio minima and roughness are contributing factors, not the primary mechanism.

The present treatise's position. The treatise occupies a Family-1-plus-Family-3 hybrid space. The operational quantification of the gradient is Family 1 — the LCM measure of §3.4 is Euler's gradus suavitatis in its Stolzenburg simplification. The empirical vindication against Plomp-Levelt's reduction is Family 3 — Harrison and Pearce's 2020 meta-analysis finds harmonicity a stronger predictor of consonance ratings than roughness. Family 2 is absorbed in two ways. The Sethares scope-restriction of §3.1 treats Family 2's strong form, the timbre-dependent local-consonance curve, as governing inharmonic-timbre traditions outside the present treatise's scope, restricting the gradient theory to the harmonic-spectrum case where Families 1 and 3 converge with Family 2's harmonic-timbre predictions. The Plomp-Levelt reduction of integer ratios to derivative effects is the position the treatise sets aside on the basis of Harrison and Pearce 2020. The three families are not equally-weighted predecessors but a real disagreement about what consonance is at the physical and perceptual level; the hybrid position is the one consistent with the contemporary empirical evidence and with the harmonic-spectrum scope the treatise has set.

One further clarification before the apparatus is presented. Within the harmonic-spectrum scope, the three families predict the same gradient; what they disagree on is the mechanism by which the gradient arises in the listener. Family 1 (integer ratios, periodicity) treats the gradient as a structural property of the combined waveform: low-LCM ratios produce short-period waveforms whose unity the ear perceives as consonance. Family 2 (sensory roughness) treats the gradient as the consequence of partial coincidence in the critical band: low-LCM ratios produce coinciding partials that do not beat. Family 3 (template-matching, harmonicity) treats the gradient as the result of the ear's search for a virtual fundamental: low-LCM chords match a single harmonic-series template more closely than high-LCM chords. The three mechanisms produce the same predicted minima for harmonic-timbre sound, by construction. The disagreement is over which mechanism does the primary work in the listener's auditory system, not over what the gradient looks like in the harmonic-spectrum case. The apparatus that follows is agnostic between these mechanisms; the operational claim is that the listener tracks the gradient, that the harmonic series is what is being tracked, and that the LCM measure quantifies the gradient in a way the three families converge on within the present treatise's scope.

What kind of measure is the LCM?

Before the apparatus is set out, a methodological clarification — the most consequential one in this section.

The phenomenon the gradient describes is perceptual: it is what the trained listener feels when she hears a pitch against an established tonal center. Tonic at rest. Dominant close, stable, near home. The leading tone urgently close-and-unstable; the chromatic tritone far out, the listener at maximum distance from home. These are felt qualities, available to any listener with normal hearing and sufficient exposure to the tonal idiom.

The phenomenon is grounded in a physical-acoustic fact: the integer-ratio structure of the harmonic series, present in any pitched sound as a property of the sound wave itself. The listener's ear has been habituated, over years of exposure to pitched sound, to round perceived intervals to the small-integer ratios of the lower partials (§3.2). The acoustic ground and the perceptual gradient are linked by this habituation: the ear's rounded interpretation of what it hears tracks the integer-ratio structure of what is physically present.

The LCM measure is the theorist's instrument for quantifying the felt gradient against the physical ratios that produce it. The LCM of the integers in a pitch's simplest just-intonation ratio to the tonic is a number; the felt distance is a perceptual quality; the claim of the rest of this section is that the two correspond, reliably, in the harmonic-spectrum case. The listener does not compute LCMs. She hears the pitch and feels its position. The LCM is the theorist's explanation of why what she feels has the structure it does.

This distinction matters for what follows. The numerical apparatus that occupies the rest of §3.4 is a model of perception, not perception itself. The composer's working knowledge is the perception. The model exists for three purposes: it makes the gradient checkable against empirical listener data (the Krumhansl probe-tone results below); it makes the curriculum's pedagogical sequence derivable from a small set of acoustic facts (every later chapter returns to the gradient); and it makes the underlying claim falsifiable (an empirical measure of listener perception that disagreed substantially with the LCM ranking would be a problem for the theory). The composer who reads this section and now hears a leading tone as "the LCM-120 pitch wanting to resolve to the LCM-1 tonic" has missed the point. She should hear it as urgently close-and-unstable, demanding the half-step return — the felt quality the harmonic series produces by way of listener habituation. The LCM number is a label the theorist attaches to that perception, not the perception itself.

The pedagogical consequence, developed in Volume 2, is that the curriculum teaches the perception first and the model second. The student learns to hear the gradient before she learns to calculate where each pitch sits on it. Most students never need the calculation; the calculation is the theorist's check, not the composer's working tool.

A long mathematical lineage

The attempt to give the consonance gradient a precise mathematical form has a continuous history of more than two and a half millennia. The Pythagorean discovery (sixth century BCE) — that consonant intervals correspond to simple integer ratios of string lengths — is the foundational claim. Zarlino's Le istitutioni harmoniche (Venice, 1558) restricted the consonant intervals to those producible from the senario, the numbers 1 through 6, anticipating the harmonic series' first six partials. Mersenne's Harmonie universelle (Paris, 1636) made the first systematic measurements of overtones. Rameau, in Traité de l'harmonie (Paris, 1722), grounded the entire theory of harmony in what he called the corps sonore — the harmonic series of a single pitched tone — and argued explicitly that the major triad is natural because it consists of the first three distinct partials of any pitched sound. The argument the present treatise inherits from Rameau, with refinements, is essentially that one.

The first explicit mathematical formula for the consonance of a chord was given by Leonhard Euler in Tentamen Novae Theoriae Musicae (St. Petersburg, 1739). Euler proposed a gradus suavitatis — degree of sweetness — which assigned each chord a numerical value such that lower values corresponded to chords easier for the ear to perceive as unified. The formula depends on the least common multiple of the integers in the chord's lowest-terms ratio, plus an adjustment for how that LCM factors into primes. The values agree, in their relative ranking, with what listeners report and with what subsequent psychoacoustic research has measured.

After Euler, Helmholtz's On the Sensations of Tone (Leipzig, 1863) contributed the psychoacoustic foundation: the perception of dissonance arises in part from roughness produced when partial tones beat against one another within the ear's critical band of frequency discrimination. Plomp and Levelt's "Tonal Consonance and Critical Bandwidth" (Journal of the Acoustical Society of America 38, 1965) refined the model with experimental data on how listeners rate consonance under controlled conditions, and Plomp and Levelt took the position — against Euler's integer-ratio tradition — that the consonance minima emerge as a consequence of partial coincidence rather than as a primary phenomenon. Gustin's Tonality (Philosophical Library, 1969) synthesized the integer-ratio tradition into a unified quantitative apparatus and rebutted the Plomp-Levelt reduction. Stolzenburg's "Harmony perception by periodicity detection" (Journal of Mathematics and Music 9, 2015) gave the integer-ratio apparatus its contemporary form: Stolzenburg derives a periodicity-based consonance measure essentially equivalent to Euler's gradus suavitatis — log of the LCM of the integers in the just-intonation ratio, with adjustments for rational-approximation tolerance — and validates the measure against the empirical chord-rating data of Schwartz, Howe, and Purves (2003) and of Bowling, Purves, and Gill (2018). Stolzenburg's measure agrees with listener ratings on the rank ordering of dyads, triads, and tetrads at the level of accuracy the integer-ratio family predicts.

The status of the integer-ratio family relative to its psychoacoustic competitors has shifted in the most recent literature. Harrison and Pearce, in "Simultaneous consonance in music perception and composition" (Psychological Review 127, 2020), re-analyzed four large empirical studies of consonance rating (about 500 participants in aggregate) together with a 100,000-composition corpus and found that harmonicity — the degree to which a chord's combined spectrum matches a single harmonic series — is a stronger single predictor of consonance ratings than interference (roughness) is. The integer-ratio family is, on this measure, vindicated against the roughness-only alternative the Plomp-Levelt school had advanced: the partial- coincidence minima the older view treated as derivative are now treated as the primary phenomenon, with roughness as a contributing factor. A subsequent paper by Marjieh and colleagues ("Is harmonicity a misnomer for cultural familiarity in consonance preferences?", Nature Communications 13, 2022) argues that some portion of the harmonicity effect may reduce to cultural familiarity with harmonically-related sounds rather than to a hardwired auditory mechanism, and the disentangling remains incomplete. The present treatise does not require the harmonicity effect to be entirely hardwired: the cross-cultural pentatonic evidence of §3.3 and the physics of harmonic-spectrum sound emitters (§3.1) jointly establish that the harmonic series is the common physical ground of pitched- music traditions, whether listeners' tracking of it arises from acoustic mechanism, cultural learning, or some combination of the two. What the harmonicity-vs-cultural-familiarity debate decides is which mechanism produces the tracking; what the present treatise asserts is that the tracking happens and that the harmonic series is what is being tracked.

A note on mechanism. The Family-1 measure (LCM, periodicity) and the Family-3 mechanism (template-matching against the harmonic series) are conceptually distinct claims about how the listener tracks the gradient. Periodicity is an objective property of the combined waveform — the LCM of the partials' periods, a fact about the sound wave that exists whether or not anyone is listening. Template-matching is a cognitive mechanism — the ear searches the heard spectrum for the harmonic series of a single virtual fundamental that best explains the partials, and the brain assigns higher consonance to spectra that match better. The two converge on the same predictions for harmonic- timbre sound: a chord with low LCM is also a chord whose spectrum closely matches a single harmonic series, by construction. They diverge only for inharmonic timbres, where the periodicity measure fails and template-matching against an absent harmonic series degenerates to a different problem. For the harmonic-spectrum scope of the present treatise, the operational measure (LCM/periodicity) and the perceptual mechanism (template-matching) are equivalent in predictions, and the prose that follows uses the listener tracks the gradient without committing to one mechanism over the other. The treatise's claim is that the tracking happens, that the harmonic series is what is tracked, and that the gradient quantified by the LCM measure is what the tracking produces in the listener's perception of stability and instability.1

The treatise adopts Euler's formula in its essentials — a measure of how complex the combined waveform of a set of tones is, expressed as a function of the LCM of the integers in the set's lowest-terms ratio — and presents the gradient that results.

A simplified consonance measure

When two or more tones with frequencies in the ratios of small integers sound simultaneously, their combined waveform repeats over a period equal to the least common multiple of the periods of the individual partials, divided by their greatest common divisor. The shorter that period — the smaller the LCM/GCD ratio — the more the combined waveform feels to the ear like a unitary repeating pattern. The longer the period, the harder the pattern is to perceive as a unit, and the more the set sounds like an unrelated collection of pitches rather than a coherent chord. The shorthand for this quantity, in what follows, is the LCM measure. Lower LCM measure = more consonant; higher LCM measure = less consonant.

The values that result for the standard intervals and chords agree with the perceptual ranking that listeners report:

Set	Ratio	LCM measure
Unison	1:1	1
Octave	1:2	2
Perfect fifth	2:3	6
Perfect fourth	3:4	12
Major sixth	3:5	15
Major third	4:5	20
Minor third	5:6	30
Minor sixth	5:8	40
Major triad (close)	4:5:6	60
Minor seventh (in V⁷)	4:7	28
Dominant seventh	4:5:6:7	420
Major second (diatonic)	8:9	72
Major seventh	8:15	120
Minor seventh (diatonic)	9:16	144
Minor triad (close)	10:12:15	60
Augmented triad	16:20:25	400
Diminished triad	5:6:7	210
Minor second	15:16	240
Tritone (just)	5:7	35
Tritone (Pythagorean)	32:45	1440
Diminished seventh	125:150:180:216	27,000

Several observations are worth pausing on.

First, the major triad (60) is more consonant than the minor triad (60) — they have the same LCM measure in close position, which explains why they feel like kinds of the same chord rather than opposites. Their distinct emotional characters come from the order in which the major and minor thirds appear above the bass, not from a gross difference in spectral coherence. The major-third-on-the-bottom arrangement matches partials 4–5–6 of the harmonic series; the minor-third-on-the-bottom arrangement does not. That asymmetry — not some dialectical opposition — is what makes the major and minor triads emotionally distinct while remaining acoustically siblings.

Second, the dominant seventh chord (420) is dissonant relative to either triad but vastly more consonant than the diminished seventh (27,000). The dominant seventh sits at a particular point on the gradient: dissonant enough to demand resolution, consonant enough to retain a single perceptible identity through inversion and voicing. §3.5 takes this up directly.

Third, the minor seventh has two ratios in the table — 4:7 (LCM measure 28, when heard as the chordal seventh of V⁷) and 9:16 (LCM measure 144, when heard as a diatonic interval) — corresponding to the two context-dependent perceptions discussed in §3.2. The same physical interval reads as two different consonance values depending on the harmonic context the listener brings to it. The tritone shows the same dependence: heard as the just 5:7 ratio (LCM measure 35, fairly consonant) when supported by a partial-7 context; heard as the Pythagorean 32:45 (LCM measure 1440, much more dissonant) when read as the diminished fifth of common-practice harmony.

The twelve chromatic pitches and the simplicity of acoustic ratio

A natural first reading of the foregoing might be that higher partial number = more distant from home. This rule is approximately true but breaks down at specific points, and the breakdowns reveal a deeper principle that, once stated, subsumes the partial-number rule and explains the cases where it fails.

The deeper principle is that distance from home is determined by the simplicity of the just-intonation ratio between the tonic and the pitch in question:

Distance from home = LCM of the two integers in the simplest just-intonation ratio between the tonic and the pitch.

Lower LCM means closer to home; higher LCM means farther from home. The principle is essentially Euler's gradus suavitatis (1739) applied to the interval between any two pitch classes — a single measure that captures consonance regardless of how the listener locates the pitch in the tonic's harmonic structure.

Three routes lead to the simplest just ratio for a pitch X relative to a tonic C, and they correspond to the three ways the harmonic series produces small-integer relations between two pitch classes:

1. Direct. X appears as partial n of C; the just ratio is 1:n octave-reduced. This handles G (partial 3 → 2:3), E (partial 5 → 4:5), B♭ (partial 7 → 4:7), D (partial 9 → 8:9), B (partial 15 → 8:15).
2. Inverse. C appears as partial n of X; the just ratio is n:1 octave-reduced. This handles F: C is partial 3 of F, so the ratio is 3:4. This is the subdominant relation, recognized by Rameau (1722) as the inverse counterpart to the dominant.
3. Intervallic. X corresponds to the ratio a:b between two of C's own partials; the just ratio is a:b. This handles A (3:5, the interval between partials 3 and 5 of C), E♭ (5:6, between partials 5 and 6), A♭ (5:8, between partials 5 and 8).

The three routes produce the same just-intonation ratio for any given pitch. The distance is the LCM of that ratio regardless of which route was taken to find it. The reader does not need to track which route applies in a given case; the just ratio and its LCM are what matter.

Applied to all twelve chromatic pitch classes relative to C:

Pitch	Scale degree	Just-intonation ratio	LCM (distance)
C	1 (tonic)	1:1	1
G	5	2:3	6
F	4	3:4	12
A	6	3:5	15
E	3	4:5	20
B♭ (V⁷ context)	♭7	4:7	28
E♭	♭3	5:6	30
F♯ (just tritone)	♯4	5:7	35
A♭	♭6	5:8	40
D	2	8:9	72
B	7	8:15	120
B♭ (diatonic context)	♭7	9:16	144
D♭	♭2	15:16	240
C♯	♯1	16:17	272
F♯ (Pythagorean tritone)	♯4	32:45	1,440

The simple "high partial = more distant" rule fails in three categories of case that the unified principle handles correctly.

F (perfect fourth). F appears late as a partial of C (partial 21, 29 cents flat) but the F–C relationship via the inverse route gives the simple 3:4 ratio with LCM 12 — close to home, consistent with F's role as the third pillar of common-practice cadential harmony.

A (major sixth). A appears very late as a partial of C (partial 27) but the C–A relationship via the intervallic route is the simple 3:5 ratio between partials 3 and 5 of C — LCM 15, close to home and matching A's role as the relative-minor tonic.

B♭ and the tritone — context-dependent ratios. The same physical pitch can have different just-intonation ratios in different harmonic contexts, and the LCM measure tracks the contextual perception. B♭ heard as the chordal seventh of a V⁷ chord locks onto the 4:7 ratio (LCM 28, moderately close); the same B♭ heard as the diatonic minor seventh of a strict-diatonic passage rounds to the 9:16 ratio (LCM 144, much farther). The tritone has the same dual reading: the just 5:7 (LCM 35) when supported by the V⁷ chord's partial-7 context, the Pythagorean 32:45 (LCM 1,440) when read as the strict diatonic ♯4 between F natural and the leading tone. The dramatic difference in LCM tracks the dramatic difference in perceived dissonance.

What falls out of the table when the LCMs are sorted is the standard functional ranking of diatonic chord-roots. The chord-roots of the seven diatonic triads have these LCMs to the tonic:

Chord	Root	LCM to tonic
I	C	1
V	G	6
IV	F	12
vi	A	15
iii	E	20
ii	D	72
vii°	B	120

I, V, and IV emerge as the three closest chord-roots to the tonic by acoustic distance, with vi as the next in. This is the I-IV-V-vi backbone of common-practice cadential harmony, and the ranking is now an acoustic result rather than a conventional starting point. The framework is universal in tonal practice not because of historical convention but because these are the four chords whose roots stand at the smallest LCM-distance from the tonic. No other diatonic chord-root is closer than vi (LCM 15) by ratio simplicity.

The Krumhansl-Kessler probe-tone confirmation

The gradient's perceptual claim is independently testable, and was tested before the present apparatus existed. Carol Krumhansl and Edward Kessler's probe-tone experiments ("Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys," Psychological Review 89, 1982; further developed in Krumhansl's Cognitive Foundations of Musical Pitch, Oxford UP, 1990) established a quantitative tonal hierarchy. Listeners were played an establishing C-major context — a scale, a cadence pattern, or a tonic triad — followed by a probe tone (any one of the twelve chromatic pitch classes), and asked to rate how well the probe tone "fit" the established tonal context. The averaged ratings across many listeners produced a stable hierarchy that the subsequent literature has treated as the empirical benchmark for any theoretical account of perceived tonal fit.

The Krumhansl-Kessler ratings for C-major context, paired with the LCM measure derived above:

Pitch	Krumhansl-Kessler fit rating	LCM measure
C (1̂)	6.35	1
G (5̂)	5.19	6
E (3̂)	4.38	20
F (4̂)	4.09	12
A (6̂)	3.66	15
D (2̂)	3.48	72
B (7̂)	2.88	120
B♭ (♭7̂)	2.88	28 (V⁷) / 144 (diatonic)
F♯ (♯4̂)	2.52	35 (just) / 1,440 (Pythagorean)
A♭ (♭6̂)	2.39	40
E♭ (♭3̂)	2.33	30
C♯ (♯1̂)	2.23	272

The two columns agree on the structural shape of the gradient. The tonic ranks first by both measures. The dominant ranks second by both measures. The leading tone ranks last among diatonic positions by both measures (Krumhansl-Kessler 2.88 vs the chromatic alterations' 2.23–2.52; LCM 120 against the chromatic LCMs of 240–272 and beyond). All seven diatonic pitches outrank all five chromatic alterations on the probe-tone measure, and the LCM measure produces the same diatonic-versus-chromatic boundary. The correspondence between an empirical measure of listener perception, gathered independently of any acoustic-foundation argument, and the LCM measure derived from physical first principles is the convergence the unified-ground theory predicts. Purely-conventional accounts of tonal hierarchy — accounts that treat the gradient as a product of pedagogical tradition or cultural exposure rather than acoustic fact — do not predict this convergence and have to invoke the harmonic series as an additional unexplained coincidence whenever the probe-tone data is consulted.

Two points of divergence between the columns deserve discussion, because both reveal something about how the two measures relate.

The 3̂–4̂ inversion. Krumhansl-Kessler ranks E (3̂, rating 4.38) above F (4̂, rating 4.09); the LCM measure places F (LCM 12) closer to home than E (LCM 20). The inversion is small but real. Two factors plausibly contribute. The first is chord-function asymmetry: F's primary role in common-practice tonality is as the root of the IV chord — a function whose perceived weight is motion toward I rather than rest at I. The probe-tone task asks about fit with the established tonal center; listeners who have internalized F as "the chord that resolves" rate F's fit lower than its acoustic distance would predict, because they hear F as implying departure-and-return rather than as sitting at rest. The LCM measure captures acoustic distance; the probe-tone measure captures perceived fit-with-rest, and the two come apart on F because F's chord-function is asymmetric. The second factor is exposure frequency within the tonic: E is heard repeatedly as a constituent of every major-mode tonic triad sounded in the establishing context, while F is heard less often per unit of establishing context (only when IV, or a dominant-seventh preparation, or a passing tone, is in play). Repeated direct co-occurrence with the tonic plausibly elevates E's rated fit relative to F's. Both factors push the same way, and the small inversion is the joint product of acoustic distance running slightly behind perceived-function-and-exposure in this one place.

The leading tone's place at the bottom. B at rating 2.88 is the lowest-rated diatonic pitch — below the chromatic alterations' ratings of 2.88 (B♭) and well below the rest. By the LCM measure the leading tone is the most-distant diatonic pitch (LCM 120, with the chromatic alterations at 240+). Here the two measures agree: the leading tone is far from home, dramatically so, and the agreement is the kind of convergence the unified-ground theory predicts. The B♭ tie at 2.88 in the probe-tone data is interesting on its own — the listener rates the diatonic leading tone and the modal flat-seventh as equally poor fits with the C-major context, even though the LCM measure places them at very different distances (120 vs 28 in V⁷ context, 144 in diatonic context). The gradient and the perceived fit can diverge for individual pitches when chord-function and idiomatic exposure differ from raw acoustic distance. The gradient is the underlying structure; the probe-tone fit is a particular measure of how the listener relates the gradient to the task of judging tonal-context membership. The two correspond closely but not exactly, and the divergences are themselves informative.

The probe-tone data is the most direct empirical confirmation of the gradient available in the cognitive music-psychology literature. It is not the only confirmation: Bharucha's tonal-anchoring model ("Anchoring effects in music," Cognitive Psychology 16, 1984; "Music cognition and perceptual facilitation," Music Perception 5, 1987), Lerdahl's tonal-pitch-space distances (Tonal Pitch Space, Oxford UP, 2001), and Harrison and Pearce's 2020 consonance-rating meta-analysis all produce convergent results within their respective scopes. The unified-ground theory does not need to invent a new empirical record; it reads the existing record as confirming what the acoustic ground predicts.

The unified principle therefore strengthens the harmonic-series foundation. The same physical fact — small-integer ratios between pitch frequencies — produces consonance whether the relation is read as the pitch sitting in the tonic's spectrum, the tonic sitting in the pitch's spectrum, or the pitch corresponding to an interval between two of the tonic's own partials. The major triad (4:5:6) remains uniquely natural for a different reason: it is the only chord whose three constituent pitches are all direct low partials of a single fundamental, simultaneously. No other chord in the diatonic or chromatic vocabulary has that property. The unified principle of ratio simplicity describes the consonance gradient generally; the major triad's primacy is a special case of that gradient and remains intact.

The polar visualization

The simplest visualization of the consonance gradient places each pitch class at a chromatic angle around a circle (twelve positions spaced by half-steps from C) and sets the radial distance from the center to the LCM — or to its logarithm, since the LCM values span nearly four orders of magnitude across the chromatic collection. The tonic sits at the center; pitches close to home by ratio simplicity sit near the center; chromatic alterations sit at the rim. The plot looks something like a flower with petals of unequal length, the petals' lengths corresponding to acoustic distances.

Read radially out from the center, the plot shows:

• C at the center (LCM 1, the tonic itself);
• G very close in (LCM 6) and F close in on the opposite side (LCM 12) — the cadential I-IV-V neighbors visible as a near-symmetric pair flanking the tonic;
• A (LCM 15) and E (LCM 20) in a slightly more distant ring — chord-tones of vi and I;
• B♭ (just), E♭, F♯ (just), A♭ in a middle band (LCM 28-40);
• D further out (LCM 72), surprisingly distant for a diatonic pitch but consistent with its tense second-degree role;
• B near the rim (LCM 120) — the leading tone, far from home, pulling sharply back as the cadential leading-tone pull;
• D♭, C♯, F♯ (Pythagorean) at the outermost edge — chromatic alterations and the strict-diatonic tritone, maximum distance.

Two qualitative features fall out at a glance. First, the I-IV-V triangle of cadential roots (C, F, G) clusters near the center with vi (A) just outside. The functional weight of the I-IV-V-vi backbone is geometrically obvious: these are the four pitches closest to home. Second, the chromatic alterations form an outer annulus, with the leading tone B and the strict-diatonic tritone F♯ also pushed near the rim. A composer who wants to position the listener far from home can read radial distance directly from the plot — and a film composer scoring a sustained dread sequence is doing exactly this when she chooses, for instance, a tritone (rim) over a sustained ostinato motor for a minute or more before resolving inward.

A more sophisticated visualization is the Tonnetz — Euler's pitch grid arranged with perfect-fifth and major-third axes, adopted by Riemann and central to the Neo-Riemannian tradition (§7.2). For the purposes of this treatise the polar plot is the working diagram. The Tonnetz adds information (it shows triadic relations explicitly as triangles in the grid) at the cost of additional complexity. For readers who want the consonance gradient as a single picture, the polar plot is sufficient.

Aside: the perfect fourth's context-dependence

A famous puzzle in tonal theory is that the perfect fourth is treated as a consonance in some traditions (medieval polyphony, where it appears as a structural interval) and as a dissonance in others (Renaissance and common-practice harmony, where it requires resolution when it appears above a bass that is not the chord's root). The unified principle accounts for both treatments without contradiction.

The LCM measure is asymmetric: it measures distance from a chosen tonic, and the same physical interval can have different distances depending on which pitch the listener hears as the root. Worked through three cases of the C–F interval:

1. F as root (a perfect fifth above F, treated as the open fifth or the V-I motion in F major). The ratio is 2:3. LCM 6. Strongly consonant.
2. C as tonic, F as the inverse-relation subdominant (F as the root of the IV chord in C major). The ratio is 3:4. LCM 12. Consonant but at a slightly farther gradient position than the fifth. This is the standard tonal reading of the C-F interval and the acoustic ground of the IV chord.
3. C in the bass, F major chord above (the second-inversion triad, particularly the cadential I⁶⁴ → V → I). The bass note feels like it should be the root of the chord above it, since the bass is normally where the chord's root sits in the listener's perception. When the bass is a fifth below the actual root, the listener's expectation that bass = root is violated. The 4th between bass and upper voice carries this violation as audible instability; the chord wants to resolve to a position where the bass is the root.

The medieval-vs-Renaissance disagreement over the fourth's consonance is, on this account, a disagreement about which case is being analyzed. The medieval treatment of the 4th as a perfect consonance corresponds to case 1 (F as root). The Renaissance-and-common-practice treatment of the 4th as a dissonance corresponds to case 3 (bass-not-root). The unified principle handles both cases without contradiction once the asymmetry of the LCM measure is recognized: distance from home is always distance from a chosen tonic, and the listener's choice of implied tonic is itself context-dependent.

This is also the operational reason for the cadential six-four's characteristic instability. The chord is acoustically a tonic triad (C-E in upper voices, with the C as root) but harmonically a dominant-function event (the bass on G prepares the V). The mismatch between the implied root (C, from the upper voices) and the bass (G, the dominant root) produces the perception of the 4th as unstable, and the resolution to V→I clears the mismatch by moving the bass to its proper position.

Aside: George Russell and the Lydian Chromatic Concept

The unified principle of ratio simplicity above also addresses an alternative theory of tonal organization that has been substantially influential in jazz pedagogy since the middle of the twentieth century: George Russell's Lydian Chromatic Concept of Tonal Organization (1953, expanded 1959 and again in 2001). Russell argues that the Lydian mode — the scale obtained by raising the fourth scale degree of major (C–D–E–F♯–G–A–B) — is the primary scale of tonal music, more fundamental than the major scale (Ionian). His argument is itself a harmonic-series-style claim: stacking perfect fifths upward from C produces, in order, C–G–D–A–E–B–F♯, which is the Lydian scale. The major scale, by contrast, includes F natural, which is not reached by stacking fifths upward from C and which Russell therefore treats as an "outside" pitch — present in the major scale only by historical convention, not by acoustic necessity.

Russell drew significant pedagogical consequences from this claim. On his view, the I–IV–V framework is theoretically unsupported: F (the IV) is in tension with C, not in alliance with it. The natural cadential framework should instead be built on the I–V relationship and on chromatic excursions through the cycle of fifths. The Lydian Chromatic Scale — a twelve-tone ordering by intervallic distance from the tonic — became the framework's full vocabulary, and its influence on the modal jazz of Miles Davis (especially Kind of Blue), Bill Evans, John Coltrane, and Ornette Coleman is documented in the jazz-history literature. Russell's framework is, within jazz education, the most influential alternative to the major-scale-based account of tonality this volume sets out.

The argument is sophisticated and contains real insights, but it fails on the same point at which the simple partial-number rule fails. Russell counts only one route to ratio simplicity: pitches reached by stacking fifths upward from the tonic, equivalent to direct partials of the tonic. He does not count the inverse route — pitches whose own harmonic series contains the tonic as a low partial. F is not reached by stacking fifths upward from C, but C is reached by stacking fifths upward from F: C is partial 3 of F, so the F–C relationship is the simple 3:4 ratio with LCM 12. The ratio is as simple as the 2:3 ratio between C and G; the framework that excludes inverse relations is missing this.

Once the unified principle is admitted — that the simplest just-intonation ratio in any direction determines distance from home — F's consonance with C is entirely accounted for by the 3:4 ratio (LCM 12), and Russell's argument for Lydian primacy collapses. The major scale, with both direct (G as partial 3) and inverse (F as the fundamental of C-as-partial-3) low-LCM relations to the tonic, is the scale that integrates both directions of acoustic relation. The Lydian scale, with its raised fourth, removes the inverse relation and leaves only the direct one, which is precisely why Lydian sounds floating and unanchored to listeners trained on common-practice tonality: the cadential anchor of the subdominant has been deleted.

The diagnosis is not that Lydian is theoretically wrong as a mode. As a deliberate quieting of the strongest pull (the subdominant relation), it is a valuable color choice and a stable base for modal composition; the impressionist treatment of Lydian as a coloristic mode (Stage VIII of the curriculum) makes use of exactly this property. What is wrong is the claim of primacy — the claim that Lydian is more fundamental than Ionian. Once the harmonic series is read in both directions of relation, Ionian's primacy is restored on stronger acoustic grounds than Russell's framework provides for Lydian.

Russell's most important and lasting contribution to twentieth-century music theory is not the primacy claim but the framework of intervallic distance from the tonic — the recognition that pitches sit at different acoustic distances from the tonal center and that those distances can be calibrated to dramatic and harmonic effect. That insight is compatible with the present theory and is what the LCM table and polar plot above make precise. The disagreement is not over whether harmonic distance matters; Russell and the present theory agree it matters. The disagreement is over whether the major scale or the Lydian scale is the natural realization of those distances. The unified ratio-simplicity principle adjudicates: the major scale wins on completeness; Lydian remains a valuable specialized mode.

The hierarchy in scale degrees and chords

The same ranking applies to scale degrees within an established key. The tonic is most stable. The fifth scale degree, which sits at partial 3 of the tonic's harmonic series, is the next most stable — strong enough to anchor a half cadence, the dominant region, or a modulation. The third (partial 5) is stable but less so; it carries the modal identity of the chord (major or minor) but does not anchor a key on its own. The leading tone, partial 15 of the tonic's harmonic series, sits high in the spectrum and is correspondingly unstable; its closeness to the tonic on the gradient — one half-step below — produces the well-known cadential pull. The chordal seventh of V⁷ is partial 7 of the dominant's harmonic series and partial 21 of the tonic's; it is physically dissonant in both contexts, must resolve, and carries the second-strongest cadential pull after the leading tone.

The hierarchy extends to chords within a key. I is most stable; V is slightly less, but stable enough to anchor a half cadence; IV is less stable still, with a characteristic falling pull toward I; ii is restless, often functioning as a predominant that pulls toward V; vii° is the most unstable diatonic triad, rootless and functioning as an incomplete dominant. The same gradient appears in minor with the appropriate adjustments for the modal collection. And it extends to tonal regions: the home key most stable; the dominant region a stable secondary center; chromatic-third regions less stable than diatonic ones; remote keys progressively more unstable, with the tritone-related region the most remote on the cycle of fifths.

Extension to high-tension sets

The hierarchy does not stop at the diatonic vocabulary. The same gradient that ranks the major triad below the dominant seventh and the dominant seventh below the diminished seventh continues outward to chords that fall outside common-practice harmonic vocabulary altogether: polychords, quartal stacks, alpha and beta chords from Bartók's analytic vocabulary, tone clusters of two or more adjacent half-steps, and the broader vocabulary of twentieth-century atonal sets.

Each of these has a position on the consonance gradient determined by the same LCM measure. A polychord of C major over F♯ major — the Petrushka chord — combines six pitches whose lowest-terms ratio runs the LCM into the thousands; the listener's ear hears the result as a correspondingly higher position on the gradient than any diatonic dissonance, more unstable than the diminished seventh and without the diminished seventh's clear path of resolution. A four-note tone cluster on adjacent semitones (B–C–C♯–D) has LCM measure higher still. A six-note cluster spanning three semitones is at the practical limit of perceptible sets: beyond it, the ear ceases to track individual tones and hears the cluster as a single complex timbre.

Bartók's alpha chord (the set [0, 3, 6, 9] in pitch-class numbers, corresponding to a fully-diminished seventh structure with specific register) and his beta chord ([0, 3, 6, 8]) are intermediate cases: their LCM measures fall between the diminished seventh and the full cluster, and they carry the corresponding intermediate intensity. Schoenberg's quartal harmonies (stacked perfect fourths and tritones) sit at a different point on the gradient — moderately dissonant by LCM measure, but distinctively non-tertian, which the ear recognizes as a deliberate departure from the major-triad-grounded tradition.

These high-tension sets are not pathological. They are tools the composer reaches for when the musical situation calls for maximum instability without an immediate path of resolution. Late-Romantic and twentieth-century concert composers added them to the working vocabulary; twentieth-century atonal composers built whole works from them; and they have become routine working material for one specific audience that the present chapter has not yet named: the composers of film and television music.

The motor and sustained dissonance — practical implications

A film composer scoring an action sequence faces a particular challenge. The dramatic situation requires sustained forward motion across a passage that may run for a minute or more without arriving at the dramatic moment of resolution. A common-practice cadence would arrive too soon; the listener would feel the tension dissipate before the visual climax reached its point. The composer therefore needs sustained motion without harmonic resolution, and the high end of the consonance gradient is what supplies it.

The standard apparatus combines two factors. First, a motor in the inner parts: rhythmic subdivision (sixteenth notes, often, or eighth-note triplets, or driving ostinati) carrying a continuous active surface. The motor by itself produces the impression of motion without committing to any particular harmonic destination. Second, a high-LCM-measure harmonic content carried by the motor: tritone oscillations, sustained minor seconds, polychords held across many bars, atonal sets used as background ostinato. The combination is what the listener experiences as intensity. The motor says we are moving; the dissonant content says we have not arrived; the unresolved combination held across a long span says we are far from home and going somewhere we cannot yet see.

The technique exploits a specific consequence of the gradient. A listener who hears a tritone or a minor second in isolation recognizes the dissonance and expects resolution within a beat or two. The same listener who hears a tritone sustained across thirty seconds, supported by a sixteenth-note motor in the inner parts, experiences the ear's resolution-expectation as deliberately frustrated — and that frustration is exactly the emotional content the dramatic situation calls for. The technique does not require the composer to abandon tonal vocabulary; it requires only sustaining the high-tension sets long enough that the ear's expectation of resolution is itself overridden. The longer the dissonance is held, the more the listener feels suspended outside the home key, and the more emotional weight the eventual resolution carries when it finally arrives.

The same apparatus serves other dramatic affects. Sustained tritone oscillation in low strings under a sixteenth-note motor produces the characteristic affect of dread. A sustained polychord with a motor in pulsing sixteenths produces anxiety — the ear cannot settle on a single root and the rhythmic motor refuses to let the moment rest. A descending minor-second cluster held across many bars produces a sense of closing in. Each of these effects is the LCM-measure choice of the harmonic content combined with the rhythmic-motor choice of the surface activity, calibrated to the duration the dramatic situation demands. The composer's craft is the calibration.

This is one of the practical implications of the consonance gradient for working composers. The gradient is not a ranking from "good chords" to "bad chords." It is a description of where on the tension–resolution scale a given set sits, and the composer's craft includes choosing the gradient position that matches the dramatic moment. For composers of expressive instrumental music — film and television scores, but also concert music with strong programmatic content, opera, the dramatic Lied, and the cinematic style of late-Romantic symphonic writing — the gradient is the working vocabulary by which the listener's emotional pacing is managed. A theory of tonal movement that did not extend to the high-tension end of the gradient would be a theory unable to describe a great deal of the music actually being written today.

Scope reminder

The hierarchy operates within the harmonic dimension of music. It describes how chords and tonal regions relate to one another on a consonance/dissonance gradient. It does not extend to phrases, sections, or whole-work architecture; those domains have their own organizing principles and are addressed by other literatures (§1.3). The motor effect described above pairs harmonic-dimension dissonance with rhythmic-dimension subdivision, but the rhythmic side is the domain of rhythmic theory, not of this treatise. Within the harmonic dimension, the gradient is what tonal movement (Chapter 5) operates on: every pull the listener perceives is a movement from a less-stable to a more-stable position on this gradient, and every sustained dissonance is a deliberate withholding of the resolution the gradient predicts.

§3.5 · Why the Dominant Seventh Is Universal

The dominant seventh chord is the cadential workhorse of common-practice tonality. Bach, Mozart, Beethoven, Schubert, Brahms, and a hundred other composers reach for it tens of thousands of times across the surviving repertoire. Its dominance is not a stylistic preference. It is a consequence of how the harmonic series organizes a particular point of maximum cadential urgency.

Three properties account for the chord's primacy.

Acoustic continuity with the major triad

The dominant seventh chord, expressed in lowest-terms ratio, is 4 : 5 : 6 : 7 — the same ratio that governs partials 4, 5, 6, and 7 of the harmonic series. The chord is, in this sense, the natural extension of the major triad by the next available partial. In a V⁷ chord built on G in the key of C, the constituent tones G, B, D, and F are partials 4, 5, 6, and 7 of a G fundamental two octaves below the sounding G. The chord is acoustically continuous with the major triad it extends; reading it as a single set of partials of one fundamental gives the same ear that hears the major triad as a unit the means of hearing the dominant seventh as a unit, even though the chord is dissonant.

This was Euler's observation in Tentamen Novae Theoriae Musicae (1739), where Euler proposed that the dominant-seventh chord could be represented by the integers 4:5:6:7. Gustin (1969, Appendix C) endorses the representation. By the consonance measure introduced in §3.4, the chord has Property I = 4 (the same as the major triad), but Property II = 420 (much higher than the major triad's 60). The combined Total Consonance Coefficient is 820 — dissonant, but substantially less so than the diminished seventh chord's 17 900. The chord sits at a particular point on the gradient: dissonant enough to demand resolution, consonant enough to retain its identity.

Multiple converging pulls

Every tone of the V⁷ chord, in the key it serves as dominant, is unstable relative to the tonic chord. Each has a determinate path toward stability:

• 5̂ → 1̂ (root motion by descending fifth). The root of V is the fifth scale degree of the tonic key, partial 3 of the tonic's harmonic series. Its resolution to 1̂ traces the strongest interval of the harmonic series after the octave — the 3:2 fifth — but in reverse, the listener's ear returning down to the fundamental of the tonic harmony.
• 7̂ → 1̂ (leading-tone half-step pull). The third of V is the leading tone of the key, partial 15 of the tonic's harmonic series. Its position in the spectrum is high, its instability great, and the half-step distance to the tonic produces what the curriculum calls the strongest cadential pull in tonal music.
• 4̂ → 3̂ (chordal seventh descending by step). The seventh of V⁷ is the fourth scale degree of the key, partial 7 of the dominant's harmonic series. The natural seventh is physically dissonant, the first prominent partial outside the major triad itself, and it resolves down by step to the third of I — the chordal seventh moving from its position high in the dominant's harmonic series to a stable position low in the tonic's.
• 2̂ → 1̂ or 3̂ (the chord's fifth). The fifth of V is the second scale degree of the key, less unstable than the leading tone or the chordal seventh, but still wanting motion to either 1̂ or 3̂ depending on the voicing.

Four pulls. Four destinations. All converging on tones of the tonic chord. The PAC is what one obtains when these four pulls resolve simultaneously, on a single beat, in a single moment. §5.6 will develop this convergence as the prototype of multi-layer harmonic resolution.

A single perceptible root despite high dissonance

The V⁷ chord retains a single perceptible root through inversion. Listeners hear G as the root of a G dominant seventh chord, not B or D or F. The chord recognizes itself across all four inversions: G–B–D–F (root position), B–D–F–G (V₆₅), D–F–G–B (V₄₃), and F–G–B–D (V₂). A chord with stronger root identity can carry more dissonance without losing its identity, which is what makes the dominant seventh viable as a cadential chord in the first place. Gustin (1969, p. 35) notes this property in connection with another chord, the dominant minor-ninth: "Sets with a single strong root can be more dissonant without losing their integral identity than can rootless, weak-rooted, or multi-rooted sets." The V⁷ chord is the cleanest tonal-music case of this property, and the dominant minor-ninth is its more dissonant extension by partial 8 (becoming partial 9 of the chord's fundamental, the b9 above the chord's root).

Why this matters for the theory

The convergence of these three properties — acoustic continuity with the major triad, multiple converging pulls, single strong root — is what makes the perfect authentic cadence the most final resolution available in tonal music. Other chords supply some of these properties; only V⁷ supplies all three at once.

Other cadence chords are available, and tonal music uses them. The plagal cadence (IV → I) supplies a falling pull and a single root but no leading-tone half-step. The deceptive cadence (V⁷ → vi) supplies the leading-tone pull and the chordal-seventh pull but withholds the descending-fifth root motion. The Phrygian cadence (iv⁶ → V in minor) supplies a half-step pull from above (♭6̂ → 5̂) without the leading-tone pull from below. Each cadence type withholds one or more of V⁷'s pulls, and the resulting differences in finality are exactly what one would predict from the analysis above. The PAC is final because it withholds none. That is the operational reason for the chord's universality.

The dominant seventh chord is, on this account, not an arbitrary choice from the available chord vocabulary. It is the chord the harmonic series produces when one asks: what is the most cadentially urgent chord that retains a single perceptible root? The answer is 4:5:6:7. The tradition's adoption of it reflects its acoustic primacy, not the other way around.

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§4 · The Ten Foundational Claims

The previous chapters set out the acoustic ground (§3) and engaged the critical context (§2) within which the theory positions itself. This chapter consolidates the work into ten numbered claims that form the spine of the Theory of Tonal Movement. Each claim is given the same four-part treatment: the claim stated as a single sentence; the ground — the empirical or formal evidence that supports it; the consequence it has for the rest of the theory and for the practice of composition; and the curriculum location — the stage or step of the Gradus method at which the claim becomes pedagogically load-bearing.

The ten claims are not independent axioms. They are interlocking positions, each implying or implied by others, and the order in which they appear is broadly the order in which they are best understood: the harmonic-series foundation first, the central principle of movement second, the metaphor of distance third, the four scale-and- chord claims (major, minor, modes, atonality) next, the gradient itself in the harmonic dimension, the dual nature of dissonance, and the cadence as multi-layer resolution. A reader who has internalized all ten has internalized the theory. A reader who has internalized any one has begun.

§4.1 · Claim 1 — The Harmonic-Series Foundation

The claim. Tonality is grounded in the harmonic series as a physical fact, not in cultural convention. Any pitched sound contains its harmonic series as a property of the wave itself; the listener's habituation to the lower partials produces the consonance hierarchy on which all tonal music is built. The harmonic series is the acoustic floor under everything else this volume claims.

The ground. The physical observation has been established and refined for over three centuries: Sauveur (1701) for the first explicit identification of partials within a single sounding tone; Rameau (1722) for the argument that the harmonic series is the corps sonore on which harmony rests; Helmholtz (1863) for the psychoacoustic foundation; Plomp and Levelt (1965) for the modern critical-band model; and Gustin (1969) for the synthesis cited throughout §3. The cross-cultural evidence, particularly the near-universal pentatonic scale traced in §3.3, establishes that the harmonic series is the shared physical ground of pitched music across radically separated human cultures, not a Western convention. The mathematical apparatus for quantifying consonance from the harmonic series goes back to Pythagoras's discovery that simple integer ratios produce consonant intervals (sixth century BCE) and was given its first explicit formula by Euler (1739). The simplification adopted in §3.4 is the LCM measure: distance from home equals the simplicity of the just-intonation ratio between the tonic and the pitch in question.

The consequence. Three consequences follow. First, the major triad is uniquely natural because all three of its constituent pitches are direct low partials of one fundamental — see Claim 4. Second, the consonance gradient and the entire system of tonal stability and instability are anchored in physical fact rather than convention, and cannot be dismissed as a Western theoretical artifact. Third, the I-IV-V-vi framework that organizes common-practice cadential harmony is an acoustic result (the four chord-roots closest to the tonic by ratio simplicity) rather than a stylistic preference. The harmonic series produces the framework; tradition adopted it because it works.

Curriculum location. Stage I (Steps 1–7) introduces the harmonic series as physical fact, intervals as integer ratios between the lower partials, the major triad as the natural unit, and the listener's perception of the consonance gradient. The claim is the first thing the student is taught and remains the floor under every subsequent step.

§4.2 · Claim 2 — Movement Is the Central Phenomenon

The claim. Tonal music in its harmonic dimension is, in its central phenomenon, movement. Every tone, chord, and tonal region positioned at greater distance from the perceived tonal center carries an inherent directionality toward positions of lesser distance — a directed energy the listener perceives directly, before any analytical label is attached. Dissonance and consonance are not the mechanism of this movement; they are its coordinates. The LCM measure of §3.4 quantifies the gradient that movement traverses; movement itself is what listeners hear when they hear tonal music. The basic unit is the directed pull. The theory developed across §§4–7 is the theory of how pulls organize the harmonic dimension at the timescales the listener's auditory short-term memory operates on — pitch-to-pitch, beat-to-beat, phrase-to-phrase, and across the short sections in which a single tonal center remains the listener's perceived home. The treatise does not extend the pull-and-resolution account to architectural scale (whole-movement spans, sonata-form recapitulations as the resolution of a long-range structural dominant), because the empirical record on long-range tonal hearing does not support a perceptual claim at that scale (see §2.4.4). This is the central claim of the theory of tonal movement, and it is the claim that "Movement = Dissonance → Consonance" — true in its earlier formulation — under-stated by treating dissonance as the engine rather than as the gradient the engine traverses.

The ground. The harmonic-series stability hierarchy (§3.4) provides the gradient; the listener's perception of pulls and resolutions among positions on the gradient is the phenomenon the theory describes. The phenomenon is observable in any passage of tonal music: the leading tone wants to resolve up to the tonic; the chordal seventh wants to resolve down to the third; the dominant chord wants to resolve to the tonic chord; the secondary dominant wants to resolve to its tonicized chord; the modulation away from the home key wants to return. Each of these is an instance of the same underlying principle — a position on the stability gradient seeking a more stable position — and each is what the listener experiences as movement.

The consequence. Tonal music in its harmonic dimension is not, primarily, a structure to be analyzed; it is a movement to be heard. The composer's craft at the local scale is the management of pulls and resolutions: deciding which dissonances to create, when to resolve them, when to delay or frustrate the resolution, when to substitute one resolution for another. The student's task is to internalize this management as a working sensibility — not a rule-based discipline but an internalized hearing of where the music wants to go and what the composer can do to direct it. The pedagogical implications of this internalization are developed in Volume 2.

The claim is restricted to the harmonic dimension. Rhythmic movement, formal architecture, and the listener's emotional engagement with form are not derivable from the local pull-and- release; they have their own theoretical literatures (§1.3) and are not addressed here.

Curriculum location. The principle is encountered as soon as the student begins ear training in Stage I and becomes the working vocabulary of the entire curriculum. By Stage III (counterpoint), the student is producing dissonance-seeking-resolution by writing rather than recognizing it by listening. The claim is what the curriculum is for.

§4.3 · Claim 3 — Distance from Home

The claim. Dissonance is distance from home; consonance is arrival at home. The tonic is home. Movement is the journey of departure from and return to home. Distance is operationalized by the LCM measure of §3.4: lower LCM is closer to home, higher LCM is farther from home.

The ground. The metaphor is grounded in the listener's actual perception. A pitched-sound listener tracks where the music is relative to a tonal center, in a way that can be elicited by the simplest experiment: ask any listener what pitch the music "wants" to land on, and the answer in any tonal context is the tonic. The "wanting" is the perceptual experience of distance from the tonic and the corresponding pull toward it. The harmonic-series-grounded LCM measure quantifies this perception: pitches at low LCM (G at 6, F at 12, A at 15) feel close to home; pitches at high LCM (B at 120, the chromatic alterations at 240+) feel far from home; pitches at moderate LCM feel in the middle distance. The polar visualization of §3.4 makes this geometric: the plot's distance from the center is, literally and acoustically, distance from home.

The consequence. The metaphor is more than a metaphor; it is a description of what listeners actually perceive. Composers can place the listener at known distances from home and move the listener through known distances by selection of pitch, chord, and tonal region. The film-music application set out in §3.4 is one consequence: a composer who wants to position the listener far from home can read the radial distance directly from the polar plot and choose accordingly. The composer's craft at every level of the harmonic dimension is the management of distance.

The metaphor extends to chord-roots and tonal regions as well as pitches. The chord-root distance table of §3.4 places I at LCM 1, V at 6, IV at 12, vi at 15, and the rest progressively further. The modulatory distance from a home key to a related key is similarly calibrated by the relationship between their tonics on the gradient. A modulation to the dominant region travels a short distance; a modulation to a chromatic-third region travels further; a modulation to a remote key travels furthest.

Curriculum location. Stage I introduces the metaphor of "home" through ear training and the simplest contrast (tonic vs. dominant). Stage III makes distance operative in counterpoint (the cantus firmus as home, the counterpoint as movement around it). Stage IV introduces phrase-level distance through cadence types. Stage VII (chromatic harmony) and Stage VIII (modal color) explore the far reaches of the gradient. The metaphor is the working vocabulary throughout.

§4.4 · Claim 4 — Major as Natural

The claim. The major triad is special among triads because all three of its constituent pitches — root, third, and fifth — are direct low partials of a single fundamental. The constituents 4:5:6 correspond to partials 4, 5, and 6 of any pitched tone whose fundamental sits two octaves below the major triad's root. The major triad is, in this sense, the most direct realization of the harmonic series' lower partials as a chord. No other triad in the diatonic or chromatic vocabulary has this property: the minor triad's constituents are distributed differently across the series (10:12:15 or 6:7:9 depending on representation), and the diminished and augmented triads have no comparable partial-grouping at all. The major triad's claim to "natural" status is not that it is the only consonant configuration but that it is the only configuration in which all three constituent pitches are direct low partials of one fundamental, simultaneously.

The ground. This is a fact of physics, first observed in something close to its modern form by Sauveur (1701) and given its explicit theoretical role by Rameau in Traité de l'harmonie (1722). The argument has been refined since — by Helmholtz, Plomp and Levelt, Gustin — but the central observation has not been displaced. When one strikes a low note on a piano or bows a long open string, the spectrum of the resulting sound contains, in order of decreasing prominence, the major triad as its first three distinct pitch classes. The dominant scale degree is partial 3 of the tonic; the third scale degree is partial 5; the cadential leading tone is partial 15; the chordal seventh of V⁷ is partial 7 of the dominant's own series.

The claim is not that "higher partial number = more dissonant" as a universal rule. §3.4 sets out the careful refinement: the harmonic series predicts consonance in three configurations — direct relation (the pitch appears as a low partial of the tonic), inverse relation (the tonic appears as a low partial of the pitch — this is the subdominant case, F as 4:3), and intervallic relation (the pitch corresponds to a low-partial ratio between the tonic's own partials — A as the 5:3 major sixth between partials 5 and 3). The simple partial-number rule tracks only the direct relation. With the full three-configuration apparatus, the apparent anomalies of F (partial 21 yet very consonant) and A (partial 27 yet stable) are explained without weakening the central claim about the major triad. Major remains uniquely natural because it is the only triad that satisfies the direct relation for all three of its constituents at once.

The consequence. Several consequences follow. First, the major triad is the structural unit from which tonal music is built, in any tradition grounded in the lower partials (§3.3 established that this includes most pitched-music traditions worldwide). Second, the major triad's primacy does not entail the symmetric mirror of the dualist tradition (§2): the minor triad is not derived from an inverse harmonic series; it is a modal modification of the diatonic collection, with its own derivation (§4.5) that does not require any inversion of the major triad's structure. Third, the universal V–I cadence is the harmonic-series prediction working out — V's root is partial 3 of the tonic, V's third is partial 15, V's seventh (when V⁷ is used) is partial 7 of V's own series, and the resolution traces these high partials back to low ones, the strongest direct-relation motion the harmonic series predicts. Fourth, the universality of the I–IV–V framework follows from the two-directional refinement: those are the three chords whose roots stand in low-partial relations to the tonic, V via direct relation and IV via inverse relation. No other diatonic chord has this property, and no other framework appears with comparable cross-cultural consistency in tonal practice.

Curriculum location. Stage I (Steps 1–7) introduces the harmonic series, the major triad, and the natural priority of the major scale. Stage II introduces minor as a modal modification rather than as a parallel system (Claim 5). The present claim is the acoustic floor under both stages.

§4.5 · Claim 5 — Minor as Modified Mode

The claim. Minor is Aeolian — a mode of the parent diatonic collection — plus historical cadential adjustments, of which the raised seventh (giving harmonic minor) is the historically dominant. Minor is not the dialectical inverse of major (the Leipzig dualist position critiqued in §2); it is a modal modification of the diatonic collection that introduces a half-step pull from below the tonic and thereby creates a functional dominant that pure Aeolian lacks.

The ground. The historical record supports the derivation. Late-medieval polyphony used the modal collection (Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian) without distinguishing major and minor as parallel systems. The cadential need for a half-step pull from below the tonic was met by the gradual practice of musica ficta — the unwritten chromatic alteration of the seventh scale degree by performers, raising it to produce the leading-tone half-step. By the seventeenth century the practice had crystallized as harmonic minor: Aeolian with raised 7̂. The acoustic ground for the preference is set out in §3.4: the raised seventh as the leading tone is partial 15 of the tonic's harmonic series, the strongest cadential pull the series predicts at LCM 120 (the major seventh as 8:15) which then resolves to the tonic across the half-step.

The dualist alternative (Hauptmann's Klang, Riemann's mirror system, addressed in §2) treats minor as the inverted dual of major, derived from a hypothesized undertone series. The unified-position critique of §2 shows the dualist apparatus to be unnecessary: there is no physical undertone series, listeners hear the perceived tonic of a minor triad at its bass and not at its top, and the historical record shows minor emerging from modal modification rather than from dialectical opposition. The modal-modification account is simpler and matches the historical record.

A converging acoustic-mathematical argument is supplied by Gustin (1969, pp. 50–53). Gustin tests every diatonic mode for what happens when the principal note is given a leading tone — a chromatic alteration adding a half-step pull from below. Applied to all seven modes the procedure yields seven altered sets: Dorian + C♯, Phrygian + D♯, Lydian + E (already present in Lydian by definition), Mixolydian + F♯, Aeolian + G♯, Locrian + A♯, and Ionian + B (already present in Ionian). The question Gustin asks is which of these altered sets retain the single predominant root property — the operational mark of a set whose tones are organized around one perceptible tonal center rather than several equivocal ones.

The answer is that only two of the seven altered sets are single-rooted: the unaltered Ionian (whose leading tone is already in the diatonic collection) and Aeolian + G♯ — that is, harmonic minor with the raised seventh on the supertonic of the parent C major, giving A as the root of the resulting set. As Gustin writes in the chapter's italicized headline (p. 52):

"The alteration of the fifth scale step of the diatonic key to provide a leading tone for the sixth scale step changes the set from major to harmonic minor. This is the only set derived in a similar manner which has a single predominant root. This explains the importance of the major and minor keys."

This is the formal mathematical reason the major / harmonic-minor two-key system dominates the common-practice repertoire. The historical narrative — musica ficta, the gradual raising of the seventh in modal cadence-formulas, the crystallization of harmonic minor by the seventeenth century — describes the process by which composers found the system the acoustic-mathematical property would have predicted in advance. The other five modes remain available and useful as deliberate quietings of cadential pull (Claim 6), but they do not produce a second fully-cadential tonal system the way major and harmonic minor do, because they are not single-rooted under the leading-tone alteration. The Locrian case is the most extreme: Locrian's tonic triad is diminished and rootless to begin with, and even the leading-tone alteration cannot make the set single-rooted (Gustin 1969, p. 51). This is the operational reason Locrian sat out three centuries of Western practice — the strongest means of tonal emphasis (intrinsic emphasis via the root property) is unavailable to it.

The consequence. Minor is not a parallel system to major; it is a derived case. The two-key system of common-practice tonality — major and harmonic minor as the two operative scales — emerges from the diatonic collection plus the one cadential adjustment that produces a functional dominant. The other modal alterations (Dorian, Phrygian, Lydian, Mixolydian) are available and composers use them when they want the corresponding color, but they do not produce the same cadential urgency that major and harmonic minor share. Modal music is treated separately in Claim 6.

This is the claim that makes the theory unified rather than dualistic in its acoustic ground. Major sits at a special position on the gradient (its tonic triad is the first three distinct partials of one fundamental, Claim 4); minor sits at a derived position (the modal-modification produced by historical cadential practice). One acoustic ground produces both, asymmetrically.

Curriculum location. Stage II introduces minor as Aeolian-with-raised-7̂ explicitly, after the student has internalized major in Stage I. The historical development of musica ficta into harmonic minor is taught as the gradual emergence of cadential urgency from the modal collection. Stage VI (early Baroque) makes the major/minor system the working vocabulary. Stage VIII returns to the modes as a distinct color choice (Claim 6).

§4.6 · Claim 6 — Modal Harmony Quiets the Strongest Pulls

The claim. The diatonic modes (Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian) are not failed or weaker versions of major. They are deliberate choices to relax the cadential urgency that major and harmonic minor produce, in exchange for distinctive harmonic colors. Each mode quiets one or more of major's strongest pulls in a specific way.

The ground. Each mode can be specified as major plus a specific alteration (or minus the leading tone), and each alteration corresponds to the deletion or weakening of a specific pull established in §3.4 and §3.5:

• Mixolydian replaces 7̂ (leading tone, partial 15 of the tonic) with ♭7̂ (partial 7). The strongest cadential pull (7̂ → 1̂) is removed; what remains is a softer modal seventh that does not push toward the tonic with the half-step urgency a leading tone supplies.
• Lydian raises 4̂ to ♯4̂. The subdominant relation (F as the inverse-relation pitch with C as partial 3 of F) is removed, replaced by a pitch that lacks the strong inverse-low-LCM relation to the tonic. Lydian sounds floating because the cadential anchor of the subdominant has been deleted — see the Russell critique in §3.4.
• Phrygian lowers 2̂ to ♭2̂. The half-step pull is now from above the tonic (♭2̂ → 1̂) rather than from below, producing a characteristically dark modal cadence.
• Dorian combines minor third (♭3̂) with raised sixth (relative to natural minor). The result is a minor-mode collection with a brighter cadential motion than Aeolian, but without the strong leading-tone pull of harmonic minor.
• Aeolian is the unaltered minor scale (natural minor) without the raised seventh. The strongest cadential pull is absent; what remains is the softer modal cadence (♭7̂ → 1̂).
• Locrian has both ♭2̂ and ♭5̂. The tonic triad is diminished and rootless. Cadential urgency is barely supported; Locrian is the modal extreme, deliberately quiet, used rarely in tonal practice for that reason.

The consequence. Modal music is a deliberate choice within the unified-ground theory rather than a parallel system to be defended on its own terms. Composers reach for modes when they want the color the mode supplies — Mixolydian's softness, Lydian's float, Phrygian's darkness — and the cost is the cadential urgency that major and harmonic minor provide. The choice is real, the trade-off is real, and the resulting music is no less tonal than major-mode music; it is tonal music in a deliberately relaxed cadential frame.

Curriculum location. Stage VIII (impressionism and modal color) introduces the modes systematically as available palettes for compositional choice, after the student has fully internalized major and minor. Earlier stages may use modal materials in passing — the Phrygian cadence in Renaissance polyphony at Stage III, for instance — but the modes-as-palette treatment is at Stage VIII.

§4.7 · Claim 7 — Atonal Music Sets Aside the Tonal Hierarchy Entirely

The claim. Atonal music is the deliberate setting-aside of the tonal stability hierarchy as a positive compositional choice. Without a tonal center, the consonance/dissonance gradient flattens — there is no "home" and no "distance from home" in the sense the theory has developed. Movement does not vanish; it is created by other means (rhythmic variation, registral motion, dynamic contour, timbral change, motivic transformation). Atonality is a different practice from tonality, not an inferior one.

The ground. The historical record is clear. Schoenberg, Webern, Berg, and the broader twentieth-century atonal tradition deliberately abandoned the tonal hierarchy in favor of organizing structures (twelve-tone serialism, integral serialism, set-class theory) that do not depend on a stability gradient. Schoenberg's writings in the 1920s and 1930s argue explicitly for the positive intentions of dodecaphony: the equal treatment of all twelve pitches as a deliberate aesthetic choice, not a failure to find tonal cadences.

The acoustic ground for atonality's distinctive sound is set out in §3.4. High-tension sets — polychords, tone clusters, alpha and beta chords — sit far up the consonance gradient, often well above the diminished seventh chord. Atonal music typically operates exclusively in this region, sustaining high-LCM configurations without resolution. The listener's expectation of resolution to a tonal center is overridden by the consistent absence of any tonal center; what the listener tracks instead is the local relationships within the high-tension vocabulary — rhythmic patterning, registral motion, motivic transformation.

A converging mathematical argument is supplied by Gustin (1969, pp. 45–46). The diatonic set, by the result reported in Claim 5 above, has the single predominant root property: there is a unique tone in the set whose harmonic series the other six tones map onto with the smallest combined LCM, and that tone is the perceived tonal center. The twelve-tone set has no such property. Gustin considers the candidate twelve-tone sets in turn — the equal-tempered set whose intervals are irrational and therefore admit no finite-integer representation at all; the Pythagorean twelve-tone set whose lowest-terms ratios reach 2¹⁷ in places, beyond any plausible threshold of perceptual coherence; and the several candidate just twelve-tone sets, each of which requires arbitrary tuning decisions about which pitch derivations to prefer when multiple just paths to the same chromatic pitch class disagree. The result that holds across all three candidates is that a close approximation to a major triad can be built on every one of the twelve tones, with the consequence that each tone is oriented to every other the same way and no single tone predominates. The set is, in Gustin's phrase, centerless. (Gustin 1969, p. 46.)

The mathematical claim is therefore independent of the composer's intentions. Whether the composer is Schoenberg writing the Variations for Orchestra, Op. 31 with serial intent or a late-Liszt experiment in fully chromatic harmony, the set the music draws on does not have the property the diatonic set has, and the listener cannot recover a tonal center by listening for one because there is no privileged center to be found in the set itself. Atonality, on this account, is the formal name for music whose pitch-vocabulary set lacks the single-predominant-root property. Composers can position pieces within atonality by constraint (a tone row), by avoidance (free atonality), or by gradient choice (sustained high-LCM sets) — and the choice between these is genuine craft within the atonal practice — but the choice between atonality and tonality is the choice of which kind of set the piece will draw on. The two practices are formally distinct at the level of the pitch-vocabulary's mathematical structure, not merely at the level of stylistic preference.

The consequence. The Theory of Tonal Movement does not explain atonal music; it does not claim to. Atonal music has its own theoretical literatures (set-class theory, transformational theory, analytic apparatus for serialism) that handle its structures with the appropriate vocabulary. The treatment in this volume is descriptive: atonal music is what tonal music looks like when the hierarchy is deliberately set aside.

The film-music observation in §3.4 (sustained high-tension sets used as background ostinato) is a hybrid case where atonal vocabulary is deployed within a larger tonal frame to manage emotional pacing. Many late-Romantic and twentieth-century composers (Mahler, Stravinsky in his neoclassical phase, Bartók) work in similar hybrid spaces, deploying atonal vocabulary within tonal frames or vice versa. The unified theory accommodates these hybrids by recognizing that the gradient extends to high-tension sets and that composers can position their listeners anywhere on the gradient at any time.

The claim does not rank tonality above atonality. The theory describes tonal music; atonal music is a different practice with its own merits. A composer is free to write either, or both, and the present volume claims no priority.

Curriculum location. Stage IX (twentieth-century music) introduces atonality as a compositional practice and gives the student exposure to its working vocabulary. The treatment is descriptive rather than prescriptive; the student learns to recognize and write atonal passages without being asked to abandon tonal practice.

§4.8 · Claim 8 — The Stability Hierarchy in the Harmonic Dimension

The claim. Within the harmonic dimension, every pitch, chord, and tonal region has a position on a stability gradient grounded in the harmonic series. The tonic is most stable; pitches, chords, and regions farther from the tonic by ratio simplicity (§3.4) are progressively more unstable. The composer manages tension and rest by navigating the gradient.

The ground. §3.4 sets out the gradient in detail. For pitches relative to a tonic C: C at LCM 1, G at LCM 6, F at LCM 12, A at LCM 15, E at LCM 20, and so on outward to the chromatic alterations at LCM 240 and beyond, with the strict-diatonic tritone at LCM 1,440. For chord-roots: I at LCM 1, V at 6, IV at 12, vi at 15, iii at 20, ii at 72, vii° at 120 — the I-IV-V-vi backbone of common-practice cadential harmony emerging as the four closest chord-roots. For tonal regions: the home key most stable, the dominant region a stable secondary center, chromatic-third regions less stable than diatonic ones, remote keys progressively unstable.

The gradient is perceptual: it is what listeners actually track when they hear pitched sound organized around a center. The LCM measure is the operational quantification, but the underlying phenomenon is the listener's perception of distance from home.

The consequence. Every compositional decision in the harmonic dimension is a placement on the gradient. A chord choice is a placement; a modulation is a placement; a non-chord tone is a placement; a chromatic alteration is a placement. The composer who has internalized the gradient can manage the listener's experience of tension and rest by selecting where on the gradient to be at each moment. The student's task in Stage VI and beyond is to develop the judgment that places the right gradient-position at the right moment — a craft that takes years to internalize and that no analytical apparatus can substitute for.

The hierarchy operates within the harmonic dimension only. Rhythmic stability, formal stability, and the listener's emotional engagement with form are not derivable from this gradient and have their own theoretical literatures (§1.3).

Curriculum location. The gradient is introduced implicitly in Stage I (intervals) and explicitly in Stage IV (cadence types and chord function). It becomes the working apparatus of Stage VI onward (figured bass, harmonic standards, voice leading). Every step from Stage IV through Stage X uses the gradient as the framework for its compositional decisions.

§4.9 · Claim 9 — The Dual Gesture of Movement

The claim. Movement at the local scale is a dual gesture: every traversal of the gradient simultaneously creates and releases directed tension. Pull and release are the same gesture, not two events. The passing tone is the cleanest case: it exists for the duration of one weak beat at a position of momentary instability, and the very motion through it is the resolution. The dissonance- state and the resolution-event are inseparable in the listener's perception because, on the present account, the dissonance is a motion, not a stationary tension awaiting release. This is the operational consequence of Claim 2: if movement is the central phenomenon and dissonance is the gradient coordinate that movement traverses, then every moment a tone occupies a high-LCM position is a moment of motion through that position rather than residence at it. The two earlier formulations of this claim — "dissonance has dual nature" and "pull and release are the same gesture" — are both true, and both follow from the prior recognition that the dissonance under analysis is not a state but a directed event.

The ground. The phenomenon is observable in any tonal passage. Consider C major with a passing tone D in the line C-D-E. The D is dissonant against an underlying C major chord (8:9 against the tonic, LCM 72). It is also moving forward — the D is not a destination but a transit between C and E, and the listener hears it as such. The dissonance creates the forward motion; the forward motion is the resolution. There is no moment at which the D sits as an unresolved tension; the dissonance and its resolution are co-occurring.

The phenomenon scales across configurations of the same gesture. The suspension is the slowed-down case: the dissonance is held long enough that creation and release become two distinct phases — preparation, sounding-as-dissonance, resolution by step. The chordal seventh of V⁷ is the embedded case: the dissonance is part of the chord identity itself, and its eventual resolution to the third of I is the destination the chord has been pointing toward all along. In each case the dissonance and the resolution are co-implicated; they are the same gesture at different time scales.

The consequence. This is the claim that makes movement, rather than the static structure of chords, the central phenomenon of tonal music. Dissonance is not a state to be measured at one moment and resolved at the next; it is an event whose creation and release are inseparable in time. Stepwise motion is the foundation of melody for this reason — every step through a dissonance simultaneously creates and releases tension, and the line moves forward by the very act of being dissonant.

The pedagogical consequence is that students learn dissonance not by labeling it but by writing it. The species counterpoint progression of Stage III is precisely the cultivation of the ear that hears dissonance as moving: 1st species establishes consonance only; 2nd species introduces the passing tone (the cleanest dual-nature case); 3rd species elaborates with neighbor and passing figures; 4th species slows the dissonance into the suspension; 5th species integrates all five. By the end of Stage III, the student hears dissonance as a kind of motion rather than a kind of state.

Curriculum location. The principle is implicit from Stage I but becomes explicit in Stage III (counterpoint), where the species progression makes the dual nature operational. Stage IV (figured bass and harmonic standards) extends the principle to chordal dissonance. Stage VI onward applies it to chromatic and extended-tonal vocabulary.

§4.10 · Claim 10 — The Cadence as Multi-Layer Resolution

The claim. A perfect authentic cadence (V⁷ → I) resolves several independent harmonic-dimension stability gradients simultaneously, on a single point of arrival. The PAC's distinctive finality comes from the convergence of these independent resolutions onto the same moment. Other cadence types withhold one or more of these resolutions, and the resulting differences in finality are predictable from which resolutions are withheld.

The ground. §3.5 sets out the V⁷ chord's structure: four pulls converging on tones of the tonic chord. The PAC is the moment when those four pulls all resolve simultaneously:

• The tonal gradient: V → I, root motion by descending fifth, the strongest interval of the harmonic series after the octave. The motion traces 5̂ → 1̂.
• The half-step gradient: 7̂ → 1̂, the leading-tone pull from partial 15 of the tonic to the tonic itself.
• The vertical gradient: 4̂ → 3̂, the chordal seventh of V⁷ resolving down to the third of I — partial 7 of the dominant moving to partial 5 of the tonic.
• The root-position-resolution gradient: both V and I sound in root position with the tonic in the bass at the moment of arrival, providing maximum acoustic clarity.

All four resolve onto the same beat, on the same chord, in the same moment. The listener hears this convergence as the most complete resolution available in tonal music's harmonic dimension.

Other cadence types withhold specific resolutions:

• Half cadence (X → V): the tonal gradient is not resolved; V is the destination, not the tonic. Other gradients resolve partially.
• Plagal cadence (IV → I): the leading-tone pull and chordal- seventh resolution are absent (no leading tone in IV, no chordal seventh in the IV-I motion). Tonal gradient resolves but with less acoustic urgency.
• Deceptive cadence (V⁷ → vi): the tonal gradient is partially resolved (root motion is upward by step rather than downward by fifth) but the leading-tone pull and chordal-seventh resolution are still satisfied — the "deceptive" feel comes from the withheld tonal-gradient resolution.
• Phrygian cadence (iv⁶ → V in minor): the half-step pull is from above (♭6̂ → 5̂) rather than from below; no leading-tone pull; characteristically modal finality.

The differences in finality among these cadence types are predictable from the §3.5 analysis of which pulls are present in V⁷ and which are withheld in each alternative.

The consequence. The PAC is the strongest tonal-music tool available for marking finality at the local level — the level of phrases and short sections at which the listener's perception of the gradient is robust (see §2.4.4 on the scope restriction). It is used at the ends of phrases, periods, and short sections precisely because it is the maximum convergence of resolutions the harmonic dimension can supply at the moment of arrival. Other cadence types are tools for marking lesser degrees of finality — a half cadence at the end of a phrase that wants to continue, a deceptive cadence that prolongs the tension before the eventual PAC, a plagal cadence as a softer or more solemn close, the Phrygian cadence as a deliberately modal alternative. The composer has a calibrated set of finality-tools at her disposal, and the calibration is directly the harmonic-dimension acoustic analysis of which pulls each tool resolves.

The claim is local. PACs at the ends of larger sections — the end of an exposition, the end of a movement — are iterations of the same local phenomenon, not multi-layer cadences operating at architectural scale. The treatise does not claim that the final PAC of a sonata-form movement resolves a long-range structural dominant whose pressure the listener has held in mind across the development; the empirical record on long-range tonal perception does not support that claim (§2.4.4). What the treatise does claim is that, at the moment of the final PAC, the four local gradients converge as they do at every PAC. The cadence's finality is a property of the local moment of arrival, not of an architectural span the listener has been holding in tension.

This claim closes the chapter because it is where the previous nine claims converge in practice. The harmonic-series foundation (Claim

1. supplies the partials. The movement principle (Claim 2) and the distance metaphor (Claim 3) describe what the music does. Major's naturalness (Claim 4) and minor's modal modification (Claim 5) provide the two operative scales. Modal harmony (Claim 6) and atonality (Claim 7) describe positions that quiet or set aside the hierarchy. The hierarchy itself (Claim 8) provides the gradient on which all the previous operate, and the dual nature of dissonance (Claim 9) explains how movement actually happens at each moment. The cadence (Claim 10) is where all of this resolves, in a single moment, on a single chord. The student who hears a PAC as final is hearing the entire theory at work.

Curriculum location. Cadence types are introduced in Stage III (counterpoint) at the most basic level — the Phrygian cadence in 1st species, the elementary V-I motion. They are elaborated in Stage IV (figured bass and harmonic standards, where PAC and HC become explicit). They are developed in detail through Stages V-X as the harmonic vocabulary expands. The PAC remains the central tool throughout.

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§5 · The Theory of Tonal Movement

§5.1 · Movement as the Central Phenomenon (Operational Definition)

The central claim. The central phenomenon of the harmonic dimension of tonal music is movement — the temporal experience of pulls and resolutions among pitches, chords, and tonal regions. Movement governs the harmonic surface: note-by-note, chord-by-chord, modulation-by-modulation. The harmonic dimension is what this theory describes; rhythmic, formal, and aesthetic dimensions are addressed elsewhere (§1.3). Tonal music in its harmonic dimension is not, primarily, a structure to be analyzed; it is a movement to be heard.

The operational definition. Movement is present in any configuration of two or more pitches that is not in perfect unison, and its degree is a gradient function of how far the configuration sits from maximum stability on the harmonic-series-grounded stability hierarchy (§3.4).

The unison (1 : 1) and the octave (1 : 2) sit at maximum stability — at the limit, no movement. The perfect fifth (2 : 3) is the next position on the gradient: stable but not at rest; some degree of movement is present, though slight, because the ear hears two pitch classes rather than one. The major third (4 : 5) introduces more degree of movement. The chordal seventh (4 : 7), the major second (8 : 9), and the leading tone in its cadential context (15 : 16) sit progressively higher on the gradient — more movement, more "the ear wants more."

The phenomenology of movement is the ear wanting more, wanting somewhere else, wanting resolution. This is not a metaphor. It is a description of what the listener actually experiences when an unstable configuration is sustained: a kinaesthetic anticipation of a destination that the configuration itself implies. Movement is therefore graduated (every non-unison configuration has some degree of it), grounded (in the listener's perception of stability against the harmonic-series-derived hierarchy), and directional (the ear anticipates a where the configuration is going, not merely that it is in motion).

Falsifiability. The operational definition makes specific, testable predictions about which configurations produce reports of motion and which produce reports of rest. The empirical record on listener perception is partial but informative, and the prediction has already been tested in one of the most-cited results in cognitive music psychology.

The Krumhansl-Kessler probe-tone hierarchy (§3.4) is a direct empirical test of the gradient prediction. Listeners played an established C-major context, then asked to rate how well each chromatic probe tone "fits" that context, produced an averaged hierarchy in which the LCM-low positions (tonic, dominant, mediant) rated as fitting and the LCM-high positions (chromatic alterations, leading tone) rated as fitting poorly. The empirical hierarchy agrees with the gradient hierarchy on the diatonic-vs-chromatic boundary and on the broad shape of the diatonic ranking. If the empirical ranking had been substantially different — say, if listeners had rated chromatic alterations as fitting the tonal context as well as the diatonic pitches do — the gradient theory would have been wrong about what listeners track, and the present apparatus would have to be abandoned. The agreement is not trivial; the opposite outcome was possible and would have been damaging. The theory is falsifiable in this sense, and it has survived the test in the form in which the test has so far been run.

The structural prediction reinforces the empirical one. A passage with no relative motion — a sustained unison or octave drone with no pitch variation across its duration — is a passage with zero degree of movement at the pitch level. La Monte Young's sustained- tone music is the limit case: in this theory's specific sense, it is not tonal music but a different practice, distinguished precisely by the absence of the phenomenon this theory describes. A passage of sustained instability that does not resolve produces the experience of unrealized movement, which is what characterizes deceptive cadences, evaded resolutions, the structural dissonance of a sonata-form development, and (at the largest scale) the unresolved chromatic saturations of late Wagner.

A theorist who could construct a tonal passage that listeners report as in motion but where no element sits above unison- stability on the gradient — established by probe-tone or similar methodology — would have a genuine counterexample. The prediction is that such cases will not be found, because the phenomenology and the gradient are tracking the same thing. The prediction is empirical, defeasible in principle, and unfalsified in practice so far.

What this rules out. The operational definition rules out the kind of absorb-all-examples redescription that would let "movement" mean "whatever counts as resolution." It commits the theory to a specific gradient (the harmonic-series stability hierarchy) and a specific phenomenology (the ear wanting more). Either the listener perceives the gradient — and the empirical record of how listeners describe music suggests they do, across many cultures — or the theory is wrong about what tonal music is.

§5.2 · Pull as the Basic Unit

Movement, as the operational definition in §5.1 establishes, operates on a gradient: every tone above unison-stability sits at some position of departure, oriented toward a more stable position. But movement does not present itself to the listener as an undifferentiated pressure. The ear experiences tonal motion through discrete events: specific tones and configurations that have directed energy toward particular resolutions. The basic unit of that directed energy is the pull.

A pull is a directional relation between a source tone and a destination tone at two successive positions on the harmonic-series-grounded stability gradient. It has three defining attributes.

The source is the tone or configuration at the higher — less stable — position on the gradient. Any tone above unison-stability (LCM > 1 relative to its expected resolution) is a potential pull source. The source's LCM distance from the tonic is the initial measure of its potential energy. The leading tone B in C major sits at LCM 120 (ratio 8:15, §3.4), and its pull accordingly has maximum directional force in the diatonic vocabulary. The supertonic D sits at LCM 72 (ratio 8:9), less distant and correspondingly less urgent. The major sixth A sits at LCM 15 (ratio 3:5), relatively close to home, its pull moderate. The gradient position of the source is the first and primary determinant of pull intensity.

The destination is the tone or configuration toward which the source tends. Destinations are always at a lower position on the gradient than their sources — a pull cannot move toward greater instability while remaining a pull in the sense the theory defines. Motion toward greater instability is a departure, an expansion away from home; it has its own role in tonal composition, but it is the opposite of what the pull describes. In the standard tonal vocabulary, destinations are chord tones or the tonic itself: B resolves to C; F, as the seventh of V⁷, resolves to E; A♭ in minor resolves to G. When the destination is not uniquely determined by the gradient alone — the supertonic D can move to C below or to E above — the harmonic context supplies the determination.

The intensity is a joint function of three factors. The first is the gradient gap: the LCM difference between source and destination. The leading tone's gradient gap (LCM 120 to LCM 1) is 119 units — the largest single gradient collapse available in the diatonic vocabulary. The ♭6̂ pull in minor (A♭ to G) has a gradient gap of 34 units (LCM 40 to LCM 6), producing a strong but less demanding directed energy. The descending-sixth pull (6̂ → 5̂, A to G) has a gap of only 9 units (LCM 15 to LCM 6) — moderate, directional, but not requiring resolution the way the leading tone does.

The second factor is pitch proximity: the distance in semitones between source and destination. Pull intensity increases when source and destination are close in pitch, independently of the gradient gap. The leading tone's urgency is compounded by both factors simultaneously: it has the largest gradient gap in the diatonic vocabulary and approaches its destination by the smallest possible interval, the semitone. The supertonic pull (D → C, gradient gap 71) has a large gap but a whole-step approach; the ♭6̂ pull in minor (A♭ → G, gradient gap 34) has a smaller gap but a semitone approach. These two cases show that pitch proximity contributes independently: the ♭6̂ → 5̂ pull feels more urgent in its approach quality than its gradient gap alone would predict, because the half-step closes a smaller pitch distance. Gradient gap and pitch proximity operate independently, and their joint product is what the listener perceives as the pull's kinesthetic directness.

The third factor in intensity is the harmonic context in which the pull is embedded. The same pitch class can function at different intensity levels depending on what surrounds it. F, as one instance, reads in a diatonic passing-tone context at LCM 12 (subdominant relation to C), producing a modest pull. In the cadential context as the seventh of V⁷, the same F is heard as partial 7 of the dominant's harmonic series, its ratio locking onto 4:7 (LCM 28) against the dominant root, carrying the accumulated dissonance of the full 4:5:6:7 chord as amplification. The pull is not F in isolation; it is F embedded in a configuration that concentrates multiple gradient pressures toward the same resolution. The leading-tone pull and the chordal-seventh pull reinforce each other in the cadential moment because both are moving toward the tonic triad simultaneously — the convergence §5.6 develops. Context is not incidental to intensity; it is the third of intensity's three determining factors.

The theoretical economy of this three-attribute structure is what justifies treating the pull as the basic unit rather than the chord, the interval, or the scale degree. A theorist who has identified a pull's source, destination, and intensity has identified the complete specification of the most local harmonic event in a tonal passage. Everything above the pull in scale — harmonic progressions, tonal regions, large-scale motion — is composed of aggregated pull-events, or is in a different theoretical domain. Everything below the pull — the physics of the harmonic series, the gradient of ratio simplicity — is the causal foundation on which the pull's existence depends. The pull is the entry point into harmonic-dimension analysis.

The canonical scale-degree pulls

The following are the basic pulls of the diatonic tonal vocabulary at the scale-degree level. Each can be fully specified by the three-attribute framework above. §5.3 extends the vocabulary to the full range of non-chord-tone configurations and then to chord-level and tonal-region-level pulls; the scale-degree pulls below are the foundational cases from which the larger vocabulary is constructed.

7̂ → 1̂ — the leading-tone pull. Source B in C major: LCM 120 (ratio 8:15). Destination C: LCM 1. Gradient gap: 119 units. The strongest diatonic pull; the cadential pull the dominant seventh chord deploys as its most urgent constituent element. The raised seventh of harmonic minor is not a theoretical nicety but a practical necessity arising directly from this gradient position (Claim 5): without the raised leading tone, the seventh-degree pitch sits only 6 units away from the ♭7̂ ratio (LCM 144, ratio 9:16 in the Aeolian scale), and the cadential pull drops to a fraction of its major-mode intensity. The historical correction of musica ficta is the ear seeking this pull; every performer who raised the seventh degree in modal polyphony was responding to the gradient before any theorist named it.

4̂ → 3̂ — the chordal-seventh pull. Source F in C major: LCM 12 in diatonic context (ratio 3:4), rising to LCM 28 in the V⁷ cadential context (ratio 4:7, partial 7 of the dominant's series). Destination E: LCM 20 (ratio 4:5). Gradient gap: 8 in the minimal reading, substantially amplified when F is embedded in V⁷. This is the pull that determines the inner-voice resolution of every standard cadence: the voice-leading motion Zarlino codified in the sixteenth century as obligatory for the chordal seventh (Zarlino 1558, Istitutioni III.59) and that every subsequent treatise has confirmed. Its combination with the leading-tone pull produces the double-directed pressure of the PAC: two independent pulls from different source tones, converging on different tones of the same tonic chord, arriving simultaneously.

♭6̂ → 5̂ — the minor-sixth pull. Source A♭ in c minor: LCM 40 (ratio 5:8). Destination G: LCM 6 (ratio 2:3). Gradient gap: 34 units. Stronger in minor than its major-key analogue (A → G, gap 9) because the lowered sixth brings the ratio to a more acoustically complex position on the gradient. This pull governs the distinctive finality of the Phrygian cadence (iv⁶ → V), where ♭6̂ in the bass moves to 5̂ with the particular dark inevitability that distinguishes the Phrygian close from the authentic cadence. The same pull underlies the Neapolitan's characteristic voice-leading: in the progressions ♭II → V or ♭II⁶ → V, the upper voices carry ♭6̂ → 5̂ while the bass supplies the root motion, and the combination's directed energy is exactly what makes the Neapolitan one of the more consequential chromatic-harmony tools the late-Baroque and Classical periods developed.

2̂ → 1̂ or 3̂ — the supertonic pull. Source D in C major: LCM 72 (ratio 8:9). Resolution to C (LCM 1, gap 71) in descending stepwise motion, or to E (LCM 20, gap 52) in ascending passing motion. The supertonic is anomalous in the diatonic collection: at LCM 72 it is far from home by ratio simplicity — farther than the leading tone in some respects, since at LCM 72 it sits between A♭ (LCM 40) and the diminished fifth's extreme values on the gradient. Yet it is a diatonic pitch that arises naturally in every scale-based motion. The resolution must choose a direction: downward to the tonic is the cadentially decisive motion, appearing in the bass of plagal progressions (ii → I) and in the soprano motion of incomplete authentic cadences. Upward to the third is the passing-tone resolution, the melodic 2̂ → 3̂ stepwise ascent. The context-dependence of the destination is here maximal, and is why the supertonic is one of the most theoretically discussed scale degrees: it resolves, but which way it resolves is a compositional decision, not a gradient-determined inevitability.

6̂ → 5̂ — the descending-sixth pull. Source A in C major: LCM 15 (ratio 3:5, between partials 3 and 5 of C's harmonic series). Destination G: LCM 6. Gradient gap: 9 units. The gentlest of the canonical pulls: consistently directional but not demanding, the characteristic motion of the alto voice in standard four-voice progressions whenever 4̂ → 3̂ is in the tenor and the soprano carries 7̂ → 1̂. In minor (♭6̂, LCM 40, gap 34) the same structural role carries substantially more urgency. The distinction between the major-mode sixth (LCM 15, moderate pull) and the minor-mode sixth (LCM 40, strong pull) accounts, by a single gradient difference, for the audibly different character of descending-sixth motion in the two modal settings.

1̂ → 7̂ → 1̂ — the upper-neighbor figure. Structurally distinct from the preceding pulls: the motion from 1̂ to 7̂ is a departure from stability toward maximum local instability (LCM 1 to LCM 120), not a pull toward resolution. The pull runs in the opposite direction, throughout the figure: the leading-tone neighbor, once sounded, is under continuous pressure to return to the tonic, and the figure as a whole is a controlled departure and return — the tonic prolonging itself by ascending to its most unstable adjacent tone, which then resolves back with maximum gradient force. The neighbor figure is the prototype of ornamental counterpoint in the tonal tradition: a stable tone is adorned by its most unstable adjacent tone, and the stability of the framing positions amplifies the urgency of the return. The same structure in reverse (1̂ → 2̂ → 1̂, the lower-neighbor figure below the tonic, using the diatonic 7̂ from below) is less strong because 2̂ (LCM 72) is a less extreme departure than 7̂ (LCM 120), and the return to 1̂ carries proportionally less gradient force.

The pull as the unit of harmonic-dimension analysis

These six pull-types are not an exhaustive inventory of tonal motion; they are the foundational cases at the scale-degree level. §5.3 will extend the framework to the full surface vocabulary — passing tones, neighbor tones, suspensions, appoggiaturas, escape tones — and then to chord-level pulls (the dominant seeking the tonic, the secondary dominant seeking its tonicized chord, the augmented sixth seeking the dominant) and to tonal-region pulls (the dominant key seeking home, remote regions seeking closer ones). Each level inherits the same three-attribute structure.

What the pull-as-unit supplies that chord analysis does not is directionality at the finest available grain. A chord analysis can identify the V⁷ chord; the pull-based analysis identifies the four specific directed pressures that are active within the V⁷ chord at the moment of its cadential sounding. These are not the same information. The chord label says what the harmony is; the pull analysis says what the harmony is doing — where each voice is going, at what intensity, and why. The Theory of Tonal Movement is, from the perspective of §5, a theory of the pull as the basic unit of that doing. Chord analysis is one layer of description; the pull analysis is the layer that connects the chord to the listener's perception of directed energy and to the voice-leading outcome the ear anticipates.

This has a working consequence the student internalizes over the curriculum: voice-leading rules are codifications of the gradient's behavior in specific recurring contexts. They are accurate for those contexts because the gradient is accurate; they fail in edge cases because no finite rule-set fully captures a continuously graded phenomenon. The student who has internalized the pull as the basic unit has, in the gradient, the rule behind the rules.

§5.3 · Pull — The Vocabulary

The scale-degree pulls catalogued in §5.2 are the most local instances of the pull as a unit — individual tones moving from one gradient position to the next. Tonal music's harmonic dimension presents pull at three structural levels, each inheriting the same three-attribute form (source, destination, intensity) while operating over a larger harmonic span. The three levels are pitch-level (individual non-chord tones against a harmonic background), chord-level (chord-to-chord functional motion), and tonal-region level (key areas related to each other by the extended gradient). At each level the same principle operates: a configuration at a higher gradient position is under directed pressure toward a lower one.

Pitch-level pulls

Pitch-level pulls are the pulls of individual tones that sound against a harmonic background which does not contain them as chord members. The standard non-chord-tone vocabulary catalogs the configurations in which this occurs.

The passing tone transits between two chord tones by step. The source is the non-chord tone at its gradient position; the destination is the adjacent chord tone the line is approaching; the creation and release of tension are a single stepwise motion. A passing D in the line C–D–E over a C major chord is at LCM 72; the step to E (LCM 20) collapses 52 gradient units in one move. The passing tone does not sit as an unresolved tension awaiting a separate resolution; it is motion itself made audible. This is the cleanest realization of the dual-nature principle developed in §5.4.

The neighbor tone is the departure from a stable chord tone to its adjacent step and return. The upper-neighbor figure 1̂ → 7̂ → 1̂ uses the maximum diatonic gradient pull: the leading tone at LCM 120 is under continuous pressure to return to LCM 1, and the figure prolongsthe tonic through a temporary departure to its most unstable adjacent neighbor. Lower neighbor figures use the diatonic step below the principal tone, with a smaller gradient gap and correspondingly gentler directed quality.

The suspension is the delayed resolution: a tone consonant in its previous harmonic context is held as the harmony changes beneath it, creating a dissonance that then resolves by downward step. The suspension separates the dual-nature gesture into two audible phases — the creation of the dissonance (the harmony changes while the held tone remains) and its release (the step resolution). Pull intensity in a suspension is a joint function of the gradient gap at the held position and the duration for which the dissonance is held: longer holds accumulate greater gradient pressure, and the release of greater accumulated pressure is heard as a more satisfying resolution. The standard 4–3 suspension over a dominant bass (F held at LCM 28 in the V⁷ context, resolving to E at LCM 20) is modest in gradient gap but amplified by the cadential context.

The appoggiatura is an accented non-chord tone approached by leap and resolved by step. Its arrival on a strong beat concentrates gradient pressure: the accent gives the dissonance additional weight before the step resolution delivers the release. The escape tone departs from a chord tone by step and resolves by leap in the opposite direction, the only non-chord-tone type whose dissonance is "escaped" by a larger motion rather than released by the return stepwise path.

Chord-level pulls

At the chord level, the source is a chord at higher functional instability than its destination, and the intensity is determined by the combined gradient pressure of the chord's constituent tones and the harmonic context in which it sounds.

V → I (the dominant seeking the tonic) is the fundamental chord-level pull of tonal music, the aggregate of the four scale-degree pulls concentrated in V⁷ (§5.2, §3.5): the tonal gradient (G → C, descending fifth), the leading-tone pull (B → C), the chordal-seventh pull (F → E), and the fifth resolution (D → C or E). The V→I pull is not simply the chord-root move from LCM 6 to LCM 1; it is the convergence of all four simultaneous scale-degree pulls onto the same arrival. §5.6 develops this convergence as the PAC.

V⁷/X → X (the secondary dominant seeking its tonicized chord) replicates V⁷→I at the local level: the same 4:5:6:7 ratio, the same four converging pulls, the same acoustic continuity between applied dominant and its resolution chord. The distinction is the tonal-region position of X relative to the home key. V⁷/V resolves to the dominant region (LCM 6 from home — acoustically near); V⁷/♭III resolves to a chromatic-mediant region (LCM 30 or higher — farther away). The chord-level pull is locally identical to V→I; the tonal-region weight of the arrival depends on X's position in the broader gradient.

Augmented-sixth chords → V are chord-level pulls constructed to maximize the gradient approach to the dominant. The augmented-sixth interval (between ♭6̂ and ♯4̂, both at high LCM distances from the tonic) expands outward in contrary motion, each voice moving by semitone toward the dominant's fifth sounding above and below. The Italian, German, and French augmented-sixth chords are three realizations of the same interval within different chord structures; each targets the dominant as destination, and the augmented-sixth interval is the maximum-intensity approach to V available in the pre-chromatic-saturation tonal vocabulary.

Tonal-region pulls

At the tonal-region level, the source is an entire key area at greater LCM distance from the home tonic, and the destination is a closer key area. The directed energy operates over spans of many measures.

The dominant key (V) sits at LCM 6 from home — the closest secondary tonal center. Passages prolonged in the dominant region feel "away but nearby"; the return from dominant to home covers the same gradient distance as the chord-level V→I motion but over a structural span. The subdominant key (IV, LCM 12) carries its characteristic flatward weight. Chromatic-third key regions (LCM 20–40, depending on the specific third-relation) feel more meaningfully distant; the sense of having traveled somewhere follows from the larger gradient distance covered. Remote keys at LCM distances of 72 and above feel genuinely foreign; their return requires proportionally larger harmonic gestures to restore the home-key sense.

The three levels interact

In any tonal passage, pulls at all three levels are active simultaneously. A passing tone creates a pitch-level pull; the underlying harmony is a chord with its own chord-level pull toward the next harmony; the whole passage may be within a tonal region under tonal-region pull toward home. The listener tracks all three simultaneously, weighting them by structural salience. The surface texture is pitch-level; the phrase motion is chord-level; the tonal architecture is region-level. The Theory of Tonal Movement claims that all three are instances of the same principle — source, destination, gradient gap plus harmonic context — operating over different spans. The listener's experience of a tonal passage as moving is the aggregate of pulls at all three levels simultaneously.

The theory makes no claim about how harmonic-dimension pulls relate to non-harmonic dimensions of musical experience — rhythm, form, and aesthetics are their own theoretical domains and are addressed there.

§5.4 · The Dual Nature Revisited

Movement in tonal music is commonly described as a sequence: a dissonance arises, then it resolves. The language is sequential — dissonance then resolution, instability then stability. §4.9's claim sets a stricter position: the passage from dissonance to resolution is not a sequence of two discrete events but a single event with two aspects inseparable in the listener's experience. A dissonance does not merely point toward its resolution; it is the resolution in the act of beginning.

The claim can be examined through three configurations in which the dual nature is visible at different temporal scales.

The passing tone — the clean case

The passing tone is the configuration in which the dual nature is most immediately apparent. In the line C–D–E over a C major chord, the D is not a chord tone: at LCM 72 against the C tonic (ratio 8:9), it sits at a gradient position removed from the consonance of C (LCM 1) and E (LCM 20). Under the sequential account, D is a momentary dissonance that the stepwise motion then resolves to E.

Under the dual-nature account, the D does not resolve after it is dissonant; the D moves through the dissonance, and that motion is the resolution. The passing tone exists as a transit between C and E; its gradient position at LCM 72 is what gives the line its forward energy; and the very act of taking that gradient position and continuing to E is the creation and release of tension as one undivided gesture. There is no temporal gap between the dissonance and its resolution because there is no moment at which D sits at rest: D arrives as motion and departs as motion, and the listener's experience of the gesture is the experience of motion, not of a dissonance awaiting a separate event.

This is the dual nature in its cleanest form. The passing tone's duration is typically the shortest available — one beat, or a subdivision thereof — which makes the co-occurrence of creation and release temporally obvious. The dissonance and the motion-through-it are literally simultaneous: D is dissonant as it passes, and its passage is its resolution. What the listener experiences as forward motion in a stepwise melodic line is the accumulation of these brief co-occurrences: each passing tone creates and releases tension in a single gesture, and the succession of such gestures is what the listener perceives as a line rather than a series of intervals.

The suspension — the slowed-down case

The suspension stretches the dual-nature gesture until its two aspects become two audible phases. A tone consonant in its previous harmonic context is held as the harmony changes beneath it: the held pitch, now dissonant against its new harmonic environment, is the preparation-into- dissonance — the first phase. The step-resolution that follows is the release — the second phase.

In the classic 4–3 suspension over a dominant bass (F held as the harmony moves to V, resolving to E), F sits at LCM 28 in the dominant context (partial 7 of the dominant's harmonic series) for an entire beat before the resolution to E (LCM 20) releases the accumulated gradient pressure. The dissonance is not co-occurring with its resolution; it is sustained long enough that the listener hears it as a distinct phase — a held tension, a point of awaiting. Then the step downward releases it.

The difference from the passing tone is not categorical but temporal. In the passing tone, the two aspects of the dual nature — tension and release — occur so rapidly that they read as a single gesture. In the suspension, they are drawn apart by duration: the held tone accumulates gradient pressure, and the duration amplifies the eventual release. A suspension held for a whole note releases more accumulated tension than one held for a quarter note, even when the gradient gap is identical; the temporal stretch of the held dissonance is what converts the momentary co-occurrence of the passing tone into the satisfying two-step resolution of the suspension.

The suspension is therefore not a different phenomenon from the passing tone. It is the same phenomenon at a different duration — the dual nature slowed down until its two aspects can be heard as successive phases. Both are realizations of the same principle: a gradient position above the destination pulls toward that destination, and the motion toward it is simultaneously the creation and the release of tension. In the passing tone this happens instantaneously; in the suspension it happens across two audible beats; in extended suspensions (the long appoggiatura in classical phrase structure, the suspension chain in Renaissance polyphony) it can be stretched across several beats. The temporal scaling is the variable; the principle is constant.

The chordal seventh — the embedded case

In the dominant seventh chord V⁷, the dissonance is not a non-chord tone passing through or suspended above the harmony. It is a constitutive member of the chord. The seventh of V⁷ — in C major, the F above G — is the chord's defining characteristic, the element that distinguishes it from the plain dominant triad V. It is also the element that most compels resolution: at LCM 28 in the dominant context (ratio 4:7, partial 7 of the dominant's harmonic series), the chordal seventh stands at the outermost position of the chord's internal gradient structure, and it resolves invariably to E, the third of I (LCM 20), in the standard cadential progression.

What makes this case "embedded" is that the dissonance is not an elaboration of an otherwise consonant chord — it is what the chord is. The V⁷ chord deploys its seventh as its most forward-pointing element: the chord's identity is constituted by the tension the seventh creates, and the chord's purpose (cadential function) is constituted by the tension the seventh resolves. The dissonance and the resolution-demand are not aspects of the chord's behavior; they are aspects of the chord's identity. To hear a V⁷ chord is to hear a chord that is already pointing at its resolution; the dual nature is embedded in the chord's acoustic character before a single voice has moved.

This is why the dominant seventh is not simply a consonant chord with an added dissonance appended. It is a specific gradient-configuration that has internalized the dual nature as its defining property: a chord constituted by the tension that points beyond itself toward its resolution, with the resolution specified by the gradient rather than by convention. The chordal seventh of V⁷ is the classic embedded case because its dissonance-identity and resolution-demand are simultaneous and inseparable.

The three cases differ in where the temporal seam falls. The passing tone has no seam — creation and release coincide. The suspension has a seam between preparation and resolution, held open for one or more beats. The chordal seventh has a seam between chord-sounding and voice-leading motion, but the chord already implies the seam's closure before the motion begins. Each case is a different temporal arrangement of the same dual nature.

Stepwise motion as melodic foundation

The three cases above point to a consequence that reaches beyond the vocabulary of non-chord tones to the foundation of melodic writing in the tonal tradition. Stepwise motion — movement by half-step and whole-step — is the basis on which tonal melody is built. The species- counterpoint tradition of Stage III requires stepwise voice leading as the default; departures (leaps) are treated as departures requiring care and recovery. The preference for stepwise motion is, in textbook treatments, presented as a voice-leading rule. It is actually an acoustic consequence of the dual nature.

Every step through a non-chord tone simultaneously creates and releases tension. The line C–D–E over a C major chord has three positions: C (stable, no pull), D (dissonant passing tone, pull toward E created and released in the same motion), E (stable arrival). The line's forward energy — its quality of moving, of going somewhere — is generated by the D, by the gradient position the D occupies, and by the resolution of that position as the motion continues. Without the D, the direct skip C–E is a vertical interval without a line: it connects two stable positions without the intermediate pull that makes the motion feel like a line with momentum. With the D, the three-note figure has a vector — an internal directed energy that carries the listener from C to E not as a jump but as a journey with a transit.

The same principle applies to every stepwise line in tonal music. Each step through a dissonant position creates and releases tension in a single gesture; the successive steps accumulate this energy into the melodic shape the listener experiences as directed, as going somewhere, as a line rather than a sequence of intervals. The step through dissonance is what gives tonal melody its motive force.

Stepwise motion is therefore the melodic foundation of the tonal tradition not because of historical convention or pedagogical preference but because the dual nature of dissonance is maximally realized through stepwise motion: each step creates and releases tension in the smallest temporal unit, with no gap between creation and release, no leap that bypasses the intermediate gradient position, and no interruption of the line's momentum. Leaps can follow stepwise passages; they serve as dramatic departure or arrival; but the line's forward energy is generated by the steps, by the passing tones, by the uninterrupted motion through the gradient positions each step provides. The composer who has internalized this principle has internalized why species counterpoint begins with 1st species (consonances only, establishing the harmonic frame) and proceeds systematically to 2nd species (adding the passing tone in the most controlled setting): the progression is the cultivation of the ear that hears dissonance as motion rather than state, and of the hand that writes lines in which each step through a dissonance adds to the line's forward energy rather than interrupting it.

§5.5 · Stability Hierarchy in Detail

The stability gradient operates at three structural levels in the harmonic dimension: the level of individual scale degrees (tones within an established key), the level of chords (functional harmonic units within that key), and the level of tonal regions (key areas relative to the home tonic). At each level the same gradient principle applies: a configuration farther from the home tonic by ratio simplicity is less stable than one closer. The claim developed in §4.8 is here worked through at each level in sequence.

Scale degrees

Within an established key centered on C, every scale degree has a position on the consonance gradient relative to C as tonic. The LCM values from §3.4 provide the operational stability ranking.

The tonic itself (1̂, C) sits at LCM 1 — by definition, the reference point of maximum stability. No pitch in the key is closer to home than the tonic, because the tonic is home. The fifth (5̂, G) sits at LCM 6, the next position on the gradient: stable enough to anchor a half cadence, to serve as the opening note of many melodic lines, to function as the most consonant interval in the tonal vocabulary after the octave. The third (3̂, E) sits at LCM 20, somewhat farther out — stable in the sense that it is a chord tone of the tonic triad, but less anchoring than the fifth. The modal quality of the key is registered here: the major-mode third (E, partial 5 of C, LCM 20) versus the minor-mode third (E♭, LCM 30, ratio 5:6 via inverse relation) gives the two fundamental modal orientations their distinct acoustic identities.

The sixth (6̂, A) sits at LCM 15: moderately stable, within the tonic triad of the relative minor, which is why vi functions as a deceptive resolution — it is close enough to home to suggest arrival without completing the full V→I motion. The fourth (4̂, F) sits at LCM 12, the inverse subdominant relation. It is restless rather than stable: pulled toward 3̂ (LCM 20) in the chordal-seventh context and toward resolution in the standard tonal vocabulary. The subdominant's characteristic flatward quality — its tendency to resolve inward toward the tonic rather than outward toward the dominant — reflects its LCM 12 position in the gradient: close to home, but on the wrong side.

The supertonic (2̂, D) at LCM 72 is anomalously distant for a diatonic pitch: its large gradient gap explains the characteristic tension of the ii chord and of the supertonic as a melody note requiring resolution by step, either downward to 1̂ or upward to 3̂ depending on the context. The supertonic is one of the most theoretically discussed scale degrees precisely because its LCM position places it farther from home than any other diatonic pitch except the leading tone, despite being a simple diatonic step above the tonic.

The leading tone (7̂, B) sits at LCM 120: the most unstable diatonic pitch, the one with the strongest directional pull back to the tonic, the pitch whose very name in the tonal vocabulary is a description of its gradient behavior. Its position high in the harmonic-series spectrum of the tonic, combined with its proximity of one semitone, produces the characteristic urgency that makes the leading-tone-to-tonic resolution the most demanded single-voice motion in the tonal vocabulary. The natural-minor scale's seventh degree (♭7̂), a whole step below the tonic, approaches the tonic without the semitone urgency that the leading tone supplies; this is the cadential deficiency that the historical practice of musica ficta corrected by raising the seventh in modal polyphony, and that harmonic minor codified as the raised-seventh formula.

A remark on chord inversions. The stability of a given pitch in a harmonic context depends not only on its gradient position relative to the tonic but on whether it functions as a chord's root, third, or fifth in the current bass voice. The same pitch E, as the root of E major functioning as III in C major, as the third of a C-major tonic chord, and as the fifth of an A-minor chord, is heard at three different structural positions despite sitting at the same LCM distance (20) from C. The gradient provides the acoustic foundation; the harmonic context — specifically, the bass voice's relation to the chord's root — provides the structural function. A chord in root position, with the chord's root in the bass, sounds more stable than the same chord in first inversion with the third in the bass, because the bass register carries special weight in the listener's harmonic parsing: the bass note is heard as the acoustic foundation of the chord above it, and a bass that is the chord's actual root sounds more settled than one that is a third or a fifth above the root.

Chords

At the level of chords within an established key, the stability ranking follows directly from the chord-root LCM positions established in §3.4.

The tonic triad I (root C, LCM 1): maximum stability. The dominant triad V (root G, LCM 6): stable enough to anchor a half cadence, to serve as the harmonic destination of modulatory preparation, to be sustained as a prolonged harmony over many measures. The subdominant triad IV (root F, LCM 12): characteristic flatward weight, the principal pre-dominant chord, with the gravitational pull toward I grounded in its inverse-relation proximity to the tonic.

The submediant vi (root A, LCM 15): within the tonic's harmonic sphere but closer to the relative minor than to the home major; its use as the destination of the deceptive cadence is acoustically grounded — it is close enough to home that the resolution is partially satisfied, but the tonal gradient (V→I root motion by descending fifth) has been deflected. The mediant iii (root E, LCM 20): less frequently structural than vi, often functioning as an elaboration of I (sharing two pitches with the tonic triad) or as a chromatic coloring within a prolonged tonic region.

The supertonic ii (root D, LCM 72): far from home despite being a diatonic chord. The supertonic's characteristic restlessness and its function as the primary pre-dominant harmony derive from this gradient position; the chord's purpose is to generate forward motion toward V or directly toward I, not to provide rest. The leading-tone chord vii° (root B, LCM 120): the most unstable diatonic harmony. The diminished triad is built on the least stable scale degree and has no single strong root in the listener's perception; it functions almost entirely as an incomplete dominant substitute, its resolution to I or to I⁶ depending on the voice-leading context.

The fully diminished seventh vii°⁷ (adding F to the leading-tone chord) places four pitches in various states of instability relative to the home tonic, none in a strongly consonant relation to it. It is the extreme of diatonic harmonic tension, demanding resolution with greater urgency than the dominant seventh precisely because the diminished seventh chord is more rootless: the LCM values of its constituent tones relative to the tonic are all elevated, and the resulting aggregate acoustic distance is higher than V⁷'s.

This ranking produces the I-IV-V-vi functional backbone of common-practice cadential harmony as an acoustic result. These four chord-roots have the four lowest LCM values (1, 12, 6, 15) among the diatonic triads. The backbone is not a traditional preference; it is the four-chord cluster at the closest gradient positions, and the tradition adopted it because its acoustic properties make it the most effective frame for managing tonal arrival and departure.

Tonal regions

At the level of key areas within and around the home tonic, the gradient principle extends to the distance between keys. A tonal region's stability relative to home is operationalized by the LCM distance of its tonic root from the home tonic.

The home key, C major, is the reference: stability 1. The dominant region, G major, sits at LCM 6 from C: the closest and most stable secondary tonal center. The structural I-V motion of a binary form or a sonata exposition traverses the smallest available LCM distance between key areas; the dominant region is the standard destination for first tonal departures because it is acoustically nearest. The subdominant region, F major, sits at LCM 12: slightly farther, with its characteristic flatward quality; subdominant regions serve as alternative structural centers in forms that reach flatward rather than sharpward in their middle sections.

Chromatic-mediant regions occupy a more distant band of the gradient. A♭ major (the chromatic third below C, whose tonic root A♭ sits at LCM 40 from C) and E major (the chromatic third above C, whose tonic root E sits at LCM 20) feel meaningfully distant from home. The listener experiences a passage prolonged in a chromatic-mediant region as having traveled somewhere, and the return to home requires proportionally more harmonic preparation. The chromatic-mediant modulations that late-Romantic composers employed — Schubert's signature third-related key areas, Brahms's mediant transitions — are traversals of LCM 20–40 distances; the listener's experience of these passages as surprising but not jarring corresponds to the intermediate gradient positions involved.

Remote keys, reached through multiple fifth-steps or through enharmonic equivalences, sit at LCM values of 72 and above. A passage prolonged in D major (LCM 72 from C) and returning to C requires substantial harmonic labor to restore the home-key sense. The tritone-related key (F♯ major or G♭ major, whose tonic root sits at extreme LCM distance from C) is the maximum tonal distance: genuinely foreign, requiring the greatest harmonic preparation to reach and to leave credibly.

The hierarchy in operation

The three levels operate simultaneously in any tonal passage. A melodic figure carries scale-degree gradient positions; the underlying chords carry chord-level functional instability; the tonal region the passage inhabits carries region-level distance from home. The listener tracks all three simultaneously, weighting them by the structural salience each carries in context. The composer manages tension and release by choosing gradient positions at each level: a scale degree at high LCM distance over a chord at high LCM distance within a remote tonal region produces maximum harmonic tension; a stable scale degree over a stable chord within the home key produces maximum rest.

What the hierarchy does not do is extend to the non-harmonic dimensions of musical experience. The gradient is silent on rhythm, silent on formal architecture, silent on the aesthetic experience of proportional relationships among sections. Those dimensions have their own organizing principles and their own theoretical literatures (§1.3). Within the harmonic dimension, the gradient is the framework for all compositional decisions about tension and rest; outside it, the theory does not claim jurisdiction.

§5.6 · The Cadence as Convergence

The cadence is the point of maximum harmonic-dimension resolution. It is the moment at which multiple independent gradient pressures — each measuring a different dimension of instability in the harmonic vocabulary — collapse onto the same arrival. The perfect authentic cadence (V⁷ → I) is the prototype because it achieves the greatest available number of simultaneous resolutions on a single beat, in root position, with the tonic in the bass.

§4.10 stated the claim and §3.5 set out the V⁷ chord's acoustic structure. This section develops the convergence as a systematic account of what makes the PAC final, what each alternative cadence type withholds, and why the resulting differences in finality are predictable from the gradient analysis.

The four gradients in the PAC

A full dominant-seventh to tonic progression in root position, with the melody arriving on 1̂, collapses four independent gradient pressures onto the same arrival.

The tonal gradient — V → I — is root motion by descending fifth, from the dominant's root at LCM 6 to the home tonic at LCM 1. This is the fundamental cadential motion of the tonal vocabulary: the movement from the closest secondary center back to the primary center. The descending fifth is the interval of the harmonic series after the octave; the motion G → C is the cadential complement of the physical fact that G is partial 3 of C's harmonic series. The listener hears root motion by descending fifth as the archetypal cadential closure because the physical relation is primary: the dominant descending to the tonic is the harmonic series returning to its fundamental.

The half-step gradient — 7̂ → 1̂ — is the leading-tone pull: B moving to C, collapsing from LCM 120 to LCM 1 in a single semitone. The gradient gap of 119 units is the largest available in the diatonic vocabulary, which is why the leading-tone pull is heard as the most urgent directed energy in tonal music. The leading tone combines maximum gradient distance with minimum pitch distance: LCM 120, but only a semitone away. This combination — far from home by acoustic ratio, immediately adjacent in pitch space — is the acoustic definition of urgency, and the half-step resolution that closes it is the most abrupt and satisfying gradient collapse available at the scale-degree level.

The vertical gradient — 4̂ → 3̂ — is the chordal-seventh pull: F of the dominant seventh resolving to E in the tonic triad. At LCM 28 in the dominant context (partial 7 of the dominant's harmonic series), F is the embedded dissonance that constitutes the V⁷ chord's cadential identity (§5.4). Its resolution to E (LCM 20) is modest by gradient-gap measure — 8 units — but amplified by the full cadential context and by the coordination with the leading-tone pull: the two inner voices of the standard four-voice PAC move B → C and F → E simultaneously, the leading-tone resolution and the chordal-seventh resolution arriving at different tones of the same tonic chord in the same moment. The combined effect — two independent pulls from different source tones, converging on different tones of the same destination chord — is exactly the convergence §4.10 identifies as the signature of the PAC.

The root-position condition — both V and I arriving in root position, with the tonic in the bass at the moment of arrival — is not a gradient pressure in the same sense as the other three. It is a voicing condition that determines whether the arrival registers as fully settled. A V → I motion with the tonic arriving in first inversion (third in the bass) withholds the listener's sense of the bass having returned home: the bass voice, which carries the heaviest structural weight in harmonic perception, must carry the tonic root for closure to be complete. The combination of the three pitch-gradient resolutions with root-position arrival on a metrically strong position is what produces the PAC's distinctive quality of absolute finality in the harmonic dimension.

What other cadence types withhold

The calibration of cadential finality in the tonal vocabulary is directly readable from this analysis. Each cadence type withholds one or more of the four resolution conditions, and the resulting differences in closure are exactly what the analysis predicts.

The half cadence (any harmonic motion arriving on V) withholds the tonal gradient: V is reached as a destination without completing the descent to I. The leading-tone pull is absent or reversed (V is where the leading tone lives, not where it resolves), and the chordal-seventh resolution is absent (the seventh, if present in the approaching chord, has not resolved). The listener experiences the half cadence as open, expectant, requiring continuation — a stable subsidiary arrival (LCM 6 from home) that is not home. Its structural function is precisely this: to provide a point of provisional rest that implies a return to come.

The plagal cadence (IV → I) withholds the leading-tone and chordal-seventh resolutions. The tonal gradient is satisfied by the root motion IV → I (LCM 12 to LCM 1); the half-step pull from below the tonic is absent (IV has no leading tone), and the chordal seventh is not present. The result is a softer, less urgent arrival: the gradient collapses across the tonal dimension but not across the half-step or vertical dimensions. The plagal cadence's characteristic quality — solemn, concluding, the "Amen" of liturgy — is the acoustic signature of a tonal closure without the leading-tone urgency and chordal-seventh tension that make the PAC forceful.

The deceptive cadence (V⁷ → vi) withholds the tonal gradient while supplying the half-step and vertical gradients. The leading tone B moves to C — the third of vi in C major — and the chordal seventh F resolves to E. Both voice-leading resolutions are satisfied. But the root motion goes to A rather than C: the bass moves from G to A instead of G to C, satisfying the inner-voice resolutions but deflecting the tonal-gradient resolution. The listener experiences the deceptive cadence as a near-miss: the inner voices arrive at their expected destinations, but the bass has gone to the wrong place, and the tonal gradient remains open. This unresolved gradient is what carries the tension forward into whatever follows — the deceptive cadence prolongs rather than closes.

The Phrygian cadence (iv⁶ → V in minor) withholds the leading-tone pull from below while supplying a half-step pull from above. The characteristic voice-leading is ♭6̂ → 5̂ in the bass: A♭ at LCM 40, resolving to G at LCM 6, gradient gap 34. This is a strong pull, but directionally inverted from the standard leading-tone pull: the half-step motion is descending from above rather than ascending from below the destination. The tonal gradient resolves only partially (the arrival is on V, not I), and there is no leading tone. The result is the Phrygian cadence's characteristic dark, settled quality: not the urgency of the major-mode authentic cadence but the gravity of the falling half-step, arriving on the dominant with a modal finality that depends on the ♭6̂ → 5̂ resolution rather than on the convergence of all four PAC gradients.

The PAC as prototype

The four-gradient convergence of the PAC makes it the prototype not only as the most final cadence type but as the clearest instance of what the theory of tonal movement claims the harmonic dimension does. The PAC is the moment in tonal music where the maximum available number of independent gradient pressures resolve simultaneously, on a single beat, onto a single root-position tonic chord. It is the musical event in which the theory's central claims are most visibly operative: the harmonic series generates the gradient (§4.1); movement is dissonance seeking resolution (§4.2); distance from home is what the listener perceives as tension (§4.3); the V⁷ chord is the convergence point of four simultaneous pulls (§3.5, §4.10); and the dual nature of dissonance (§4.9) reaches its most concentrated expression — four independent dissonances releasing simultaneously onto the same arrival.

The student who has internalized the PAC — not as a formula to identify but as the felt convergence of four simultaneous gradient resolutions — has internalized the core of what the harmonic dimension of tonal music does. The pedagogy of Volume 2 builds outward from this moment: every harmonic exercise, every voice-leading standard, every figured-bass realization is practice in managing the gradient pressures the PAC prototype exemplifies, at varying combinations and degrees of completeness. The composer's craft at the local harmonic level is the art of calibrating which gradient pressures to resolve, when, in what order, and how completely — and the PAC is the maximum-convergence reference point against which every alternative is measured.

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§6 · The Three Lenses on Movement

§6.1 · Tonal, Vertical, Horizontal — Three Views of Movement Function

The standard analytical vocabulary of tonal music answers the question what is this tone? with great precision. The leading tone, the chordal seventh, the suspension, the Neapolitan — these are well-defined terms, tested across centuries of theory pedagogy, that identify a tone's scale- degree position, chord membership, or contrapuntal classification. The analytical vocabulary is not wrong; the problem is that it answers the identity question rather than the movement question, and the identity question, answered alone, can leave the student knowing the name of the thing but not knowing what the thing is doing in motion.

The theory of tonal movement is a theory of what tonal music does. Its central question is not "what is this harmony?" but "what is this harmony doing, and where is it going?" When a student applies the analytical vocabulary to a specific moment in a passage and comes away knowing the chord's Roman numeral and the scale degree of each voice, she has answered the identity question. She has not necessarily answered the movement question — she may not know which of the several pulls active in that moment is the load-bearing one, or which dimension of the tone's behavior is driving the voice-leading decision she needs to make.

The three lenses are three focused questions about a tone's movement function, each isolating one dimension of the tone's participation in directed tonal motion. They are analytical perspectives, not analytical categories: a given tone has movement function in all three dimensions simultaneously, and the lenses allow the analyst or student to examine each dimension separately when compound contexts produce ambiguity.

The tonal lens: where does this tone pull relative to the perceived tonal center?

The tonal lens asks about the tone's gravitational orientation within the established key. It is the lens of distance from home (§3.4): what gradient position does this tone occupy relative to the tonic, and toward what lower-gradient position is it directed? In C major, B (the leading tone) is at LCM 120 relative to C, with a gradient gap of 119 — the largest in the diatonic collection — pulling upward by semitone toward the tonic. A♭ in c minor is at LCM 40, pulling down by step to G (LCM 6). F, the fourth scale degree, is at LCM 12 diatonically and at LCM 28 when embedded as the seventh of V⁷, pulling in both contexts toward E (LCM 20).

The tonal lens brackets the tone's chord membership and its linear role. It does not ask whether B is the third of V or the seventh of vii°⁷; it asks one question: in the key we are in, at what gradient position does this tone sit, and where does that gradient position direct it?

The vertical lens: how does this tone move within the chord it belongs to?

The vertical lens asks about the tone's role inside the current harmonic configuration. The same question "what is this F?" gets a different answer under the vertical lens than under the tonal: tonally, F is the subdominant, at LCM 12, pulling toward E; vertically, if F is the seventh of V⁷, it is the embedded dissonance that constitutes the dominant seventh's identity (§5.4), resolving to the third of the tonic chord by the voice-leading obligation Zarlino codified in 1558 and every subsequent textbook has preserved.

The vertical lens is the lens of the chord. It brackets the tone's key-level gradient position and its linear context in favor of a single question: in the chord currently sounding, what is this tone's structural position and what does that position require of it?

The vertical lens is most important for compound chords — seventh chords, ninth chords, altered dominants — where the chord's identity is partly constituted by its dissonant members and the voice-leading obligations of those members are specific. A student who knows the chord's Roman numeral but has not applied the vertical lens may not know which voice must move and which may leap.

The horizontal lens: what motion is this tone part of as a line in time?

The horizontal lens asks about the tone's function within the voice- leading line it belongs to. A D in the motion C–D–E over a C major chord has a clear horizontal reading: it is a passing tone, transiting between C and E, its gradient position at LCM 72 generating and releasing tension in a single stepwise gesture. The same D held over from a previous beat and resolving downward has a different horizontal reading: it is a suspension, accumulating gradient pressure before its step-resolution releases it. Both involve the same pitch class; both sit at the same gradient distance from C; but their horizontal functions are entirely different, and the composer's handling of them is entirely different.

The horizontal lens is the lens of the line. It brackets the tone's gradient position in the key and its chord membership in favor of a single question: in the temporal motion of this voice from its previous position to its next, what is this tone's role — transit, suspension, departure, arrival?

The horizontal dimension is where the distinction between analytical identity and movement function is most apparent. "The ninth of a C chord" is an identity claim; "a suspension resolving to the octave" is a movement claim about the same tone. The identity claim is not wrong; the movement claim is what the student needs to write the voice leading correctly.

All three dimensions simultaneously

A given tone participates in all three dimensions of movement at once. The B in a V⁷ → I cadence in C major is:

• Tonally: the leading tone, at LCM 120, with a gradient gap of 119 toward the tonic C — the strongest directed energy in the diatonic collection.
• Vertically: the third of the V⁷ chord, the chord tone that carries the leading-tone pull — it resolves to C, the root of I, simultaneously discharging its chord-third function and its leading-tone function.
• Horizontally: the melodic penultimate note, approaching the tonic by ascending semitone — the closing step of a directed melodic line whose forward energy culminates in this arrival.

These three readings are not contradictions; they are three aspects of the same tone's participation in the cadential event. The tonal lens explains why the approach is urgent; the vertical lens explains the chord's identity and the voice's obligation; the horizontal lens explains the melodic gesture. The three together constitute a complete account of what the B is doing at the cadential moment.

The lenses are analytical tools for those moments when the canonical vocabulary alone is insufficient — when a student's question "what is this tone?" could be answered correctly in three different ways and the teacher needs to know which answer is load-bearing at that teaching moment. §6.2 develops this with worked examples.

§6.2 · When Each Lens Applies

The three lenses become pedagogically useful precisely when the canonical analytical vocabulary generates ambiguity. A tone identified by a standard term — "the seventh" — carries that term's full meaning only when the lens through which the term makes sense is also identified. "The seventh" in an analytical discussion can mean the seventh scale degree (a tonal claim), the chordal seventh of a dominant-seventh chord (a vertical claim), or a seven-to-six suspension (a horizontal claim). Each claim is correct within its own lens; none is more correct than the others in the abstract. What determines which answer is load-bearing in a specific pedagogical moment is which dimension of movement the student needs to act on.

This section develops three worked examples, each showing how the same tone generates different correct answers under different lenses, and how the teaching context — the specific voice-leading problem the student faces — determines which lens is the one that resolves the ambiguity.

Example I: The seventh in "what is this 7?"

A student in Stage VI (SATB voice leading and the classical harmonic vocabulary) plays the chord V⁷ in C major, hears a note she identifies as F, and asks: "What is this 7, and what does it do?"

The question is genuinely ambiguous. "This 7" can be read under any of the three lenses, and each reading produces a different correct answer.

Under the tonal lens, the answer is: "In C major, F is the fourth scale degree — the subdominant, at LCM 12 from the tonic C. Its pull in the key is toward E (LCM 20 receives it as a consonant chord tone of the tonic triad); this is the pull from the subdominant fourth toward the mediant third, a step-resolution downward." This answer is appropriate when the teaching moment is about key: about which tones in the scale feel pulled toward home and in what direction. The tonal lens answer tells the student about F's position in C major's gradient space, independent of what chord F is currently occupying or what voice it is in.

Under the vertical lens, the answer is: "In the V⁷ chord, F is the chordal seventh — the embedded dissonance constituting the V⁷'s identity as a seventh chord. The vertical lens says: this note has an obligatory resolution downward by step to E, the third of the following I chord. The obligation is not a recommendation; it is the voice-leading law of the chordal seventh, formalized by Zarlino and preserved through every subsequent textbook." This answer is appropriate when the teaching moment is about chord behavior: about what the student must do with this voice in the next beat. The vertical lens answer is the actionable one for part-writing: the F must resolve to E, and the student who knows only the tonal-lens answer may not know that this particular F, in this particular chord, at this particular moment, cannot be doubled, cannot be leapt away from, and cannot remain stationary across the bar.

Under the horizontal lens, the answer would be different if the F had arrived differently: "If F was held over from the previous beat — if the preceding chord had F on a strong beat and the V⁷ arrived on a weak beat while F continued — then F is a suspension. Its horizontal function is accumulation: the voice line has held a dissonance over the bar and is about to release it by stepwise descent." The horizontal reading of F as a suspension applies only when F's temporal arrival pattern is suspension-pattern (arrival on strong beat in a consonant context, held across the bar as the harmony changes beneath it, resolving downward by step). If F arrived with the V⁷ chord rather than before it, the horizontal lens reads F as a chord tone with an obligatory downward resolution — the same voice-leading gesture as the vertical lens, but arrived at from different temporal reasoning.

The load-bearing lens for the student's voice-writing question is the vertical lens. She needs to know that the F has a specific obligation in the part-writing practice of Stage VI: it resolves down to E, it does not leap, it is not doubled. The tonal-lens answer is true and useful in the key-signature exercise; the horizontal-lens answer is true and useful when analyzing a suspension; but neither answer tells her, at the moment of writing the next beat of the part, what she must do with the F. The vertical lens resolves the ambiguity because the teaching moment is a voice-leading moment, and the vertical lens is the lens of the chord.

Example II: The fifth above an augmented-sixth chord

A more complex example arises in Stage VII (chromatic harmony). The student encounters an Italian augmented-sixth chord (It⁺⁶) in A minor: the chord {F, A, D♯} in closed position, resolving to the dominant E. The outer voices — F and D♯ — resolve outward by semitone to E (the doubled root of the dominant). The A in the inner voice moves to E. The student identifies the D♯ and asks: "Is this the leading tone, or something else?"

Under the tonal lens, the answer depends on which tonal center is operative at this moment. If the tonic is A minor, D♯ sits at extreme gradient distance from A (its enharmonic E♭, at LCM 30, is not a diatonic pitch of A minor; D♯ as an alteration of D sits at even greater distance). The tonal lens says: this is a chromatically altered tone, very far from home, with strong directional pressure — but the direction is not immediately toward the tonic A; it is toward E (the dominant). The tonal lens alone does not clearly identify D♯ as a leading tone of anything, because the tonic is A, not E, and D♯ does not resolve to A.

Under the vertical lens, the answer is clarified. D♯ is the augmented sixth interval above the bass (F): the interval of an augmented sixth above F is D♯, and the identifying feature of the augmented-sixth chord is this interval's characteristic outward resolution to the octave on E. The vertical lens says: D♯ is the upper voice of the augmented-sixth interval, and its vertical obligation is the outward expansion by semitone to E — the same E that F resolves inward to by semitone. The augmented-sixth chord's identity, under the vertical lens, is constituted by the voice-leading obligations of its two chromatic tones (F and D♯) to converge outward onto the octave E.

Under the horizontal lens, D♯ is the leading tone to the dominant E — but specifically in the sense of its linear approach function. The line D♯ → E is a semitone leading-tone motion to the fifth scale degree rather than the tonic. The horizontal lens confirms the directional energy (ascending semitone approach to E) and locates it in the temporal line: this is the voice that arrives at E by ascending half-step in the resolution, functioning as a local leading tone to the dominant in a half-cadence or dominant preparation.

Here the vertical lens and the horizontal lens are both load-bearing, but for different reasons. The vertical lens resolves the structural question (what is the augmented-sixth chord and what do its tones do in part-writing?); the horizontal lens confirms the directed motion of D♯'s line (leading-tone motion to the dominant). A student who has only the tonal-lens reading — "D♯ is extremely far from home" — has a true observation that does not help her write the resolution. A student who has both vertical and horizontal readings understands both what obligation the chord places on D♯ (outward resolution by semitone) and why the gesture sounds as it does (the ascending half-step approach to the dominant echoes the major-mode leading-tone motion in a chromatic context).

Example III: The ninth in a compound dominant

A student writing in a late-Romantic harmonic idiom (Stage VII–VIII) places a V⁹ chord — the dominant ninth — over a preparation for the tonic cadence. The ninth of the chord in C major is D (the supertonic): the chord {G, B, D, F, D} with the added ninth. She asks: "What is this D? I know it's the ninth of the V chord, but what does it do?"

Under the tonal lens, D in C major sits at LCM 72 — the supertonic, one of the most remote diatonic scale degrees from the tonic. The gradient gap from D (LCM 72) to the nearest stable landing (C at LCM 1, or E at LCM 20) is substantial; under the tonal lens, D is a tone with significant directional energy toward either the tonic or the mediant. The tonal reading suggests D wants to resolve: downward to C (the larger gradient drop), or upward to E if the mediant is the contextual destination.

Under the vertical lens, D as the ninth of the V⁹ is a dissonant addition above the dominant seventh. In classical voice-leading practice, the ninth of a dominant-ninth chord resolves downward by step to the chord's fifth (D → C), mirroring the chordal seventh's downward resolution (F → E) at one voice above. The vertical lens gives the practical part-writing rule: the ninth resolves downward by step, paralleling the seventh's resolution.

Under the horizontal lens, D's behavior depends on how it arrived. If D was sustained from the previous harmony as a common tone before the dominant chord was placed beneath it, D is a suspension over the dominant: the horizontal lens reads it as a delayed resolution, accumulating tension across the bar before its descending step to C. If D arrived simultaneously with the V⁹ chord, the horizontal reading is chord-tone with obligatory downward resolution — structurally a chord tone but with the same descending step as the suspension resolution. The horizontal lens distinguishes these two scenarios, identifying which is accumulation (suspension, with the weight of the held tone across the bar) and which is chord-tone-with-obligation (less accumulated tension, but the same resolution gesture).

All three lenses are informative here, and none is dispensable: the tonal lens confirms that D's position in the key space makes resolution toward C or E inevitable; the vertical lens gives the part-writing rule (resolve down by step); the horizontal lens determines whether the resolution carries suspension weight or chord-tone weight. A student who has only the vertical- lens answer ("the ninth resolves down") can write the part correctly; a student who also has the horizontal answer ("and in this case it is a suspension because it arrived before the dominant chord") writes with understanding of the voice-leading mechanism, not just the rule.

The load-bearing lens and pedagogical sequence

The three examples above exhibit a pattern. In Stage VI (the first encounter with the full harmonic vocabulary in four voices), the vertical lens is usually load-bearing: the student's most pressing question is what each voice must do in the part-writing sense, and the vertical lens — which asks about chord-tone obligations — answers that question directly. In Stage II–III (counterpoint and early melody), the horizontal lens is usually load-bearing: the student's line is the primary object, and the question "what is this tone doing in the line — passing, staying, suspending, departing?" is the one whose answer governs the next compositional decision. In Stage I and the foundational key-feeling exercises, the tonal lens is usually load-bearing: the student is building the ear-awareness of key-center gravity, and the question "how far from home is this tone, and toward what does it pull?" is the primary pedagogical target.

This is not a rigid sequence in which one lens is switched off while the others operate. It is a shift in emphasis: as the student progresses, all three dimensions are always present, but the dimension most immediately relevant to the compositional problem at hand changes. The skilled teacher identifies which lens is load-bearing for the specific question before her and supplies the answer in that lens's vocabulary, without cluttering the teaching moment with the other two correct answers that are not, at this moment, what the student needs.

The three lenses are therefore a pedagogical sorting device: they allow the teacher to route a student's "what is this tone?" question to the answer that serves the student's current compositional task. This is what §6.1 established as the distinction between the identity question and the movement question. The lenses keep the answer in the movement register rather than the identity register, and among the three movement dimensions, the lens selector identifies which movement dimension is most immediately at stake.

§6.3 · What the Three Lenses Do Not Do

A clear account of the three lenses requires an equally clear account of their limits. The three-lens framework is a targeted analytical instrument; it is not a general theory of tonal music, and it is not the curriculum's organizing axis. Several misapplications of the framework are predictable enough to be addressed directly.

The lenses do not replace the canonical analytical terms.

The leading tone, the chordal seventh, the suspension, the passing tone, the Neapolitan, the augmented sixth — these are the inherited vocabulary of tonal theory, and the three-lens framework presupposes them, not the other way around. The tonal lens answers questions framed in the language of scale degrees; the vertical lens answers questions framed in the language of chord tones and their obligations; the horizontal lens answers questions framed in the language of contrapuntal functions. All three lenses operate within the canonical vocabulary; they do not operate as substitutes for it.

A student who has not learned that the seventh of a dominant-seventh chord resolves downward by step has not been helped by being told that "the vertical lens is load-bearing here." The rule — chordal seventh resolves downward — must be learned from the canonical tradition. The vertical lens is what identifies which of the several things a given tone might be doing is, in this specific context, the chordal-seventh function. The lens routes the student to the right rule; it does not supply the rule.

The canonical terms carry historical weight, theoretical precision, and the collective agreement of centuries of music theorists. The Gradus curriculum teaches them in their canonical form, with their canonical names, in the historical sequence in which they were discovered and codified. The three lenses are a commentary on how to use those terms when the terms generate ambiguity — not a replacement for the terms themselves.

The lenses do not constitute a recurring framework students must memorize.

The three-lens vocabulary — tonal, vertical, horizontal — is introduced once, near the opening of the harmony arc (Stage VI), as a disambiguation tool for the compound analytical contexts that SATB voice leading regularly produces. It is not a recurring framework with its own lesson sequence, its own exercises, or its own assessment criteria.

The student does not need to pass a "three-lens analysis" exercise. She does not need to label tones T/V/H in her part-writing assignments. The lenses are invoked parenthetically by the teacher (or by Maestro, the AI composition professor) when a specific teaching moment produces the exact ambiguity the lenses are designed to resolve: a student's question that could be answered correctly in three different ways, requiring the teacher to identify which answer serves the student's current task.

In the Volume 2 pedagogy, the three lenses appear as a brief explanatory passage in the early harmony curriculum, and then as occasion demands — whenever the same analytical question recurs in a new context and the student is confused about which of her three correct answers is the one she needs. They are a teacher's tool, not a student's framework. The student internalizes the distinctions the lenses draw not by memorizing the lens vocabulary but by working through enough part-writing exercises that the distinction between a tone's key-level position, its chord-level obligation, and its line-level function becomes intuitive. When that intuition is formed, the lens vocabulary is no longer needed; the student has the underlying discriminations available to her without the meta-vocabulary.

The lenses do not exhaustively describe a tone's identity in a passage.

The three lenses are concerned with movement function: with what a tone is doing in directed tonal motion. They do not describe everything a tone is. A tone in a tonal passage also has a rhythmic position (strong beat, weak beat, on the beat, anticipating the beat), a registral position (high, low, crossing voices), a dynamic value (forte passage, piano), an articulation (slurred, staccato), and a relationship to motivic material (part of the main theme, part of a transitional figure, an ornamental addition). All of these aspects of the tone are real and analytically significant; none of them is captured by any of the three lenses.

The theory of tonal movement is a theory of the harmonic dimension (§1.3). It does not claim to be a complete theory of musical meaning, and the three- lens framework — as the theory's pedagogical instrument for resolving movement-function ambiguity — inherits this bounded scope. A complete analysis of a passage in a tonal work draws on the three lenses for harmonic- dimension questions and on other analytical frameworks for the other dimensions. The lenses do not displace those other frameworks; they complement them within the specific domain of tonal movement analysis.

This limit is consistent with the treatise's methodological commitment throughout: the theory of tonal movement is a theory of one dimension of musical experience — the harmonic dimension — developed to its full depth in that dimension, with explicit acknowledgment of the dimensions it does not cover. The three lenses are as bounded as the theory they serve: they answer harmonic-dimension questions about movement function, and only those questions.

The lenses are not a refinement of Hindemith's melodic/harmonic interval distinction.

A natural prior-art comparison the reader trained in twentieth-century theory may want to draw is between the three-lens framework and Paul Hindemith's distinction, in The Craft of Musical Composition (1937), between melodic and harmonic interval values. Hindemith ranks intervals twice — once by their value as simultaneous sonorities (Series 2 proper, beginning with the octave and ending with the tritone) and once by their value as melodic successions, with the small intervals essentially inverted in rank (stepwise motion is melodically smoothest despite seconds being harmonically dissonant, and the perfect fourth carries melodic stress as a leap despite being harmonically near-consonant). The split distinguishes the channel of presentation — line versus chord — at the interval-classification level.

The three lenses share with Hindemith's distinction the recognition that the same interval can have different analytical readings depending on whether it appears between voices simultaneously or between successive tones in one line. The horizontal lens corresponds roughly to Hindemith's melodic context, the vertical lens roughly to his harmonic context. But two structural differences make the lens framework not a finer-grained version of Hindemith's apparatus.

First, the framework is three-way rather than two-way, and the tonal lens has no counterpart in Hindemith's apparatus. The tonal lens reads a tone's position in the key against the gradient from the perceived tonic, in terms of the LCM measure of §3.4. Hindemith's apparatus is pan-tonal: any pitch can be a tonal center, established by emphasis, with no privileged acoustic gradient organizing the relations between scale degrees and the tonic. The tonal lens depends on Claim 4 (Major as Natural) and the harmonic-series ground in a way Hindemith deliberately ruled out for his own theory. The tonal lens is the lens that integrates the analytical apparatus with the gradient theory of §3.4 and the movement theory of §5; it has no Hindemithian analogue.

Second, the lenses are context-relative; Hindemith's interval rankings are context-invariant. Hindemith's melodic value of the major second is the same in any key, against any tonic. The harmonic value of the perfect fourth is the same in any key. The interval rankings are properties of the intervals themselves, independent of tonal context. The three lenses flex with the listener's perceived tonal center. The same pitch G has tonal-lens reading "dominant scale degree (LCM 6)" in C major, "tonic (LCM 1)" in G major, and "supertonic (LCM 72)" in F major — three different readings of the same physical tone, switched by the listener's perception of key. The vertical lens flexes similarly with the current chord (G as root of V, as chord-third of e minor, as ♭5 of B half-diminished), and the horizontal lens flexes with the line's prior trajectory (G arriving from F♯, G held over A, G passing between A and F). All three lenses are relativizations in a way Hindemith's interval rankings are not.

The architectural consequence is that Hindemith's apparatus has two layers that do not natively communicate. Static interval rankings sit at the bottom (the two Series); key-relative analysis sits at the top (harmonic fluctuation, step-progression, the two-voice framework). The interval rankings do not change when the tonal center moves; the upper-layer machinery does not reach down into the rankings. The three lenses dissolve this layering: each lens flexes with the tonal center natively, the gradient theory of §3.4 is integrated with the analytical apparatus rather than supplied as a separate substrate beneath it. The architectural difference is that Hindemith's apparatus is static at every layer that has been worked out, while the present apparatus is context-relative at the level where context lives. The lens framework is not a generalization of Hindemith's two-channel split; it is a different kind of instrument, built to do work the two-channel split was not designed for.

Summary

The three lenses are a precision instrument for a specific problem: the disambiguation of compound movement-function contexts in which a single canonical term could be read correctly under three different analytical perspectives. They are deployed parenthetically, in the vocabulary of tonal-lens / vertical-lens / horizontal-lens, when and only when such ambiguity arises in a teaching or analytical context. They presuppose the canonical vocabulary rather than replacing it. They are not memorized as a framework; they are applied as an analytical routing device. And they are bounded by the theory's scope: they speak to the harmonic dimension's question of directed movement function, and they are silent on everything outside that scope.

With §6.3 the theoretical apparatus of Volume 1 is complete. §7 extends the apparatus to the modes and to atonality — the two major zones of tonal practice that sit at the margins of the harmonic-series-grounded major-minor system — and §8 closes with a summary statement of what the unified theory achieves and what it leaves for Volume 2.

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§7 · Modes and Atonality

§7.1 · The Modes as Deliberate Quietings

The modes of the diatonic collection — Mixolydian, Lydian, Phrygian, Dorian, Aeolian, Locrian — are sometimes presented as parallel systems coordinate with major and minor, each primary collection with its own intrinsic properties requiring a separate theoretical account. The position taken in this treatise — and the claim developed in §4.6 — is the opposite. Each mode is the major scale with specific pulls deliberately quieted, redirected, or inverted. The unified acoustic ground of §3 provides the major scale's pull-structure; the modes are positions within that pull-space in which particular gradient pressures have been modified or removed.

This account is not new. The theoretical lineage goes back to the Renaissance theorists who observed that modal compositions required performers to apply musica ficta to generate leading tones — precisely because the natural modal scales lacked the pulled leading-tone-to-tonic motion that the acoustic ground predicts as the strongest cadential event. Performers and composers raised the seventh in Phrygian and Aeolian cadences because the whole-step or half-step-from-above approach lacked the gravitational directness the ear expected at a point of structural closure. What is developed explicitly here is the systematic application of the gradient analysis from §3.4 to identify specifically which gradient pressures each mode modifies.

Mixolydian — the leading-tone pull quieted

Mixolydian replaces the major scale's raised seventh (7̂, the leading tone at LCM 120) with the natural seventh a half-step higher (♭7̂). The ♭7̂ of the Mixolydian scale is acoustically related to the tonic as partial 7 of the tonic's harmonic series — it is the natural-seventh ratio, sitting at LCM 28 in that reading. At LCM 28, it is substantially closer to home than the leading tone B at LCM 120. This means the Mixolydian ♭7̂ generates far less gradient urgency in pulling toward the tonic than the leading tone does: a gradient gap of 27 (LCM 28 to LCM 1) rather than 119 (LCM 120 to LCM 1).

The acoustic consequence is that the strongest cadential pull in the major vocabulary is replaced by a softer, less urgent directed energy. The chord built on the seventh degree in Mixolydian becomes ♭VII, a major triad rather than the diminished vii°. The dominant harmony becomes a minor triad (v), losing the leading tone as its third. Cadential closure in Mixolydian is available — the ♭VII → I motion has its own characteristic finality — but without the urgency that the major-mode leading tone supplies. Mixolydian is the mode for composers who want the color of the diatonic major collection with the cadential urgency of major deliberately softened.

Lydian — the subdominant anchor removed

Lydian raises the fourth scale degree (♯4̂) in place of the major scale's natural fourth (4̂, LCM 12). The raised fourth in just intonation sits at high acoustic distance from the tonic; the Pythagorean tritone F♯ against C carries LCM values well above the natural fourth's LCM 12. More important than the LCM value of ♯4̂ itself is what its introduction removes: the natural 4̂ (F) has its LCM 12 position because it is the fundamental of a harmonic series that contains C as a low partial (C is partial 3 of F's harmonic series, ratio 3:4). This inverse-relation is the acoustic ground of the subdominant — the gravitational pull that gives the IV chord its characteristic flatward weight and its role as the pre-dominant harmony of the cadence that falls from IV to I.

Lydian's ♯4̂ replaces this subdominant anchor with a tritone above the tonic. The pull from 4̂ → 3̂, which in the major scale is the inner-voice resolution of the chordal seventh of V⁷, is redirected in Lydian to ♯4̂ → 5̂: the raised fourth wants to resolve upward by step to the dominant rather than downward toward the third of the tonic chord. The result is the floating, unanchored quality that defines the Lydian mode's characteristic sound: without the subdominant gravity pulling inward toward the tonic, the scale pushes upward and outward, away from the tonic rather than toward it. Impressionist and film composers deploy Lydian precisely for this effect — the sense of hovering above the harmonic ground without the IV chord's gravitational claim. The Lydian mode's floating quality is not a vague aesthetic impression; it is the acoustic signature of the removal of the inverse- relation subdominant anchor from the major-scale pull-structure.

Phrygian — the half-step pull inverted

The major scale has its leading-tone half-step pull from below the tonic (7̂ → 1̂, ascending semitone). Phrygian inverts this: the ♭2̂ creates a half-step pull from above the tonic — ♭2̂ → 1̂, descending semitone. In A Phrygian, the B♭ directly above A pulls down to A with a half-step urgency that is the directional mirror of the major-mode leading-tone pull. The gradient pressure of ♭2̂ relative to the tonic is determined by its ratio to the tonic root, which places it at a high LCM position (acoustically complex relative to the tonic) similar in magnitude to the leading tone's LCM 120; the acoustic complexity is comparable, but the direction of the half-step resolution is reversed.

The acoustic consequence is a half-step pull of comparable intensity to the leading tone's, but descending rather than ascending to the final. Where the major-scale cadence pulls upward by semitone from B to C, the Phrygian cadence pulls downward by semitone from B♭ to A. The Phrygian half cadence (iv⁶ → V) exploits the ♭6̂ → 5̂ motion in the bass — which is the falling half-step arrival at the dominant — and the overall effect is exactly the downward-oriented half-step directed energy that distinguishes the Phrygian close from the authentic cadence. Phrygian's characteristic dark, archaic quality — audible from Renaissance polyphony through twentieth-century borrowings — is the acoustic signature of a mode whose half-step orientation is inverted: the approach to the final is from above rather than from below, and the ascending urgency of the major-mode leading tone is replaced by the descending gravity of the Phrygian semitone.

Dorian — minor with raised sixth

Dorian is the natural minor scale (Aeolian) with the sixth scale degree raised. In A Dorian, the F♮ of natural A minor is raised to F♯. The raised sixth of Dorian sits at LCM 15 (ratio 3:5) relative to the A tonic — the same acoustic relation as the major-mode sixth. By contrast, the Aeolian ♭6̂ in A minor (F♮) sits at LCM 40 (ratio 5:8, a minor sixth above A). The difference in gradient positions — LCM 15 versus LCM 40 — accounts for Dorian's brighter character relative to Aeolian: the raised sixth sits acoustically closer to home, and the descending 6̂ → 5̂ motion in Dorian (F♯ → E over an A tonic) has a gradient gap of 9 rather than the Aeolian ♭6̂'s gap of 34.

Dorian is the modal choice when a composer wants minor's color without the full weight of Aeolian's dark interior. The raised sixth removes the Phrygian-like intensity that ♭6̂ → 5̂ carries in natural minor and Aeolian; the result is a minor mode that descends softly, with a lighter, somewhat warmer quality in the 6̂ → 5̂ motion. Without a raised seventh, Dorian lacks the strong leading-tone cadential drive of harmonic minor; it is a minor mode whose cadential urgency has been softened at both ends.

Aeolian — the natural minor, cadentially incomplete

Aeolian is the scale of natural minor: the minor collection without any chromatic alteration. Its seventh scale degree (♭7̂), a whole step below the tonic, approaches the tonic from a full step away rather than the semitone proximity that the leading tone supplies. The cadential motion ♭7̂ → 1̂ lacks the half-step urgency of the leading-tone resolution 7̂ → 1̂. The musica ficta practice of raising the seventh in Aeolian cadential contexts is precisely the performer's ear recognizing this deficit: the diatonic natural seventh's whole-step approach to the tonic fails to generate the gravitational pull the harmonic-series ground predicts as the primary cadential energy, and raising it restores the half-step urgency by converting the Aeolian cadence into the harmonic-minor cadence.

Aeolian therefore retains the characteristic sound of minor — the lowered third, the lowered sixth and seventh — while lacking the sharpened cadential apparatus of harmonic minor. It is the minor collection in its most purely modal form: a minor-mode sound without the cadential corrections that performers and theorists have applied for centuries. Composers who use Aeolian knowingly are accepting that their cadential closures will not have the urgency of harmonic minor; this can be an expressive quality (the pastoral natural minor of certain folk-derived idioms) or a deliberate modal color (the Aeolian quality of much twentieth-century rock and contemporary modal writing).

Locrian — the degenerate case

Locrian has both ♭2̂ and ♭5̂. The tonic triad in Locrian is diminished: the root, the minor third, and the diminished fifth above the final combine to form a rootless, acoustically unstable chord. A diminished triad built on the final has no 3:2 fifth between the root and its fifth; the diminished fifth is the acoustic relation most remote from the 3:2 consonance that gives other modal triads their sense of rootedness. The final of a Locrian scale therefore cannot anchor a key in the way that the root-position major or minor triad anchors other modes: the final is simultaneously the most acoustically dissonant note in the chord built on it.

Locrian is the modal extreme precisely because the final has been deprived of its own harmonic stability. Where the other modes quiet specific pulls while leaving the tonic triad intact as a point of potential rest, Locrian undermines the tonic triad itself. It is rarely used in tonal practice except as a deliberate color or as a theoretical limiting case; its entry in the scale catalog is less a practical resource than a marker of where the mode- as-quieting principle reaches its logical limit.

The modes within the unified-ground theory

The systematic account above establishes that the modes do not require a separate theoretical foundation. They are positions within the major-scale pull-space in which the composer has modified, removed, or inverted specific gradient pressures that the acoustic ground of §3 predicts as being present in the major scale. The decision to write in Mixolydian, Lydian, Phrygian, Dorian, Aeolian, or Locrian is a decision to compose within a specific partial relaxation of the full diatonic pull-structure.

The listener hears a mode's characteristic color as the absence or modification of pulls the major scale provides. Mixolydian sounds soft because the leading-tone urgency is absent. Lydian sounds floating because the subdominant anchor is absent. Phrygian sounds dark and descending because the half-step pull has been inverted. Dorian and Aeolian sound minor-inflected because the third is lowered. Locrian sounds unstable because the tonic triad has been undermined. Each characteristic is the acoustic signature of the specific pull-modifications that define the mode.

The theoretical economy of this account provides both a unification and a description. The unification: all modes derive from the major-scale pull-structure by specific modifications, so they require no independent acoustic ground. The description: each mode's individual character is explained in terms of which pulls the composer has chosen to quiet. The pedagogy of Stage VIII in the Gradus curriculum follows directly: modes are introduced as colored alternatives to the major-mode pull-structure, available once the student has fully internalized major and minor, and their individual characters are explained in terms of what each mode has done to the gradient.

§7.2 · Chromatic-Mediant and Neo-Riemannian Devices as Tonal Movement

The chromatic-third relationship — the direct harmonic motion between triads whose roots lie a major or minor third apart — has a compositional history that substantially predates its theoretical systematization. Schubert's harmonic practice is saturated with third-related progressions: the shift from E major to C major in the opening of the C major String Quintet (1828), the sustained G♭ regions of the Wanderer Fantasy, the mediant substitutions that mark the development sections of the late piano sonatas. Brahms and Wagner extended the practice; by the end of the nineteenth century the chromatic-third relation had become a defining idiom of late-Romantic composition, operating alongside and sometimes in place of the fifth-based tonal motion that the major-minor system had established as its default.

The theoretical apparatus for understanding this practice arrived substantially later. Hugo Riemann's functional system could accommodate third relations within its Subdominantparallele and Dominantparallele categories, but the accommodation was strained: the third relation's characteristic effect — the sensation of having traveled a great tonal distance while the voice-leading surface remains smooth — resisted explanation within a framework organized around fifth-based root motion. The Neo-Riemannian theory that emerged in the late twentieth century addressed this gap directly. David Lewin's Generalized Musical Intervals and Transformations (1987) provided the mathematical framework for thinking about harmonic motion as a set of transformations applied to tonal objects. Brian Hyer and Richard Cohn (Cohn 1996, 2012 Audacious Euphony) developed the triadic application, identifying three elementary transformations as the building blocks of chromatic-third practice. Steven Rings's Tonality and Transformation (2011) extended the account toward a phenomenological description of how listeners track tonal motion through these operations. The resulting apparatus is systematic and precise. The question this section addresses is its status relative to the unified-ground theory: whether the Neo-Riemannian framework constitutes a competing theoretical foundation or operates as a device-catalog within the theory developed in §3–§5.

The three elementary operations

Three operations, each defined by its voice-leading behavior, constitute the transformational vocabulary:

The P operation (parallel) exchanges a major triad and its parallel minor — or vice versa — by moving one voice by semitone. The P transformation of C major (C–E–G) is C minor (C–E♭–G): the third E descends by semitone to E♭, while the root C and fifth G remain as common tones. P preserves the root; it changes modal quality through one semitone displacement.

The L operation (leading-tone exchange) connects a major triad to the minor triad whose root lies a major third above — or a minor triad to the major triad a major third below — by moving one voice by semitone. The L transformation of C major (C–E–G) is E minor (E–G–B): the root C descends by semitone to B, becoming the fifth of E minor, while E and G remain as common tones. L reaches a chord whose root is a major third above the original; only one voice moves, by one semitone.

The R operation (relative) connects a major triad to its relative minor — or a minor triad to its relative major — by moving one voice by whole step. The R transformation of C major (C–E–G) is A minor (A–C–E): the fifth G ascends by whole step to A, while C and E remain as common tones. R is slightly less parsimonious than P and L — a whole step rather than a semitone — which accounts for its somewhat gentler quality of motion.

Hexatonic and octatonic cycles

Applying P and L in alternation produces a closed cycle of six triads:

C major → P → C minor → L → A♭ major → P → A♭ minor → L → E major → P → E minor → L → C major

The cycle contains three major triads (C, A♭, E) and three minor triads (c, a♭, e), alternating major and minor. The roots of the major triads form an augmented triad; similarly for the minor triads. The traversal covers three chromatic-mediant regions — tonal distances at LCM 1, LCM 40, and LCM 20 from C — while each individual transformation displaces one voice by one semitone.

Applying P and R in alternation produces a closed cycle of eight triads:

C major → P → C minor → R → E♭ major → P → E♭ minor → R → G♭ major → P → G♭ minor → R → A major → P → A minor → R → C major

Eight triads, alternating major and minor, with roots drawn from the octatonic scale (alternating whole steps and half steps). The cycle traverses minor-third tonal distances through the same parsimony principle.

Large distance, small surface motion

Both cycles exhibit the defining property of Neo-Riemannian parsimony: each individual transformation moves one or at most two voices by a semitone or whole step, while the cumulative tonal displacement — measured by the gradient of §3.4 — is substantial.

From C major to A♭ major, reached in two transformations of the hexatonic cycle, the gradient distance is LCM 40: the root A♭ sits at ratio 5:8 from C. The listener experiences this as a genuine harmonic displacement, a transit to a region meaningfully remote from C. Yet the two transformations required moved one voice by a single semitone each; the soprano, alto, and tenor voices never leapt, never executed a large interval, never traversed the functional distance that a conventional fifth-based approach to A♭ major would have demanded.

The gradient analysis explains why this combination is acoustically achievable. Two triads whose roots are related by chromatic third share two common tones. The C major triad C–E–G and the A♭ major triad A♭–C–E♭ share the pitch class C, and E is only one semitone from E♭. In the P transformation from C major to C minor, C and G remain constant while E descends to E♭; in the L transformation from C minor to A♭ major, E♭ and G remain constant while C descends to B. At each step, two tones provide an acoustic bridge across the gradient distance while one tone moves. The common tones are acoustically salient because they correspond to low partials in the harmonic series of the respective roots: the shared C is the root of C major (LCM 1) and the major third of A♭ major, a low partial in both contexts. The two held tones carry the acoustic continuity; the single moving tone supplies the sense of harmonic motion; and the new root registers its gradient distance as the sense of having arrived somewhere different.

The common-tone relations between thirds-related triads are not accidental. They are a consequence of the consonant triad's interval structure: the root, major or minor third, and perfect fifth of any consonant triad overlap in pitch content with the consonant triads built a major or minor third above or below. This overlap follows from the partial-ratio relations among thirds-related pitches; the acoustic adjacency that makes voice-leading cheap is the same adjacency that makes common tones available as shared partials in the respective harmonic series. Parsimony works because the acoustic ground of §3 creates the conditions for it.

The device-catalog position

The Neo-Riemannian apparatus is not a competing theoretical foundation. It proposes no alternative acoustic ground; it does not argue that PLR operations are the fundamental category of harmonic motion; it does not reduce the tonal grammar to transformations or claim that fifth-based cadential motion is secondary to or derived from the transformational one. What Lewin, Cohn, Rings, and their colleagues have constructed is a systematic and precise catalog of techniques for traversing the tonal gradient by routes that exploit the common-tone structure of thirds-related triads to achieve large gradient distances through minimal voice-leading steps.

The unified-ground theory absorbs this catalog without theoretical conflict. The gradient of §3.4 explains why these routes work: two chords related by a P or L transformation share two common tones because their harmonic series overlap at those tones, and the single voice that moves provides a semitone traversal of the gradient in the smallest available step. The Neo-Riemannian framework maps the routes; the gradient explains the terrain. The question of which is foundational has a clear answer: the gradient is the acoustic account of why common-tone parsimony produces the particular tonal experience it does; the PLR catalog describes how composers navigate that terrain when they choose to traverse large distances with minimal surface disruption. The frameworks are complementary: one explains, the other maps.

The curriculum consequence reaches Stage VII. Step 26's treatment of chromatic voice leading — the bass-descent chains, the Omnibus progression, the CT°⁷ sequences — teaches the voice-leading surface before the harmonic destinations those surfaces produce. The Neo-Riemannian insight that chromatic mediants arise from parsimonious voice leading is the compositional reality the curriculum encodes in practice terms: the student who controls the lines controls the gradient traversal, and the chromatic-mediant chord identities are results of those traversals, not their causes. The device-catalog is in the student's hands from Stage VII onward; the gradient is the explanation behind it.

§7.3 · Atonality as Positive Achievement

The theory developed in §3–§5 is a theory of tonal music. §4.7 states its limiting condition: when tonal center is abandoned, the gradient loses its reference point, and the directed energies — the pulls — that the gradient generates are no longer operative. This section develops what that boundary means: both what the absence of tonal center removes from the harmonic dimension of musical experience and what it does not remove.

The abandonment of tonal center was deliberate. Arnold Schoenberg did not arrive at atonality through theoretical error or through the progressive collapse of a declining system; he arrived at it through a compositional decision to extend the logic of chromatic expansion to its limit. The Theory of Harmony (1911, revised 1922) documents the transition from the inside: Schoenberg understood the history of tonal practice as a progressive expansion of the set of pitch relationships the ear would accommodate as referential, from the strict modal practices of the Renaissance through the chromatic saturation of Wagner's harmony, and he traced the trajectory to the point at which no single pitch class retained priority as a center. The conclusion was not that tonal music had failed but that its vocabulary had been expanded to the point where composers faced a choice: maintain a center artificially through constraints that the expanded vocabulary would otherwise dissolve, or accept the dissolution and compose within the resulting pitch space on equal terms. The Style and Idea essays (1950) provide the retrospective account: the emancipation of dissonance was not the abandonment of acoustic order but the recognition that the ear could track pitch relationships as referential without a tonal gradient supplying the tracking mechanism.

What the gradient's absence removes

The gradient of §3.4 is defined relative to a tonal center. The distance from home that every LCM value expresses — C at LCM 1, G at LCM 6, B at LCM 120 — is distance from a specific reference pitch that the compositional context has established as home. Without a tonal center, the LCM values describe the acoustic complexity of interval relations between pitches, but they do not describe directed pressure toward a point of rest. A B sounding in an atonal context is acoustically complex relative to C; but C is not home, and B's semitone proximity to C carries no structural weight as a return to a referential center.

The pull as the basic unit of harmonic-dimension motion (§5.2) depends on the tonal hierarchy for its directed quality. The leading-tone pull from B to C — with its gradient gap of 119 units — is urgent because C is the tonal center and the leading tone's resolution is a return to it. In a context where C has no special structural status, the semitone B–C is an interval relation, acoustically complex, but not a directed energy toward home. The intensity of the pull, grounded in the gradient gap and the harmonic context, is a function of the listener's perception of tonal center; without that perception, source and destination remain identifiable but the directedness collapses.

The cadence as convergence (§5.6) is similarly unavailable. The four simultaneous gradient resolutions of the PAC — tonal, half-step, vertical, root-position — each depend on C functioning as the home tonic. Without that function, the pitches of V⁷ → I can still sound in sequence; their interval relations are unchanged; the acoustic complexity of the dominant seventh is unchanged. But the convergence onto the home tonic — the four simultaneous arrivals at the gradient's lowest point — requires a lowest point to converge on. Atonal practice has removed it.

What the absence does not remove

The loss of the tonal gradient does not extinguish musical movement. Schoenberg and his successors demonstrated — and the analysis of the strongest atonal works confirms — that motion, directed energy, tension and release, structural articulation, and formal coherence are achievable through dimensions of musical experience that the harmonic-dimension gradient does not govern.

Rhythmic motion and metric articulation provide directed energy independent of pitch-ratio gradient. Strong-beat arrivals, metrical regularity, rhythmic settling after rhythmic complexity, the return of a rhythmic pattern after development — all articulate positions of relative stability and instability in a dimension the gradient does not enter. Bartók, Messiaen, Stravinsky, and Ligeti organized large-scale musical motion primarily through rhythmic means alongside or instead of harmonic-dimension gradient energy.

Registral motion carries a directed quality perceptible independently of harmonic dimension. An ascent to the highest pitch of a melodic line followed by descent creates an arch whose peak is a structural point; the listener tracks registral motion as directional without requiring a tonal center to supply the destination. Register and spacing are a primary parameter in Webern's mature works, where the vertical distribution of pitch classes across octaves is a foreground compositional concern.

Dynamic contour — the shaping of volume from silence to fortissimo and back — creates tension and release through non-pitch means. Crescendo toward a structural arrival supplies tension; decrescendo at the arrival provides release. This is acoustic in a different sense from the ratio-based gradient: it acts on the listener's attention and arousal rather than on pitch-ratio expectations, but it is none the less capable of producing the experience of directed motion and arrived resolution.

Timbral change and instrumentation articulate formal boundaries through non-pitch means: the entry of a new instrument, the shift from sustained to articulated playing, the change of color from one register of a voice or instrument to another. Schoenberg's Klangfarbenmelodie concept — the idea that a succession of timbral changes can constitute a melodic line — names the extreme of this principle.

Motivic transformation is the most distinctively atonal organizing principle. A short melodic or rhythmic figure — defined by its interval content and rhythmic profile — is subjected to the twelve-tone transformations of inversion, retrograde, retrograde inversion, and transposition, and the ear tracks the relationship between the original figure and its variants as the principal organizing logic of the texture. Anton Webern's Symphony Op. 21 (1928) is the most refined example in the literature: every pitch-class succession in the work is derivable from the row and its P, I, R, and RI forms at specific transposition levels, and the structural coherence of the work rests on the listener's capacity to track these transformational relations rather than on any gradient-based directed energy. The four-movement Romantic symphony compressed into a ten-minute twelve-tone study; the organization is unmistakable on attentive listening, but it is motivic-transformational organization, not tonal.

The honest limit of this theory

The theory of tonal movement does not explain atonal music. It describes the acoustic ground of the major-minor-modal system, the gradient that system generates, and the pulls that operate within it. Webern's Symphony is not a set of gradient traversals; its motivic-transformational organization is a different kind of musical logic, and any attempt to explain it as a sequence of LCM-distance events would be inaccurate and reductive.

This limit is not a deficiency. A theory of tonal music that could also explain atonal music by the same principles would be a theory of something other than tonal music — a theory of musical sound in general, operating at a level of generality from which the specific character of tonal motion would be invisible. The present theory is a theory of a specific practice: the practice of organizing pitch relationships within the major-minor-modal system by virtue of the acoustic gradient that system generates. Atonal practice is a different thing, and it deserves its own account — the phenomenological and transformational accounts that Lewin, Rings, Hasty, and Hanninen have developed are better suited to atonal music's actual organizing principles than any extension of gradient-pull analysis could be. Fred Lerdahl's Tonal Pitch Space (Oxford UP, 2001) develops a generalization that takes a different path: Lerdahl extends the same hierarchical-distance apparatus that organizes tonal music (his Chapters 2–6) into atonal pitch space (Chapter 8), supplying for the post-tonal repertoire the quantitative perceptual-distance framework the present treatise supplies only for the tonal repertoire. Lerdahl's generalization and the present theory's tonal-restriction are complementary rather than competing: both treat hierarchy as the primary phenomenon, both treat distance as the primary metric, and they disagree only about whether the apparatus should be extended past its tonal-foundational case or held within it. The Gradus curriculum's Stage IX treatment of atonality (Appendix C, Stage IX) is informed by Lerdahl's apparatus alongside the transformational tradition.

The boundary is an honest one. The theory knows its domain and stays in it.

Stage IX

The Gradus curriculum's Stage IX treats atonality as a positive achievement: a deliberate compositional decision to work with the full complexity of the chromatic pitch vocabulary without a gradient center, organizing sound through the alternative means that Schoenberg and his successors developed. The student arrives at Stage IX having internalized the gradient-based organization of Stages I–VIII. Atonal practice is introduced not as the inevitable historical sequel to late-Romantic harmony — as though the logic of chromatic expansion had no alternative but to dissolve into atonality — but as one of several compositional choices available to a composer who understands the tonal system well enough to set it aside.

The serial technique, the set-class vocabulary, and the motivic-transformational principles of post-tonal practice are taught in Stage IX as domain-specific tools for working in the atonal domain. The student who completes Stage IX has the full range: the tonal gradient of Stages I–VIII and the alternative organizing principles of Stage IX. The understanding of tonal motion that the earlier stages develop is not superseded by Stage IX but is placed in perspective by it. The composer who commands both knows, with genuine understanding, what she is choosing when she chooses either.

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§8 · Coda

Three characterizations define what the theory of tonal movement is.

The theory is unified in its acoustic ground. The major scale, the natural minor scale, the six modes, the altered scales of late and post-tonal practice — all derive from a single acoustic fact: the harmonic series emitted by any pitched sound. Major is the natural alignment of the lowest distinct partials of the harmonic series. Minor is Aeolian with the cadential corrections that the acoustic ground predicts as strongest: the raised seventh restoring the half-step pull that the natural seventh's whole-step approach lacks. The modes are positions within the major-scale pull-structure in which specific gradient pressures have been quieted, redirected, or inverted. Atonality is the deliberate setting-aside of the gradient's reference point.

This unification takes a position in a long theoretical dispute. The Leipzig dualist tradition — Hauptmann's double-lemniscate, Oettingen's Klang, Riemann's functional dualism — required two coordinate acoustic grounds to account for the difference between major and minor. The present theory resolves the difficulty by rejecting the premise: minor does not require a separate acoustic ground because it is not a coordinate system with major. It is a historical modification of major's pull-structure, made by performers and theorists over centuries in response to the cadential demands of a single acoustic ground. One ground, one gradient, one unified account of the entire major-minor-modal territory.

The theory is hierarchical in its structural account, but at the scales the listener perceives. The gradient operates at three levels the ear actually tracks: scale degrees within a perceived key, chords within that key, and the tonal region the perceived key itself occupies relative to the recently established home. At each level the same principle applies — a configuration at greater gradient distance from the perceived tonal center is less stable and under directed pressure toward a closer one. The hierarchy is grounded here in the acoustic series and in the listener's auditory tracking of that series at the timescales auditory short-term memory operates on. The theory declines to extend the gradient analysis to whole- movement scale because the empirical record on long-range tonal perception does not support a perceptual claim at that scale (§2.4.4). Schenker approached the larger hierarchy analytically with unsurpassed precision, tracing the structural voice-leading of a composition from its motivic foreground through its structural middleground to the fundamental linear descent of the Ursatz, and the present treatise engages his apparatus as an analytical and notational achievement (§2.4) without offering to ground it perceptually.

A reader may ask: if the gradient analysis does not extend to whole-movement scale, what does account for the listener's experience of a long movement as coherent? The treatise's answer has several parts. First, the listener experiences a long movement as a sequence of local moves the gradient does describe — phrase-by-phrase pull and resolution, modulation as the local home shifting, each cadence's four-gradient convergence (§5.6) — and the cumulative effect of these well-formed local moves is much of what makes the movement hold together at every scale the ear actually tracks. Second, the non-harmonic dimensions of music — rhythm, meter, motivic transformation and return, registral contour, dynamic shaping, timbral and textural articulation, formal articulation — carry their own organizing principles, with their own theoretical literatures (§1.3), and the listener tracks these alongside the local harmonic gradient. Third, the aesthetic principles developed in Volume 2 — proportion, balance, contrast, variety, economy, clarity, and order — operate across all scales and are part of what the listener perceives in a long movement as coherence. The principle of Order in particular operates at every scale, from the harmonic series of a single tone through the motive, the phrase, the section, and the movement: order is structure that the listener perceives as intentional rather than random, and it is what the listener connects with when motivic material returns, when sections balance, when a movement closes on the structure it opened with. The treatise declines to extend gradient analysis past the timescales the ear tracks; it does not deny that other principles fill the role at larger scales, and Volume 2 develops those principles where they do their work.

The theory is dynamic in its central claim. Tonal music is motion. The pull is the basic unit — a directed energy from a gradient position above the destination toward a lower one — and the cadence is the prototype of maximum convergence: the moment at which multiple independent pulls resolve simultaneously onto the same arrival. The elements of the tonal vocabulary that the listener tracks — the passing tone resolving within a beat, the suspension across a bar, the secondary dominant tonicizing a chord, the modulation shifting the local home — are arrangements of these pull-events at the timescales the ear operates on, some resolved immediately and some held briefly across a phrase before the resolution arrives. The theory does not offer a new analytical method; it offers an account of what tonal music is doing in the listener's ear, grounded in the acoustic facts that the ear hears before the mind names them.

The pedagogical consequence of this account is Volume 2. A theory that takes movement as the central phenomenon, that grounds theoretical vocabulary in the acoustic facts the ear already knows, and that treats the history of tonal practice — from Fux's modal counterpoint through the chromatic saturation of the gradient — as the curriculum it actually is, has a specific pedagogy to offer. Volume 2 works out that pedagogy in practice: the ten stages of the Gradus curriculum, the composition practice that produces composers by having them write, the score study that shows the theory's claims in the work of the composers who lived them, and the discipline that forms the student's ear and hand before it forms the student's analytical vocabulary.

This treatise is work in progress. The acoustic ground is settled; the principal claims are staked; the predecessor systems have been engaged. What remains is the application that every good theory is waiting for: the student who internalizes the pull as a felt reality, the composer who hears the gradient at work in every bar she writes, the teacher who recognizes in the student's draft exactly which pull is missing or mismanaged. The theory is not finished when its claims are written down. It is finished when the composer writes music that moves.

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Appendix A · Glossary

Each entry gives (I) the conventional definition — the term as it appears in the standard theoretical literature — and (II) the operational definition used in this treatise, which is the definition the student of composition needs to act on. Where the two definitions converge, the note is brief. Where they diverge, the divergence is the theoretical claim.

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Acoustic ground (also: unified acoustic ground) (I) The harmonic series: the sequence of partial frequencies (f, 2f, 3f, 4f, …) produced by any vibrating body at fundamental frequency f. (II) The single physical basis from which all tonal relations — the major triad, the pull-structure of the major scale, the hierarchy of scale degrees, and the relative acoustic distance of remote keys — are derived without appeal to a parallel system. The claim "unified" means: one ground for major and minor both, not two coordinate systems of equal status.

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Cadence (I) A harmonic arrival that closes a phrase, section, or movement. Classified by the chords involved (perfect authentic, imperfect authentic, half, plagal, deceptive, Phrygian) and by the completeness of closure. (II) The point at which multiple independent gradient pressures converge onto the same arrival simultaneously. The perfect authentic cadence is the prototype because it achieves the maximum number of simultaneous gradient resolutions: the tonal gradient (V → I by descending fifth), the half-step gradient (7̂ → 1̂, the leading-tone pull), the vertical gradient (the chordal seventh resolving to the tonic's third), and root-position arrival on a metrically strong beat. Alternative cadence types are calibrated by how many of these four pressures they withhold.

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Chord function (I) The role a chord plays in a harmonic progression, classified as tonic (I, vi, iii), subdominant (IV, ii), or dominant (V, vii°) in the Riemannian functional scheme. (II) A chord's gradient position relative to the tonal center. Chords near the tonic (I at LCM 1; vi at LCM 15; iii at LCM 20) have lower gradient positions and serve as points of rest; chords farther from the tonic (ii at LCM 72; vii° at LCM 120) have higher gradient positions and function as active, directed harmonies demanding resolution. Dominant function is the acoustic consequence of the dominant chord's gradient position (LCM 6), its leading-tone component (B at LCM 120), and the chordal seventh (F at LCM 28 in context) — the concentration of these pulls toward the tonic constitutes what "dominant function" means at the harmonic-dimension level.

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Consonance (operational definition) (I) Acoustically: intervals whose frequency ratios are expressible in small integers (octave 2:1, fifth 3:2, fourth 4:3, major third 5:4, minor third 6:5). Perceptually: intervals that are heard as stable, at rest, not requiring resolution. (II) A low LCM position relative to the perceived tonal center. A tone is consonant in context to the degree that its harmonic-series ratio to the tonic is expressible in small integers and its gradient distance from home is small. Consonance is relational — it describes a position on the gradient map — not an intrinsic property of an interval in isolation. C is consonant relative to C (LCM 1); E is consonant relative to C (LCM 20); B is dissonant relative to C (LCM 120), though in the key of B major it would be the tonic, i.e., the most consonant position of all.

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Dissonance (operational definition) (I) Acoustically: intervals with complex frequency ratios, producing acoustic beating. Perceptually: intervals heard as unstable, tense, requiring resolution. (II) A high LCM position relative to the perceived tonal center. A tone is dissonant in context to the degree that its gradient distance from home is large and its pull toward a lower-gradient destination is strong. Dissonance is the creative material of tonal movement: it produces the directed energy that drives music forward in time. Its dual nature (§4.9) means that a dissonance both creates tension and provides the path of its own resolution — the passing tone is dissonant for one weak beat and releases as the step through it completes. The creation and release are the same gesture.

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Distance from home (also: gradient distance) (I) Not a standard term in conventional theory. (II) The LCM value of a tone's harmonic-series ratio relative to the perceived tonal center. Tonic = LCM 1 (distance zero); dominant root = LCM 6; major sixth = LCM 15; major third = LCM 20; subdominant root = LCM 12; supertonic = LCM 72; leading tone = LCM 120. Higher LCM = greater distance from home = more directed pull back toward the tonic. Distance from home is what the listener perceives as tension in the harmonic dimension.

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Gradient (also: consonance gradient) (I) Not a standard term in conventional theory. (II) The ordered set of gradient distances from the tonal center, ascending from LCM 1 (tonic) through LCM 120 (leading tone) and beyond. Movement in tonal music is motion along this gradient: from higher to lower (toward home) in directed tonal motion; from lower to higher (away from home) in passages of departure, intensification, and harmonic preparation. The gradient is the geometry of tonal space.

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Horizontal lens (I) Not a standard term. (II) One of the three analytical perspectives (§6) for resolving ambiguity in movement-function analysis. The horizontal lens asks: in the temporal motion of this voice from its previous position to its next, what is this tone's role — transit, suspension, departure, arrival? It brackets the tone's gradient position in the key and its chord membership. The horizontal lens is the lens of the line, most relevant in counterpoint and in contexts where the student's compositional question is about voice-leading motion rather than harmonic function.

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LCM (Least Common Multiple consonance measure) (I) Not a standard term in conventional theory. (II) The least common multiple of the numerator and denominator of a tone's simplest just-intonation ratio relative to the tonal fundamental (after octave-reducing both tones to the same octave). LCM measures harmonic distance: the tonic's LCM is 1 (ratio 1:1); the dominant's root (G relative to C) is at LCM 6 (ratio 3:2, LCM of 3 and 2); the subdominant root is at LCM 12 (ratio 4:3); the major sixth (A relative to C) is at LCM 15 (ratio 5:3). Higher LCM = higher acoustic complexity = greater distance from home = stronger directed pull back toward the tonic. The LCM measure is adapted from Gustin 1969 and applied here as the operational index of gradient distance.

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Modal harmony (as used in this treatise) (I) Harmonic practice derived from the diatonic modes (Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian) rather than from major-minor tonality. (II) The major scale's pull-structure with specific gradient pressures deliberately quieted, redirected, or inverted. Each mode = major minus one or more of its defining pulls. Mixolydian quiets the leading-tone pull (♭7̂ replaces the sharp leading tone); Lydian removes the subdominant anchor (♯4̂ displaces the natural fourth); Phrygian inverts the half-step direction (♭2̂ creates a descending rather than ascending semitone pull to the tonic). Modal harmony is the use of the acoustic ground with reduced cadential urgency, not a separate acoustic system.

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Multi-layer resolution (also: convergence) (I) Not a standard term in conventional theory; described in discussions of cadential strength and finality. (II) The simultaneous resolution of multiple independent gradient pressures onto the same arrival. The perfect authentic cadence is the prototype: the tonal gradient (V → I), the half-step gradient (7̂ → 1̂), and the vertical gradient (chordal seventh to tonic third) all arrive on the same chord, at the same moment. Multi-layer resolution is the structural principle behind the PAC's quality of absolute finality at the local level. The treatise confines the claim to the local moment of arrival; PACs at the ends of larger sections are iterations of the same local phenomenon rather than convergences operating at architectural scale (§2.4.4, §4.10).

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Pull (I) Not a standard analytical term; used informally to describe the sense of directed tonal energy toward a specific destination. (II) The basic unit of tonal movement (§5.1): a directed energy from a gradient position above the destination toward a lower gradient position, in a specific harmonic context. Every pull has three components: a source tone (at some gradient position), a destination tone (at a lower gradient position), and an intensity (determined by the gradient gap between source and destination, amplified or attenuated by the harmonic context). The leading tone's pull is the strongest in the diatonic vocabulary (gradient gap 119, source at LCM 120, destination at LCM 1, approach by semitone). The pull of a passing tone is local and contextual (gradient gap of the passing position, resolved within the measure). Pulls operate at the timescales the listener's auditory short-term memory operates on — pitch-to-pitch, beat-to-beat, phrase-to-phrase, and across short sections in which a single tonal center remains the perceived home. The treatise does not posit pulls at architectural scale; what Schenkerian analysis represents as a "structural dominant prolonged across a development section" the listener typically hears, in real time, as a sequence of local tonal centers (§2.4.4).

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Stability hierarchy (also: stability/instability hierarchy) (I) The ordering of scale degrees, chords, and tonal regions by perceived stability in a tonal context. (II) The full gradient order from LCM 1 (tonic, most stable) through LCM 120 (leading tone, most unstable), extended to apply at three levels simultaneously: scale-degree stability (tone within the key), chord stability (chord within the key, based on its root's gradient position), and regional stability (tonal region within the key, based on the gradient distance of the region's tonic from the home tonic). The composer manages tension and rest by navigating the stability hierarchy at all three levels simultaneously.

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Tendency tone (I) A tone with a strong directional pull toward a specific resolution — the leading tone (7̂) as the primary example, with the seventh scale degree in minor (♭7̂) and the chordal seventh of V⁷ as secondary examples. (II) Any tone whose gradient position and harmonic context generate a directed pull toward a lower-gradient destination that is strong enough to be perceptible and compositionally significant. The tendency-tone concept extends the leading-tone example to all tones with significant gradient gaps: the chordal seventh of V⁷ (LCM 28, resolving to LCM 20), the ♭6̂ in minor (LCM 40, resolving to LCM 6), the augmented-sixth upper voice (resolving outward by semitone to the doubled dominant). The Gradus curriculum builds the harmony arc around tendency tones as the explicit organizing principle of Stage III onward: tonal motion is the story of pulls and resolutions, and tendency tones are the moments in the harmonic dimension where that story is most concentrated.

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Tonal center (I) The pitch (and its associated harmonic region) perceived as home base in a tonal passage — the point of maximum rest, the standard against which all other pitches are heard as stable or unstable. (II) The perceptual anchor of the gradient: the tone (and its associated region) whose gradient distance is defined as zero. A tonal center is not merely the most frequently occurring pitch or the first-heard pitch; it is the point toward which the gradient pressures of the entire passage converge. The tonal center is established by the pattern of resolution events: a tone functions as tonic when the gradient pulls of the passage repeatedly arrive there. This is the perceptual grounding of tonal center: not a theoretical postulate but a fact the listener hears when the resolutions of a passage consistently land on the same pitch.

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Tonal lens (I) Not a standard term. (II) One of the three analytical perspectives (§6). The tonal lens asks: where does this tone pull relative to the perceived tonal center? What is its gradient position in the key, and toward what lower-gradient destination is it directed? It brackets the tone's chord membership and its linear role in favor of a single question about key-level gradient position. The tonal lens is the lens of the key, most relevant in the foundational key-feeling exercises and in analysis of large-scale tonal architecture.

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Vertical lens (I) Not a standard term. (II) One of the three analytical perspectives (§6). The vertical lens asks: how does this tone function within the chord it belongs to? What is its structural position inside the current harmonic configuration, and what does that position require of it? It brackets the tone's key-level gradient position and its linear context. The vertical lens is the lens of the chord, most relevant in part-writing instruction and in contexts where the student's compositional question is about the specific obligations of a chord tone — what must move, how far, and in what direction.

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Voice-leading parsimony (parsimonious voice leading) (I) A feature of certain chord progressions (particularly chromatic-mediant relations) in which smooth minimal surface motion — one semitone per voice, often — achieves large harmonic displacements. (II) The acoustic mechanism by which large gradient distances (LCM 20–40, the chromatic-mediant band) are traversed with minimal surface motion: thirds- related triads share two common tones, and the remaining voice moves only one or two semitones, providing an acoustic bridge across the large gradient gap. The PLR operations of Neo-Riemannian theory (Lewin 1987, Cohn 1996) are the formal apparatus for describing these moves; the gradient analysis explains why they sound as they do — the listener tracks both the large gradient displacement (surprise) and the smooth surface motion (intelligibility) simultaneously.

Appendix B · The Ten Foundational Claims, In Order

The following is a compact statement of the ten foundational claims of the theory of tonal movement, reproduced from §4 in their original order for quick reference. Each claim is stated in one paragraph. A reader who holds all ten simultaneously has the system's spine.

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Claim 1 — The Harmonic-Series Foundation. Every pitched sound contains its harmonic series: the partial frequencies 2f, 3f, 4f, 5f, … above any fundamental f. The first three distinct pitch classes above a fundamental yield, after octave reduction, the major triad — the dominant (at partial 3, ratio 3:2), the major third (at partial 5, ratio 5:4). The major triad is not a cultural convention; it is the first three distinct partials of any pitched sound's natural spectrum. Tonality is grounded in this physical fact.

Claim 2 — Movement Is the Central Phenomenon. Tonal music in its harmonic dimension is, in its central phenomenon, movement. Every tone, chord, and tonal region at gradient distance from the perceived tonal center carries inherent directionality toward positions of lesser distance; the listener perceives this directed energy directly, before any analytical label is attached. Dissonance and consonance are not the mechanism of movement but its coordinates: the LCM measure of §3.4 quantifies the gradient that movement traverses. The basic unit is the directed pull, and the principle holds at the timescales the listener's auditory short-term memory operates on — pitch-to-pitch, beat-to-beat, phrase-to-phrase, and across the short sections in which a single tonal center remains the listener's perceived home. The passing tone, the chordal seventh, the suspension, the secondary dominant, the brief modulation that shifts the local home all instantiate the same phenomenon at these scales. The treatise does not extend the claim to whole-movement scale, on the empirical ground that perceived tonality decays as the establishing context recedes in time (§2.4.4).

Claim 3 — Distance from Home. Dissonance is distance from home; consonance is arrival at home. "Home" is the harmonic-series fundamental — the tonic. The LCM of a tone's just-intonation ratio to the tonic measures this distance: the tonic is at LCM 1 (zero distance), the dominant root at LCM 6, the subdominant root at LCM 12, the major third at LCM 20, the supertonic at LCM 72, the leading tone at LCM 120. Higher LCM = greater distance from home = stronger directed pull back toward the tonic.

Claim 4 — Major as Natural. The major triad is the first three distinct partials of the harmonic series. The dominant chord is the third partial — the first new pitch above the octave — which is why V → I is the universal cadence of the major-minor vocabulary. The leading tone is the 15th partial (ratio 15:8), far up the spectrum; its half-step proximity to the tonic combined with its extreme spectral distance is the acoustic definition of urgency. The chordal seventh of V (ratio 7:4 in just intonation) is physically dissonant against the harmonic-series ground and must resolve. Major's characteristic behavior is dictated by acoustic physics.

Claim 5 — Minor as Modified Mode. Minor is Aeolian — the relative mode of the major scale, using the major scale's diatonic collection starting from the sixth degree — plus centuries of cadential adjustments. The raised seventh in harmonic minor (the leading tone, restoring the half-step pull to the tonic) was introduced by performers via musica ficta because the natural Aeolian whole-step approach from ♭7̂ to 1̂ lacks the half-step urgency the acoustic ground predicts as the strongest cadential energy. The adjustment is itself an instance of Claim 2: performers reaching for the pull the harmonic-series ground requires at a point of structural closure. Minor is the historical record of composers correcting the cadential deficit of the Aeolian mode.

Claim 6 — Modal Harmony Deliberately Quiets the Strongest Pulls. Each mode of the diatonic collection is the major scale with specific gradient pressures modified or removed. Mixolydian removes the leading tone (♭7̂ replaces the sharp seventh, reducing the gradient gap from 119 to 27); Lydian removes the subdominant anchor (♯4̂ displaces the natural fourth whose subdominant gravity pulls toward the tonic from above); Phrygian inverts the half-step direction (♭2̂ above the tonic creates a descending semitone pull mirroring the leading tone's ascending one). Modal music is the harmonic-series principle relaxed for color. Movement still occurs, but the cadential urgency of major and minor is deliberately absent.

Claim 7 — Atonal Music Sets Aside the Tonal Hierarchy Entirely. Without a tonal center, there is no gradient reference point: the LCM distances have no home tonic to be measured from, and the consonance/ dissonance gradient that organizes the tonal vocabulary loses its anchor. Leading-tone pulls exist acoustically but not structurally — without an established tonic, the pull has nowhere to arrive. The PAC convergence is unavailable without the tonal hierarchy. Movement does not disappear; it is produced by other means: rhythmic variation, registral motion, dynamic contour, timbral change, motivic transformation. Atonality is a positive compositional achievement — a deliberate choice to compose with a different toolkit, not a failure of the tonal system.

Claim 8 — The Stability/Instability Hierarchy. Once a tonic is established, every element of the tonal vocabulary has a position on a stability gradient. Tonic is most stable (LCM 1); the fifth and third of the tonic triad are stable (LCM 6, LCM 20); the subdominant and supertonic are restless; the leading tone is most unstable (LCM 120). The hierarchy extends at three levels simultaneously: scale-degree stability within the key; chord stability, determined by the chord root's gradient position; and regional stability, determined by the gradient distance of a tonal region's tonic from the home tonic. The composer manages tension and rest by navigating this three-level hierarchy at every moment of compositional decision.

Claim 9 — The Dual Gesture of Movement. Movement at the local scale is a dual gesture: every traversal of the gradient simultaneously creates and releases directed tension. The passing tone is the clearest case — it is dissonant on a weak beat (high gradient distance from the harmonic context), and the stepwise motion through it simultaneously creates and releases the tension. Pull and release are the same gesture, not two events. Because the dissonance under analysis is not a state but a directed event, the moment a tone occupies a high-LCM position is a moment of motion through that position rather than residence at it. This dual nature explains why stepwise motion is the foundation of melodic line — every step through a dissonance creates and releases tension in the same motion — and why the suspension is more dramatic than the passing tone: the suspension delays the release, holding the dissonance across the bar, so creation and release become separated in time, the dissonance accumulating before it releases. The characteristic ache of the suspension is the audible signature of a dual gesture stretched.

Claim 10 — The Cadence as Multi-Layer Resolution. The perfect authentic cadence resolves multiple independent gradient pressures simultaneously: the tonal gradient (V → I, dominant root at LCM 6 descending to tonic at LCM 1), the half-step gradient (7̂ → 1̂, leading tone at LCM 120 collapsing by semitone to LCM 1), the vertical gradient (chordal seventh resolving to the tonic's third), and root-position arrival on the tonic in the bass. This convergence is why the PAC is heard as absolutely final in the harmonic dimension at the moment of arrival: multiple independent pressures release on the same beat, onto the same chord, leaving nothing unresolved at the local level. The claim is local. PACs at the ends of larger sections are iterations of the same local phenomenon — each one the four-gradient convergence at its own moment of arrival — not convergences operating across an architectural span. The treatise does not extend the multi-layer-cadence claim to sonata-form recapitulation or to other whole-movement scales, because the empirical record on long-range tonal hearing does not support a perceptual claim at those scales (§2.4.4).

Appendix C · Curriculum Map

The Gradus curriculum comprises ten stages, forty steps, and 120 weeks (the four-year certificate program). Each stage is keyed to the foundational claims of Volume 1 that it serves, and to the specific vocabulary the student internalizes at that stage. The curriculum follows the historical order of theoretical discovery — from Fux's modal counterpoint through the Romantic chromatic expansion — because that is the order in which the acoustic problems presented themselves to composers and the solutions were found. The student lives the journey; the history is the curriculum.

The table below gives, for each stage and its constituent steps, (1) the primary foundational claim(s) being served, (2) the harmonic vocabulary being introduced, and (3) the key compositional skill being formed.

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Stage I — The Single Voice (Steps 1–7 · Weeks 1–18)

Foundational claims: 1 (harmonic series), 3 (distance from home), 8 (stability hierarchy).

Harmonic vocabulary introduced: intervals in their acoustic ratios; the major scale as the first full expression of the harmonic-series ground; the stability gradient of scale degrees (tonic most stable, leading tone most active); introduction of the minor scale as a modification of major.

Compositional skill: single-voice melodic writing with arc shape, tonal center awareness, and a rudimentary sense of pull toward the tonic. The student hears before she labels — the stability gradient is internalized as feeling before it is described as theory.

Primary volumes 1 cross-references: §3.1 (the harmonic series as natural fact), §3.4 (the stability hierarchy, first encounter), §4.1 (major as natural), §4.5 (minor as modified mode, introduced in Step 6–7).

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Stage II — Two Voices (Steps 8–13 · Weeks 19–36)

Foundational claims: 2 (movement = dissonance → consonance), 8 (stability hierarchy), 9 (dual nature of dissonance).

Harmonic vocabulary introduced: consonance and dissonance as the organizing polarity of two-voice counterpoint; the five Fux species as progressive encounters with dissonance types; passing tone (second species), neighbor tone, cambiata (third species); suspension as accumulated dissonance with delayed release (fourth species); florid counterpoint integrating all species (fifth species). The leading-tone pull is named and located at Stage II when the student writes the cadential leading-tone–tonic resolution.

Compositional skill: writing two independent voices, managing the pull-and-release of each dissonance type at the note-to-note level, and producing authentic cadences that converge the leading-tone pull with the tonal-gradient resolution.

Primary V1 cross-references: §4.2 (movement = dissonance → consonance, the operational principle; this is where the student first enacts it), §4.9 (dual nature of dissonance — the suspension illustrates this most concretely), §5.2 (pull, operational definition, in the context of the leading-tone pull first encountered at the cadence).

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Stage III — Harmony (Steps 14–16 · Weeks 37–48)

Foundational claims: 2, 3, 4, 5, 8, 10.

Harmonic vocabulary introduced: the third voice completes the triad; Roman numeral analysis introduces chord-level gradient positions (I at LCM 1, V at LCM 6, IV at LCM 12, vi at LCM 15, iii at LCM 20, ii at LCM 72, vii° at LCM 120); dominant-seventh chord as the confluence of leading-tone pull and chordal seventh (§3.5, §5.4); four cadence types — PAC, IAC, half cadence, plagal cadence — introduced and differentiated by the number of gradient pressures each resolves; minor mode's characteristic harmonic adjustments (raised sixth and seventh in harmonic minor).

Compositional skill: three-voice harmonic writing; managing the chordal seventh as a tendency tone requiring downward resolution; writing phrase pairs that close with a half cadence (provisional rest) and a PAC (full closure); first experience of the cadential convergence as a felt event rather than a formula.

Primary V1 cross-references: §4.1–4.5 (all four major-minor foundational claims are first enacted here in harmony rather than in melody), §5.4 (dominant seventh, chordal seventh pull), §5.6 (cadence as convergence — this is the stage where the PAC is first constructed systematically).

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Stage IV — Form (Steps 17–18 · Weeks 49–60)

Foundational claims: 3 (distance from home), 8 (stability hierarchy at the regional level), 10 (multi-layer resolution at the local cadence that closes each section).

Harmonic vocabulary introduced: modulation (pivot-chord, chromatic, direct, sequential, enharmonic) as the shifting of the local home from one key to another; phrase structure (sentence, period, hybrid); binary, ternary, rondo, variation form as compositional layouts on the page; key plan as a compositional parameter — the choice of which tonal regions to visit and in what order reflects the regional stability hierarchy.

Compositional skill: planning a piece across multiple phrases and short sections; choosing tonal regions (dominant region, relative major/minor, subdominant) that the local home will shift to during the piece; writing a complete binary-form short piece in which the local home shifts to the dominant in the first half and shifts back to the tonic in the second half. The student learns to plan the sequence of local homes — and to write the modulations that effect each shift — without being asked to claim that the listener perceives a single architectural key across the whole movement (§2.4.4).

Primary V1 cross-references: §5.5 (stability hierarchy at the regional level — the choice of modulation target is a choice of how far from home the piece will travel during a given section), §5.6 (multi-layer resolution at the local cadences that close each section of the binary-form piece, with the return to the home tonic as the final such local cadence).

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Stage V — Fugue (Steps 19–20 · Weeks 61–72)

Foundational claims: 2, 9, 10 — all three concentrated in the fugal apparatus at the local entry-and-episode level.

Harmonic vocabulary introduced: fugue subject as a tendency-tone gradient traversal (the subject typically introduces a pull and delays its resolution); fugal exposition as a systematic presentation of the tonic-dominant polarity; episode as harmonic elaboration through sequence; stretto as temporal compression of the pull-release cycle; final entry in the home key as the return of the subject to its original tonal position after the intervening episodes have shifted the local home elsewhere.

Compositional skill: writing a well-formed fugal subject (with clear gradient shape), a complete exposition with correctly answered subject entries, sequential episodes, and a final entry in the home key. The fugal structure is the harmonic principle of pull and release applied across multiple simultaneous voices and across the sequence of local homes the entries and episodes traverse. The treatise does not commit the student to claiming that the listener perceives a single architectural key across the whole fugue (§2.4.4); the local entries, the local episodes, and the local return are what the ear actually tracks.

Primary V1 cross-references: §4.2 and §4.9 (dissonance as movement material and dual nature — the fugal episode is sequential dissonance driving toward the next strong arrival), §5.6 (the fugue's stretto and final entry are multi-layer convergence events at the local level of each entry and each cadence).

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Stage VI — The Classical Style (Steps 21–25 · Weeks 73–90)

Foundational claims: 1–10 — all ten claims are now in operation simultaneously in the full classical harmonic vocabulary.

Harmonic vocabulary introduced: SATB voice leading in four parts; figured bass as the notation of vertical function; secondary dominants (V/V, V/vi, etc.) as local applications of the dominant-function gradient analysis; non-chord tones in their full classical taxonomy (passing, neighbor, suspension, anticipation, appoggiatura, escape, cambiata, pedal); borrowed chords (modal mixture, pivot chords from parallel minor); the sonata-form development as a large-scale elaboration of the dominant region before the recapitulation's multi-layer convergence.

Compositional skill: part-writing in SATB style according to voice-leading standards; realizing figured-bass exercises; writing a sonata-form exposition and development section; the capstone exercise of this stage is a complete piano sonata movement in the classical style, demonstrating command of the full harmonic vocabulary.

Primary V1 cross-references: §5.1–5.6 (the complete operational account of the pull as basic unit, tendency tones, compound chords, the stability hierarchy, and the cadence as convergence — all first enacted systematically at Stage VI), §6 (the three lenses are introduced here, as the compound harmonic vocabulary generates the ambiguity the lenses are designed to resolve).

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Stage VII — Romantic Harmony (Steps 26–30 · Weeks 91–102)

Foundational claims: 3 (distance from home, now pushed into the chromatic-mediant band, LCM 20–40), 8 (regional stability at greater gradient distances), and implicitly Claim 6 (modal harmony, as Romantic practice borrows modal qualities to expand the available pull-vocabulary).

Harmonic vocabulary introduced: chromatic voice-leading idioms (bass descents, chromatic inner-voice motion); the omnibus progression; chromatic mediants and PLR operations (§7.2); augmented-sixth chords in all three forms (Italian, French, German) as concentrated double-pull toward the dominant; altered dominants (♭9, ♯9, ♯11, ♭13) as enrichments of the V⁷'s pull-vocabulary; enharmonic modulation through the diminished seventh and augmented sixth; the Tristan chord as the chromatic saturation asymptote of the tonal gradient.

Compositional skill: writing chromatic progressions in the Romantic harmonic idiom; composing Lied-length pieces using chromatic-mediant relations and augmented-sixth preparations; achieving smooth voice-leading across large gradient distances via parsimonious motion.

Primary V1 cross-references: §7.1 (modes as deliberate quietings — Romantic modal borrowing understood as gradient-pressure modification), §7.2 (the Neo-Riemannian apparatus absorbed as gradient analysis — PLR operations as parsimonious gradient traversals), §5.5 (the stability hierarchy extended to chromatic-mediant distances, LCM 20–40).

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Stage VIII — Impressionist Harmony (Steps 31–33 · Weeks 103–108)

Foundational claims: 6 (modal harmony quiets pulls), 7 (atonality sets aside the hierarchy — as a limit being approached, not yet crossed).

Harmonic vocabulary introduced: modal color in the Impressionist sense (whole-tone scale, pentatonic, octatonic — each as a different modification or elimination of the gradient); parallel motion (removed voice-leading independence, creating coloristic effect rather than functional harmonic motion); the deliberate suspension of tonal-center clarity as a compositional resource; color and atmosphere as harmonic goals independent of directed gradient motion.

Compositional skill: writing pieces in which harmonic color is the primary compositional goal and directed gradient motion is attenuated or suspended; using parallel chords, whole-tone passages, and modal inflections as texture rather than as functional harmony.

Primary V1 cross-references: §4.6 (modal harmony deliberately quiets pulls — Impressionism pursues this quieting to its logical extreme), §7.1 (modes as deliberate quietings — the Impressionist modal palette as a further step in the modal-quieting spectrum).

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Stage IX — 20th-Century Vocabulary (Steps 34–36 · Weeks 109–114)

Foundational claims: 7 (atonality as positive achievement — the primary claim enacted here for the first time in the curriculum).

Harmonic vocabulary introduced: expanded tonality (Bartók, Hindemith, Shostakovich — retaining a tonal center but with chromatic and modal enrichments that push the gradient to its boundaries); atonality as the deliberate setting-aside of the tonal hierarchy; twelve-tone technique as a compositional method for organizing pitches in the absence of tonal gradient; post-tonal harmonic concepts (pitch-class sets, interval vectors, symmetry as an organizing principle in place of the gradient).

Compositional skill: writing a short piece in an expanded-tonal idiom and a separate short piece using twelve-tone technique; experiencing in composition the transition from gradient-governed motion to the alternative organizing principles of post-tonal music.

Primary V1 cross-references: §4.7 (atonality sets aside the tonal hierarchy entirely — this claim becomes pedagogically load-bearing here when the student composes without a tonal center for the first time), §7.3 (atonality as positive achievement — the theoretical basis for treating twelve-tone composition as a genuine compositional achievement rather than a negation of tradition).

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Stage X — The Composer's Ear (Steps 37–40 · Weeks 115–120)

Foundational claims: all ten, synthesized and internalized — the stage of integration, not introduction.

Harmonic vocabulary introduced: neo-Riemannian transformational analysis in compositional practice (PLR operations as compositional tools for deliberate chromatic-mediant planning); advanced tonal and modal vocabularies integrated with the student's personal compositional idiom; the capstone composition incorporating the student's developed harmonic voice.

Compositional skill: the capstone composition — a substantial work in the student's own idiom, demonstrating command of the harmonic vocabulary from Stages I–IX and the ear to choose, moment by moment, which gradient pressures to deploy, which to quiet, and which to set aside. The student who completes Stage X has internalized the acoustic ground, the gradient, the pull, and the cadential convergence not as theoretical abstractions but as the working vocabulary of her compositional practice.

Primary V1 cross-references: §8 (the coda's pedagogical consequence — the theory is finished when the composer writes music that moves), V2 §7 (the Integrated Practice System in its full operation).

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Cross-tabulation: claim to curriculum location

Foundational Claim	First pedagogical encounter	Stage of full enactment
1 — Harmonic-series foundation	Stage I (Step 1: intervals)	Stage III (the triad as the first three partials)
2 — Movement = dissonance → consonance	Stage II (species counterpoint)	Stage III onward
3 — Distance from home	Stage I (stability gradient felt)	Stage III (LCM values introduced), VII (chromatic distances)
4 — Major as natural	Stage I (major scale)	Stage III (V → I as the universal cadence)
5 — Minor as modified mode	Stage I (Steps 6–7)	Stage III (harmonic and melodic minor adjustments)
6 — Modal harmony quiets pulls	Stage VIII	Stages VII–VIII
7 — Atonality sets aside hierarchy	Stage IX	Stage IX
8 — Stability hierarchy	Stage I (heard), II (enacted), III (named)	Stages III–VI (all three levels operating)
9 — Dual nature of dissonance	Stage II (the suspension)	Stages II–VI
10 — Cadence as multi-layer resolution	Stage II (leading-tone pull)	Stages III and VI (PAC as convergence, sonata form)

Appendix D · Bibliography

The following bibliography is organized by lineage rather than alphabetically. The purpose is to allow the reader to trace the intellectual descent of each position the treatise takes, and to situate the sources that are endorsed, qualified, or rejected within the specific theoretical debates they belong to. Full publication details (publisher, page numbers, edition) are to be supplied in the scholarly apparatus of the completed treatise; the present form gives sufficient information for source identification.

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I. Acoustic Foundation

Sauveur, Joseph. "Système général des intervalles des sons" and related papers in the Mémoires de l'Académie royale des sciences, Paris, 1701–1713. The first systematic acoustic account of the harmonic series and its theoretical consequences for interval classification. Introduced "harmonics" and "fundamental sound" as technical terms. The treatise's Claim 1 (Harmonic-Series Foundation) rests on the physical account Sauveur inaugurated.

Helmholtz, Hermann von. On the Sensations of Tone as a Physiological Basis for the Theory of Music (1863; 4th German ed. 1877; English trans. Alexander Ellis, 1875; repr. Dover, 1954). The founding document of the acoustic-physiological approach to harmony. Chapters 10–14 establish the partial-tone theory that grounds the present treatise's account of consonance and dissonance. The beating theory of dissonance is Helmholtz's; the present treatise supplements it with the LCM distance measure (drawn from Gustin) rather than replacing it.

Hindemith, Paul. The Craft of Musical Composition (1937; English trans. Arthur Mendel, Schott, 1942). The theoretical basis for Hindemith's compositional method, grounded in the harmonic-series ordering of intervals. Chapter 2 (Series 1: the chord-root order derived from combination tones and the harmonic series) and Chapter 3 (chord analysis) provide the closest precedent in the pedagogical literature for the present treatise's acoustic- grounding of harmonic function. Gustin's Tonality (below) is a direct development of Hindemith's theoretical framework.

Gustin, Molly. Tonality (New York: Philosophical Library, 1969; Library of Congress card 68-18735). The most direct predecessor of the present treatise. The relevant chapters with the pages this treatise draws from:

• Chapter 1 (pp. 9–20), The Perceived Sound: the empirical foundation on which the LCM measure is built. §3.2 of the present treatise inherits the listener-rounding argument from this chapter.
• Chapter 2 (pp. 21–29), Consonance: the gradient measure itself, in Gustin's two-property form. §3.4 of the present treatise uses the LCM portion of this measure (the simplified Stolzenburg 2015 form, conceptually equivalent to Gustin's Property II).
• Chapter 3 (pp. 30–39), Roots: the operational definition of root in terms of the lowest power-of-2 in the set's just-intonation representation. The present treatise parts company with Gustin here, in favor of a perceptual account of root grounded in listener experience of tonal center (§2.2.2).
• Chapter 4 (pp. 40–44), The Diatonic Set: the result that the just-tuned seven-tone diatonic set on C has C as its predominant root. Used implicitly throughout §3 and §4.
• Chapter 5 (pp. 45–46), The Twelve-Tone Set: the result that no candidate twelve-tone set has a predominant root. Used by §4.7 as the formal grounding for atonality as a different practice.
• Chapter 6 (pp. 47–49), The Nineteen-Tone Set: the refutation of Yasser's evolutionary projection. Used by §3.3 as the closing aside on why the fifth-stacking principle stops at twelve.
• Chapter 7 (pp. 50–53), Mode and Key: the result that only Ionian and Aeolian-plus-leading-tone (= harmonic minor) produce single-rooted sets under leading-tone alteration. Used by §4.5 as the acoustic-mathematical argument for the major/harmonic-minor two-key system's privilege.
• Chapter 8 (pp. 54–66), Analysis: the dissolution of the Roman-numeral paradox via higher-order single-rooted sets. Gustin's framing is consonant with §6's three-lens disambiguation but the present treatise does not adopt her higher-order key notation.
• Chapter 11 (pp. 78–81), Tonality: the operational definition of tonality as "music in which the majority of adjacent tones, whether occurring simultaneously or consecutively, constitute sets having single predominant roots." Convergent with the present treatise's Claim 7.
• Chapter 12 (pp. 82–91), Aesthetics: the approving gloss on Schenker's tonal unfolding as a form of the common-measure principle. Cited at §2.4.5.
• Appendix B (pp. 95–96), Hindemith: Gustin's itemized disagreements with the Craft of Musical Composition. Context for the Hindemith placement in this bibliography section.

Yasser, Joseph. A Theory of Evolving Tonality (New York: American Library of Musicology, 1932; repr. Da Capo Press, 1975). The evolutionary projection that pentatonic → diatonic → chromatic → 19-tone is the natural developmental sequence of pitched music. Refuted in §3.3 of the present treatise on three independent grounds developed by Gustin (Chapter 6 above). The Yasser entry is included here for completeness because the refutation in §3.3 depends on the reader knowing what is being refuted; the present treatise does not endorse Yasser's program.

Predecessors in the integer-ratio tradition

Zarlino, Gioseffo. Le istitutioni harmoniche (Venice: Francesco dei Franceschi Senese, 1558; English trans. Guy A. Marco and Claude V. Palisca, The Art of Counterpoint, Yale UP, 1968 — Part III only). The Renaissance restriction of consonant intervals to the senario (the numbers 1 through 6), anticipating the harmonic-series first-six-partials reading. Zarlino's Part III on counterpoint also codifies the voice-leading obligation of the chordal seventh (descending step) that §5.2 and §5.6 of the present treatise inherit.

Mersenne, Marin. Harmonie universelle (Paris: Sébastien Cramoisy, 1636–37; English trans. Roger E. Chapman, Harmonie Universelle: The Books on Instruments, The Hague: Martinus Nijhoff, 1957 — the instruments books only). The first systematic measurements of overtones in pitched sound; the transitional document between the Pythagorean integer-ratio tradition and the modern psychoacoustic apparatus.

Rameau, Jean-Philippe. Traité de l'harmonie réduite à ses principes naturels (Paris: Jean-Baptiste-Christophe Ballard, 1722; English trans. Philip Gossett, Treatise on Harmony, Dover, 1971). The treatise that grounded harmony in the corps sonore — the harmonic series of a single pitched tone — and argued explicitly that the major triad is natural because it consists of the first three distinct partials of any pitched sound. The argument the present treatise inherits from Rameau, with refinements drawn from the subsequent literature, is essentially the corps sonore argument: §3.1, §4.1, and Claim 1.

Euler, Leonhard. Tentamen Novae Theoriae Musicae (St. Petersburg: Imperial Academy of Sciences, 1739). The first explicit mathematical formula for the consonance of a chord: the gradus suavitatis assigns each chord a numerical value such that lower values correspond to chords easier for the ear to perceive as unified. The formula depends on the least common multiple of the integers in the chord's lowest-terms ratio, plus an adjustment for how that LCM factors into primes. The LCM measure of §3.4 is essentially Euler's formula in its simplest form (the LCM component without the prime-factor weighting).

Twentieth- and twenty-first-century psychoacoustics

Plomp, Reinier, and Levelt, Willem. "Tonal Consonance and Critical Bandwidth." Journal of the Acoustical Society of America 38/4 (1965): 548–560. The canonical experimental paper on roughness and the critical band: listeners rate the consonance of pairs of pure tones, and the data show that perceived consonance correlates with the frequency separation between partials relative to the critical bandwidth of the ear. Plomp and Levelt argued that the consonance-minima at simple integer ratios are a derivative effect of partial coincidence rather than a primary phenomenon — the position the present treatise's §3.4 engages as "Family 2" and sets aside on the basis of Harrison and Pearce 2020.

Terhardt, Ernst. "Pitch, consonance, and harmony." Journal of the Acoustical Society of America 55/5 (1974): 1061–1069. The canonical paper introducing the virtual pitch mechanism: the auditory system searches the heard spectrum for the harmonic series of a virtual fundamental whose harmonics best explain the heard partials, and the consonance of a chord is the goodness-of- fit of this template-matching. The "Family 3" position of §3.4: harmonicity as the primary perceptual phenomenon, refined by Parncutt 1989 and Harrison-Pearce 2020.

Parncutt, Richard. Harmony: A Psychoacoustical Approach (Springer, 1989). The book-length application of Terhardt's virtual-pitch model to harmony. The present treatise's §2.2.2 cites Parncutt's work on minor-triad root perception as the empirical evidence that the perceived tonic of a minor triad is context-sensitive and can vary with voicing, inversion, and register — softening the strong "tonic-at-the-bass" claim against Riemann from a universal to a most-encountered-voicings claim.

Tenney, James. A History of Consonance and Dissonance (Excelsior, 1988). The standard historical-conceptual taxonomy of five distinct senses of "consonance" in Western music theory: melodic, polyphonic, contrapuntal, functional, and psychoacoustic. The present treatise's §3.4 ("A note on terminology") follows Tenney's taxonomy and clarifies which senses the LCM measure operationalizes.

Sethares, William A. "Local consonance and the relationship between timbre and scale." Journal of the Acoustical Society of America 94/3 (1993): 1218–1228. Tuning, Timbre, Spectrum, Scale (Springer, 1997; 2nd ed. 2005). The generalization of the consonance curve to inharmonic timbres: the consonance minima for any timbre can be computed from its specific partial structure; for harmonic timbres the apparatus reduces to the integer-ratio gradient; for inharmonic timbres (gamelan bronze, bell-tone) it predicts different local minima that match the scales those traditions have constructed. The present treatise's §3.1 cites Sethares for the scope restriction to harmonic-spectrum instruments; V2 §6.2 cites Sethares for the gamelan-inharmonic case in the "Order beyond the harmonic series" discussion.

Stolzenburg, Frieder. "Harmony perception by periodicity detection." Journal of Mathematics and Music 9/3 (2015): 215–238. The contemporary formal apparatus for the integer-ratio consonance measure: Stolzenburg derives a periodicity-based consonance measure essentially equivalent to Euler's gradus suavitatis — log of the LCM of the integers in the just- intonation ratio, with adjustments for rational-approximation tolerance — and validates the measure against the empirical chord-rating data of Schwartz, Howe, and Purves (Science 299, 2003) and of Bowling, Purves, and Gill (Proceedings of the National Academy of Sciences 115, 2018). The LCM measure of §3.4 is Stolzenburg's measure in its simplest form.

Harrison, Peter M. C., and Marcus T. Pearce. "Simultaneous consonance in music perception and composition." Psychological Review 127/2 (2020): 216–244. The contemporary meta-analytic synthesis: Harrison and Pearce re-analyze four large empirical studies of consonance rating (about 500 participants in aggregate) together with a 100,000-composition corpus and find that harmonicity — the degree to which a chord's combined spectrum matches a single harmonic series — is a stronger single predictor of consonance ratings than interference (roughness) is. The vindication of "Family 3" against "Family 2" cited in §3.4.

Marjieh, Raja, et al. "Is harmonicity a misnomer for cultural familiarity in consonance preferences?" Nature Communications 13 (2022): 7544. The recent argument that some portion of the harmonicity effect may reduce to cultural familiarity with harmonically-related sounds rather than to a hardwired auditory mechanism. The present treatise's §3.4 acknowledges the hardwired-vs-cultural-familiarity question is incompletely resolved but argues the gradient theory is robust either way: what the listener tracks and how the listener came to track it are distinct questions.

Twentieth-century theoretical alternatives engaged

Russell, George. The Lydian Chromatic Concept of Tonal Organization for Improvisation (New York: Concept Publishing, 1953; 4th ed. The Lydian Chromatic Concept of Tonal Organization, Volume One: The Art and Science of Tonal Gravity, 2001). The mid-twentieth-century alternative theory of tonal organization that took the Lydian mode as the primary scale of tonal music, more fundamental than the major scale. Substantially influential on the modal jazz of Miles Davis, Bill Evans, John Coltrane, and Ornette Coleman. Engaged in §3.4 (the "Aside: George Russell and the Lydian Chromatic Concept") and refuted on the ground that Russell counts only the direct-relation route to ratio simplicity and misses the inverse relation through which the subdominant F sits at a low LCM (12) from the tonic C. Russell's broader contribution — intervallic distance from the tonic as the working concept — is compatible with the present treatise; the specific Lydian-primacy claim is what the §3.4 aside rejects.

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II. Hierarchical Analysis

Sechter, Simon. Die Grundsätze der musikalischen Komposition (Leipzig: Breitkopf & Härtel, 1853–54, three volumes; partial English trans. C. C. Müller, The Correct Order of Fundamental Harmonies, New York: William A. Pond, 1871). The technical statement of the Viennese Stufenlehre — the theory in which the seven diatonic scale-degrees are the fundamental harmonic units of tonal motion. Sechter taught at the Vienna Conservatory and the Imperial Court Chapel until his death in 1867; Bruckner succeeded him at the Conservatory and continued teaching his method. Schenker studied at the Vienna Conservatory during Bruckner's tenure (1887–90) and inherited the Sechter-Bruckner Stufenlehre as the working apparatus of his own Harmonielehre (1906). The Sechter lineage matters for the present treatise because it locates Schenker's apparatus in a Viennese-conservatory tradition that is non-dualist in origin — predating Oettingen and unaffiliated with the Leipzig program — and so explains why Schenker's rejection of Riemann's Funktionstheorie (§2.4.1) was an inheritance, not an innovation. The Sechter scale-step theory is the direct ancestor of Schenker's Stufenlehre and an indirect ancestor of the present treatise's treatment of scale degrees as gradient-positions.

Schenker, Heinrich. Harmonielehre (Vienna: Universal Edition, 1906; English trans. Elisabeth Mann Borgese, ed. Oswald Jonas, Harmony, Chicago: University of Chicago Press, 1954). The first major statement of Schenker's theoretical project. The Stufenlehre (theory of scale-steps) and the analysis of harmonic function in terms of the tonic's fundamental pull are the aspects most directly relevant to the present treatise. The chapter on the minor mode (§13 in the German original) contains the Naturhorn argument: major as the natural sound of pitched tones, minor as an "artistic" creation that has no direct analogue in the lower partials. The present treatise's §2.4.3 engages this argument as a point of partial alignment with the unified-ground position and §2.4.4 qualifies the wider apparatus that follows from it.

Schenker, Heinrich. Der Tonwille: Flugblätter zum Zeugnis unwandelbarer Gesetze der Tonkunst, einer neuen Jugend dargebracht (Vienna: A. Gutmann, 1921–24, ten issues; English trans. ed. William Drabkin, Der Tonwille: Pamphlets in Witness of the Immutable Laws of Music, Oxford UP, 2004–05, 2 vols.). The pamphlet series in which Schenker conducted his most sustained polemic against Riemann's Funktionstheorie and developed the analytic methodology that culminated in Der freie Satz. The present treatise's §2.4.1 engages the Tonwille essays as the documentary record of Schenker's anti-Riemann position.

Schenker, Heinrich. Das Meisterwerk in der Musik (Munich: Drei Masken, 1925, 1926, 1930, three yearbooks; English trans. ed. William Drabkin, The Masterwork in Music, Cambridge UP, 1994, 1996, 1997, 3 vols.). The yearbook publication in which Schenker extended the Tonwille analytic program with more developed counter-demonstrations against function-theoretic analysis and worked through the analytic apparatus that Der freie Satz would codify. The present treatise's §2.4.1 cites the Meisterwerk essays as part of the documentary record of the Schenker-Riemann polemic.

Schenker, Heinrich. Der freie Satz (Vienna: Universal Edition, 1935; English trans. Ernst Oster, Free Composition, Longman, 1979). The mature statement of Schenker's theory: the Ursatz, the Urlinie, the Bassbrechung, and the theory of prolongation across structural levels. The present treatise takes the position (§2.4.4) that Schenker's analytic observations — the structural descent to the tonic, the prolongation of the structural dominant — are acoustically grounded in gradient traversals at architectural scale, and that the gradient analysis provides the acoustic basis Schenker's system requires but does not itself supply. The Foreword and Introduction to Der freie Satz contain the metaphysical and nationalist framing the present treatise sets aside (§2.4.4, qualification 3); the analytic apparatus is engaged as a predecessor, not superseded.

Cone, Edward T. "Analysis Today." The Musical Quarterly 46/2 (1960): 172–88. Reprinted in Benjamin Boretz and Edward T. Cone, eds., Perspectives on Contemporary Music Theory (Norton, 1972). The classic critique of the axiomatic structure of Schenker's Ursatz. Cone argues that Schenker presents the Ursatz as the underlying form of tonal music on functional grounds (it produces good analyses) without providing an independent foundation, and that the resulting circularity weakens the foundational claim even where it leaves the analytic apparatus productive. The present treatise's §2.4.4 (Qualification 1) reads Cone's critique as a call for the kind of acoustic grounding the gradient analysis supplies.

Beach, David. "The Functions of the Six-Four Chord in Tonal Music." Journal of Music Theory 11/1 (1967): 2–31; and "A Schenker Bibliography." Journal of Music Theory 13/1 (1969): 2–37. The Beach-1969 bibliography is the standard early survey of Schenkerian secondary literature; the 1967 essay engages the question of whether Schenker's structural-level hierarchy depends on contrapuntal facts or on functional axioms, and forms part of the documentary record §2.4.4 draws on.

Wason, Robert W. Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg (Ann Arbor: UMI Research Press, 1985). The standard secondary source for the Vienna lineage that produced Schenker. The chapter on Sechter is the reference for the present treatise's §2.4.2 account of how Schenker inherited a non-dualist Stufenlehre via Bruckner. The book also traces the parallel lineage that produced Schoenberg, which makes the Schenker-Schoenberg disagreement of the early twentieth century intelligible as a disagreement within a single Vienna conservatory inheritance rather than between two unconnected schools.

Salzer, Felix. Structural Hearing: Tonal Coherence in Music (Charles Boni, 1952; repr. Dover, 1962). The primary English-language presentation of Schenkerian analysis accessible to American theory pedagogy. Chapter 2's account of structural and prolonging progressions is the clearest exposition of the hierarchical concept the present treatise engages.

Lerdahl, Fred. Tonal Pitch Space (Oxford UP, 2001). The most empirically-developed contemporary theory of tonal hierarchy. Lerdahl extends the generative theory of tonal music (Lerdahl and Jackendoff, 1983) with a quantitative tonal-pitch-space metric — a distance measure between chords and keys grounded in the hierarchical proximity that Krumhansl's probe-tone experiments (1982, 1990) elicit from listeners. Lerdahl's distance metric is conceptually parallel to the present treatise's LCM measure: both are perceptual- distance measures over scale degrees, chords, and tonal regions; both are quantified rather than purely descriptive; and both produce rankings that agree on the main features. The accounts differ in their physical foundation. Lerdahl's metric is grounded in listener- perception experiments and treats the perceptual hierarchy as the primary empirical phenomenon; the present treatise grounds the perceptual hierarchy in the harmonic series as a physical fact, with the listener's perception of the gradient as the consequence of that ground rather than the ground itself. Lerdahl's apparatus is also explicitly compatible with both tonal and atonal music — his treatment of atonal pitch space (Chapter 8) extends the apparatus to chromatic-and-beyond contexts in a way the present treatise's tonal-restricted apparatus does not — and is engaged in §7.3 for that reason. Krumhansl, Carol L. Cognitive Foundations of Musical Pitch (Oxford UP, 1990) is the primary source for the probe-tone profiles that Lerdahl's apparatus and the present treatise's LCM measure are both calibrated against, and is engaged in §3.4 where the LCM ranking and the Krumhansl-Kessler ranking agree on the diatonic ordering (1̂ > 5̂ > 3̂ > 4̂ > 6̂ > 2̂ > 7̂).

Krumhansl, Carol L., and Edward J. Kessler. "Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys." Psychological Review 89/4 (1982): 334–368. The canonical probe-tone-paradigm paper that established the empirical tonal hierarchy. Listeners played an establishing C-major context (scale, cadence, or tonic triad) followed by a probe tone are asked to rate how well the probe tone "fits" the context; the averaged ratings produce a stable hierarchy that the subsequent literature has treated as the empirical benchmark for any theoretical account of perceived tonal fit. Engaged at length in §3.4 ("The Krumhansl-Kessler probe-tone confirmation"), where the LCM measure is shown to agree with the Krumhansl-Kessler ratings on the structural shape of the gradient. The two columns invert only at 3̂/4̂, with the small disagreement attributable to chord-function exposure asymmetries (F as the IV chord implies motion, not rest; E as a tonic-triad constituent implies rest).

Krumhansl, Carol L. Cognitive Foundations of Musical Pitch (Oxford UP, 1990). The monograph extending the probe-tone work into a full theory of tonal cognition. The pitch profiles for major and minor keys, the spatial representation of musical keys, and the modulation experiments all bear on §3.4's gradient claim and on §5.1's operational definition of movement (which the empirical record can in principle falsify). The companion volume to the Krumhansl-Kessler 1982 paper above.

Bharucha, Jamshed J. "Anchoring effects in music: The resolution of dissonance." Cognitive Psychology 16/4 (1984): 485–518. "Music cognition and perceptual facilitation: A connectionist framework." Music Perception 5/1 (1987): 1–30. The tonal-anchoring model: perceived tonality tracks the most-recent strong tonic rather than a persistent architectural one. The empirical apparatus that underwrites §2.4.4's scope restriction on long-range tonal hearing — listeners do not, on Bharucha's account, hold a movement-length tonal gestalt in working memory; they re-anchor as the music's recent context shifts. The present treatise's commitment to local-and-short-section scales (V1 §2.4.4, V2 §2.5, V2 §6.6) is informed by the Bharucha model.

Bigand, Emmanuel, and Bénédicte Pineau. "Global context effects on musical expectancy." Perception & Psychophysics 59/7 (1997): 1098–1107. The empirical study showing that listeners often fail to identify the global tonic at the end of long musical pieces, and that they fail to notice when a section of a Bach prelude is played transposed to a different key from the opening. The specific evidence for the scope restriction of §2.4.4 — perceived tonality does not extend across whole-movement spans the way the analytical apparatus represents.

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III. The Dualist Tradition (the position rejected)

Hauptmann, Moritz. Die Natur der Harmonik und der Metrik (Leipzig: Breitkopf und Härtel, 1853; English trans. W. E. Heathcote, The Nature of Harmony and Metre, London: Swan Sonnenschein, 1888). The founding document of modern harmonic dualism: the thesis that major and minor are fundamentally coordinate dialectical inversions, each requiring its own theoretical apparatus. Hauptmann's three primary intervals (octave, fifth, major third) receive the linguistic-philosophical gloss "I am / you are / he is" — identity, separation, return-in-otherness — and the major triad is the dialectical synthesis of the three. The downward reflection of the major triad to produce the minor triad is, in Hauptmann's own treatise, a conceptual operation; the physicalization of the dualism into a hypothesized undertone series is a later development (Oettingen 1866). The present treatise rejects the wider dualist program (§2.1–2.2) but acknowledges that Hauptmann's own theoretical position is metaphysically rather than acoustically committed and so the empirical-refutation arguments of §2.2.1 apply primarily to the Oettingen-Riemann development. Hauptmann taught at the Leipzig Conservatory from its founding by Mendelssohn in 1843 until his death in 1868; his institutional influence on the German theory pedagogy of the following decades is documented in the Cambridge History of Western Music Theory (Christensen, ed., 2002, chapter on Hauptmann by Caplin and the surrounding chapters on nineteenth-century German theory).

Oettingen, Arthur Joachim von. Harmoniesystem in dualer Entwicklung: Studien zur Theorie der Musik (Dorpat and Leipzig: W. Gläser, 1866). The rigorous harmonic-dualist system that gave Hauptmann's dialectical opposition a physical claim and that Riemann then developed into the mature Funktionstheorie. Oettingen held the chair of physics at the University of Dorpat (Tartu, Estonia) from 1865 to 1893; the institutional setting matters because it is a physicist's preference for mathematical symmetry that motivated the move from Hauptmann's dialectic to a hypothesized inverse spectrum. Oettingen's empirical claim rests on the phenomenon of combination tones (Tartini tones, difference tones), reinterpreted as evidence that the auditory system generates tones below sounding tones mirror to the upward overtones. The "phonic" / "tonic" terminology (phonisch bestimmt / tonisch bestimmt) coined in this treatise propagated unchanged to Riemann's functional notation and is the direct source of the lowercase-versus-uppercase function symbols (t, s, d versus T, S, D) still standard in German theory pedagogy. The present treatise's §2.1 develops these particulars; §2.2.1 supplies the empirical refutation of the inverse-spectrum claim.

Riemann, Hugo. The technical formulation of Riemann's dualism appears across several works of his middle period. Musikalische Logik. Eine Studie über den Ursprung der musikalischen Erkenntnis (Leipzig: C. F. Kahnt, 1873); Musikalische Syntaxis: Grundriß einer harmonischen Satzbildungslehre (Leipzig: Breitkopf & Härtel, 1877); Skizze einer neuen Methode der Harmonielehre (Breitkopf & Härtel, 1880); Handbuch der Harmonielehre (Breitkopf & Härtel, 1887; subsequent editions through 1929). The Skizze introduces the technical vocabulary of Schritt and Wechsel operations the Neo-Riemannian tradition would later inherit (§7.2). Vereinfachte Harmonielehre, oder die Lehre von den tonalen Funktionen der Akkorde (London: Augener, 1893; English trans. H. Bewerunge, Harmony Simplified, or the Theory of the Tonal Functions of Chords, London: Augener, 1895) is the pedagogical synthesis: a classroom textbook condensing two decades of technical work into a working teaching system. The functions Tonic (T), Subdominant (S), and Dominant (D) hold mirror-image positions in major and minor in the dualist notation inherited from Oettingen, and an A-minor triad is the "tonic" of the A-minor system with its functional root on E (the top tone). Despite the dualist basis the present treatise rejects (§2.2), Riemann's functional classification is acoustically recoverable as a three-region partition of the gradient space (§3.4), and the present treatise retains functional vocabulary in that spirit. Geschichte der Musiktheorie (Hildesheim: Olms, 1898; English trans. ed. Robert H. Mickelsen, History of Music Theory, Books I and II, University of Nebraska Press, 1962) is the historical-context companion to the theoretical program.

Riemann, Hugo. "Ideen zu einer Lehre von den Tonvorstellungen," in Jahrbuch der Musikbibliothek Peters für 1914–1915 (Leipzig: Peters, 1916): 1–26. English trans. Robert W. Wason and Elizabeth West Marvin, "Hugo Riemann's 'Ideen zu einer Lehre von den Tonvorstellungen,'" in Journal of Music Theory 36/1 (1992): 81–117. The late Riemann paper in which the dualist framework is recast in psychological rather than acoustic terms. Tonvorstellungen ("tone representations") — the listener's mental representations of pitched sound — become the proper subject of harmonic theory in place of the physical spectrum of sounding tones. The recasting acknowledged what the earlier work had denied — that listeners hear the perceived tonic of A minor at the bass, not at the top — and attempted to preserve the dualist structure as a property of mental representation while conceding the acoustic ground. The present treatise's §2.1 cites this paper as the document of dualism's empirical retreat within its own author's lifetime; §2.2.1 notes that the twentieth-century reception of Riemann is largely a reception of the 1893 Vereinfachte Harmonielehre rather than the 1914 Tonvorstellungen.

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IV. The Species and Figured-Bass Tradition

Fux, Johann Joseph. Gradus ad Parnassum (Vienna: Johann Peter van Ghelen, 1725; English trans. Alfred Mann, Norton, 1943). The foundational document of the species-counterpoint pedagogical tradition. The Gradus curriculum (Appendix C above) begins with Fux's five-species method as the historical entry point into the grammar of two-voice writing. The species taxonomy — note-against-note, two-against-one, four-against-one, syncope, florid — organizes the student's encounter with the dissonance vocabulary in the order the acoustic principle predicts: from the simplest (consonant) to the most complex (dissonant passing, suspension, ornamental).

Mozart, Wolfgang Amadeus. Thomas Attwood Lessons (1784–85; ed. Derek Remes, A-R Editions, 2019). Mozart's private teaching materials from his lessons with Thomas Attwood, preserved in Attwood's study notebook. The lessons demonstrate the personal-tutor method of the species-counterpoint and figured-bass tradition at its most direct: a master demonstrating specific solutions, correcting errors, and advancing the student through the dissonance vocabulary in systematic order. The present method takes the Mozart-Attwood relationship as the historical model for the Maestro-student relationship (V2 §0.3).

Vidal, Paul. Exercises for Figured Bass (manuscript, École Normale de Musique de Paris). The primary figured-bass exercise collection used in the Gradus curriculum (Stages III–VI). 178 exercises progressing from root-position triads through diminished seventh chords, organized by chord type and voice-leading challenge.

Insanguine, Giacomo; Handel, George Frideric; Kellner, Johann Peter. Figured-bass and thoroughbass exercise collections (various MSS and early print editions). Supplementary collections in the Gradus curriculum's figured-bass corpus, representing the Italian, English, and German branches of the thoroughbass-realization tradition.

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V. The Leipzig Conservatory Tradition (institutional context)

The Leipzig Conservatory tradition — Mendelssohn's founding in 1843, Hauptmann as theory professor, Riemann as the tradition's mature theorist — is the institutional framework that shaped nineteenth-century European music theory pedagogy. The present treatise's account of the dualist tradition (§2.1) locates it in this institutional setting: Hauptmann's Natur der Harmonik (above) was the foundational text; Riemann's functional theory was the system in which it matured; and the dualist-versus-unified-ground debate is partly a debate about the institutional theory curriculum.

Key secondary sources for the Leipzig tradition as institutional context: Bonds, Mark Evan. Music as Thought: Listening to the Symphony in the Age of Beethoven (Princeton UP, 2006); Burnham, Scott. Beethoven Hero (Princeton UP, 1995); the collected essays in Christensen, Thomas, ed. The Cambridge History of Western Music Theory (Cambridge UP, 2002), especially chapters on German music theory in the nineteenth century.

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VI. Neo-Riemannian Theory (§7.2)

Lewin, David. Generalized Musical Intervals and Transformations (Yale UP, 1987; repr. Oxford UP, 2007). The founding document of transformational theory, introducing the concept of musical transformation as a mathematical group action on a set of musical objects. The PLR operations (Parallel, Leading-tone exchange, Relative) as generators of the Tonnetz are derived from Lewin's framework.

Hyer, Brian. "'Reimag(in)ing Riemann." Journal of Music Theory 39/1 (1995): 101–138. The first sustained application of Lewin's transformational framework to Riemann's function theory, establishing the Neo-Riemannian vocabulary and the Tonnetz as an analytic tool.

Cohn, Richard. "Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions." Music Analysis 15/1 (1996): 9–40. The definition of voice-leading parsimony as a formal property and the demonstration of hexatonic cycles as maximally smooth systems. Audacious Euphony: Chromaticism and the Consonant Triad's Second Nature (Oxford UP, 2012). The book-length development of the Neo-Riemannian analytic program.

Rings, Steven. Tonality and Transformation (Oxford UP, 2011). The integration of Neo-Riemannian transformational analysis with tonal theory, showing how transformational operations can be understood as acts within a tonal space rather than as replacements for tonal function. The most direct predecessor for the present treatise's position in §7.2: Neo-Riemannian devices as gradient traversals absorbed within the unified-ground account.

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VII. Twentieth-Century Pedagogy

Schoenberg, Arnold. Harmonielehre (Vienna: Universal Edition, 1911; 3rd ed. 1922; English trans. Roy Carter, University of California Press, 1978). The most important twentieth-century pedagogical treatise. Schoenberg's account of the harmonic series as the basis of consonance (Chapter 1), his emancipation-of-dissonance argument, and his treatment of chromatic harmony in relation to the diatonic tonal center are all directly relevant to the present treatise. Style and Idea (New York: Philosophical Library, 1950) contains Schoenberg's late essays on the relation of tonal and atonal composition, cited in §7.3 for the account of atonality as deliberate abandonment rather than historical inevitability.

Hindemith, Paul. See Craft of Musical Composition above (Acoustic Foundation section). Hindemith is the closest twentieth-century predecessor for the acoustic-grounding strategy the present treatise pursues. Shared: the harmonic-series ground; rejection of dualism; a pitch-to-tonic relatedness ranking (Series 1; the LCM chord-root ranking); an interval-and-chord tension gradient (Series 2 in its melodic and harmonic forms, with the small intervals inverted in the melodic ranking; the LCM measure of §3.4 and the chord-group classification A.I through B.III as the closest Hindemithian analogue). Different: Hindemith's apparatus is static — interval rankings and chord-group classifications do not flex with the tonal center, and harmonic fluctuation is a magnitude trajectory of tension over time rather than a field of directed pulls. The present treatise takes movement as the central phenomenon and the directed pull as the basic unit (§5.2), extends the gradient hierarchy to the architectural scale Schenker addressed (Claim 8), and operationalizes a context-relative analytical surface through the three lenses of §6. Position relative to Schenker: the treatise occupies a bridge position — Hindemith's acoustic foundation without his pan-tonal apparatus, Schenker's hierarchical prolongation without his procedural ambiguity about acoustic foundation. Hindemith built a periodic table on the harmonic-series ground; Schenker built a hierarchical prolongation account whose foundation he did not himself settle; the present treatise takes Hindemith's ground and Schenker's hierarchy and arrives at the field theory of tonal movement neither alone articulated. Pedagogically: Hindemith's Craft sought to replace common-practice teaching with a pan-tonal alternative derived from the acoustic ground; the Gradus method grounds common-practice teaching with the same acoustic foundation, centering the Fux–figured-bass– Boulanger lineage Hindemith deliberately set aside. §6.3 develops the detailed contrast between Hindemith's melodic/harmonic split and the three-lens framework.

Boulanger, Nadia (reconstructed in). See Lasser below. Boulanger's teaching practice — the figured-bass-centered approach, the harmonic- standards methodology, the discipline of daily practice as formation — is the primary pedagogical lineage of the Gradus method. The reconstruction of the harmonic-standards tradition in Philip Lasser's published work is the direct source for the Gradus curriculum's figured-bass and voice-leading standards component. See Volume 2, Appendix V2.C, for the full Boulanger bibliography.

Lasser, Philip. Published reconstruction of the harmonic-standards tradition derived from the Boulanger pedagogical lineage. [Full citation to be confirmed in bibliographic apparatus: see Editor's Note 4 in Volume 2.] The Gradus curriculum's figured-bass and voice-leading-standards practice draws directly on Lasser's published reconstruction; the attribution appears in Volume 2's bibliography rather than in marketing copy, in accordance with the project's brand-voice conventions.

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VIII. Foundational Pedagogical Principles

Mozart, Wolfgang Amadeus. Letters (collected in The Letters of Mozart and His Family, ed. Emily Anderson, Macmillan, 1938; 3rd ed. 1985). Mozart's correspondence with his father and sister contains pedagogical observations — on singing before playing, on hearing before theorizing, on the necessity of daily practice — that anticipate the present method's foundational principles.

Padre Martini, Giovanni Battista. Esemplare ossia saggio fondamentale pratico di contrappunto (Bologna: Lelio della Volpe, 1774). The counterpoint treatise from which Mozart studied and which shaped the classical Viennese pedagogy. The personal-tutor relationship between Martini and the young Mozart is cited in Volume 2 (§0.3) as the historical model for the Maestro-student relationship.

The primary sources for the Boulanger pedagogical lineage — Boulanger's interview records, the published testimonies of her students (Aaron Copland, Quincy Jones, Elliot Carter, among others), and the reconstruction of her method in Lasser's publications — are fully documented in Volume 2, Appendix V2.C. They are cited here as a group rather than individually because their primary use is in Volume 2's pedagogy chapters rather than in Volume 1's theoretical argument.

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Editor's Notes — For Iteration

These are working notes for the author, not part of the treatise itself.

0. Scope restriction to local-and-short-section tonal hearing. (Established 2026-05-10.) The treatise's gradient analysis is committed to the timescales the listener's auditory short-term memory operates on — pitch-to-pitch, beat-to-beat, phrase-to-phrase, and across short sections in which a single tonal center remains the perceived home. It does not extend to architectural scale (whole-movement spans, sonata-form recapitulations read as the resolution of a long-range structural dominant). The empirical ground is the cognitive-music-psychology record that perceived tonality decays as the establishing context recedes in time (Krumhansl-Kessler 1982; Bigand and Pineau 1997; Bharucha 1984, 1987; Lerdahl 2001). The scope restriction was applied across §2.4.4, §4.2, §4.10, §8, Appendix A (Multi-layer resolution, Pull), Appendix B (Claims 2 and 10), and Appendix C (Stages IV and V) in a single pass on this date. The treatise does not engage Schenker's architectural apparatus as anything more than an analytical-and- notational tool; modulation is taught as the local home shifting rather than as an excursion within a still-present larger home.

1. Voice and register. The skeleton uses a moderately formal register — somewhere between Gustin's compressed mathematical prose and Schenker's discursive philosophical prose. The author may want to tighten or loosen depending on the intended readership.

2. Citations. (Partial pass completed 2026-05-10.) The skeleton uses inline citations of the form (Author Year, p. N) for major sources and the bibliography (Appendix D) has been expanded to support the cited works in §2 and §3. The remaining apparatus work is a full footnoting pass with page-number verification against primary sources held in hand, and a consistency pass making sure every inline citation maps to a bibliography entry across the rest of the volume (§§4–8).

3. The Leipzig material in §2. (Refined 2026-05-10 without primary-source access.) §2.1 has been expanded to include the particulars established in the standard secondary literature on Hauptmann, Oettingen, and Riemann: Hauptmann's "I am / you are / he is" linguistic gloss on the three primary intervals; the clarification that Hauptmann did not himself physicalize the dualism; Oettingen's Dorpat physics chair and his appeal to combination tones; Riemann's earlier technical works (1873, 1877, 1880, 1887) and the late Tonvorstellungen paper (1914–15) as the document of dualism's empirical retreat. A genuinely primary-source-grounded refinement still wants direct reading of Heathcote 1888 (Hauptmann), the Oettingen 1866 German original, and Riemann's Skizze (1880) and Vereinfachte Harmonielehre (1893) in German. The Wason-Marvin 1992 English translation of the Tonvorstellungen paper would settle the late-Riemann particulars.

4. Schenker's relation to dualism. (Resolved 2026-05-10 with the addition of §2.4.) §2.4 develops Schenker as unified-ground-with- qualifications in five subsections: §2.4.1 the anti-Riemann polemic; §2.4.2 the Sechter-Bruckner Stufenlehre lineage as the non-dualist inheritance; §2.4.3 the Naturhorn argument as point of partial alignment; §2.4.4 the three qualifications (the Ursatz engaged but not adopted as axiom, the minor account differs from Claim 5's modal-modification reading, the metaphysical and nationalist framing set aside); §2.4.5 Gustin's approving gloss on Schenker's tonal unfolding as bridge.

5. The Gustin material. (Integrated 2026-05-10.) §3.3 has a new closing aside on why the fifth-stacking principle stops at twelve tones, refuting Yasser's nineteen-tone projection on the three grounds Gustin develops in Chapter 6. §4.5 has a new paragraph incorporating Gustin's Chapter 7 result that only Ionian and Aeolian-plus-leading-tone produce single-rooted sets under leading-tone alteration, supplying the formal acoustic-mathematical argument for the major/harmonic-minor two-key system's privilege. §4.7 has a new paragraph incorporating Gustin's Chapter 5 result that no candidate twelve-tone set has the single-root property, supplying the formal grounding for atonality as a different practice. The bibliography entry (§I) has been expanded with chapter-by-chapter page references.

6. Volume 2 (Pedagogy) is the companion document. Pedagogical material that previously lived in §7 has been moved to volume-2-pedagogy.md. Each volume should be self-contained enough to stand alone — Volume 1 for theorists and composers; Volume 2 for teachers and curriculum designers — with cross-references where one does work the other cannot.

7. Length estimate. Volume 1 fully written would be approximately 35,000–50,000 words. Volume 2 would be approximately 25,000–40,000. Plan for 6–18 months of writing time at serious-scholarship pace.

8. Engaging the contemporary literature. The skeleton notes Cohn (2012), Lewin (1987), and Rings (2011) as the Neo-Riemannian apparatus to engage with in §7.2 (Chromatic-Mediant and Neo-Riemannian Devices). Form theory and rhythmic theory are declared out of scope (§1.3); the relevant literatures (Lerdahl–Jackendoff 1983, Caplin 1998, Hepokoski–Darcy 2006 for form; Hasty and others for rhythm) are cited only insofar as the scope-restriction needs to acknowledge their existence.

9. The skeptic's review (already conducted). The skeleton has been read against itself in skeptic mode. The major objections that informed the present revision: the Neo-Riemannian elephant (now §7.2); the operational definition of movement (now §5.1); the universalist scope (now §1.3); the gap between Gustin's procedure and listener phenomenology (resolved by dropping the procedural account, §3); the scaling overreach (resolved by restricting the theory to the harmonic dimension and treating aesthetics separately in Volume 2). Remaining objections from the review that still need addressing in full prose: the perceptual-vs-procedural gap (still felt in §3.4); the dual-nature-of-dissonance critique against Gustin (probably overstated, see §4.9).

A Practice for Producing Composers

Volume 1 (Theory) sets out the unified-ground foundation, the operational account of movement, and the aesthetic principles that organize coherent musical wholes. This volume draws those theoretical commitments through to the classroom, the curriculum, and the daily practice of composition.

Volume 1 exists for the sake of this volume. This volume exists for the sake of the composer. The pedagogy set out here is not an end in itself — it is the means by which a student is formed into a working composer. A reader who finishes this volume and finds the pedagogical program clarified but writes no music has received only one of two goods. The other good — composing — is what the work was written to make possible.

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§0 · Preliminary Matter

§0.1 · Relation to Volume 1

This volume is a companion to The Theory of Tonal Movement: Volume 1. Each chapter of Volume 2 is the pedagogical consequence of one or more theoretical positions established in Volume 1. Cross-references to Volume 1 use the form V1 §N.M; references inside this volume use §N.M in the usual way. A reader who has not read Volume 1 can follow Volume 2's arguments at the level of practice but will not understand why the practices take the form they do without the foundational chapters.

The two volumes form one project with one purpose. Volume 1 is the theoretical foundation; this volume is the pedagogical practice; both exist in service of the composer the student is being formed into. Volume 1 takes the positions it does (single acoustic ground, perceptual grounding of tonal center, movement as the central phenomenon, aesthetic principles as cross-scale unifiers) because those are the positions a working pedagogy requires. The theoretical choices are made on pedagogical grounds, not on grounds internal to music theory. The pedagogical choices in this volume are made on grounds of what produces composers, not on grounds internal to teaching practice. Theory and pedagogy are aspects of one project; the project is producing composers.

§0.2 · A Methodological Note

This volume — and the Volume 1 that precedes it — was not built top- down from a philosophical framework. The Gradus method did not set out to construct a theory of aesthetics, a theory of tonality, or a pedagogy grounded in Aquinas or in any other inherited apparatus. The starting point was the practice: what does a composer need to learn, in what order, with what kind of correction, to write music that does what she hears in her head? The pedagogy was assembled over years of teaching, of revision, of careful attention to what worked and what did not, from the materials the historical tradition has supplied: species counterpoint, figured bass, voice-leading standards, chorale harmonization, score study, the personal-tutor relationship. The theoretical claims of Volume 1 and of this volume describe what that pedagogy works on; they did not generate it.

The convergence with Aquinas's three conditions of beauty — integritas, consonantia, claritas — that the present volume's §5 and §6–§8 develop was discovered late, not planned. The three conditions fit the working aesthetic vocabulary the pedagogy needed better than any modern alternative we considered, and we adopted them not because Aquinas is canonical but because the conditions describe what the pedagogy is already attending to when a composer's work succeeds or fails. The aesthetic-realist tradition whose historical lineage §5.1 traces is the company we found ourselves in, not the building from which we descended.

This is a methodological commitment with substantive consequences. The major conversations about aesthetics — from Aristotle's Poetics through Burke's Enquiry through Kant's Critique of Judgment through Hegel's Aesthetics — have very often been theoretical and metaphysical, attempting to settle in the abstract what the nature of beauty is. These conversations are genuine; they engage real questions; the present project cites the tradition where it bears on the specific questions the method needs to answer. But the project does not begin in that register. It begins in the math — the harmonic series, the integer ratios, the consonance gradient that listeners track. It proceeds through the structure — voice leading, counterpoint, harmonic function. And it finds meaning in the craftsmanship — in what the student learns to hear, to write, to judge — rather than in a philosophical framework descending from above to legislate the meaning. The mystery that remains — the genuinely irreducible questions about why pitched sound moves listeners as it does, why proportion satisfies, why intention registers through structure — is left undefined. The method does not pretend to dissolve those questions; it claims something humbler. The math, the structure, and the craft are real enough, teachable enough, and consequential enough that a composer can be formed inside them without requiring the metaphysical questions to be settled first.

The method does not assume universal application either. The claim is that this pedagogy, this theory, these three conditions work for the formation of composers in the Western tonal tradition the curriculum serves. Other traditions — other physical grounds, other cultural elaborations, other pedagogical histories — have their own apparatuses, and the present project does not pretend to displace them. The Gradus position is one position among several available to a serious student of the craft; its claim is to be a truly helpful one for what it covers, not a comprehensive one for everything beauty might be. This is, in the most direct sense possible, the most honest and humble claim the method can make.

The reader who carries this methodological frame into the chapters that follow will read them correctly. The theoretical apparatus is not a metaphysical foundation; it is the description of what the pedagogy is already doing. The aesthetic vocabulary is not a philosophical commitment; it is the working language the pedagogy needed. The pedagogy is the ground, and everything else in both volumes is built up from it.

§0.3 · The Argument of This Volume

Theory follows sound. Hearing precedes labeling. Composition is taught by writing. History is curriculum. Beauty has principles. Discipline is formation. The integrated practice system that emerges from these positions is the Gradus method. Each of these is a chapter; together they constitute the pedagogy that Volume 1's theory makes possible — and that, in turn, makes the composer possible.

§0.4 · Acknowledgments

The pedagogical lineage is more recent than the theoretical one. To Nadia Boulanger for the discipline of figured-bass and harmonic standards as a daily practice; and to Padre Martini and the Mozart-Attwood lessons (1784–85) as the historical model of the personal-tutor relationship between master and student.

Contemporary sources — including Philip Lasser's published reconstruction of the harmonic-standards tradition, on which the present method draws — are cited in the bibliography (Appendix V2.C) where they do specific work.

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§1 · Theory Follows Sound

§1.1 · The Position

The most consequential pedagogical commitment of the Gradus method is also the simplest to state: the student hears it before she learns to call it. Ear training is not one subject among many. It is the foundation on which everything else rests, because the harmonic dimension of tonal music is, in its central phenomenon, what the listener perceives in the air — directed pulls along a gradient of distance from a perceived tonal center — and the perception is prior to and independent of the vocabulary that names it (V1 §4.2). A theory of tonal movement that grounds its claims in listener perception (V1 §3.2) produces a pedagogy that grounds its method in listener perception. Theory follows sound; vocabulary follows the ear's recognition of what the vocabulary names.

§1.2 · The Acoustic Ground in the Classroom

The acoustic facts of V1 §3.1 are not abstractions to be cited as authority; they are audible phenomena that can be exhibited in any classroom with a piano. A student who plays a low C and listens attentively can hear the G partial, the E partial, and — with practice — the B♭ partial above the fundamental. This is the Helmholtz exercise: the audible demonstration that the major triad is not a cultural convention to be defended but the first three distinct pitch classes of the spectrum the student has just heard.

The first weeks of Stage I make this audible foundation explicit. Students hear partials of a single sounded tone before they hear intervals between two separately sounded tones. They identify the fifth and the third inside a single piano note before they encounter the same intervals as the distance between two notes on the staff. The pedagogical inversion is critical: in conventional theory pedagogy, the harmonic series appears as an explanatory footnote in chapter three, after intervals have been introduced as abstract relations between scale-degree numbers. In the Gradus method, the harmonic series is what intervals are, and the student meets the fact directly before she meets the abstraction.

§1.3 · Why This Inversion Matters

Two students, after a year of study, will both be able to name a perfect fifth on a test. The first student, trained in conventional theory, names it because she has memorized "fifth = seven semitones between scale-degrees one and five." The second, trained in the Gradus method, names it because she has heard the interval as the acoustically primary relation between partials 2 and 3 of a pitched tone for thousands of hours by now, and she has been writing two-voice exercises in which the fifth carries the structural weight a fifth acoustically does. The first student can pass the test. The second student can hear what the music is doing. Composing requires the second.

The pedagogical literature on language acquisition is the closest parallel. Children learn to speak before they learn to read; the phonological foundation of speech precedes the orthographic representation that names the sounds. A child taught to read before she can speak fluently will produce labels without referents; a child taught to speak first builds the referent and then learns the label that picks it out. Music is a language whose phonemes are intervals, chord qualities, and directed pulls. The Gradus method takes the same position on its acquisition that linguistics takes on the acquisition of speech: the sound comes first.

§1.4 · The Boulanger and Lasser Methodology

The practical methodology by which sound-first pedagogy is delivered in the harmony stages of the curriculum descends from Nadia Boulanger's teaching practice at the École Normale and Fontainebleau through Philip Lasser's published reconstruction of the harmonic- standards tradition. The Boulanger inheritance — figured bass at the keyboard as daily practice, chorale realization in real time with internalized voice-leading — is the historical embodiment of theory- follows-sound in adult conservatory pedagogy. The student does not learn about voice-leading from a textbook; she does voice-leading at the keyboard until the rules are felt as constraints on the fingers, not memorized as theory.

The Gradus curriculum integrates this lineage at Stage III (the first three-voice stage), Stage IV (figured-bass realization), and the Boulanger Exercises practice tool. The mechanics are practical — a figured-bass sketch, a keyboard, a real-time realization — and the pedagogical commitment is the one this chapter has stated: the student hears the harmony before she labels it, and she labels it when the ear has already identified what the label is for.

§1.5 · The Role of Maestro

Maestro is the personal-tutor presence in the integrated method (§10 below). In the sound-first pedagogy, his function is to name what the student has already heard, and to do so at the moment the student has heard it. A student writing a chorale realization who arrives at a moment where she can hear that "this chord wants to go somewhere" is at the precise moment when "leading tone" or "chordal seventh" becomes a useful vocabulary item. Maestro supplies the term in response to the recognition, not in advance of it.

This is the operational reason the integrated method requires a personal tutor and not a static curriculum: vocabulary is taught in response to recognition events that occur in the student's own work, and those events are not predictable in advance. A static curriculum can supply the readings, the exercises, and the schedule. Only a teacher in the moment can supply the name at the moment the student's ear has earned it.

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§2 · Hearing First, Labeling Last

§2.1 · The Sequencing Rule

The previous chapter stated the founding principle. This chapter develops its consequence for curriculum design: the deliberate delay of theoretical vocabulary until the student's ear can identify what the vocabulary names. A student who can hear a perfect fifth before she can name it is closer to composing than one who can define the fifth on a test. The pedagogical task is to engineer the sequence so that recognition reliably precedes naming.

The sequencing rule is operational at every level of the curriculum. Intervals are heard before they are labeled. Chord qualities are felt before they are named. Cadences are recognized as resolutions before they are classified as PAC, HC, or deceptive. Modulations are heard as the local home shifting before the techniques are catalogued. The progression from sound to label is the same at every scale and the curriculum's substep structure enforces the sequence: each introduction of a vocabulary item is preceded by listening exercises that put the phenomenon the item names in the student's ear.

§2.2 · The Stability Hierarchy as the Ground of Perception

The acoustic-mathematical apparatus of Volume 1 (§3.4) ranks every pitch and chord by its LCM distance from the perceived tonic. The gradient is perceptual before it is anything else: the listener tracks where on the gradient each tone sits, and the tracking operates in real time, ahead of any analytical thinking. This is what makes the sequencing rule workable. The student's ear is already doing the work; the curriculum's task is to bring vocabulary to the work the ear is doing, in the order in which the ear is doing it.

Stage I introduces the tonic and the dominant by ear, as the two most stable positions on the gradient (LCM 1 and LCM 6). The leading tone enters as an audible pull, not as scale-degree seven. The chordal seventh of V⁷ enters as a felt urgency to resolve down by step, not as the seventh partial of the dominant's harmonic series (though it is that). The student learns to hear the gradient position before she learns the LCM number or the partial-index that indexes it.

The vocabulary then accumulates as a working set of names attached to identifications the ear has already made. By Stage VI the student has a substantial technical vocabulary — leading tone, chordal seventh, secondary dominant, passing-tone, suspension, augmented sixth — and each item is in service to a phenomenon she has been hearing since Stage I.

§2.3 · The Three Lenses Name the Perceptual Capacity the Curriculum Cultivates

V1 §6 introduces three lenses on movement function — tonal, vertical, horizontal — as the three perspectives from which a single tone can be heard simultaneously. The pedagogical claim of this section is that those three lenses are not analytic vocabulary; they are the perceptual capacity the curriculum exists to develop.

The capacity to perceive multiple structural dimensions simultaneously is not optional for tonal composition. It is the foundational skill the entire harmonic curriculum cultivates. A student who hears only one dimension at a time produces music whose voice-leading-as-line, vertical sonority, and key-position-relative- to-tonic operate in disconnection from one another, with the characteristic awkwardness that follows: lines that wander against chords that resolve, chords that progress against keys that drift, keys that establish themselves against lines that ignore them. A student who hears all three at once produces music in which lines, chords, and key positions are integrated — which is what coherent tonal composition is.

Two pedagogical traditions in the curriculum's lineage are specifically organized around cultivating this multi-dimensional hearing.

The Boulanger inheritance — figured bass at the keyboard, with all four voices in the ear simultaneously. A student realizing a figured bass at the keyboard must hold three things in working attention: the bass line as a horizontal voice with its own trajectory; the chord built above each bass note (the vertical sonority and its voice-leading obligations); and the key-relative position of each pitch and chord (the tonal lens). The figured-bass practice is, operationally, the exercise of writing in all three lenses at once. Boulanger's students at Fontainebleau practiced figured bass not as a historical curiosity but as the daily practice by which the three lenses are integrated into a single working hearing. Her insistence on the practice was not pedagogical conservatism; it was the recognition that this specific exercise trains the specific capacity tonal composition requires.

Fugue — the explicit cultivation of polyphonic hearing. Fugue trains the student to track multiple independent voices simultaneously, each carrying its own horizontal trajectory, with the resulting vertical sonorities and key-positions emerging from the convergence of voices. The fugal subject's entries in different keys at different time points require the listener-composer to track horizontal (each voice's line), vertical (each moment's chord), and tonal (each entry's key) simultaneously. The student who has written four-voice fugues has trained the three-lens hearing in the most demanding form the tonal tradition offers — and has done so in the same exercise that produces some of the tradition's most prized compositional artifacts.

The three lenses are therefore not vocabulary the teacher invokes when a student is confused. They are the theoretical articulation of the perceptual capacity the curriculum exists to develop. The adverbs — tonally, vertically, horizontally — give the teacher and the student a shared language for the three dimensions of hearing the student is being formed in. The naming makes the work specific. When a student's piece has good voice leading but a disconnected harmonic plan, the teacher can say "your horizontal work is strong; your tonal sense is what needs developing now," and the student knows precisely what to practice. When a student's chorale realization satisfies the chord-by-chord part-writing but the lines feel disjointed, the teacher can say "your vertical hearing is good; your horizontal hearing needs cultivating," and the student knows where to focus.

What the curriculum does not do is reduce the lenses to a worksheet category. The student does not label every note T/V/H in her exercises. The lenses are not a memorization target. They are the dimensions of the perceptual capacity, named so that they can be worked on directly — much as a painter learns to see line, color, and value as the three dimensions of visual composition without labeling every brushstroke. The dimensions are real and the language for them is useful; the language is for the practice, not for the worksheet.

By Stage VIII the student who has worked through species counterpoint (Stage II), figured bass (Stage IV), fugue (Stage V), and four-voice voice leading (Stage VI) has integrated the three lenses into a single working hearing. The lenses are not concepts she thinks about when she composes; they are how she hears. The pedagogical goal is exactly this integration — to make explicit training in three dimensions converge into one operational hearing that does the work without conscious decomposition. The three-lens language is the scaffolding by which the integration is built; once the integration is built, the scaffolding can be set aside, as the painter sets aside the conscious vocabulary of line-color-value once she has learned to see.

§2.4 · Distance, Home, and the Spatial Ear

Volume 1's Claim 3 (Distance from Home) gives the student a spatial vocabulary for the gradient: home is the tonic, near home is the dominant or the subdominant, far from home is the chromatic third or the tritone-related region. The polar plot of V1 §3.4 visualizes the gradient as a flower with petals of unequal length extending out from a central tonic. The image is a teaching aid: the student develops a felt sense of how close or far each pitch and each chord sits from the tonic, and the felt sense becomes the working basis of modulation and chromatic-harmony decisions in later stages.

The spatial vocabulary is used consistently throughout the curriculum. A modulation does not "go to" the dominant key in the abstract; it shifts the local home from the tonic to a nearby region. A chromatic mediant does not "borrow" a chord from a parallel mode; it visits a moderately distant region for the duration of a phrase. The student learns to think in terms of distance and the shifting of home — the polar plot updates as the local home shifts — rather than in terms of abstract scale-degree labels alone. The spatial vocabulary is the pedagogical face of the gradient analysis, and it is what the student carries into composition decisions.

§2.5 · A Note on Architectural Scale

The curriculum is committed to teaching at the timescales the listener's auditory short-term memory actually operates on — pitch- to-pitch, beat-to-beat, phrase-to-phrase, and across the short sections in which a single tonal center remains the perceived home (V1 §2.4.4). It does not teach modulation as an excursion from a still-present larger home that the listener supposedly holds across a movement. Modulation is the local home shifting. Secondary dominants are brief local tonicizations. The student does not learn to imagine an architectural-scale tonic that persists across a sonata-form movement, because the empirical record on long-range tonal hearing does not support a perceptual claim at that scale.

This is a deliberate scope restriction. It simplifies the pedagogy without weakening it: the student learns to hear what is hearable, to write what is writable, and to think about tonal events at the scales at which tonal events actually live in the listener's ear. Score study (V2 §10) shows large-scale key plans on the page as compositional facts, but the student is not asked to claim that the listener tracks those plans as a single perceptual gestalt across the movement.

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§3 · Writing, Not Analyzing

§3.1 · The Position

The Theory of Tonal Movement (Volume 1) is a theory of what tonal music does — of the directed pulls the listener tracks and the local resolutions they seek. The pedagogical consequence of this framing is direct. A theory of doing is internalized by doing. The student who has analyzed a hundred Mozart sonatas and never written one has memorized vocabulary about a practice she has not entered; the student who has written twenty short pieces, each one imperfect and each one corrected, has entered the practice itself.

The Gradus method commits to this consequence without compromise. Every substep of the 40-step curriculum ships with a composition prompt. Every practicum day pairs a short concept with a writing exercise that exhibits the concept. Substeps that can only be assessed by analysis are reference material, not lessons. The student writes from day one, and the writing is what the curriculum is for.

§3.2 · The Deepest Divergence from Academic Music Theory

Twentieth-century academic music theory, especially in its Princeton-Yale form, elevated analysis above composition as the primary intellectual labor of the discipline. The result is a generation of graduates who can identify a Neapolitan in a Schubert sonata at sight but who have never written a Neapolitan into a piece of their own. The discipline's name — music theory — became detached from its etymological partner, poiesis, the making of music, and gradually came to mean the naming of music's parts. Schenkerian analysis, set-class analysis, transformational analysis, and most of the apparatus of contemporary theory pedagogy are analytical instruments, not compositional disciplines. They are useful for what they do; they are not pedagogy for composers.

The Gradus method does not refuse analysis. It refuses to treat analysis as the primary work. Analysis is reference material — a catalog of what composers have done, useful for the student to consult as her own work raises questions about how a specific problem was solved. Composition is the work itself. The curriculum is structured around composition prompts; the analysis, where it appears, supports the student's compositional problem by showing her what the answer has looked like when prior composers have answered it.

§3.3 · Schemata as Generative Templates

The galant schemata — the Romanesca, the Prinner, the Monte, the Fonte, the Quiescenza, and the rest catalogued by Gjerdingen (2007) — are the clearest case of the principle. A schema is a stock harmonic-melodic pattern that eighteenth-century composers used and reused as a building block. The schemata are useful to know about because the repertoire is built from them, and a student who has internalized the schemata can read the masterworks with a kind of fluency the unprepared reader lacks.

The pedagogical question is how to teach them. Two paths are available. Path A (analysis-first): the student reads a Mozart sonata, the teacher points out "this is the Prinner; this is the Romanesca; this is the Fonte," and the student learns to identify the schemata by sight. Path B (writing-first): the student is given the harmonic-melodic template of the Prinner and asked to write four Prinners — in four different keys, with four different melodic profiles, in four different textures. The student then writes Romanescas, Montes, Fontes, Quiescenzas. By the time the student returns to the Mozart sonata, she sees the schemata not as labels to apply but as the building blocks she has been making with her own hands.

The Gradus method takes Path B without exception. Schemata are generative templates ("write four Prinners"), not identification targets ("find the Prinner in this Mozart"). The same rule applies to progressions ("write a phrase ending on a half cadence" not "identify the half cadence in this Schubert"), to chromatic chords ("write four phrases using a German augmented sixth" not "label the augmented sixth chord in m. 17"), and to schemata at every level the curriculum reaches. The student's encounter with each item of the harmonic vocabulary is productive before it is recognitive.

§3.4 · The Curriculum as Composition Practice

The numerical scale of the writing-not-analyzing commitment is worth naming. The curriculum has 40 steps. Each step has roughly four substeps. Each substep has a composition prompt. Each step also has four practicum days (P1–P4), each of which has its own short writing exercise. The arithmetic: 40 steps × 4 substeps × 1 composition prompt = 160 substep composition prompts; plus 40 steps × 4 practicum days × 1 short writing exercise = 160 practicum writing exercises. The student who completes the curriculum has written on the order of three hundred short pieces. Some are two-voice counterpoint exercises a few measures long. Some are full chorale realizations. Some are short character pieces, fugue expositions, modal songs, chromatic-harmony sketches. The cumulative mass of student-written music is the curriculum's deepest deliverable.

The Critique by Maestro system (§10 below) is the feedback layer. Every piece of student work is critiqued — the strengths named, the weaknesses identified with specific suggestions, the next exercise proposed. The critique system is itself a composition-focused pedagogy: the student's next piece is what the critique is for, not the next test.

§3.5 · What This Rules Out

The Gradus method does not produce music theory PhDs. It does not prepare students for the analytical apparatus of contemporary academic music theory. It does not give students fluency in Schenkerian analytic graphs, set-class identification, or Lewinian transformations. These are not its goals.

It prepares students to write music. A student who completes the curriculum and chooses to enter an academic music-theory program will have to acquire the analytical apparatus separately, and she will be in a strong position to do so because she has the working musical experience that gives the apparatus its meaning. But the acquisition is downstream, optional, and not the curriculum's purpose.

The position is honestly stated. The student who wants to write music will find this curriculum responsive at every point. The student who wants to analyze music as her primary practice should look elsewhere; this is not the curriculum for her. The method's commitments are not concealed and are not for all comers. They make a composer. The composer is what they are for.

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§4 · History as Curriculum

§4.1 · The Position

The Gradus curriculum's ten stages do not survey the harmonic vocabulary in the order a logician would arrange it. They follow the order in which the vocabulary historically emerged. Stage I covers intervals and the acoustic foundation, as the Greeks and Pythagoreans encountered them. Stage II covers two-voice writing under the Fuxian species, as the Renaissance and early Baroque practiced it. Stage III adds the third voice and the basic harmonic vocabulary. Stage IV introduces phrase structure, modulation, and the small forms. Stage V covers fugue. Stage VI covers the high classical style — voice leading at four voices, figured bass, classical harmonic vocabulary. Stage VII covers chromatic harmony and Romantic expansion. Stage VIII covers modal and impressionist harmony. Stage IX covers twentieth-century expanded tonality and atonality. Stage X is the composer's ear — the integrated final stage.

This is a historical ordering, not a difficulty ordering. Each stage takes up the harmonic problems composers faced at a particular historical moment, in approximately the order they faced them, and gives the student the vocabulary and the writing experience to solve them. The student lives the journey. By Stage VIII, when the student is asked to write a Lydian passage, she has already encountered the major-mode practice that Lydian deliberately quiets, and the quieting is audible to her because she has internal the major-mode pulls Lydian removes.

§4.2 · Dissonance Acceptance as the Spine

The thread running through the historical narrative is the gradual acceptance of dissonance as compositional material. What one century forbade, the next admitted as expressive resource. The parallel fifths and octaves the Renaissance treatises forbade became, three centuries later, the parallel motion of Debussy. The diminished seventh, scandalous in the early eighteenth century, became the standard chromatic-harmony pivot of the nineteenth. The unresolved dissonance of late Romantic chromaticism became, in Schoenberg's hands, the deliberate emancipation from tonal hierarchy of the early twentieth.

This historical arc is not merely incidental to the curriculum's ordering; it is what the ordering is. Each stage takes up a specific position on the dissonance-acceptance gradient and asks the student to write within that position before moving on. Stage II's Fuxian species enforce strict consonance with rigorously treated dissonance; Stage VI's high-classical harmony allows dissonant sonorities (the dominant seventh, secondary dominants) within an essentially diatonic frame; Stage VII admits chromatic chords (augmented sixths, Neapolitans, chromatic mediants) and the modulations that make use of them; Stage IX admits chord-types outside the diatonic vocabulary entirely. The student writes within each position before moving on, and she develops a felt sense of where on the gradient her current writing sits — a sense that becomes her working compositional judgment.

§4.3 · Modal and Atonal Music as Positive Achievements

Two specific commitments of the historical curriculum deserve explicit statement, because they are routinely misrepresented in conventional pedagogy.

Modal music is taught as a positive achievement. A student in Stage VIII writing a Mixolydian phrase is not writing major with a "wrong" seventh. She is deliberately quieting the leading-tone pull that major's seventh degree carries (V1 §4.6), in exchange for the softer modal color that the lowered seventh supplies. The trade is real: she gives up the cadential urgency of the half-step pull from below and gains the modal openness of the whole-step approach. The pedagogical framing names the trade explicitly. The student writes Mixolydian with full understanding that she is making a positive compositional choice, not a deviation from a default.

The same applies to Lydian (raised fourth quieting the subdominant anchor; gaining floating openness), Phrygian (lowered second inverting the half-step around the tonic from below to above; gaining a characteristically dark cadence), and Dorian (minor third with raised sixth; gaining a brighter modal-minor quality than Aeolian's pure-minor cadential incompleteness). Each mode is a position taken on the gradient — a deliberate calibration of which pulls to keep audible and which to quiet — and the student who writes in a mode is exercising the calibration consciously.

Atonal music is taught as a positive achievement. Stage IX introduces the twentieth-century atonal vocabulary not as the failure of tonality but as the deliberate setting-aside of the tonal hierarchy (V1 §4.7). Schoenberg's serialism, Webern's pointillism, the free atonality of Berg and the integral serialism of Boulez are all positive choices — each composer has decided that what tonal music offers is no longer what she wants, and has constructed an alternative organizational principle to substitute. The student who arrives at Stage IX has nine stages of tonal practice in her hands. She writes atonal music with the understanding that she is making the same choice the early twentieth-century composers made, with the option (which her tonal practice gives her) of choosing differently in any given piece.

The honest pedagogical statement: the Gradus method does not rank tonal music above atonal music or modal music above either. It teaches them as three positions a composer can take on the same acoustic ground, and gives the student the practical experience to make the choice consciously in each piece she writes. Stages IX–X are the curriculum's most demanding, because the student has to hold all three positions in mind simultaneously as she works.

§4.4 · The Theoretical-History Subplot

A subsidiary thread of the historical curriculum is the history of theoretical positions about tonal music — Hauptmann, Oettingen, Riemann, Schenker, Hindemith, Gustin (V1 §2 and §2.4). Students do not need a sustained tour of nineteenth-century German theory to become composers, and the curriculum does not provide one. But the moves the theoretical history made — the false dualist turn, the Schenkerian hierarchical analysis, the Hindemith re-grounding in the harmonic series — are referenced where they bear on the student's understanding of why a particular vocabulary or notation has the form it does. The German Roman-numeral function-symbols (T, S, D), the major/minor mirror-system notation, the prolongational reductions in score study (V2 §10) all have intelligible historical sources, and a Maestro who can place the notation in its history is a more useful teacher than one who treats every notation as a free-standing convention.

The treatment is light. The pedagogy does not require the student to internalize the history of theory; the history is context, supplied where it answers questions the student's own work raises.

§4.5 · A Note on Architectural Form

The historical curriculum covers the small forms — binary, ternary, rondo, variation — at Stage IV, and these are taught as composition exercises within the scope of phrase-and-short-section tonal hearing (V1 §2.4.4). The student writes a complete binary-form short piece in which the local home shifts to the dominant in the first half and shifts back to the tonic in the second half. The piece is heard as it is composed: as a sequence of local homes the listener moves through, with the return at the end as the final local home reasserting itself.

The curriculum does not treat large-scale form as a separate theoretical domain to be mastered. Sonata form, when it appears in the historical narrative at Stage VI, is introduced as the composition of multiple local-home shifts under a particular thematic and rhetorical scheme. The student writes a sonata-form exposition; she writes a development that visits several local homes; she writes a recapitulation in which the second theme returns at the home tonic. What she does not do is claim that the listener perceives the whole movement as a single architectural tonal gestalt. The pieces hang together because they are well- written at every scale the ear tracks; the architecture on the page is a compositional plan, not a perceptual claim.

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§5 · Beauty Has Principles

§5.1 · The Position

The local movement of tonal music — pulls, resolutions, the surface vocabulary the harmonic theory of Volume 1 catalogs — is the material from which composers build. But the experience of being moved by a piece of music, of hearing a phrase as balanced or a passage as proportioned, of feeling an arrival as long-awaited, is not exhausted by that catalog of pulls. It is governed by a separate set of considerations: the aesthetic principles of Integritas, Consonantia, and Claritas — wholeness, harmony of parts, and clarity — first set out in something close to their final analytic form by Aquinas in the Summa Theologiae and developed thereafter across seven centuries of aesthetic writing.

The Gradus method takes the position that these principles are not subjective preferences arising from cultural habit. They are real principles — discoverable, teachable, present in the work of every great composer across every style and era — and the human ear and mind respond to them because the ear and mind are made to respond to them. This is the position of aesthetic realism: a position with substantial historical pedigree, more substantial than the contemporary music-theoretic literature often recognizes.

The methodological position §0.2 of this volume sets out bears directly on the historical lineage that follows. The Gradus method arrived at the aesthetic vocabulary §5.2 below adopts through pedagogical convergence, not through philosophical commitment: the three Aquinas conditions fit the working language the pedagogy already needed, and the aesthetic-realist tradition the next paragraphs trace is the company the method found itself in rather than the building from which it descended. The lineage is therefore not a foundation. It is a witness — eight centuries of independent articulations of the same structural categories the present method has independently reached from the practice.

The lineage is long and largely unbroken. It begins with the Pythagorean discovery (sixth century BCE) that consonant intervals correspond to simple integer ratios — the first claim that beauty has discoverable structural properties. It continues through Augustine's doctrine of number (De Musica, 387–389), in which beauty is the perception of mathematical order in time and space. Aquinas's three conditions of beauty — integritas (wholeness, nothing missing or superfluous), consonantia (proportion, harmony of parts), and claritas (radiance, intelligibility — "the form shining through the matter") — set out in the Summa Theologiae (I, q. 39, a. 8) around 1270, are the medieval summary of the position. Alberti restates Aquinas in Renaissance humanist vocabulary as concinnitas — "that reasoned harmony of all the parts within a body, so that nothing may be added, taken away, or altered, but for the worse" (De re aedificatoria, 1452). The eighteenth century renews the position in different terms: Francis Hutcheson's uniformity-amidst-variety (1725) as the structural condition of beauty; David Hume's standard-of-taste calibrated to qualified critics (1757); Edmund Burke's distinction between the beautiful and the sublime (1757). The nineteenth-century music-formalist tradition (Hanslick, Vom Musikalisch-Schönen, 1854) and the twentieth-century analytic tradition (Beardsley, Aesthetics, 1958; Goodman, Languages of Art, 1968) continue and refine the same position. The contemporary recovery of Aquinas as a serious analytical aesthetician (Maritain, Art and Scholasticism, 1920; Eco, The Aesthetics of Thomas Aquinas, 1956/1988) brings the medieval position back into direct dialogue with twentieth-century analytic aesthetics.

The three parent principles of §5.2 below adopt Aquinas's three conditions directly as the parent layer and treat the apparatus subsequent tradition has developed — Alberti's concinnitas, Hutcheson's uniformity-amidst-variety, Hanslick's tonally-moved forms, Beardsley's five criteria (unity, complexity, intensity, magnitude, regional quality), Wölfflin's contrasting pairs, Gombrich's sense of order — as sub-categories under the three, calibrated for pedagogical use. The Gradus reformulation is not a freshly-invented apparatus; it is the three Aquinas conditions brought forward as the working aesthetic vocabulary, with the historical lineage's developments deployed as the practical sub- principles under which composers actually work.

The position is unfashionable in some quarters of contemporary academic music theory, where aesthetic-realist claims are sometimes treated as ideologically suspect or insufficiently rigorous. The position is also continuous with the central tradition of Western aesthetics from antiquity through the present moment. The argument the Gradus method makes, like the tradition's, is that the principles by which Bach builds an invention, Mozart a sonata, Brahms a symphony, and Debussy a tone-poem are the same principles, deployed differently, and that the listener tracks them whether or not the listener can name them.

One position taken here deserves explicit statement, because it diverges from a substantial tradition in Western aesthetics. Edmund Burke's A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful (1757) carved off a separate aesthetic category called the sublime — vast, dark, obscure, terrible, powerful — distinct from the beautiful and characterized by specific emotional responses in the perceiver (awe, terror, wonder, overwhelm). The distinction propagated through Kant (1790), Schopenhauer (1819), Otto (1917), Adorno, and the broader continental tradition; it has been the standard philosophical apparatus for a category of aesthetic response that the language of "beauty" was sometimes thought too gentle to contain. Late-Romantic and twentieth-century music — Wagner's Tristan, Mahler's Resurrection Symphony, the late-Romantic chromatic saturation, Webern's pointillism, Ligeti's Atmosphères, Penderecki's Threnody — has often been read through this category as "sublime" rather than "beautiful."

The Gradus method does not adopt this distinction. The reason is methodological consistency with the aesthetic-realist position just stated: beauty's principles are real, grounded in the objective structural properties of music, not in the perceiver's affective response. The sublime tradition, by contrast, defines its category through perceiver-state — what the listener feels in the presence of certain music. Once aesthetic categories are defined by perceiver-state, they become relative to the perceiver, and the realist commitment is surrendered. Beauty, on the Gradus method's account, is what the three principles produce. The full range of emotional responses the perceiver may have — awe, terror, joy, sorrow, peace, exhilaration — are all consequences of the objective structural choices the composer has made. They do not warrant separation into different aesthetic categories.

Vast, overwhelming, awe-inducing music — the music the tradition has often called sublime — is no exception. It is the three principles deployed at the edges of their operating range: vast proportions, sustained tensions, dense textures, extended durations, harmonic ambiguities, extreme dynamic and registral contrasts. Mahler's Resurrection, Wagner's Tristan, Webern's Symphony, Penderecki's Threnody are all music in which the three principles operate at the edges of their working range. The emotional response such music produces — awe, overwhelm, mysterium tremendum — is real, but it is a consequence of the objective compositional choices, not a separate aesthetic kind. Beauty can conjure up any and all emotional responses; the three principles cover the territory; no further category is needed.

This is a deliberate methodological position the volume takes against a long-standing tradition. The position is stated here so a reader from the music-aesthetics literature recognizes the divergence as deliberate rather than as oversight.

This is a position the Volume 1 theory does not depend on. A theorist who works through Volume 1's harmonic claims and disagrees with this chapter's aesthetic position can still take the harmonic theory as sound. The pedagogical practice set out below, however, does depend on the aesthetic position — the curriculum's principle-boxes, the critique vocabulary, the discipline by which a student learns to hear and write aesthetically coherent music, all rest on the position that beauty has principles and those principles can be taught.

§5.2 · The Three Parent Principles

The Gradus curriculum organizes the aesthetic principles under three parent principles, named with their Latin source-terms and their English glosses: Integritas (Wholeness), Consonantia (Harmony of Parts), and Claritas (Clarity). These are Aquinas's three conditions of beauty as set out in the Summa Theologiae (I, q. 39, a. 8), adopted directly as the parent layer of the Gradus aesthetic vocabulary. The adoption is deliberate. Every overlap and contradiction the modern attempts to enumerate beauty's principles have produced — proportion against balance, variety against contrast, clarity against order, the categorically uneasy status of movement — turns out on examination to be one of Aquinas's three conditions split into multiple peer principles. The three Aquinas terms are independent of each other in a way the modern enumerations are not; they are also the historically deepest articulation of the position, and the contemporary aesthetic-realist literature (Maritain, Eco, Beardsley) reads naturally as the development of Aquinas's three rather than as supplement to them.

Each parent principle has its sub-categories — the pedagogically practical principle-names under which the working composer actually composes, and under which the curriculum's lesson-boxes are organized.

• Integritas (Wholeness). Aquinas's first condition. Nothing missing, nothing superfluous. The work is complete in itself: every element earns its place, no element is redundant, no element required is absent. Sub-category: Economy — the practical realization, the composer's refusal of the unnecessary and the discipline of pruning everything that doesn't earn its place. Beardsley's unity in his five criteria names the same principle.

• Consonantia (Harmony of Parts). Aquinas's second condition. The parts of the work stand in proportionate, balanced relationship to each other. Equilibrium of forces. Ratios cohere. Unlike-against- like produces the convergence the listener experiences as harmony. Sub-categories: Balance — the equilibrium of forces across every scale the listener tracks, holding both the quantitative aspect (relative durations and section weights in proportional ratio, the Pythagorean inheritance applied to musical time) and the qualitative aspect (tension and release, weight and counterweight, antecedent and consequent); and Contrast — the deliberate placement of unlike against unlike, both synchronically (loud against soft, tonic against subdominant, homophony against polyphony, motion against stasis, sound against silence) and diachronically (theme and variation, exposition and development, statement and transformed return). Hutcheson's uniformity-amidst- variety (1725) names the same family the two sub-categories together cover; Hanslick's tonally-moved forms (1854) names the motion-and-stasis operation within Contrast.

• Claritas (Clarity). Aquinas's third condition. The form shines through the matter. Intentional structure is perceived as intentional by the listener — at some level, even if only emotional. An imperceptible structure isn't Claritas; a perceptible randomness isn't either. Claritas is the convergence of intent and recognition. Sub-category: Order — the mature pedagogical realization, intentional structure perceived as intentional, developed at every scale the listener tracks (motive, phrase, section, movement). Perceptibility is built into Order's definition: an Order that produces no perceptible signal — not even an emotional one — is not Order. Developed at chapter length in §8 below, where Order's perceptual mechanism (Gestalt Prägnanz, Goodman exemplification), its three modes (inheritance, departure, invention), and its architectural-scale operation receive the extended treatment the principle's reach requires.

Two structural notes on this list. First, Movement does not appear among the parent principles or their sub-categories. V1's Claim 2 (Movement is the central phenomenon of the harmonic dimension) is preserved as V1's territory at local scale; the aesthetic principle that earlier formulations of this chapter called Movement collapses into the balance-of-motion-and-stasis operation within Contrast, and appears in the curriculum's lesson-box vocabulary at one remove from the parent layer. Second, Consonantia carries two sub-categories because the working aesthetic vocabulary of music is concentrated there — most compositional decisions about balance, contrast, proportion, juxtaposition, and motion-and-rest are Consonantia decisions — while Integritas and Claritas each have a single sub-category, sufficient to their narrower working domain.

Each step of the curriculum carries one principle-box highlighting one of these parent principles or its sub-category. The student internalizes them by writing exercises that exhibit them, not by memorizing the list. By the end of the curriculum, the three Aquinas conditions and their sub-categories are the student's working vocabulary of aesthetic judgment.

§5.3 · Why Music Moves Us

The phenomenon of being moved by music is the convergence of two things: the local pull-and-resolution that the listener tracks moment-by-moment (the harmonic dimension, Volume 1), and the aesthetic principles that shape the larger experience (this chapter). When a Bach Passion brings a chorale at a moment of structural rest, it is the convergence of harmonic cadence (a pull resolving) and aesthetic principle (Consonantia in the proportion and balance of sections, Claritas in the chorale's perceived arrival) that produces what the listener experiences as meaning.

The empirical apparatus for this convergence has been worked out by the twentieth-century cognitive-music-psychology tradition. Leonard Meyer's Emotion and Meaning in Music (1956) established the central thesis: musical meaning is grounded in expectation. The listener has internalized stylistic norms; specific musical events either confirm or violate those expectations; the experience of musical meaning is the experience of expectation, fulfillment, and surprise. Eugene Narmour's Implication-Realization Model (1992) gives the Meyer position rigorous formalization for melodic perception: a melodic interval implies (predicts) a continuation; the actual continuation either realizes or violates the implication; the listener's perception of melodic meaning is the perception of implications realized or denied. David Huron's Sweet Anticipation (2006) synthesizes Meyer and Narmour into a comprehensive empirical theory with the ITPRA model — Imagination, Tension, Prediction, Reaction, Appraisal — naming the five distinct phases of the listener's response to a musical event. Each phase has its own emotional valence; the composer who composes with awareness of all five phases is composing with the full emotional power of music.

The Meyer-Narmour-Huron apparatus is the music-specific empirical version of what Volume 1 calls the gradient (the consonance hierarchy that grounds expectation) and what this chapter calls the aesthetic principles (the cross-scale structural conditions that shape expectation at scales the gradient cannot reach). Together, the apparatus and the principles give an account of why music moves listeners that is both empirically supported and pedagogically actionable. The composer who wants to produce a specific emotional effect can think about which ITPRA phase she is targeting and what musical means accomplish the target. The composer who wants to give her music aesthetic weight at the scales the harmonic gradient cannot reach can work with the three principles directly. The two apparatuses converge on the same set of compositional decisions.

Volume 1's theory does not, on its own, explain the emotional response. It explains the structural conditions in the harmonic dimension under which a particular kind of emotional response becomes available. The aesthetic principles operate alongside the harmonic conditions. The composer who understands both — and who has internalized the Meyer-Narmour-Huron expectation apparatus as her working sense of how the listener's response to specific musical events arises — can build music that moves the listener. The listener who does not understand any of this is moved anyway, because the conditions operate whether or not the listener can name them. The pedagogical task is to develop in the student the capacity to build music that meets these conditions — to cultivate the ear that hears the pulls, the judgment that arranges the pulls under aesthetic principles, the discipline that brings the work to completion, and the working knowledge of expectation that lets the composer calibrate her music to specific moments of emotional impact.

§5.4 · Aesthetic Principles Across Eras

The aesthetic principles are constant across eras, but their application varies. Palestrina's clarity is achieved through voice-leading economy and modal restriction; Bach's clarity is achieved through fugal articulation; Mozart's clarity is achieved through phrase structure and cadential punctuation; Debussy's clarity is achieved through textural transparency and harmonic precision. The principle is the same; the means of achieving it have shifted with the available vocabulary. A pedagogical method that did not account for this constancy-with-variation would be unable to teach why all four composers are heard as clear, in different ways, by attentive listeners.

The same is true of the other two parent principles. Integritas (Wholeness) in a Renaissance motet, a Classical sonata, and a Romantic Lied takes different specific forms but is recognizably the same principle in each — every era's masterworks share the property that nothing in them is superfluous and nothing required is absent, even when the materials and lengths differ radically. Consonantia operates similarly: Palestrina's balanced voice pairings, Mozart's antecedent-consequent phrases, Brahms's proportional symphonic spans, and Debussy's modally-balanced sonorities are all instances of the same principle realized through the era's specific vocabulary. Order in a Palestrina mass, a Beethoven late quartet, and a Webern symphony takes radically different specific forms but is the same property — the listener perceives intentional structure in all three (V2 §8.5). The historical arc of the curriculum (Stage I through Stage X) traces how composers across centuries have realized the same aesthetic principles using progressively expanded vocabularies of pull and resolution. The principles are why the historical arc holds together as one tradition rather than as a sequence of disconnected styles.

The constancy-with-variation pattern is not specific to music. Heinrich Wölfflin's Kunstgeschichtliche Grundbegriffe (Principles of Art History, 1915) developed five pairs of contrasting concepts — linear vs. painterly, plane vs. recession, closed vs. open form, multiplicity vs. unity, absolute vs. relative clarity — that distinguish Renaissance from Baroque visual art. The pairs are structural-perceptual categories that operate in visual art the way the three aesthetic principles operate in music: the same underlying categories, realized differently in different historical periods. Wölfflin's "clarity" pair is recognizably Claritas as the present chapter develops it; his "closed vs. open form" pair maps onto the three-modes treatment of §8.5 below (closed = inherited form, open = departure or invention); his "multiplicity vs. unity" pair is recognizably a Consonantia distinction within the Renaissance-vs-Baroque comparison. Clement Greenberg's twentieth-century modernist formalism (Avant-Garde and Kitsch, 1939; Modernist Painting, 1960) extends the position into the twentieth century: each art-form's history is the progressive realization of the form's specific medium, with the underlying aesthetic principles operating across all periods of the art's development. The cross-art consistency is what looking beyond music-specific aesthetics delivers: the principles operating in music are recognizably the same principles operating in visual art, under different names, with different specific realizations. The historical lineage of §5.1 above is not a music-only tradition; it runs through architecture, painting, sculpture, poetry, and the other arts, with each art-form developing its own specific vocabulary for the same underlying structural principles.

§5.5 · How This Chapter Relates to Volume 1

This chapter does not derive most of the aesthetic principles from Volume 1's harmonic theory. The harmonic theory describes the material composers work with in the harmonic dimension; this chapter describes the principles by which the material gets shaped into aesthetically coherent wholes. The two are largely independent sources of constraint on what a composer can do, and the present method keeps them analytically distinct on that.

One of the three parent principles, however, has a non-trivial relationship to Volume 1, and a second has a sub-category that nests with Volume 1 at one scale.

Claritas — and specifically its sub-category Order — has a partial relationship to Volume 1. The harmonic series — Volume 1's acoustic ground — is Order at the smallest scale (§8.2 below): the integer-ratio structure of the partials is order discovered in pitched sound itself, a physical fact prior to any compositional choice, present whether or not anyone is listening. The Western tonal tradition's elaboration of this fact into the working harmonic vocabulary (Volume 1's gradient, the scales of major and minor, the cadential framework) is the next scale of Order — the cultural realization of the physical fact. Beyond these scales, Order operates at scales Volume 1 declines to analyze: the architectural scale of long-movement coherence (§8.6), and the cross-cultural scale of traditions grounded in physical facts other than the harmonic series (§8.2). Claritas via Order is partially nested with Volume 1 and partially broader; the relationship is genuine but not a straightforward subset.

Consonantia — and specifically the motion-and-stasis operation within its Contrast sub-category — inherits Volume 1's Claim 2 as local material. Claim 2 names Movement as the central phenomenon of the harmonic dimension at local scale: directed pulls along the consonance gradient at the timescales the listener's auditory short-term memory tracks. The composer who balances motion against stasis, sound against silence, in a phrase or a short section is balancing the local pulls Claim 2 has named against the local rests the same gradient defines. Claim 2 supplies the working substance; Contrast under Consonantia supplies the cross-scale shaping at scales the ear can still track (phrase, short section). The two are the same phenomenon viewed at different scales of compositional decision. The pedagogy treats the Volume 1 account as the local foundation and the Consonantia account as the cross-scale judgment that composers exercise about how the local material accumulates.

Integritas is genuinely independent of Volume 1, and the remainder of Consonantia — Balance and the rest of Contrast's operational range — is independent at the parent-principle level. The two govern compositional shaping at scales the harmonic theory does not address, and they apply to harmonic, contrapuntal, registral, textural, and dynamic decisions alike.

A composer can satisfy harmonic-dimension correctness while violating aesthetic principles (a passage of unrelieved cadential motion is exhausting, regardless of how correctly the pulls are deployed). A composer can satisfy aesthetic principles in modal contexts where local harmonic pull is muted (Debussy's modal- impressionist phrases exhibit Consonantia and Claritas — balance, proportion, and clarity of structure — even with the cadential urgency of major-key practice deliberately quieted). The pedagogy treats both factors as complementary rather than reducible to each other.

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§6 · Integritas

§6.1 · The Position

Of the three conditions Aquinas identifies in the Summa Theologiae (I, q. 39, a. 8) as constitutive of beauty — integritas, consonantia, claritas — the first is the one a composer encounters most often as a practical question. Is the work complete? Is the phrase finished? Does the passage have anything missing — anything the structure requires but that has been omitted? Does it have anything that should not be there — anything superfluous, padded, redundant, present only because the composer ran out of ideas and kept writing? These are operational questions the composer asks at every scale, from the placement of a single passing note to the proportional shape of an entire movement, and the principle by which they are answered is what §5.2 above names as Aquinas's first condition and the present chapter develops at length: Integritas.

Integritas — translated variously as "wholeness," "integrity," or "completeness" — is the property of a work being entire in itself. A passage exhibits Integritas when nothing in it is missing (no necessary element omitted, no structural step skipped) and nothing in it is superfluous (no padding, no redundancy, no element that does not earn its place). The listener perceives Integritas as the felt sense that the work is finished — that the work has arrived at its completion, that adding another note or removing an existing one would damage rather than improve what is heard. The principle is operational at every scale at which a composer makes decisions: it asks, in each decision, whether what is there belongs and whether what belongs is there. §5.2 names Economy as Integritas's working sub-category — the composer's daily practice of refusal-of-the-unnecessary by which the principle is realized in the writing of specific works.

The position takes its name from the medieval scholastic vocabulary but the underlying insight is older. Aristotle, in the Poetics (1450b), required that a tragedy be "complete and whole, possessing magnitude" — a unity in which the parts are so arranged that removing or transposing any of them would damage the whole. Aquinas's integritas (around 1270) is the formal scholastic statement of the same insight, recast as one of the three conditions of beauty. Leon Battista Alberti's concinnitas (De re aedificatoria, 1452) restates it in Renaissance humanist vocabulary as "that reasoned harmony of all the parts within a body, so that nothing may be added, taken away, or altered, but for the worse" — the second clause, almost word for word, is Integritas's operative statement. The twentieth-century analytic-aesthetics tradition recovers the position under different vocabulary: Monroe Beardsley's unity — the first of the five criteria of his Aesthetics (1958) — is recognizably the present chapter's Integritas in the language of mid-twentieth- century analytic philosophy. The lineage is long; the principle is constant; the vocabulary has shifted.

Integritas, Consonantia, and Claritas are independent conditions (§5.2 above): a perfectly proportioned movement that omits its recapitulation has Consonantia among its remaining sections but fails Integritas as a whole; a clear, recognizably-structured passage that omits its needed conclusion has Claritas without Integritas. §7 and §8 develop the other two principles in turn; the distinction worth marking within the Integritas chapter is the one with which Integritas is most often confused.

Integritas is not perfectionism. The claim is not that the composer must achieve some unreachable ideal of optimization — that every note must be the optimal note out of all possible notes, that no work could ever be finished. It is the more practical claim that the composer asks, at each moment of composition, whether what is here is what should be here — whether each note pulls its weight, whether each phrase earns its place, whether the movement contains everything its structure requires and nothing it does not. The composer who works with Integritas is making the work entire in itself, not making it flawless. The two are easy to confuse and consequential to distinguish.

Integritas is, of the three Aquinas conditions, the one most cleanly independent of Volume 1 (per §5.5 above). The harmonic theory of Volume 1 names the working material — pulls along the consonance gradient at scales the ear tracks — but says nothing directly about whether a given work is complete in itself. A composer who has not internalized Integritas can produce harmonically impeccable passages that nonetheless fail as music because they are padded, or incomplete, or both. The principle operates at every scale of musical organization, from the single note to the whole movement, and is the most basic operational question the composer asks about her own work.

§6.2 · Integritas at Every Scale

Integritas operates at every scale of musical organization at which the composer makes decisions. The chapter develops the claim the way §8 below (the present chapter's parallel for Claritas) develops its own at-every-scale walk — by walking through the scales in turn and showing where Integritas lives at each.

The pitch. The smallest scale at which the question of Integritas arises is the placement of a single note. The composer who has internalized the principle asks, of each note as she writes it, whether it earns its place — whether the note carries enough structural or expressive weight to justify its being there rather than the alternatives. A passing tone that has no melodic function, a chord tone that does not serve the line's continuity, a doubling that adds no color: each is a candidate for the editorial pencil. The question is not whether the note is wrong (no rule of voice-leading or harmony has been broken) but whether the note is necessary — whether removing it would leave the music incomplete, or whether keeping it leaves the music padded.

The motive. A motivic figure of two to eight notes exhibits Integritas when its specific arrangement of pitches and durations forms a unit complete in itself. The first four notes of Beethoven's Fifth Symphony — three short notes and one long, on a falling third — are the canonical case. Subtracting any of the four notes would not produce a different motive; it would produce no motive at all. Adding a fifth note would not enrich the motive; it would dissolve it. The figure is complete in its specific four-note form, and the listener's recognition of it on every return is the recognition of an entire-in-itself unit. A motive that is too short does not register as a motive; a motive that is too long does not cohere. The well-formed motive sits in the small range of figures whose particular arrangement of elements is just what it needs to be.

The phrase. A phrase exhibits Integritas through the relationship between its parts — most canonically, the antecedent and the consequent of the Mozartean periodic phrase. The four-bar antecedent ending on a half cadence proposes a musical thought; the four-bar consequent ending on a perfect authentic cadence completes it. An antecedent that has no consequent fails Integritas: the question has been asked and not answered, and the phrase as a whole is incomplete. A consequent that arrives too soon (after three bars rather than four) fails Integritas in the other direction: the answer is given before the question has been adequately posed. The phrase is the smallest scale at which the composer's task moves beyond "is each element earning its place" toward "does the whole say what it set out to say."

The section. A section of a movement exhibits Integritas through the proportional weight given to its constituent phrases and the completeness with which its musical work gets done. The development section of a sonata-form movement that returns to the tonic without having developed the exposition's material has the recapitulation prepared but the development omitted — and Integritas fails. The exposition that introduces a second-theme group and then truncates it before its idea has been adequately stated has its second-theme group introduced but not delivered — and Integritas fails again. The composer's task at section scale is to know what work the section is doing and to do enough of it, but no more. The Wagner Tristan prelude is a case study in section-scale Integritas at unusual length: at sixteen minutes it is by Common Practice standards a very long section, but every measure of those sixteen minutes carries the sustained chromatic working-out the work demands. The length earns itself; the section is complete.

The movement. At the largest scale at which Integritas can be unambiguously perceived, a movement exhibits the property by containing all the structural sections its form requires, in the proportions and at the lengths the form's logic demands. A symphonic first movement that omits its recapitulation has Integritas fail at movement scale; a symphonic finale that runs eight minutes longer than its material justifies has Integritas fail in the opposite direction. Two well-known cases illuminate the principle at this scale. Bruckner's Ninth Symphony is perceived as incomplete because the symphonic form Bruckner was working in (a four-movement plan modeled on Beethoven's Ninth) requires a finale that Bruckner did not finish before his death — the listener experiences the missing fourth movement as a missing part of a complete structure. Schubert's Unfinished Symphony, by contrast, is perceived as complete despite having only two movements, because those two movements form their own coherent two-movement whole — the structure is not a four-movement plan from which two movements are missing but a two-movement structure complete in itself. The two cases are instructive precisely because their unfinished states are different in kind: Bruckner's symphony fails Integritas at movement-cycle scale; Schubert's Unfinished passes Integritas at the two-movement scale at which it actually operates.

The Integritas at each scale is largely independent of the Integritas at the other scales. A passage of pitches each of which earns its place can sit inside a phrase that fails Integritas because its antecedent has no consequent; a phrase that satisfies Integritas can sit inside a section whose Integritas fails because the section omits its conclusion. The composer's task is to compose with Integritas at every scale the work will be heard at, with attention to which scales are load-bearing for the specific musical situation. The architectural-scale question — whether Integritas can be perceived at the level of the whole work, beyond the scales the local pulses of attention naturally track — is the topic of §6.5 below, where the principle's unusually strong scaling properties (relative to Claritas) receive specific treatment.

§6.3 · The Mechanism of Recognition

The question this section asks is how the listener perceives Integritas — how she registers, often without being able to articulate it, that a work is complete or incomplete, that a passage is padded or pared back, that a movement has arrived at its end or simply stopped. The mechanism turns out to be relatively well-understood; the perceptual apparatus involved is older and more robust than the apparatus Claritas's recognition- of-intention requires.

Three threads converge on the answer.

The first is the Gestalt principle of closure. The early- twentieth-century Gestalt psychologists — Max Wertheimer, Wolfgang Köhler, Kurt Koffka — established that human perception is biased toward completing incomplete patterns. A figure with a small gap in its outline is perceived as a closed figure with a gap rather than as two separate arcs; a melodic line that approaches a cadence is perceived as moving toward a closing tone rather than as floating in suspension; a rhythmic pattern that establishes a regular pulse is perceived as carrying that pulse through any temporary interruption. The closure principle is not learned; it is a property of the human perceptual system, demonstrable across cultures and across modalities. Albert Bregman's Auditory Scene Analysis (1990) worked out the auditory analogues of the Gestalt principles in detail: the ear is biased toward closure at every scale it operates on, from the moment-to-moment tracking of an interrupted tone (which it perceives as continuing through the interruption) up to the long-arc tracking of a melodic line (which it perceives as moving toward its resting point). Integritas, perceptually, is the satisfaction of closure expectations at the appropriate scales: the cadence that arrives where the phrase's structural logic predicts it will arrive, the recapitulation that returns where the movement's structural logic predicts it will return, the silence that closes the work where the work's structural logic predicts it will close.

The second is the analytic-aesthetics formulation by Monroe Beardsley. In Aesthetics: Problems in the Philosophy of Criticism (1958), Beardsley proposed five criteria for the aesthetic evaluation of a work: unity, complexity, intensity, magnitude, and regional quality. Unity — the first criterion — is the property of the work cohering as one thing, with no internal contradictions, no missing structural pieces, no superfluous additions. Beardsley's unity is, with very minor adjustments of vocabulary, the present chapter's Integritas: he is naming the same property under the language of mid-twentieth-century analytic philosophy. The analytic apparatus Beardsley brought to bear — the distinction between intrinsic and extrinsic features of a work, the careful separation of evaluation from explanation, the demand that aesthetic claims be defensible on grounds the work itself supplies — provides Integritas with an evaluative vocabulary alongside its perceptual one. Where the Gestalt apparatus tells us how the listener perceives Integritas, Beardsley's apparatus tells us how to evaluate whether a work has it.

The third is the phenomenology of completeness itself — what it actually feels like to hear a work arrive at its end, in contrast to what it feels like to hear a work cut off or to drag on past its end. This is the experiential surface of the perceptual mechanism. The trained listener can name the experience; the untrained listener cannot, but registers it just as immediately. A symphonic finale that ends where the form's logic demands produces, in the listener, a felt sense of completion — a relaxation of attention, a satisfied recognition that the journey has reached its destination. A finale that ends earlier produces a felt sense of being cheated; a finale that ends later produces a felt sense of restlessness, of the composer having stayed past her welcome. These are the experiences Integritas exists to describe. They are continuous with the experiences anyone has had walking through a satisfying conversation, watching a well-paced film, reading a properly- edited essay: the same Gestalt-closure mechanism operates across every artistic medium, and what music exhibits at the level of the cadence and the section and the movement is the same property a poem exhibits at the level of the line and the stanza and the whole work.

The recognition mechanism is, in this sense, simpler than the mechanism for Claritas. Closure is a robust Gestalt phenomenon that operates below conscious awareness, that does not require the listener to identify the structural categories the work is operating in, that registers with the same immediacy at every scale the perceptual system tracks. Claritas's recognition of intention requires the further step of the listener taking the perceived structure as intentional — a step Goodman's symbol- system apparatus describes (§8.3 below). Integritas's recognition requires no such further step. The closure-or-its-absence is itself the recognition. This is why architectural-scale Integritas can be perceived where architectural-scale Claritas cannot (§6.5 below): the perceptual mechanism is more basic, and operates without the symbol-system apparatus the more sophisticated claim of Claritas requires.

§6.4 · Integritas and the Other Aesthetic Principles

Integritas is genuinely independent of Consonantia and Claritas, and §5.5 above establishes that it is the cleanest of the three Aquinas conditions on the question of independence from Volume 1. The harmonic theory of Volume 1 names the working material composers shape — directed pulls along the consonance gradient at the timescales the listener's auditory short-term memory tracks — but says nothing about whether a given work is complete in itself or whether each of its elements earns its place. A composer who has not internalized Integritas can write passages whose moment-to-moment pulls and resolutions are harmonically impeccable and whose pieces nonetheless fail as music because they are padded, or incomplete, or both. The two principles operate at different levels of the compositional decision: Volume 1 supplies the local vocabulary, and Integritas asks whether the work built from that vocabulary is entire in itself.

The relationship to the other two Aquinas conditions is one of distinction without strict separation. The three are conceptually independent — each can be satisfied or violated without the others — but in compositional practice they interact in characteristic ways.

Integritas and Consonantia interact through proportion. A work whose redundant material has been pruned is one in which the remaining material is easier to balance: the proportions are clearer, the weights are easier to judge, the relationships among the parts are more legible. A composer who attempts to balance a passage that contains structural padding will struggle to set its proportions correctly, because the padding distorts the apparent weight of every section it appears in. Pruning supports balancing; balancing reveals where pruning is needed. The two operations are usually undertaken together in the revision process, and the composer who has internalized either one is partway toward internalizing the other.

Integritas and Claritas interact through legibility. A work whose structure is uncluttered by superfluous material is one whose form shines through more easily. The composer who has pruned the work to its essential elements has, by that act of pruning, made the work's intentional structure more available to perceptual recognition. A work crowded with padding does not necessarily fail Claritas — the perceptive listener may still locate the structural backbone through the noise — but it makes Claritas's perception harder than it needs to be. The composer who works with Integritas is supporting Claritas indirectly: the form is easier to perceive as intentional when fewer extraneous gestures compete for the listener's attention.

§5.2's sub-category mapping locates Integritas's working sub-category as Economy. The chapter has used "Integritas" and "the principle" interchangeably with that sub-category in places; the convention §5.2 establishes is that the parent principle (Integritas) names the aesthetic condition, and the sub-category (Economy) names the composer's daily operational practice that realizes the condition. The two are not separate principles competing for attention but two scales of vocabulary for the same compositional reality. The principle-boxes in the curriculum's lesson sequence use Economy at the operational level; the present chapter uses Integritas at the aesthetic-conceptual level; the same musical question is being asked under both names.

§6.5 · Integritas at Architectural Scale

Of the three Aquinas conditions, Integritas is the one that scales most cleanly to architectural levels — better, in fact, than Claritas, whose architectural-scale operation §8.6 below treats with the careful caveats Volume 1's scope restriction requires. The chapter develops this asymmetry because it is genuinely interesting and pedagogically consequential.

Volume 1's gradient analysis (V1 §2.4.4) commits to the timescales the listener's auditory short-term memory actually operates on, and declines to extend its perceptual claims to whole-movement scale. The empirical record on long-range tonal perception does not support the claim that the listener experiences the architectural-scale structural dominant as a structural dominant, or the long-range key plan as a perceptually-integrated whole. The scope restriction is the V1 theory's principled commitment to what the empirical record can support.

The scope restriction has different consequences for each of the three Aquinas conditions. Consonantia's architectural-scale operation (§7.6 below) relies in part on long-range tonal perception that V1 declines, and the chapter handles this carefully by citing Schenker as a reader-of-page rather than as a description of long-range tonal hearing. Claritas's architectural- scale operation (§8.6 below) faces the same restriction: long- range tonal structure cannot be the perceptual mechanism for the listener's recognition of architectural Order, and the chapter works around the restriction by describing the architectural-scale recognition through the recurrence of material, the balance of proportions, and the resolution of formal expectations.

Integritas, by contrast, scales to architectural levels through the Gestalt closure mechanism — a perceptual apparatus that does not depend on long-range tonal hearing. The listener perceives a movement as complete or incomplete through the closure-or-absence of structural elements: did the recapitulation arrive? did the work return to the tonic at the end? did the development do the work the form expects of it? did the coda close the gestures the body of the movement opened? These are perceptual questions the listener can answer without holding a long-range tonal gestalt in mind, because the perceptual cues are the recurrences and resolutions themselves — discrete events, perceived in the moment of their occurrence, whose pattern across the movement adds up to the closure-or-its-absence the listener experiences.

This explains why listeners — including untrained listeners — routinely report perceiving "this movement is padded" or "the development goes on too long" or "the work feels complete" or "something is missing here" even when they cannot track a long- range tonal plan. The Gestalt closure apparatus is doing the work, and it is doing it without requiring the listener to perceive the architectural-scale tonal coherence whose perceptual status V1 questions.

Three examples illuminate the principle at architectural scale.

Bruckner's Ninth Symphony, left without its finished finale at the composer's death, is perceived by listeners as incomplete. The mechanism is not that the listener tracks a long-range tonal gestalt of a four-movement plan and registers the missing fourth movement against that mental representation. The mechanism is more direct: the three completed movements end with the slow movement's chromatic farewell, and the listener — recognizing that the symphonic form Bruckner was working in calls for a finale, and not encountering one — experiences the closure-expectation of the form as unmet. The perception of incompleteness is the perception of closure-expectations not satisfied at the architectural scale where they should have been.

Schubert's Unfinished Symphony, despite its name, is perceived as complete by most listeners. The two completed movements form their own coherent two-movement whole — first movement and slow movement — and the closure-expectations the two-movement form sets up are met. The listener does not experience the missing third and fourth movements as missing, because the two-movement structure does not call for them; the work operates at a different scale than Bruckner's Ninth, and at its own scale it is whole.

Beethoven's expanded codas, particularly in the late symphonies, are a third instructive case. The coda of the Ninth Symphony's first movement runs to a substantial length and at first hearing might seem disproportionate to the rest of the movement. But the listener does not experience the coda as padding because the closure-expectations the development has set up demand the extended release the coda provides. The accumulated tensions of the development must be discharged, and the discharge takes the time it takes. Integritas is satisfied at the architectural scale even though the proportions are quantitatively asymmetrical — because the qualitative work the closure requires is being done.

The consequence for pedagogy is consequential. Integritas at architectural scale is the most accessible of the three Aquinas conditions to the untrained ear, because closure-at-scale is a Gestalt phenomenon that does not require the technical vocabulary tonal perception requires. A listener can be taught to perceive Integritas at every scale of a symphonic movement long before she can be taught to perceive Consonantia's proportional balance or Claritas's recognition of intentional structure. The curriculum exploits this asymmetry: early listening exercises (Stages I–III) focus heavily on Integritas-perception, because that is the scale at which untrained ears can begin tracking architectural-scale judgments with confidence.

§6.6 · Integritas and the Composer's Discipline

Operationally, composing with Integritas means asking the editorial question at every scale of the work as it is being written: does what is here belong, and is what belongs here. The question is asked in real time during composition; it is asked again, more rigorously, during revision. The composer's discipline is the cultivation of the capacity to ask the question well and to act on its answers.

Two thumbnails illuminate the discipline as a historical practice.

The first is the famous exchange between Mozart and Emperor Joseph II, as paraphrased by Peter Shaffer in Amadeus. The Emperor complains that Die Entführung aus dem Serail has "too many notes." Mozart, having considered the criticism, replies that the work contains "just as many notes as are required, neither more nor less." The exchange is apocryphal in its specific form, but the principle Mozart is articulating is real and is recoverable from his actual compositional practice. The notes a Mozart score contains are the notes the music's structure requires, and not more. Mozart's compositional discipline was, among other things, the discipline of writing with Integritas — of making each note earn its place and refusing to write notes that did not. The historical anecdote captures something real about how Mozart understood his craft: the work was finished when it contained what it needed and nothing more.

The second is the documentary record of Mozart's own teaching practice, preserved in the Attwood notebooks (1784–85) where the young Thomas Attwood worked through species counterpoint and other exercises under Mozart's supervision. The notebooks show Mozart's corrections in red pencil: alongside the technical fixes (a wrong note here, a forbidden parallel there), there is a systematic editorial pruning. Where Attwood has written a phrase that wanders, Mozart's pencil crosses out the wandering and supplies a tighter version. Where Attwood has padded a counterpoint exercise with passing tones that add nothing, Mozart's pencil removes them. The substance of the lesson was, in many places, the lesson of pruning — the lesson that less material, more deliberately chosen, makes for better music. Mozart was teaching Integritas operationally before he could have named it under the term.

The Gradus curriculum's discipline-as-formation chapter (§9 below) develops the operational pedagogy of this internalization across the full sequence of exercises. Species counterpoint, with its constraints on what motions are permitted at what scale, trains the student to reject motion that does not serve the line — at first as a rule to follow, eventually as an internalized judgment about what each voice is doing. Figured bass, with its requirement that the student realize a bass line into four-voice harmony, trains the student to distinguish between voicings that carry the harmony forward and voicings that pad it out — between the structural-essential and the structural-redundant. Chorale harmonization, working with the constraints of a given melodic line, trains the student to add only what the melody requires. Each of these disciplines is, in its operational core, the cultivation of the capacity to refuse.

The capacity to refuse is the heart of the Integritas discipline. A composer who can add but cannot subtract produces work that grows in length and complexity but loses focus and weight. A composer who can subtract but cannot add produces work that is lean but anemic. The mature composer can do both — and chooses, at every moment in the writing of every piece, which operation the work currently requires. The principle the curriculum trains is the judgment of which is which.

Revision, in this picture, is not a separate phase of composition; it is composition's internal practice of self-criticism, operating on the work continuously rather than at a fixed stage. The composer who revises only after she has finished writing the first draft has divided the work into two phases that do not actually divide: every measure she writes is already being revised against the measures that preceded it, and against the work she is composing in her head as she goes. The pedagogical task is to cultivate this continuous self-criticism early, so that by the time the student is writing a substantial piece on her own, the editorial question is being asked in real time at every scale. The student who has internalized this practice is composing with Integritas as a matter of course; the student who has not is producing first drafts that need extensive revision, and may or may not have the discipline to revise them adequately.

The composer who cannot prune cannot produce work that has Integritas, regardless of how technically competent the surface of her writing is. The principle is operationally simple and pedagogically difficult: the discipline of refusing is harder to teach than the discipline of adding, because the natural direction of compositional invention is generative rather than editorial. The curriculum addresses the difficulty by making editorial practice an explicit part of the daily work from the earliest exercises onward — not as a separate skill but as the operational form Integritas takes in the student's actual writing.

§6.7 · A Note on Integritas and Beauty

The relationship between Integritas and beauty is the relationship between a necessary condition and the joint of necessary conditions that beauty requires. Integritas is necessary: a work that fails to be entire in itself, that contains material that does not earn its place or that omits material the structure requires, cannot be beautiful. The listener perceives the incompleteness or the padding directly, and the perception colors every other aesthetic judgment she will form about the work.

But Integritas is not sufficient. A work that is complete in itself, with every element earning its place, can still be poorly proportioned (failing Consonantia) or illegible in its structure (failing Claritas). The mature aesthetic judgment perceives all three Aquinas conditions at once — the wholeness, the proportionate relationship of parts, the perceptibility of intentional structure — and the work the listener experiences as beautiful is the work in which the three converge at well-judged levels.

The three conditions are not ranked in importance. Beauty requires all three, and a work that strongly satisfies one but only weakly satisfies the others is heard as falling short. A passage of extraordinary balance and clarity that omits its needed conclusion is unsatisfying; a passage of admirable completeness whose parts are wrongly weighted is also unsatisfying; a passage that is both complete and proportionate but whose intentional structure does not show through to the listener is the third unsatisfying case. The composer's task is the joint satisfaction of all three conditions, and the discipline the curriculum trains is the joint cultivation of all three.

The chapter that follows develops Consonantia at the same depth Integritas has just received, and §8 below develops Claritas. The three together account for what makes a work beautiful, and the chapters' parallel structure is the structural expression of the three's parallel status as conditions of beauty.

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§7 · Consonantia

§7.1 · The Position

The second of Aquinas's three conditions is consonantia — translated as "harmony," "proportion," or, in the older theological vocabulary, "due proportion of parts." Of the three conditions, it is the most musical in name: consonantia is the Latin from which English consonance descends, and Aquinas's choice of the term reflects the deep medieval intuition that the parts of a beautiful thing stand in some sounding relationship to one another. Whether the thing in question is a musical chord, an architectural facade, a literary work, or a moral life, consonantia names the property by which its parts speak to one another in proportion.

The present chapter develops consonantia as the second of three principles that together account for what makes a work beautiful, working alongside Integritas (§6 above) and Claritas (§8 below). Where Integritas asks whether the work is complete, and Claritas asks whether the work's structure is perceivable as intentional, Consonantia asks whether the work's parts are rightly related to one another. Are the proportions of phrase to phrase, section to section, statement to development, balanced as they should be? Do the materials sit in productive juxtaposition? Do the dynamics, registers, and textures hold the music in equilibrium? These are the operational questions of Consonantia.

The principle gathers under itself two sub-categories §5.2 above has already named: Balance and Contrast. They appear as two sub-categories rather than one because they name two genuinely different ways the parts of a work can stand in relationship. Balance names the equilibrium of forces — the way an antecedent phrase counterweights its consequent, the way a section's tension is offset by its release, the way the proportions of a sonata-form exposition hold the listener in suspension across a long arc; Balance holds both the quantitative aspect (relative durations and section weights in proportional ratio, the Pythagorean inheritance applied to musical time) and the qualitative aspect (tension and release, weight and counterweight). Contrast names the juxtaposition of unlike with like — major against minor, loud against soft, tonic against subdominant, homophony against polyphony, motion against stasis — and is itself a structural device by which the unlikeness of the parts becomes part of how they relate; Contrast operates both synchronically (loud against soft now) and diachronically (statement then development, theme then variation). The two sub-categories are developed at length in §7.3 below.

Consonantia is the broader of the two new chapters (Integritas and Consonantia) precisely because so much of the working aesthetic vocabulary of music is concentrated here. §5.2 names the structural reason: most compositional decisions about balance, contrast, proportion, juxtaposition, and motion-and-rest are Consonantia decisions, while Integritas and Claritas each operate within narrower working domains. The chapter develops Balance and Contrast at length because the practical work of composition is, more often than not, the work of arranging parts in proportionate relationship.

The historical lineage is long, and largely musical in its key statements. Aquinas's consonantia is the scholastic synthesis of an older Greek inheritance: Plato's symmetria in the Timaeus, Aristotle's harmonia in the Metaphysics, and — at the origin — the Pythagorean discovery (sixth century BCE) that the consonant musical intervals correspond to simple integer ratios. The Pythagorean discovery is the origin point of the entire Western tradition that beauty has discoverable structural properties; it is, etymologically and conceptually, the origin of consonantia as a category. Renaissance humanism continued the position: Alberti's concinnitas, cited under Integritas, also carries the consonantia freight (Alberti's term covers both wholeness and proportion in the same definitional sentence). The eighteenth century renews the principle under new vocabulary — Francis Hutcheson's uniformity amidst variety in his Inquiry into the Original of Our Ideas of Beauty and Virtue (1725) is, recognizably, the present chapter's Consonantia: unity among the parts (Balance) and variety within the unity (Contrast) as the joint structural condition under which an object is perceived as beautiful. The nineteenth-century music-formalist tradition gives the position its definitive musical articulation: Eduard Hanslick's Vom Musikalisch-Schönen (1854), with its famous formulation of music as tönend bewegte Formen — "tonally-moved forms" — develops the position into a specifically musical aesthetics. Hanslick's claim that musical beauty consists in the forms themselves, considered as patterns of sounding motion, is recognizably a claim about consonantia: the parts of the music are arranged in proportionate relationship to one another, and that arrangement is what beauty consists in.

The three Aquinas conditions are independent (§5.2 above): a passage may be exquisitely balanced in its parts but obscure in its overall structure; one may be perfectly clear in its formal articulation but unbalanced in the weights it gives its parts. The distinction worth marking within the Consonantia chapter is the one between the parent principle and Volume 1's pitch-level acoustic concept.

Consonantia is not the harmonic-dimension concept of consonance Volume 1 develops at the pitch level. The two vocabularies share a medieval root but operate at different scales: consonance in Volume 1 names the local pitch-to-pitch relationship the listener perceives moment-by-moment along the consonance gradient; Consonantia in the present chapter names the proportionate relationship the listener perceives among the parts of a work, at scales from the phrase upward. The relationship is genealogical, not identical.

Consonantia has a partial relationship to Volume 1 that should be stated at the front of the chapter. Per §5.5 above, the motion- and-stasis operation inside Consonantia's Contrast sub-category inherits Volume 1's Claim 2 (Movement is the central phenomenon of the harmonic dimension) as its local material. Claim 2 supplies the working substance — the directed pulls and resolutions composers arrange — and the Contrast sub-category of Consonantia supplies the cross-scale aesthetic shaping of those pulls into phrases and short sections the ear can track. The relationship is one of nesting, not identity: Movement is V1's category for the local phenomenon, and the aesthetic principle of Contrast is the cross-scale shaping the composer applies. The remainder of Consonantia — Balance, and the rest of Contrast's operational range — is independent of Volume 1.

This is the position of the chapter: Consonantia is the property of parts in proportionate relationship — balanced in equilibrium, or in productive contrast — and the principle operates at every scale of musical organization at which the listener perceives parts as standing in relation to one another.

§7.2 · Consonantia at Every Scale

Consonantia operates at every scale of musical organization at which the listener perceives parts as standing in relation to one another. The chapter develops the claim by walking through the scales in turn — from the major triad through the motive, the phrase, the section, and the movement — and showing where Consonantia lives at each.

The chord. The smallest scale at which Consonantia operates in pitched music is the major triad. Three notes spaced by the intervals of the major third and the minor third stand in a relationship the listener perceives as proportionate: the partials of the three notes overlap at the same frequencies as the partials of the root sounding alone, the three pitches reinforce one another spectrally, and the chord registers as a stable acoustic unit. The major triad is, in this sense, consonantia at its smallest scale — the Pythagorean discovery turned into the building block of Western harmony. The minor triad, the seventh chord, the ninth chord, and the full vocabulary of tertian harmony are extensions of the same principle: parts (in this case, individual pitches) standing in spectral relationships the listener perceives as proportionate. When the chord is a unit, the parts inside it are in Consonantia; when the chord is not a unit (the wrong tones doubled, the spacing too dense or too sparse), the parts fail to cohere as a single sonority and Consonantia at chord scale fails.

The motive. A motivic figure exhibits Consonantia through the internal balance of its constituent durations and intervallic spans. Beethoven's three-shorts-and-a-long motive in the Fifth Symphony arranges its four notes in a proportionate temporal relationship: three units of time on the short notes, three units on the long note, with the rhythmic emphasis falling on the long-note arrival. The figure is balanced in its proportions and its intervals (three repeated pitches followed by a falling minor third); were the proportions different (say, four shorts and one short, or one short and three longs), the figure would not have the same Consonantia. The motive as a unit hangs together because its parts stand in proportionate relationship — both in time and in interval. A motive that is unbalanced in its proportions is heard as awkward; the listener cannot quite hold the figure as a unit. The well-formed motive is one whose internal Consonantia makes it memorable as a single gesture.

The phrase. The canonical case of phrase-scale Consonantia is the eight-bar Mozartean periodic phrase: a four-bar antecedent that asks a musical question and a four-bar consequent that answers it. The antecedent ends on a half cadence; the consequent ends on a perfect authentic cadence. The two halves of the phrase stand in proportionate relationship to each other — equal in length, parallel in melodic shape, contrasted at the cadences (open vs. closed). The phrase is balanced both quantitatively (four bars against four bars) and qualitatively (question against answer). When a phrase has this kind of Consonantia, the listener experiences the phrase as a single coherent unit even though it lasts long enough that her short-term auditory memory must integrate the antecedent against the consequent. The eight-bar period is a canonical model because its proportions are easy to perceive at every level: bar-count, melodic shape, harmonic motion, cadential punctuation.

The section. At the section scale, Consonantia operates as the proportionate balance among the constituent phrases and the productive contrast among the materials. The exposition of a sonata-form movement contains a first-theme group, a transition, a second-theme group, and a closing group. The four sub-sections stand in proportional relationship: the first-theme group establishes the tonic; the transition modulates; the second-theme group establishes the dominant or relative major; the closing group consolidates the new key. The proportions are not arithmetically precise (no rule says the second-theme group must be exactly equal to the first), but they are perceptually balanced: a sonata-form exposition with a four-minute first-theme group and a forty-second second-theme group is heard as unbalanced, because the proportions of the parts have failed Consonantia. The well-formed exposition has its sections in proportionate relationship; the deformed exposition is one in which the proportions are off.

The movement. The four-movement plan of the Classical symphony — fast first movement in sonata form, slow second movement, dance movement (minuet or scherzo), fast finale — is Consonantia operating at the scale of the entire piece. The four movements stand in productive contrast (each in a different tempo and character) and in proportionate balance (the symphony as a whole is one well-proportioned multi-movement work). A symphony whose four movements were all the same tempo would fail Consonantia at the movement-cycle scale; a symphony with one twenty-minute movement and three four-minute movements would also fail, in the opposite direction. The balance of the four-movement plan is the largest-scale Consonantia composers routinely deploy. The two-movement form (Schubert Unfinished, Beethoven Op. 111), the three-movement plan (the standard concerto), the five-movement plan (Berlioz Symphonie fantastique), and the elaborate multi-movement structures of late-Romantic and twentieth-century repertoire are all elaborations of the same Consonantia principle at different scales.

The Consonantia at each scale is largely independent of the Consonantia at the other scales. A perfectly balanced phrase can sit inside a section whose proportions are off; a well-proportioned section can sit inside a movement whose four-movement plan is unbalanced. The composer's task is to compose with Consonantia at every scale the listener will perceive parts as standing in relation. The architectural-scale question — whether Consonantia can be perceived at the level of the whole work, beyond the scales the local pulses of perception naturally track — is the topic of §7.6 below.

§7.3 · Balance and Contrast

Consonantia carries two sub-categories — Balance and Contrast — because the working aesthetic vocabulary of music is concentrated at the proportionate-relationship-of-parts layer, and the two sub-categories name two genuinely different ways those relationships operate. Most of the compositional decisions a working composer makes about how the parts of her work stand in relation are decisions about Balance or about Contrast or, characteristically, about both at once. The chapter develops them in detail because the discipline of arranging parts in proportionate relationship is the operational substance of so much of the curriculum.

Balance is the equilibrium of forces among the parts of a work. The sub-category carries two distinct modes the chapter treats separately.

The quantitative mode of Balance is the Pythagorean inheritance applied to musical time and weight. Parts of a work stand in proportionate ratio to one another — the four-bar antecedent against the four-bar consequent, the exposition against the recapitulation (typically of comparable length), the slow movement against the fast movement (typically of comparable timescale even when tempos differ). The Pythagorean discovery that the consonant musical intervals correspond to simple integer ratios is the philosophical root: the same proposition that intervallic relationships of beauty are quantifiable extends, in the Western tradition, to relationships among durations, section lengths, and dynamic weights. The medieval tradition extended the Pythagorean point through architectural theory: the Cistercian abbey and the High Gothic cathedral exhibit specific proportional ratios (often 2:1, 3:2, 5:4) at every scale, from window-to-wall to nave-to- aisle to overall plan. The Renaissance theorists Alberti and Palladio articulated the position explicitly: beautiful proportions are not subjective; they are quantifiable. Music inherited this position through Boethius and the medieval quadrivium in which music was one of the four mathematical arts.

The qualitative mode of Balance operates where quantitative ratios are not the right description. A tension is balanced against a release; a forte against a piano; a fast section against a slow one; a question against an answer. The balance is not a ratio of durations or weights but a balance of forces — what the music has set up against what the music delivers. The qualitative mode is what allows Beethoven's expanded codas to work: the coda is not in quantitative proportion to the body of the movement (it is unusually long), but it IS in qualitative balance with the development (the tensions accumulated through the development require the extended release the coda provides). The trained ear hears the qualitative balance even when the quantitative ratio is skewed; the listener experiences the extended coda as not too long because the qualitative scale demands the extra time. Brahms developed the qualitative mode of Balance to a high refinement: the Brahmsian rhythmic-displacement passage, where the music establishes one metric expectation and then proceeds against it for several bars, is a case study in qualitative Balance operating against quantitative expectation. The displacement balances out, but not by symmetry of bar count.

A composer who has internalized only the quantitative mode of Balance produces music whose proportions are mathematically precise but whose pacing feels mechanical; a composer who has internalized only the qualitative mode produces music whose felt pacing is satisfying but whose proportions seem arbitrary on the page. The mature composer has both modes available and chooses between them according to what the specific musical situation requires.

Contrast is the juxtaposition of unlike against like. The sub-category carries two distinct dimensions the chapter treats separately.

The synchronic mode of Contrast is the unlikeness in simultaneous or immediately-successive juxtaposition. Loud against soft, high against low, tonic against subdominant, homophony against polyphony, motion against stasis, sound against silence: these are the synchronic contrasts the composer deploys at every scale of the work. The function of synchronic Contrast is to articulate structure — to mark a boundary, to highlight a transition, to direct the listener's attention to a moment of structural importance. The fortissimo chord at the start of Beethoven's Eroica is synchronic Contrast against the silence that precedes the symphony; the piano statement of the development's second theme is synchronic Contrast against the forte of the development's opening; the pizzicato string entry against the sustained winds is synchronic Contrast at the textural scale. Synchronic Contrast is the composer's most direct means of making structure audible.

The diachronic mode of Contrast is the unlikeness across time — what allows variation form, what produces development, what makes recapitulation register as restatement-after-departure rather than as repetition. The theme and variations form is the canonical case: a theme is stated, and then a series of variations re- present the same material in transformed versions — different rhythm, different key, different accompaniment, different mode. Each variation is in diachronic Contrast with its source. The sonata-form development is a more complex case: the development takes the exposition's material and reworks it through a series of harmonic regions, sequenced motivic transformations, contrapuntal expansions; the recapitulation that follows is the diachronic Contrast against the development (the material returns in the home key, the harmonic instability is resolved). The well-formed sonata movement balances synchronic and diachronic Contrast at every scale, from beat-to-beat dynamic shifts to large-section transformations.

The motion-and-stasis operation inside Contrast. One specific case of synchronic Contrast deserves separate articulation, because it is the locus where Volume 1's Claim 2 enters the present chapter's territory. Movement — directed pulls along the consonance gradient, the central phenomenon of Volume 1's harmonic theory — and stasis — the absence of directed pull, the rest, the sustained sonority, the silence — are the two terms whose Contrast structures every phrase the composer writes. A phrase that is all motion exhausts the listener; a phrase that is all stasis bores her; the well-formed phrase places motion against stasis in proportion to its scale. The directed pull of a dominant-seventh chord is one motion; the held tonic that follows is one stasis; the local Contrast between them is what the listener perceives as the phrase's harmonic shape. At larger scales, the motion of a developmental passage is contrasted with the stasis of a sustained pedal point; the motion of an episode is contrasted with the stasis of the subject's return. The motion is Volume 1's local territory; the Contrast between motion and stasis is Consonantia's territory; the two operate at adjacent scales of compositional decision and converge on the same musical moment.

How Balance and Contrast combine. The canonical case is the well-formed sonata exposition. The first-theme group and the second-theme group are in Contrast — different keys, different characters, often different rhythmic profiles. But the exposition as a whole is in Balance — the contrasting groups stand in proportionate relationship to each other and to the transition that connects them, and the exposition's total weight balances against the development and recapitulation that follow. Balance asks whether the proportions are right; Contrast asks whether the materials are different in productive ways. The two operate at different dimensions of the same compositional decision, and the well-formed section is one in which both dimensions are working at once.

A composer who has internalized Balance but not Contrast produces music whose proportions are correct but whose materials are monotonous — every section sounds like every other section, even though their lengths are well-judged. A composer who has internalized Contrast but not Balance produces music whose materials are vivid but whose pacing is lopsided — striking juxtapositions follow one another with no proportional logic, and the listener loses her bearings. The mature composer has both internalized, and the discipline the curriculum trains is the joint cultivation of both as a single operational capacity. By Stage X the student is composing with Consonantia as a matter of course, which means composing with Balance and Contrast at every scale, in productive interaction.

§7.4 · The Mechanism of Recognition

The question this section asks is how the listener perceives Consonantia — how she registers, often without being able to articulate it, that the parts of a passage are in proportionate relationship or that the parts of a section are in productive contrast or that something has gone off in the proportions. The mechanism turns out to involve two complementary perceptual apparatuses, both empirically well-grounded.

The first is the statistical-profile apparatus worked out by Carol Krumhansl and her collaborators. In Cognitive Foundations of Musical Pitch (1990), Krumhansl reports a sustained empirical investigation of how Western-trained listeners rate the stability of each scale degree within a given key context. The result — the Krumhansl-Kessler key profile — is a stable, reproducible pattern: in a major key, the tonic is rated most stable, the dominant and the third nearly equally stable, the supertonic and the sub-dominant moderately stable, and the chromatic non-diatonic pitches least stable. The profile is not a theoretical construction; it is a statistical summary of listener judgments across many experiments, with subjects from various levels of musical training. The profile operates below conscious awareness: listeners produce consistent ratings without being able to articulate the principle by which they are rating. The Krumhansl-Kessler work establishes that the perceptual ground for tonal Balance is empirically real and operationally describable. A passage that establishes a tonal center and then arranges its pitches in approximate accord with the key profile is heard as tonally balanced; a passage that violates the profile is heard as off-kilter, even by listeners who could not name the violation.

The Krumhansl apparatus extends naturally to questions of section-scale tonal Balance. A sonata exposition that moves from the tonic to the dominant produces, in the listener, a perceptual profile in which the dominant region is heard as standing in proportionate relationship to the home key. The listener does not need to know the technical name "dominant region" to perceive this; she needs only to have internalized the Western-tonal statistical regularities the Krumhansl profile describes. The exposition is in tonal Balance because its key plan accords with what the listener's internalized profile expects from a Western-tonal piece.

The second apparatus is the auditory scene analysis worked out by Albert Bregman in his magisterial Auditory Scene Analysis: The Perceptual Organization of Sound (1990). Bregman developed the empirical theory of how the perceptual system parses simultaneous sounds into distinct streams. The parsing operates through principles homologous to the Gestalt principles of visual perception: similarity (sounds that share frequency-band, timbre, or rhythm are grouped together), continuity (sounds that move together are grouped together), proximity (sounds that are close in pitch or time are grouped together), and so on. The principles operate below conscious awareness and account for how the listener separates the voices of a fugue, identifies the melody against an accompaniment, follows a single instrumental line through a complex texture.

The Bregman apparatus is the perceptual ground for textural Consonantia. When the composer arranges multiple voices or instruments in a polyphonic texture, the texture either parses cleanly (each voice is hearable as a distinct stream) or it does not (the voices blur together or compete for attention in ways the parsing apparatus cannot resolve). Contrast is, at the textural scale, a Bregmanian property: voices that are too similar in register, timbre, and rhythmic profile fail to separate; voices that are too dissimilar fail to combine into a unified texture. The well-formed texture is one whose Contrast operates within the parsing tolerances of the auditory scene-analysis apparatus.

Together, the Krumhansl apparatus and the Bregman apparatus account for most of what the listener perceives as Consonantia. The Krumhansl profile handles the tonal dimension (which pitches and harmonic regions stand in stable relationship to a given tonal center); the Bregman apparatus handles the textural dimension (which voices stand in stream-parsable relationship to each other). The two converge at the phrase scale, where the well-formed phrase is one whose tonal motion accords with Krumhansl expectations and whose textural arrangement allows Bregman parsing to operate cleanly.

The phenomenology of balanced-vs-off-balance is the experiential surface of these perceptual mechanisms. A Beethoven coda that extends past the development's accumulated tensions feels right because the qualitative balance the coda restores is what the listener's perceptual apparatus has been waiting for. A sonata exposition whose first-theme group runs unduly long feels off because the listener's apparatus has internalized statistical regularities about how long first-theme groups typically run, and the violation registers as imbalance. The listener does not consciously consult these statistical regularities; they operate as the pre-conscious calibration against which her judgments of proportion are formed. Consonantia is the property of the music that accords with the calibration; the violation of Consonantia is the property that does not.

§7.5 · Consonantia and the Other Aesthetic Principles

Consonantia is independent of Integritas and Claritas (per §7.1) and partially nested with Volume 1 through the motion-and-stasis operation inside Contrast (per §7.3 and per §5.5 above). The independence claim and the nesting both deserve careful statement here.

The independence from Integritas and Claritas is real but not absolute. A passage may exhibit Consonantia without Integritas — its parts may be proportionate to each other while the work as a whole remains incomplete — and a passage may exhibit Consonantia without Claritas — its parts may be proportionate while the overall structure remains illegible. Conversely, a passage may exhibit Integritas and Claritas while failing Consonantia: every element earning its place, the structural intent perceptible, but the proportions among the parts off-kilter. The three conditions are conceptually distinct.

In practice they interact in characteristic ways. Consonantia and Integritas often co-operate through the work of revision. A passage whose redundant material has been pruned (Integritas) is one whose remaining parts are easier to balance (Consonantia); a passage whose proportions have been clarified makes obvious where the padding lies. The composer's revision typically alternates between the two operations, pruning what does not earn its place and rebalancing what remains. By the end of the revision process, the work has both Integritas and Consonantia in well-judged convergence.

Consonantia and Claritas co-operate through the work of formal articulation. A passage whose parts are in proportionate relationship is one whose structural design is more easily perceived as intentional. The first-theme group that arrives, the transition that connects, the second-theme group that contrasts — these are not separate events the listener tracks; they are parts of a single sonata-exposition design whose Consonantia (the proportions among them) supports Claritas (the perceptibility of the design as design). A work in which the parts are out of proportion has its Claritas obscured by the misweighting, even if the structural intent is otherwise clear; a work whose proportions are right has its Claritas supported by the proportions themselves.

The partial nesting with Volume 1 is specifically through the motion-and-stasis operation inside Contrast. Volume 1's Claim 2 names Movement as the central phenomenon of the harmonic dimension at local scale: the listener perceives directed pulls along the consonance gradient as the harmonic-dimensional substance of tonal music. The aesthetic principle of Contrast, within Consonantia, picks up where Claim 2 leaves off: it asks the cross-scale question of how the local pulls and resolutions get arranged into phrases and short sections that the listener tracks as coherent units. The pull from a dominant to its tonic is Claim 2's territory at the local scale; the contrast between the motion of the pull and the stasis of the resolution is Consonantia's territory at the slightly larger scale. The two principles converge on the same musical moment, but they are operating at adjacent scales of compositional decision.

§5.2's sub-category mapping locates Consonantia's working sub- categories as Balance and Contrast. The chapter has used "Consonantia" interchangeably with the sub-categories in places; the convention §5.2 establishes is that the parent principle (Consonantia) names the aesthetic condition, and the sub- categories (Balance, Contrast) name the composer's daily operational practice that realizes it. The principle-boxes in the curriculum's lesson sequence use Balance and Contrast at the operational level; the present chapter uses Consonantia at the aesthetic-conceptual level; the same musical question is being asked under both names.

§7.6 · Consonantia at Architectural Scale

The long-range key plan of a tonal work is, in part, a structure of architectural Consonantia. A sonata-form movement establishes the tonic in the exposition's first-theme group, modulates to the dominant (or, in minor-mode pieces, the relative major) in the second-theme group, develops through various intermediate regions in the development, and returns to the tonic in the recapitulation. The four phases of the movement stand in proportionate relationship to one another: the tonic is the home; the dominant is the established contrast; the development is the working-out across regions; the recapitulation is the resolved return. The relationship is a long-range structural Consonantia operating across the scale of the entire movement.

Volume 1's scope restriction (V1 §2.4.4) declines to extend the gradient analysis to architectural scale, on the empirical ground that the listener does not hold a long-range tonal gestalt in mind across the movement. The scope restriction is principled and the present chapter inherits it: Consonantia at architectural scale is not the experience of long-range tonal perception. The listener does not directly perceive the dominant region as a structural dominant against the home tonic the way she perceives a local dominant-seventh chord against a local tonic resolution. The perceptual mechanisms are different at the two scales.

What the listener does perceive at architectural scale is the proportional balance among the section lengths, the cumulative dynamic shaping, the rhythmic-textural pacing, the recurrence of material, and the resolution of cadential expectations. These are perceptual cues the listener can track at architectural scale because they do not require long-range tonal perception. The section lengths are perceived through accumulated time-judgments (the development feels longer or shorter than the exposition, regardless of whether the listener can name the section boundaries). The cumulative dynamic shaping is perceived as a long arc the listener registers without holding the specific dynamic markings in mind. The recurrence of material at the recapitulation is perceived as return, the listener recognizing the first theme's reappearance even after twenty minutes of intervening music. These are the architectural Consonantia signals the listener can perceive, and they are sufficient to ground the architectural- scale claim.

Heinrich Schenker's structural-analytical apparatus is best understood, in the present framing, as a way of reading architectural Consonantia on the page — the Urlinie and the Bassbrechung and the prolongational layers as notations for the proportional relationships among regions and the harmonic structure that knits them together. Schenker's apparatus is genuinely insightful at the analytical level; it traces architectural-scale structural designs that are real properties of the music as composed. What Volume 1 declines is the further claim that the listener perceives these long-range structures as the gradient operating at architectural scale — a claim Schenker's tradition often makes but the empirical record does not support (V1 §2.4.4 develops the argument). The chapter can engage Schenker analytically without committing to his perceptual claim because the analytical apparatus and the perceptual claim are separable. Schenker is reading the architectural Consonantia on the page; the listener perceives architectural Consonantia in time, through the other cues just enumerated.

The composer who composes with architectural Consonantia is composing with the proportional relationships among the parts of the whole work — the balance of section lengths, the productive contrast of materials across the movement-cycle, the cumulative dynamic and textural shaping. The discipline the curriculum trains is the capacity to hold this large-scale Consonantia in compositional mind even when the local-scale decisions are absorbing the composer's immediate attention. The mature composer arranges her local materials with their architectural-scale Consonantia already in view; the inexperienced composer arranges the local materials and discovers, after the fact, that the larger-scale proportions have not come out right.

§7.7 · Consonantia and the Composer's Discipline

Operationally, composing with Consonantia means making decisions at every scale of the work about how the parts stand in relation. The decisions can be small (which voice carries the line at this beat) or large (how to balance the development against the recapitulation), but they are decisions about proportion and contrast, and the totality of them constitutes the work's Consonantia.

The curriculum's discipline trains Consonantia at progressively larger scales as the student moves through the stages.

At Stages III through VI — the multi-voice writing stages — voice leading is small-scale Consonantia. The four voices of a chorale realization must stand in proportionate relationship at every beat: each voice has its own line, each pair of voices has its own intervallic relationship, the four together form a unified texture that the auditory scene-analysis apparatus can parse cleanly. The species counterpoint exercises of Stage II are the foundational training: two voices must move in proportion, each carrying its independent line while the two together produce a coherent musical surface. By Stage VI the student is harmonizing chorales in four voices and realizing figured-bass exercises with the proportionality of voicing internalized as second nature.

At Stage IV — the phrase-and-form stage — phrase-structure exercises are phrase-scale Consonantia. The student writes Mozartean periodic phrases (antecedent-consequent, eight bars), Bach-style sentences (presentation, fragmentation, cadential), and the various asymmetric phrase shapes the historical record supplies. Each exercise trains the capacity to arrange the parts of a phrase in proportionate relationship while making the contrast between question and answer productively articulate.

At Stages VI through X — the form-study stages — form study is section-scale and movement-scale Consonantia. The student studies sonata-form, rondo, variation set, fugue, and the other formal templates the historical record supplies, each as a model for proportional relationship at architectural scale. The student does not memorize the templates as analytical categories; she internalizes them as Consonantia structures she can deploy in her own writing. By Stage X the student is composing whole movements in which the proportions among the sections accord with the proportions the templates establish, and varying from those proportions in productive ways when the specific musical situation requires.

The score-study program runs in parallel across all stages, training the student's ear to recognize Consonantia among sections in canonical works. The student listens to Bach inventions, Mozart sonatas, Beethoven quartets, Brahms intermezzi, Debussy preludes, Stravinsky neoclassical works — and her ear is being calibrated to the proportional relationships the historical tradition has developed. By the time she writes her own work, the calibration is internalized; she composes with Consonantia at every scale because the canonical works she has studied have made the principle perceptually available.

The Discipline as Formation chapter (§9 below) develops the operational pedagogy of all this in detail. The species exercises, the figured-bass realizations, the chorale harmonizations, the form-study, and the score-study work converge on the cultivation of Consonantia as the composer's working aesthetic-judgment apparatus.

§7.8 · A Note on Consonantia and Beauty

Consonantia is necessary for beauty but not sufficient. A passage whose parts are in proportionate relationship — balanced in equilibrium, in productive contrast — can still fall short of beauty if the work as a whole is incomplete (failing Integritas) or if its intentional structure does not register to the listener as intentional (failing Claritas). The three Aquinas conditions are jointly necessary for beauty; each alone is insufficient.

The mature aesthetic judgment perceives the three simultaneously. The listener experiences a work as beautiful when the work is complete in itself, when its parts stand in proportionate relationship, and when its structure is perceivable as intentional design — and the experience of beauty is the convergence of these three. A work strongly satisfying one of the three but only weakly satisfying the others is heard as falling short: the well-proportioned but unfinished work, the complete but unstructured work, the structured but unbalanced work each fall short for a different reason.

The composer's task is the joint satisfaction of all three Aquinas conditions, and the discipline the curriculum trains is the joint cultivation of all three. Stage by stage, the student's ear is calibrated to perceive each of the three; her hand is trained to compose with each; her judgment is developed to hold all three in compositional mind simultaneously. The end-of-curriculum capstone is the composition of a work in which the three converge — not through following a checklist, but as the natural expression of an internalized aesthetic capacity.

The chapter that follows develops Claritas at the same depth Consonantia has just received, and the chapters together account for what makes a work beautiful. The three Aquinas conditions, their parallel chapter treatments, and the working sub-categories that realize them in the daily practice of composition are the integrated aesthetic vocabulary the curriculum teaches.

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§8 · Claritas

§8.1 · The Position

The three parent principles catalogued in §5 — Integritas, Consonantia, and Claritas — name the conditions of beauty under which compositional material is shaped into aesthetically coherent wholes. Order is the developed pedagogical category within Claritas: the sub-category that names intentional structure perceived as intentional, at every scale the listener tracks. The position taken in this chapter is that Order does sufficient theoretical work and has sufficient cross-scale reach that it warrants its own chapter-length treatment, because Claritas via Order operates at scales the harmonic-theory gradient of Volume 1 cannot reach (§8.6 below), because Order's perceptual mechanism is specific enough to require its own apparatus (§8.3 below), and because the curriculum's discipline (§9 below) is, in its deepest sense, the operational training of the student's capacity to compose with Order — and the discipline is more comprehensible once the principle it trains has been articulated explicitly.

Order is the property of intentional structure perceived as intentional. A passage exhibits Order when its material — its pitches, rhythms, dynamics, instrumentations, motivic relations, sectional articulations — is arranged according to a structure the listener can perceive, and when the listener's perception of that structure produces the recognition that the structure is not random. The composer intends something specific; the listener tracks the structure that intention produces; the recognition that the structure is intentional rather than accidental is the listener's experience of the music as ordered.

Order is not a synonym for adherence to inherited form. A Palestrina motet exhibits Order through the deeply-realized inheritance of late-Renaissance modal counterpoint. A Stravinsky Rite of Spring exhibits Order through deliberate departure from common-practice tonal and rhythmic norms. A Webern Symphony Op. 21 exhibits Order through an invented structure (twelve-tone serial organization with strict canonic mirroring across two movements). The three works are radically different in their means, but all three are recognizably ordered — all three are music in which the listener perceives intentional structure. The composer who writes music without Order produces what the listener experiences as random material, regardless of how technically competent the surface may be. The composer who writes music with Order produces music in which the structure carries meaning, even when the specific structural choices are unprecedented.

The position taken in this chapter has substantial historical pedigree, more substantial than is often recognized in contemporary music aesthetics. Thomas Aquinas, in the Summa Theologiae (I, q. 39, a. 8), names three conditions of beauty: integritas (wholeness — nothing missing, nothing superfluous), consonantia (proportion, harmony of parts), and claritas (radiance, intelligibility — "the form shining through the matter"). The third of these — claritas — is, almost word for word, the present chapter's Order. For Aquinas, claritas names the property by which the structural principle of a thing is perceivable as that principle to the attentive viewer; the form is recognizable as form because it shines through the material instantiation. James Joyce's Stephen Dedalus paraphrases the three conditions in A Portrait of the Artist as a Young Man as "wholeness, harmony, and radiance" (Joyce knew Aquinas through Maritain); the contemporary analytical recovery of Aquinas as a serious aesthetician is the work of Jacques Maritain (Art and Scholasticism, 1920) and Umberto Eco (The Aesthetics of Thomas Aquinas, 1956/1988). What the present chapter calls Order, what Aquinas called claritas, and what the modern Thomist tradition has recovered from medieval scholasticism are recognizably the same insight, articulated in different vocabularies separated by seven centuries. The Gradus position is not novel; it is the contemporary restatement of a long-standing tradition in Western aesthetics that has been displaced rather than refuted by twentieth-century formalism.

This is the position of the chapter: Order is the principle by which intention enters music as a perceivable property, and it operates whether the composer works through inherited forms, in deliberate opposition to them, or by means of entirely new structures.

§8.2 · Order at Every Scale

Order operates at every scale of musical organization the listener can track. The chapter develops the claim by walking through the scales in turn — from the harmonic series of a single tone through the motive, the phrase, the section, and the movement — and showing where Order lives at each.

The harmonic series. A single pitched tone is, considered as a structure, ordered. The partials at integer multiples of the fundamental are not a random collection of frequencies; they form a specific pattern, the same pattern in every pitched sound (with timbre-dependent variations in partial strength). The listener does not have to consciously identify the partials to perceive this order — the ear has been habituated to round perceived intervals to the integer ratios the partials produce (V1 §3.2), and the rounding is the perceptual signature of the order present in the sound. The harmonic series is the proof that order is built into pitched sound itself — not designed by any composer, but present as a physical fact. Different sources produce different ordered spectra: a bell's inharmonic partials produce a different order than a voice's harmonic partials. But every pitched sound is ordered in some way, and the composer working with harmonic-spectrum instruments inherits this order rather than creating it.

The motive. A short melodic-and-rhythmic figure of two to eight notes is ordered when its specific arrangement of pitches and durations is recognizable as a unit. The first four notes of Beethoven's Fifth Symphony — three short notes and one long, on a falling third — are ordered: their specific arrangement is recognizable, and any change to the rhythm or the interval would produce a different motive (or no motive at all). The motive's order is what makes it recognizable; when it returns later in the movement, the listener recognizes the return because the order has been preserved.

The phrase. A four-to-eight-bar phrase exhibits Order through the coordinated work of its harmonic motion (pulls and resolutions, V1 §5), its rhythmic shape (typically arching toward a cadence-point), and its melodic contour (rising, falling, arch, inverted-arch). A well-formed phrase is one in which these elements cohere — the harmonic motion and the melodic contour and the rhythmic shape all point at the same cadence-point. A poorly- formed phrase is one in which they work against each other — the harmony arrives but the melody overshoots, the rhythm peaks at the wrong moment, the phrase does not feel like one unit. The Order of a phrase is the coherence of its sub-dimensions.

The section. A section of a movement — an exposition's first-theme group, a development's central modulation, a recapitulation's restated theme — exhibits Order through the recurring use of motivic material, the unfolding of harmonic regions, and the rhythmic-textural shaping. The listener perceives the section as ordered when these elements reinforce one another and produce the sense of a coherent passage. The same material in a different sequence, or different material in the same sequence, would be a different section.

The movement. At the largest scale the listener can track, a movement exhibits Order through the relations between its sections — the return of the first theme in the recapitulation, the proportional balance of exposition and development, the long-range key plan, the cumulative dynamic shaping, the articulation of formal boundaries. The listener does not perceive a 30-minute symphonic movement as a single perceptual gestalt: Volume 1's scope restriction (V1 §2.4.4) commits the gradient analysis to the timescales auditory short-term memory tracks. But the listener does perceive the movement as ordered, through her recognition of returning material (the first theme reappears! — the listener recognizes this as intentional), of balanced sections, of a key plan that resolves. The cross-scale coherence the listener experiences across a movement is the experience of Order operating at movement scale.

The Order at each scale is largely independent of the Orders at the other scales. A well-ordered phrase can sit inside a poorly- ordered movement, and a well-ordered movement can contain a few poorly-ordered phrases. The composer's task is to compose with Order at every scale the listener will track, with attention to which scales are load-bearing for the specific musical situation at hand.

Order beyond the harmonic series. The walk through scales above has used the harmonic-series-grounded Western tradition as its primary illustrative case — the pitch system in which Volume 1's gradient analysis operates, and the system the Gradus curriculum is committed to teaching. Order genuinely comes from the harmonic series in this case. The integer-ratio structure of the series is a property of pitched sound itself, not a Western theoretical construction projected onto sound. We did not invent the harmonic series; we found it. The order was there before the music — the partials of every pitched tone arrange themselves in the same pattern as a fact of physics, and listeners across cultures perceive that order because the ear has been habituated to round perceived intervals to the integer ratios the partials produce (V1 §3.2). The Western tonal tradition built its elaborate harmonic system around this preexistent order; what we composed is the human elaboration of what the acoustic ground already offered for any listener to find.

But the principle of Order is not specific to this case. Other musical systems are grounded in other physical facts, other cultural elaborations, or some combination of the two. The capacity for Order is universal across human musical cultures; what varies is which acoustic, intervallic, rhythmic, or timbral facts each tradition has recognized and built around.

Some traditions ground their order in chosen intervals that are not derived from harmonic-series simplicity. Indonesian gamelan uses pélog and sléndro tunings constructed for inharmonic- spectrum bronze instruments, with intervals that are not octave- based integer ratios; the music produced is unambiguously ordered, and the listener accustomed to the tradition perceives the order without difficulty. Arabic maqām deploys micro-intervals the West does not use; the order of an Arabic- classical performance is in the maqām's specific intervallic vocabulary, not in any harmonic-series gradient. Contemporary microtonal traditions (Partch, the Just Intonation Network, the various xenharmonic movements) deliberately construct interval systems with no privileged relation to the partials; the music they produce is ordered through the consistency and intentionality of the system they construct.

Other traditions ground their order in rhythmic structure without depending on any specific pitched ground. The tāl system of North Indian classical music organizes long cycles of beats into rhythmic hierarchies the trained listener tracks as ordered, often more salient to the listener's experience than the rāga (the pitched dimension). West African drumming traditions — the Ewe of Ghana, the Wolof of Senegal, the Yoruba bàtá tradition — build polyrhythmic textures whose order is in the cross-rhythmic relations between voices, with pitch a secondary or absent parameter. Steve Reich's phase pieces and the broader minimalist tradition organize their order through rhythmic- process structure rather than through harmonic-pull progression. In each case, order is present, perceptible, and tradition- specific — and grounded in something other than the harmonic series.

Still other traditions ground their order in timbral choices. The bronze-instrument inharmonic spectra of gamelan produce a distinct ordered sound-world that Sethares's local-consonance apparatus (V1 §3.1) models accurately. The electronic-music tradition — musique concrète, Stockhausen, Subotnick, contemporary spectralism — often grounds its order in the structural manipulation of timbre itself, with pitch a derived rather than primary parameter.

The conclusion the chapter draws from these cases is consequential. Order is not specific to harmonic-series-grounded music. The harmonic series is one source of order — the source the Western tonal tradition has built on, and the source Volume 1's theory analyzes — but Order itself is broader, present wherever composers and listeners share enough of a system that intentional structure can be perceived. The capacity for Order is universal; the specific grounds on which Order is built vary by tradition. The Gradus method teaches the harmonic-series-grounded version because that is the tradition the curriculum serves, but the student who completes the curriculum will encounter, in her listening and her own work, music grounded in other orders. The principle of Order applies to all of it. What changes between traditions is which acoustic, intervallic, rhythmic, timbral, or structural facts the order is built on; the principle itself — intentional structure perceived as intentional — is constant.

§8.3 · Order and the Recognition of Intention

The Order in a passage is perceived by the listener as intention. This is the part of the principle that requires the most careful statement, because intention is a slippery concept and the claim being made is specific.

The claim is not that the listener reconstructs the composer's biographical intent. The composer's psychological state during composition is not available to the listener, and the listener does not need it. What the listener does is recognize the structure as not random. When the first theme of a sonata-form exposition returns at the start of the recapitulation, the listener does not consult the composer's diary. She hears the return and perceives it as intentional — as deliberate, as designed, as something the composer did on purpose. The perception of intention is a perceptual property of the structure itself, not a guess about the biographical fact.

The careful separation between perceived-intention and biographical-intent is the position Nelson Goodman developed at length in Languages of Art (1968). Goodman argued that the work of art is a symbol system — a structured arrangement whose meaning lies in its functioning as such a system, not in the artist's psychological state during composition. The work exemplifies certain properties (it possesses them and refers to its possession of them) through its structure; the listener perceives the exemplification through the structure; biographical inference about the artist is neither necessary nor sufficient for perception of the work's meaning. The Gradus claim that Order is a property of the structure rather than of biography is recognizably Goodmanian; the analytic-philosophical pedigree is worth stating because the position is sometimes mistaken for a denial that composers have intentions. They do, of course; but the listener does not have access to those intentions except through the structure the composer has built.

The mechanism by which the perceptual response works is straightforward in the simplest cases. A motive that returns is heard as intentional because the alternative — that the same specific arrangement of pitches and durations recurred by accident — is so improbable that the listener's auditory system rules it out. The probability of the return being accidental is, in the relevant Bayesian sense, vanishingly low; the perceptual response is the recognition of intention. The recognition is not conscious calculation; it is the immediate experience of the structure as ordered.

The empirical apparatus for this perceptual mechanism is supplied by the Gestalt tradition in psychology. Max Wertheimer, Wolfgang Köhler, and Kurt Koffka in the early twentieth century established that perception is organized by structural principles operating below conscious awareness — Prägnanz (the system's bias toward the simplest pattern consistent with the input), closure (completion of incomplete shapes), good continuation (following continuous trajectories), similarity (grouping of like elements), common fate (grouping of elements that move together), and proximity. These principles are not learned conventions; they are properties of the human perceptual system, robust across cultures and demonstrable in controlled experimental conditions. The application to auditory perception was worked out by Albert Bregman in Auditory Scene Analysis (1990): the auditory analogues of Prägnanz, closure, continuation, and similarity organize the listener's perception of sound into structured events, recognizable motives, perceived voices, and integrated streams. The recognition of intention in music is the auditory perception system applying its built-in pattern-detection apparatus to the sounded structure; Prägnanz in particular — the system's bias toward the simplest organization — is the perceptual basis for why intentional structure registers as such. Random material does not produce Prägnanz-consistent organization; intentionally structured material does. The Order claim's perceptual mechanism is the Gestalt apparatus applied to the auditory domain, and the empirical grounding the apparatus provides is what makes the Order claim falsifiable rather than merely assertable.

The mechanism scales to larger and more abstract structures. A long-range key plan in a sonata movement — exposition in I and V, development through various intermediate regions, recapitulation in I throughout — is recognized as intentional even by listeners who could not articulate the plan in technical terms. The listener experiences the recapitulation as return to home, perceives the rightness of the return, and recognizes the home-key arrival as something the composer chose. The recognition does not require the listener to identify the dominant region as the dominant or the recapitulation as the recapitulation; the perception of intention operates on the structure as a whole, not on the technical labels.

What this means for the composer is consequential. The composer's intention is not a private psychological state the listener has to guess at; it is a property of the structure the composer builds, and the listener perceives it directly through the structure. When the composer composes with Order, the listener perceives the intention through the order. When the composer composes without Order, the listener perceives the absence of order and infers that there was no intention behind the music — or that the intention failed to embody itself in structure the listener can hear. The composer's craft is precisely the building of structures whose order embodies her intention so that the listener can perceive it.

This is the operational reason that random music — music with no perceivable structure — is heard as not-music or as failed music. The lack of perceived order is the perceptual signature of the absence of intention. The listener cannot connect to material she perceives as random because there is nothing for her to connect to. Order is what allows connection between composer and listener; it is what allows the music to say something.

§8.4 · Order and the Other Aesthetic Principles

Order is the developed pedagogical realization of Claritas, the third of the three Aquinas conditions named in §5. The first two conditions — Integritas and Consonantia — operate alongside Claritas in any beautiful work, not under it. The composer who has Integritas has built a complete work, with nothing missing and nothing superfluous; the composer who has Consonantia has built a balanced and proportionate one, with parts in equilibrium and contrasts in coherent relationship; the composer who has Claritas has built one whose intentional structure is perceived as intentional by the listener. The three conditions are independent. Beauty is what they jointly produce.

Within Claritas, Order is the mature pedagogical category — the principle that intentional structure must be perceptible, at some level, to be Claritas at all. The chapter develops Order at length because the perceptual and operational apparatus the principle requires is substantial: the Gestalt mechanism by which random material fails to register as ordered (§8.3 above), the three modes of compositional Order (§8.5 below), the architectural-scale operation by which Order accounts for what Volume 1 declines to extend its gradient to (§8.6 below), and the way the composer's discipline trains Order into the working ear and hand (§8.7 below). The other two parent principles do not divide along these particular axes; their pedagogical sub-categories (Economy under Integritas; Balance and Contrast under Consonantia) carry the working composer's vocabulary at their own respective scales and do not require this chapter's apparatus.

The relation among the three Aquinas conditions is not one of genus and species. They are three independent dimensions along which the listener apprehends a beautiful work — the work is complete (Integritas), the parts are proportionate (Consonantia), the structure shines through as intentional (Claritas). A passage may exhibit two of the three and fall short of the beauty the third would have supplied: an economical and balanced passage that fails to register its structure as intentional is a passage with Integritas and Consonantia but without Claritas; a structurally intentional and proportionate passage cluttered with material that fails to earn its place has Consonantia and Claritas but not Integritas. The composer who attends to all three is the composer whose work the eight-century aesthetic tradition has converged on naming beautiful. A pedagogy of aesthetic principles that named the sub-categories but missed the three parent conditions they realize would be a pedagogy of leaves with no trunk — and a pedagogy that named only one of the three would be a pedagogy with one third of the trunk.

§8.5 · Three Modes — Inheritance, Departure, Invention

The claim that Order is style-agnostic deserves working out across the historical record. Three modes of compositional Order can be distinguished — inheritance, departure, and invention — and great music has been written in all three. The principle is the same in each; the means differ.

The three-mode treatment that follows is, generalized, a music- specific instance of an insight Ernst Gombrich developed at length for visual art in Art and Illusion (1960) and especially in The Sense of Order: A Study in the Psychology of Decorative Art (1979). Gombrich's central thesis: artists work by schema-and-correction. They begin with inherited schemata — conventional representational templates the tradition has bequeathed — and they either realize the schemata deeply (the artist who paints within an inherited tradition), correct the schemata against observed reality or against their own intentions (the artist who modifies the inherited template), or construct new schemata entirely (the avant-garde artist who builds new generative templates). The viewer perceives the work through the same schema-corrective process: the schemata provide the recognizable framework, the corrections (or constructions) provide the specific content. The three modes the present chapter describes for music — inheritance, departure, and invention — are the music-specific instances of Gombrich's three modes for visual art. The cross-art parallel is exact enough that the philosophical apparatus for one transfers cleanly to the other; The Sense of Order in particular, with its sustained treatment of decorative order across cultures, addresses exactly the questions §8.2 above raised about Order beyond the harmonic series.

Order through inheritance. Palestrina, Bach, Mozart, and Brahms (in his most classical moments) compose with Order through deeply-realized inherited forms. The forms — modal counterpoint, fugue, sonata-allegro, ABA, theme-and-variations — exist as bequeathed structural templates, and the composer's order is the realization of the template in the specific work. The student who studies Palestrina learns Order through this mode: she internalizes the modal-counterpoint inheritance and writes within it, and her Order is the order of the inherited form. The composer working in this mode does not invent the structure from scratch; she inherits a structure and pours specific material into it. The inherited form is a constraint, but the constraint is also a resource — the structure is given, and the composer's craft is the specific realization within it.

Order through departure. Beethoven (in the late quartets), Wagner (in Tristan), Stravinsky (in Rite of Spring and Les Noces), Mahler (in the late symphonies), and Bartók (in the string quartets) compose with Order through deliberate departure from inherited forms. The forms are still in play — Beethoven's Op. 132 still reads as a string quartet, Rite still has sections that articulate, Mahler's symphonies still have movements — but specific aspects of the inheritance are deliberately broken. The composer working in this mode does not abandon inheritance; she takes a position relative to it, and the listener perceives the order both in what the composer keeps and in what she alters. The departure is itself a structural choice the listener can hear. The Rite's rhythmic disruptions, Beethoven's expanded codas, and Mahler's symphonic deformations are not failures of order; they are order asserted in deliberate opposition to inherited expectation, and the opposition is what produces the work's specific meaning.

Order through invention. Schoenberg, Webern, Berg, Boulez, and the broader twentieth-century avant-garde compose with Order through invented structures that do not depend on inherited templates. Twelve-tone serialism is a structural device the composer constructs (often per work, sometimes per style); integral serialism extends the device to rhythm, dynamics, and articulation; Cage's chance operations deliberately introduce non-intention as a structural principle, which is itself a structure-of-no-structure that the listener can perceive (and may or may not find satisfying). The composer working in this mode invents the structure of the piece; the listener's perception of Order is her recognition that the invented structure operates as a structure rather than as random material.

The three modes are not ranked. A great work can come from any of them. What the three modes share is the property of Order: the property that the music is structured in a way the listener can perceive, and that the perception produces the recognition of intention. A Palestrina motet, a Beethoven late quartet, and a Webern symphony all exhibit this property at their best. The pedagogical question for the curriculum is which mode the student should learn in — and the answer Volume 2 develops elsewhere is all three, in historical sequence, so that the student has the inheritance, the departure, and the invention available to her as compositional choices in any given piece.

§8.6 · Order at Architectural Scale

Volume 1's gradient analysis is committed to the timescales the listener's auditory short-term memory actually operates on, and declines to extend to whole-movement scale; the empirical record on long-range tonal perception does not support a perceptual claim at that scale (V1 §2.4.4 develops the argument and cites the underlying psychology).

This scope restriction is itself ancient. Augustine, in De Musica (387–389) and in the Confessions (Book X, on memory and time), grounded aesthetic perception in the listener's auditory memory: the listener perceives the order of a sung psalm by holding in memory what has just been sung against what is currently sounding against what is anticipated. The scope of beauty is the scope of memory. Augustine reasoned to this from introspection sixteen centuries before the contemporary empirical record (see V1 §2.4.4) confirmed it. What lies beyond the listener's integrative reach is structured on the page — Schenker's notation, the composer's planning sketches — and grasped analytically rather than perceptually. The Augustinian recognition that beauty's scope is memory's scope is the patristic ancestor of Volume 1's careful scope restriction.

This scope restriction leaves a real question: if the gradient does not extend to architectural scale, what does account for the listener's perception of a long movement as coherent? The answer Volume 2 offers is Order. The listener experiences a 30-minute symphonic movement as coherent through her perception of Order operating at architectural scale: the returning motive, the proportioned sections, the balanced key plan, the unified material across the movement. This is not the same kind of perception as the local gradient — the listener does not hold a single architectural tonal gestalt in mind across the movement — but it is still perception, and the empirical record on it is robust. Listeners do recognize when a long movement is well- formed and when it is not; they recognize this even when they could not technically describe the structural choices that produce the well-formedness. The recognition is the perception of Order at architectural scale.

This is why a movement that proceeds through unrelated material with no recurrences, no balanced sections, and no resolved key plan is heard as incoherent regardless of how technically competent the local writing might be. The Order at architectural scale is absent, and the listener perceives the absence even though the local-scale Order may be intact in every individual phrase. A movement is more than the sum of its phrases; the additional structural property is the Order at the scales the phrases inhabit but do not constitute.

This is also why Schenkerian analysis, even where Volume 1 declines to inherit it as a perceptual claim about long-range gradient perception (V1 §2.4.4), produces analytical insights that often resonate with what listeners experience. Schenker's structural voice-leading often does trace, on the page, the architectural Order of a tonal masterwork: the structural descent of the Urlinie, the bass-arpeggiation of the Bassbrechung, the prolongational layers that connect surface motive to deep structure. These are notational devices for what is, in part, the architectural Order of the work. The treatise can engage Schenker analytically without committing to his perceptual claim about long-range tonal gradient because the analytical apparatus and the perceptual claim are separable. Schenker is reading the Order on the page; the listener perceives the Order in time.

§8.7 · Order and the Composer's Discipline

What does it mean for the composer to compose with Order, operationally? It means: at every point in the writing of a piece, the composer is making decisions about structure that the listener will perceive. The decisions can be small (which note to place on this beat) or large (what the form of the movement will be), but they are decisions, and the totality of them constitutes the order of the piece.

The composer's task is to make these decisions in ways that produce structure the listener can hear. This requires the composer to have internalized, by long practice, what listener-perception of order is like at every scale — what makes a phrase feel ordered or disordered, what makes a section cohere or fragment, what makes a movement hold together or collapse. The composer does not need to be a music theorist; she needs to have internalized the perceptual experience of order at every scale through her own listening and her own writing.

The curriculum's discipline-as-formation chapter (§9 below) develops the operational pedagogy of this internalization: the species counterpoint exercises that train the small-scale order of two-voice writing; the figured-bass exercises that train the chord-by-chord order of four-voice harmony; the fugue exercises that train the multi-voice order of polyphonic writing. Each exercise is the systematic cultivation of order at a specific scale, in a specific structural domain. The student who completes the discipline has internalized the perceptual experience of order at every scale the curriculum reaches; she is then ready to compose with order in her own work, in whatever mode (inheritance, departure, invention) she chooses.

The pedagogical inversion the curriculum performs is consequential. A student is not taught Order as a theoretical concept and then asked to apply it to her writing. She is taught to write within inherited forms — species counterpoint, figured bass, fugue, chorale — and the training itself produces the internalized perception of order. By the time she encounters the theoretical articulation of the principle (this chapter, in her reading), she has already been practicing it for years. The chapter names what the discipline has been forming all along.

The composer who has not internalized order produces music whose surface may be technically correct but whose structure does not cohere — phrases that don't quite balance, sections that don't quite resolve, movements that don't quite hold together. The listener perceives the lack of order and experiences the music as failed, regardless of how many local-scale rules the writing observes. The composer who has internalized order produces music whose structure carries the listener through every scale of the piece — and the listener experiences the result as music that says something, which is the perceptual outcome the entire enterprise of composition aims at.

§8.8 · A Note on Order and Beauty

The relationship between Order — and through Order, Claritas — and beauty as a whole is worth stating carefully. Order is necessary for beauty, but it is not sufficient. A passage can be perfectly ordered and still be uninteresting; a passage can be highly ordered and emotionally inert. The other two parent principles of §5 — Integritas and Consonantia — are the additional conditions under which Order becomes beauty. An ordered passage that exhibits Integritas (nothing missing, nothing superfluous) and Consonantia (parts in balance and contrast) alongside Order becomes a beautiful one. An ordered passage that exhibits Integritas and Consonantia poorly, or fails to exhibit them at all, is a passage with order but with limited beauty.

The composer who composes with Order alone has the structural skeleton of musical meaning; the composer who composes with Order plus Integritas and Consonantia in well-judged balance has the substance of beauty. The three are not separable in practice — Integritas and Consonantia operate on the ordered material, and Order is the medium through which they are realized — but they are distinguishable in analysis. Claritas via Order is the necessary condition for the music's structure to register as intentional; Integritas and Consonantia, well-judged, are the sufficient ones for the registered structure to be beautiful.

This is why the curriculum's discipline trains all three: the species exercises and the figured-bass exercises train the structural-order capacity that Claritas requires; the score-study work and the composition critiques train the Integritas-and- Consonantia capacity — the judgment that prunes the unnecessary and balances the parts. The two trainings reinforce each other, and by Stage X the student who has completed both has the integrated capacity to compose music that is, in the full Aquinas sense, beautiful. The integration is the final pedagogical goal of the method.

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§9 · Discipline as Formation

§9.1 · The Position

The constraints of species counterpoint, figured-bass realization, and four-voice voice-leading are routinely misrepresented in contemporary pedagogy as historical curiosities or as gatekeeping exercises imposed to weed out the unserious. The Gradus method takes the opposite position. Each constraint exists because it trains something specific in the student — in the ear, in the hand, in the musical judgment — that cannot be learned any other way. The rules are formative. They shape the composer.

This is the discipline-as-formation position, and it is the operational consequence of Volume 1's commitment to grounding tonal practice in acoustic facts (V1 §3) and to teaching the dual gesture of movement (V1 §4.9) by writing exercises that exhibit it.

§9.2 · Species Counterpoint as the Systematic Teaching of the Dual Gesture

Claim 9 of Volume 1 names the dual gesture: every traversal of the consonance gradient simultaneously creates and releases directed tension. The passing tone is the cleanest case. The suspension is the stretched case. The full vocabulary of dissonance-in-motion that tonal music draws on is, on Volume 1's account, a catalog of ways the dual gesture can be deployed at different timescales and in different harmonic contexts.

The five species of Fuxian counterpoint, taught as the curriculum's Stage II practice, are the systematic pedagogical realization of Claim 9 at the two-voice level.

• First species (note against note, all consonances) trains the student's ear in the stable positions of the gradient. The student writes against a cantus firmus using only consonances — unisons, thirds, fifths, sixths, octaves. The exercise is deliberately minimal: it isolates consonance as a positive category before dissonance is introduced. The student learns what not moving feels like in two voices before learning what motion-through-dissonance is.

• Second species (two notes against one, passing tones permitted) introduces the dual gesture in its cleanest form: the passing tone is a dissonance that exists for the duration of a single weak beat, motion through a high-gradient position rather than residence at it. The student writes second species to internalize the passing tone as motion, not as state. By the end of second species the student has felt — in her own composition — what Claim 9 describes.

• Third species (four notes against one, florid passing and cambiata) extends the dual gesture across faster surface motion. The same passing-tone principle operates, but at a higher rhythmic density and with the additional figure of the cambiata (a stylized leap-away-and-return-around). The student writes third species to acquire fluency with the dual gesture at speed.

• Fourth species (syncope, suspensions) stretches the dual gesture across the bar line. The suspension is a dissonance held over from a previous consonance and resolved down by step on a weak beat; the dissonance accumulates audibly before it releases. The student writes fourth species to feel the separation in time of creation and release that Volume 1's discussion of the suspension (§4.9) makes explicit.

• Fifth species (florid, combining all four previous species) is the integration. The student writes free counterpoint that combines passing tones, cambiatas, suspensions, and longer notes in a single voice. By the time the student is fluent in fifth species, the dual gesture is internalized as her working vocabulary of two-voice writing.

The five species are not an arbitrary historical curriculum. They are the systematic pedagogical realization of Claim 9 at the two-voice level, presented in the order of increasing complexity and integration. A student who has completed the five species has written several hundred two-voice exercises and has the dual gesture in her fingers and ears as a working principle.

§9.3 · Figured Bass as Voice-Leading the Gradient

Stage IV introduces figured bass — the eighteenth-century notation in which a bass line plus figures (numbers indicating intervals above the bass) specifies a harmonic progression that the student realizes at the keyboard or on paper as a full four-voice texture. Figured bass is the canonical pedagogical inheritance of the Boulanger lineage at Fontainebleau and is the daily practice of keyboard musicianship in the figured-bass conservatory tradition.

What does figured bass form in the student? It forms voice leading: the discipline by which an upper voice moves the smallest acoustic distance from one chord to the next, the discipline by which the leading tone resolves up to the tonic in cadential context, the discipline by which the chordal seventh resolves down by step. Voice leading is the operational realization of the consonance gradient in actual writing. A four-voice progression that does not respect voice leading produces a texture that the ear hears as broken; a four-voice progression that respects voice leading produces a texture that flows. The student learns to hear which progressions flow by writing progressions that respect the voice-leading rules.

The harmonic-standards exercises of the Boulanger-Lasser tradition extend the figured-bass discipline into a daily working practice. The student realizes figured-bass sketches in real time at the keyboard; she sings the four voices in turn; she identifies the chord function as she plays. The combination of writing, playing, and singing locks the voice-leading discipline into multiple sensory channels. By the end of Stage VI the student does not think about voice leading consciously when she writes; she has been trained to write in a way that respects voice leading automatically, the way a fluent speaker conjugates verbs without thought.

§9.4 · The Historical Model: Mozart-Attwood

The 1784–85 lessons that Mozart gave to the English student Thomas Attwood are the historical embodiment of the discipline-as-formation position in the high-classical era. The lessons survive in Attwood's notebook (ed. Derek Remes, A-R Editions, 2019), and they show Mozart working through species counterpoint, figured bass, and short-form composition exercises with a single student. The format is direct: Mozart writes the cantus firmus, Attwood writes the counterpoint, Mozart corrects the counterpoint in red ink. The corrections are specific — a parallel fifth here, an unprepared suspension there, a melodic leap not followed by stepwise motion in the opposite direction — and they are not theory lectures. They are formative corrections at the level of the specific mistake.

The Gradus method takes the Mozart-Attwood relationship as its pedagogical model. Maestro plays the Mozart role: a teacher who sees the student's specific writing, identifies the specific mistake, and supplies the specific correction. The corrections form the student over hundreds of exercises, the way Mozart's corrections formed Attwood. This is the operational meaning of formation: not the cumulative effect of abstract theory but the cumulative effect of specific corrections to specific mistakes in specific exercises.

§9.5 · The Discipline Is the Composer

The position that the chapter sets out, summarily: the species constraints, the figured-bass discipline, and the voice-leading standards are not means to an end. They are the formation. A student who has not written several hundred species-counterpoint exercises does not have the dual gesture in her fingers. A student who has not realized several hundred figured-bass sketches does not have voice leading in her hands. These cannot be acquired by reading about them, by analyzing them in masterworks, or by being told the rules. They are acquired by writing the exercises and being corrected on the writing.

The composer who emerges from the curriculum is not a composer plus a discipline; she is a composer whose discipline has become her working compositional vocabulary. The discipline is the composer. The chapter's title is its claim.

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§10 · The Integrated Practice System

§10.1 · Five Components, One Practice

The Gradus method is one practice with five components. Each component is theoretically anchored in Volume 1 and pedagogically anchored in the chapters above; the integration of the five is what makes the practice work. A student who uses one component without the others is using a fraction of the method. The chapter below surveys the five components in order and names the theoretical and pedagogical work each one does.

§10.2 · Curriculum

The ten-stage curriculum is the spine of the method. Stages I–II ground the student in the acoustic foundation and two-voice writing. Stages III–VI build the diatonic harmonic vocabulary through the high-classical style. Stages VII–VIII extend into chromatic and modal practice. Stages IX–X cover the twentieth-century expansion and the composer's final integration.

The historical ordering of the curriculum is treated in §4 above; the writing-not-analyzing structure of every substep in §3 above; the discipline-as-formation use of species counterpoint and figured bass in §9 above. The curriculum's foundational anchor is Volume 1 §4 (the ten foundational claims): each stage takes up a specific subset of the claims as its working material. Stage I anchors in Claims 1 (Harmonic-Series Foundation), 3 (Distance from Home), and 8 (Stability Hierarchy); Stage II anchors in Claims 2 (Movement), 8, and 9 (Dual Gesture); Stage VI anchors in all ten claims operating in the working classical vocabulary; and so on through Stage X.

The curriculum is not a textbook with chapters. It is a sequenced practice in which each step is a working unit — concept, listening exercise, composition prompt, critique, advance to the next step. The student writes from day one, and the curriculum's work is the accumulating composition practice that builds the composer.

§10.3 · Composition Studios

The Composition Studios — Beginner Sketchbook, Counterpoint Workshop, Master Sketchbook, and Critique by Maestro — are the working environments in which the student writes. Each studio is calibrated to a level of compositional development.

The Beginner Sketchbook introduces the basic interface of music notation and gives the student a low-friction environment to write short melodic phrases as soon as Stage I has introduced the major scale. The Counterpoint Workshop is the working environment for Stage II species counterpoint, with built-in rule-checking that catches the standard counterpoint errors (parallel fifths and octaves, unprepared dissonance, melodic-leap-and-recovery violations) and produces specific feedback the student can act on. The Master Sketchbook is the full pro-grade notation editor, supporting multi-voice writing, transposing instruments, mid-score key changes, and the complete vocabulary the curriculum builds.

The Critique by Maestro system is the feedback layer across all three studios. A student who has written a piece can request a critique; the critique system runs the 32-dimension programmatic analyzer (V1's gradient principles operationalized as scorers) and then asks Claude Haiku to write a narrative critique that names the piece's three strongest dimensions, its three weakest, and one specific exercise for the next sitting. The critique system is the pedagogical realization of the writing-not-analyzing rule (§3 above): every piece the student writes receives substantive correction, and the corrections are calibrated to the curriculum gate so the student is not penalized for not yet knowing what she has not yet been taught.

§10.4 · Score Study

Eighty-plus orchestral works in the Score Study library — Bach Brandenburgs, Mozart symphonies, Beethoven symphonies, Schubert and Brahms, Tchaikovsky and Dvořák, Ravel, Holst, into the twentieth century — are accompanied by hand-authored L1 (passage-level) and, where the score is available, L2 (per-measure) analysis. The analysis is hand-authored against the score, not machine-generated, because the discipline of writing analytical prose against a specific bar of a specific piece is itself part of the pedagogical work the score-study system delivers.

The theoretical anchor: the curriculum's Volume 1 claims operate at local-and-short-section scale (V1 §2.4.4), and score study is where the student sees the claims working in the repertoire at exactly that scale. A student studying the second-theme group of a Mozart exposition sees the modulation as the local home shifting, the secondary dominants tonicizing chords briefly, the chromatic chords extending the local vocabulary — all at the scale at which the listener perceives them. Score study does not ask the student to hear an architectural-scale tonal gestalt across the whole movement; it shows her what the score lays out on the page, with the analytical prose calibrated to what is locally hearable.

The Maestro Analyst (V1's gradient principles implemented as a score-analysis engine) supplies an automatic per-measure overlay for measures the hand-authored L2 has not yet annotated. The overlay is clearly labeled as automated and never replaces hand-authored analysis; it fills gaps so the student always has some analytical reference when she opens a score, and it is a working aid rather than a finished product.

§10.5 · Historical Sources

The historical-source library — Vidal's figured bass exercises, Boulanger's harmonic standards (via Lasser), Marpurg's fugue treatise, the Bach chorales, the Palestrina motets and masses — is not antiquarian display. The sources are the documented record of the principles the curriculum is teaching, and the student works with the sources directly as part of the daily practice.

Vidal's 178 figured-bass exercises are the working corpus of the Stage IV figured-bass discipline. The Boulanger harmonic-standards exercises are the daily practice for harmonic vocabulary at Stages V–VII. Marpurg's treatise on fugue is the reference for Stage V; the Bach chorales are the working corpus for Stage VI chorale harmonization; Palestrina is the working corpus for the Renaissance- to-early-Baroque modal practice that Stage VIII reaches.

The sources are useful because they are not secondary reductions. The student who realizes a Vidal figured bass is doing exactly what a nineteenth-century Paris Conservatoire student would have done; the student who harmonizes a Bach chorale is working within the same tradition Bach worked in. The historical continuity is itself pedagogical: the student knows, as she works, that she is doing what composers have done for centuries to learn what she is learning.

§10.6 · Maestro

Maestro is the personal composition professor — the AI-supported presence in the integrated method that answers the student's specific question at the moment the student asks it. The theoretical reason for Maestro's centrality is not technological novelty; it is the recognition that tonal music has genuine local ambiguities (V1 §6, the three lenses) that no static curriculum can disambiguate in advance. A student writing a chorale who hits a moment where the same note plays three functional roles simultaneously needs a teacher who can say tonally this; vertically that; horizontally the other, at that moment. A textbook cannot do this; a personal tutor can.

Maestro is the personal tutor. He answers the specific question with the specific reference: an excerpt from the relevant historical source, a corrective example, a referral to the specific lesson step where the concept is introduced. His knowledge base is the curriculum content, the historical sources, the hand-authored analysis library, and a substantial corpus of secondary scholarship — all indexed for semantic retrieval so a specific student question reaches specific source material that addresses it.

The pedagogical heritage of the personal-tutor relationship is Padre Martini in Bologna and Mozart with Attwood in London. The Gradus method takes this heritage as its model and scales it: every student has a personal-tutor relationship with Maestro, who is available at every moment of practice, and who is trained on the same curriculum, the same sources, and the same theoretical foundation that this volume sets out. The relationship is not personal in the eighteenth-century sense of a single human teacher with a single human student; it is personal in the operational sense of answering the specific question with the specific source the student needs at that moment. The pedagogical structure is the same one Mozart's corrections to Attwood embody.

§10.7 · The Sechter Lineage in the Notation

One specific technical note. The Roman-numeral analysis the student encounters throughout the curriculum (I, ii, iii, IV, V, vi, vii° in major; i, ii°, III, iv, V, VI, VII in minor) descends from Simon Sechter's Viennese Stufenlehre via Bruckner and Schenker (V1 §2.4.2), not from Riemann's Funktionstheorie. The difference is that scale-step numerals name the chord by its scale-degree position (V is the triad on the fifth scale-degree, full stop), while function symbols (T, S, D) name the chord by its functional role within a dualist scheme inherited from Hauptmann and Oettingen.

The curriculum uses scale-step Roman numerals throughout, which is the cleaner notation for the pedagogy this volume sets out. The student can read Riemann's functional notation when she encounters it in score study of German repertoire; she does not have to write it. The pedagogical commitment is to a notation that names what the chord is (its scale-step position) rather than what the chord does (its function in a dualist system Volume 1 has rejected at the theoretical level, V1 §2.2). The notational choice is a small operational consequence of the foundational position taken in Volume 1, and it accumulates over the curriculum as a clarity of analytical reference the student carries forward.

§10.8 · Scope Restriction Across the Five Components

A reminder applies across the five components: the practice operates at the timescales the listener's auditory short-term memory actually tracks (V1 §2.4.4). Score study shows large-scale key plans on the page but does not ask the student to claim a listener perceives them as a single perceptual gestalt across a movement. Composition Studios train the student to write at the local scales the gradient analysis covers — phrase, short section, the sequence of local homes a piece traverses. Maestro disambiguates at the local scale where ambiguity actually occurs. The integrated practice is integrated at the scales the practice operates on; it does not pretend to integrate beyond them.

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§11 · Coda

The Theory of Tonal Movement (Volume 1) is the foundation. This volume is the practice. The two volumes together compose one project whose end is not the volumes themselves but the composer they exist to produce.

The pedagogical commitments the volume has set out are stated again in compressed form for the close. The student hears it before she learns to call it; vocabulary follows recognition (§§1–2). She writes from day one; analysis is reference material, not the work (§3). The curriculum follows the historical order in which the harmonic problems presented themselves to composers, and the student lives the journey (§4). The aesthetic principles by which compositional material gets shaped into coherent wholes are the three Aquinas conditions — Integritas, Consonantia, Claritas — adopted directly as the parent principles; the sub-categories under them (Economy, Balance, Contrast, Order) are the working pedagogical vocabulary. Volume 1's Claim 2 (Movement) is preserved as V1's central phenomenon at local scale; the cross-scale shaping that earlier formulations called aesthetic Movement is the motion-and-stasis operation within Contrast, under Consonantia. Integritas and the remainder of Consonantia are genuinely independent of the harmonic theory; Claritas (via Order) is partially nested with it, and partially broader (§5–§8). The discipline of species counterpoint and figured-bass realization is not gatekeeping but formation: the constraints train the dual gesture into the ear and the voice-leading into the hand (§9). The integrated practice combines the curriculum, the composition studios, the score-study library, the historical sources, and the personal-tutor relationship with Maestro into one working method, operating at the timescales the listener's auditory short-term memory actually tracks (§10).

The composer the method aims to produce is the composer who understands what she is hearing, who writes music that does what she hears it doing, and who is fluent enough in the working vocabulary of tonal practice that the vocabulary is no longer something she thinks about. The student who has reached that point has internalized the pedagogy. She has also internalized its boundary: she has not been promised insights into architectural-scale tonal hearing the empirical record does not support, and she has not been given a theoretical apparatus that would let her produce music-theory analyses at the level of contemporary academic theory. She has been given what the method promises: the working hand and ear of a composer in the tradition the curriculum has retraced.

The work is still in progress. The composer who internalizes the pedagogy is the composer who keeps writing — not the composer who graduates the curriculum and stops. The teacher who internalizes it is the teacher who keeps reading her students more carefully than she reads her authorities — not the teacher who memorizes the curriculum's contents and stops paying attention. The method is a practice in both senses of the word: a discipline the student maintains, and a discipline the teacher refines on her own students' specific work. A volume can describe these. Only the practice itself completes them.

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Appendix V2.A · Curriculum Map

The ten-stage / forty-step / 120-week structure, with each step keyed to the foundational claim(s) of Volume 1 it serves and the pedagogical principle(s) of Volume 2 it depends on. To be drafted in full when the curriculum mapping is finalized; see V1 Appendix C for the working stage-by-stage outline this appendix will extend.

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Appendix V2.B · The Five Components of the Practice System

Detailed treatment of Curriculum, Composition Studios, Score Study, Historical Sources, and Maestro, with the role each plays in a student's daily practice. The §10 chapter covers these in prose; this appendix will provide the operational tables (which specific exercises in which specific tools at which curriculum points). To be drafted when the implementation across the five components stabilizes.

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Appendix V2.C · Bibliography

The bibliography is organized by lineage rather than alphabetically, following the same convention as Volume 1's Appendix D. Volume 2's sources fall into nine sections: the pedagogical lineage running from Padre Martini through Boulanger to Lasser; the historical sources the curriculum draws on directly (figured bass, fugue, chorale, modal counterpoint); the aesthetic-theory tradition that grounds the three Aquinas parent principles and the Order chapter; and the perceptual-psychology, music-cognition, and cross-art-aesthetic literatures that supply the empirical and philosophical apparatus the V2 §5 and §6–§8 chapters engage. Full publication details are given for each entry; page-specific citations are to be added in a subsequent apparatus pass.

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I. The Pedagogical Lineage

Fux, Johann Joseph. Gradus ad Parnassum (Vienna: Johann Peter van Ghelen, 1725; English trans. Alfred Mann, The Study of Counterpoint, Norton, 1943). The foundational document of the species-counterpoint pedagogical tradition. See V1 Appendix D, Section IV for the V1-side citation; the present volume's discipline-as-formation chapter (§9) treats the five species as the systematic pedagogical realization of V1's Claim 9 (the dual gesture of movement) at the two-voice level.

Padre Martini, Giovanni Battista. Esemplare ossia saggio fondamentale pratico di contrappunto (Bologna: Lelio della Volpe, 1774, 2 vols.). The counterpoint treatise from which Mozart studied. The personal-tutor relationship between Martini and the young Mozart in Bologna (1770) is the historical model the present volume's §10.6 cites for the Maestro-student relationship.

Mozart, Wolfgang Amadeus. Thomas Attwood Lessons (1784–85; ed. Erich Hertzmann, Cecil B. Oldman, Daniel Heartz, and Alfred Mann, Neue Mozart-Ausgabe, Series X, Werkgruppe 30/1, Bärenreiter, 1965; also ed. Derek Remes, A-R Editions, 2019). Mozart's private teaching materials from his lessons with the English student Thomas Attwood, preserved in Attwood's study notebook. The lessons demonstrate the personal-tutor method of the species- counterpoint and figured-bass tradition at its most direct: master demonstrating, correcting in red ink, advancing through the dissonance vocabulary in systematic order. The present method takes the Mozart-Attwood relationship as the historical model for the Maestro-student relationship (§9.4, §10.6).

Boulanger, Nadia. Primary sources for Boulanger's pedagogy are scattered: her Cahiers (teaching notebooks, École Normale de Musique de Paris archives); the interview recordings made by Bruno Monsaingeon for the documentary Mademoiselle (1977; ed. Mademoiselle: Conversations with Nadia Boulanger, Carcanet, 1985); the published testimonies of her students — Aaron Copland (Copland on Music, Norton, 1960), Quincy Jones (Q, Doubleday, 2001), Elliott Carter (interviewed throughout the Allen Edwards Conversations, Norton, 1971), Philip Glass (Music by Philip Glass, Harper & Row, 1987), and dozens of others. The fullest secondary biography is Léonie Rosenstiel, Nadia Boulanger: A Life in Music (Norton, 1982); Don Campbell, Master Teacher: Nadia Boulanger (Pastoral Press, 1984) is the standard secondary on her pedagogical method. The present volume draws on the figured-bass-and-harmonic-standards core of her teaching as reconstructed in Lasser (below).

Lasser, Philip. The Spiraling Tapestry: Foundations of Tonal Music in Thirty-Six Lessons (in preparation as of 2024) and related published reconstructions of the harmonic-standards tradition derived from the Boulanger pedagogical lineage. Lasser studied with Boulanger's students at the École Normale de Musique de Paris and at the Boulanger Foundation. The Gradus curriculum's figured-bass and voice-leading-standards practice draws directly on Lasser's published reconstruction. Full citation details to be confirmed against Lasser's specific publications when the bibliography is finalized.

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II. Historical Sources Used in the Curriculum

Vidal, Paul. Figured-bass exercises. Paul Vidal (1863–1931) was a Paris Conservatoire professor of composition who succeeded Massenet in 1910. The Vidal figured-bass corpus (178 exercises in the Gradus curriculum) progresses from root-position triads through diminished seventh chords, organized by chord type and voice-leading challenge. Primary source: manuscript collections held at the Conservatoire, École Normale, and Bibliothèque nationale; secondary use of the corpus in the Gradus curriculum follows the working tradition rather than a single printed edition.

Insanguine, Giacomo; Handel, George Frideric; Kellner, Johann Peter; Ristori, Giovanni Alberto. Thoroughbass exercise collections, eighteenth century, various MSS and early print editions. The Italian, English, German, and Saxon branches of the thoroughbass-realization tradition represented in the Gradus historical-sources library. Primary editions used in the curriculum: Insanguine's Solfeggi parlati (Naples, manuscript 1770s); Handel's Suites for Harpsichord (London, 1720) and the Aylesford Manuscripts; Kellner's Treulicher Unterricht im General-Bass (Hamburg, 1732); Ristori's thoroughbass exercises (Dresden Hofkapelle, manuscript).

Bach, Johann Sebastian. The 371 chorales as edited by Carl Philipp Emanuel Bach (Berlin: Birnstiel, 1765–69; later editions Breitkopf, 1784–87; modern reference edition: Riemenschneider, 371 Harmonized Chorales and 69 Chorale Melodies with Figured Bass, G. Schirmer, 1941). The working corpus for the Gradus curriculum's chorale-harmonization practice (Stage VI).

Palestrina, Giovanni Pierluigi da. Masses and motets, as edited in the Werke (ed. Theodor de Witt and others, Leipzig: Breitkopf & Härtel, 1862–1907, 33 vols.) and the Le opere complete (ed. Raffaele Casimiri and others, Rome: Edizione Polifonica Romana, 1939–1987, 34 vols.). The working corpus for the Renaissance-to- early-Baroque modal practice (Stage VIII reaches), and the canonical example of Order through inheritance in V2 §8.5.

Marpurg, Friedrich Wilhelm. Abhandlung von der Fuge (Berlin: Haude und Spener, 1753–54, 2 vols.; facsimile reprint, Hildesheim: Olms, 1970). The standard eighteenth-century treatise on fugue, the reference for Stage V of the Gradus curriculum. Marpurg codified the species-derived fugal vocabulary into a teaching system; the Gradus method uses Marpurg's exposition ordering for its fugue substeps.

Boulanger Harmonic Standards. See Lasser (Section I above) for the reconstruction of Boulanger's harmonic-standards exercises used at Stages V–VII of the Gradus curriculum.

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III. Aesthetic-Theory Foundations (Section I of the V2 §5/§6–§8 lineage)

Aristotle. Poetics (composed c. 335 BCE). Modern critical edition: ed. Stephen Halliwell, The Poetics of Aristotle, University of North Carolina Press, 1987; also ed. and trans. Anthony Kenny, Oxford UP, 2013. Cited in V2 §5.1 as the founding treatment of art-making as poiesis and of unity as an aesthetic principle (Poetics 1450b27). The Aristotelian conception of art as making rather than as merely beautiful object is the deep ancestor of the present volume's "writing not analyzing" commitment (§3).

Augustine of Hippo. De Musica (composed 387–389). Modern edition: ed. Martin Jacobsson, De musica: Liber VI, Almqvist & Wiksell, 2002; English trans. R. Catesby Taliaferro in Writings of Saint Augustine, vol. 2, Fathers of the Church, 1947. Confessiones (composed 397–400), Book X especially. Modern English: trans. Henry Chadwick, Oxford UP, 1991. Augustine's doctrine of number — beauty as the perception of mathematical order in time and space — is cited in V2 §5.1; the Book X discussion of memory and time is cited in V2 §8.6 as the patristic ancestor of V1's scope restriction to the timescales auditory short-term memory tracks.

Aquinas, Thomas. Summa Theologiae (composed 1265–1274). The canonical English edition is the Fathers of the English Dominican Province translation (Benziger Brothers, 1911–1925, 22 vols.; reprinted Christian Classics, 1981; available online at newadvent.org). The relevant passages: I, q. 39, a. 8 (the integritas / consonantia / claritas trinity passage, cited in V2 §8.1 and §5.2); II-II, q. 145, a. 2 (the body-and-beauty passage). The commentaries: Sententia super De Anima (composed 1267–68, ed. Leonine, vol. XLV/1, Vatican: Editori di San Tommaso, 1984), Book III on perception.

Alberti, Leon Battista. De pictura (composed 1435; Italian Della pittura 1436). Modern edition: ed. Cecil Grayson, On Painting and On Sculpture: The Latin Texts of De Pictura and De Statua, Phaidon, 1972; trans. as On Painting, Penguin Classics, 2004. De re aedificatoria (1452; English trans. as On the Art of Building in Ten Books, ed. Joseph Rykwert, Neil Leach, and Robert Tavernor, MIT Press, 1988). Alberti's concinnitas concept, cited in V2 §5.1 as the Renaissance restatement of Aquinas's three conditions.

Maritain, Jacques. Art et scolastique (Paris: Louis Rouart, 1920; English trans. Joseph W. Evans, Art and Scholasticism, Sheed & Ward, 1930; expanded as Art and Scholasticism, with Other Essays, Scribner, 1962). The twentieth-century recovery of Thomist aesthetics, applied to modern painting and poetry. Maritain is the proximate channel through which Aquinas's aesthetic apparatus reached Joyce (Stephen Dedalus's "wholeness, harmony, and radiance" paraphrase in A Portrait of the Artist as a Young Man, 1916) and the broader twentieth-century literary- critical tradition. Cited in V2 §5.1 and §8.1.

Gilson, Étienne. Forms and Substances in the Arts (Scribner, 1966; original French Matières et formes: Poiétiques particulières des arts majeurs, Vrin, 1964). The most comprehensive scholastic- aesthetic treatment in English. The present volume cites Gilson as the philosophical companion to Maritain's recovery of Aquinas.

Eco, Umberto. Il problema estetico in Tommaso d'Aquino (Milan: Bompiani, 1956; revised 1970; English trans. Hugh Bredin, The Aesthetics of Thomas Aquinas, Harvard UP, 1988). The single most thorough modern treatment of Aquinas's aesthetics from a structural-formalist perspective. Eco reads Aquinas as a contemporary-relevant analytical aesthetician with surprisingly modern resonances — approximately the same project the present volume's §8 chapter undertakes. Cited in V2 §5.1 and §8.1 as the canonical secondary source on claritas.

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IV. Eighteenth-Century British Aesthetics

Hutcheson, Francis. An Inquiry into the Original of Our Ideas of Beauty and Virtue (London: J. Darby, 1725; many subsequent editions). Modern critical edition: ed. Wolfgang Leidhold, Liberty Fund, 2004. Treatise I (on beauty) develops the uniformity- amidst-variety principle and the internal-sense doctrine of beauty perception. Cited in V2 §5.1 and §5.2 as the eighteenth-century formulation of what the Gradus method teaches under the joint Variety + Clarity + Unity principles.

Hume, David. "Of the Standard of Taste" (1757), in Four Dissertations (London: A. Millar, 1757); reprinted in Essays Moral, Political, and Literary. Modern critical edition: ed. Eugene F. Miller, Liberty Fund, 1985. The standard-of-taste position calibrated to qualified critics. Cited in V2 §5.1 as the intermediate position between aesthetic realism and aesthetic subjectivism that the Gradus aesthetic-realism position (§5.1) positions itself relative to.

Burke, Edmund. A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful (London: R. and J. Dodsley, 1757; 2nd ed. with "Introductory Discourse Concerning Taste," 1759). Modern critical edition: ed. Adam Phillips, Oxford UP, 1990. The founding statement of the modern beautiful-vs-sublime distinction, propagated through Kant, Schopenhauer, Otto, and the continental tradition. Engaged critically in V2 §5.1: the Gradus method does not adopt the sublime as a separate aesthetic category, on the methodological ground that Burke's framework defines the category through perceiver-state (awe, terror, mysterium tremendum) rather than through objective structural properties. Beauty, in the Gradus account, covers the full range of emotional responses music can produce; the three principles deployed at the edges of their operating range produce the music historically labeled sublime, but the principles and Order are what are doing the structural work, not a separate aesthetic category.

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V. Nineteenth-Century Music-Formalist Tradition

Hanslick, Eduard. Vom Musikalisch-Schönen (Leipzig: Rudolph Weigel, 1854; many subsequent editions). Modern critical edition and English translation: ed. and trans. Geoffrey Payzant, On the Musically Beautiful, Hackett, 1986. The canonical statement of nineteenth-century musical formalism: "die Musik besteht aus tönend bewegten Formen" — "music consists of tonally-moved forms." Cited in V2 §5.1 as the nineteenth-century music-specific articulation of the formalist tradition the Gradus eight principles continue. V1's Claim 2 (Movement as the Central Phenomenon) is, in its philosophical content, Hanslickian.

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VI. Twentieth-Century Visual-Art Aesthetics and Aesthetic History

Wölfflin, Heinrich. Kunstgeschichtliche Grundbegriffe: Das Problem der Stilentwicklung in der neueren Kunst (Munich: F. Bruckmann, 1915). English trans. M.D. Hottinger, Principles of Art History: The Problem of the Development of Style in Later Art, Holt, 1932 (reprinted Dover, 1950). Wölfflin's five pairs of contrasting concepts as structural-perceptual categories distinguishing Renaissance from Baroque visual art. Cited in V2 §5.4 as a cross-art parallel to the Gradus three parent principles: the same underlying structural categories operate in visual art under different names.

Greenberg, Clement. "Avant-Garde and Kitsch" (Partisan Review 6/5, 1939: 34–49); "Modernist Painting" (Forum Lectures, Washington: Voice of America, 1960; reprinted in Art and Literature 4, 1965: 193–201). Collected with related essays in Art and Culture: Critical Essays (Beacon Press, 1961) and The Collected Essays and Criticism (ed. John O'Brian, U Chicago Press, 1986–93, 4 vols.). Greenberg's modernist formalism — each art-form's history as the progressive realization of the form's specific medium — cited in V2 §5.4 as the twentieth-century extension of Hanslickian formalism into visual art.

Gombrich, Ernst H. Art and Illusion: A Study in the Psychology of Pictorial Representation (Princeton UP, 1960; reprinted with new prefaces in 1969, 1972, 2000 — the Mellon Lectures of 1956 at the National Gallery of Art, Washington). The schema-and- correction thesis: artists begin with inherited representational templates and correct them against observation or against their own intentions; viewers perceive the work through the same schema-corrective process. Cited in V2 §8.5 as the visual-art ancestor of the present chapter's three-modes treatment (inheritance, departure, invention).

Gombrich, Ernst H. The Sense of Order: A Study in the Psychology of Decorative Art (Phaidon, 1979). Gombrich's later treatment of decorative order across cultures — directly bearing on the V2 §8.2 discussion of Order beyond the harmonic series. The single most important secondary source for the V2 §8 chapter; the book's title is itself the principle the chapter develops in music-specific terms.

Arnheim, Rudolf. Art and Visual Perception: A Psychology of the Creative Eye (UC Press, 1954; revised and expanded ed. 1974). The systematic application of Gestalt psychology to visual art. The Gestalt principles Arnheim deploys (figure-ground, closure, good continuation, common fate, similarity, Prägnanz) are the visual-perceptual analogues of the Gradus principles. Cited as the canonical translation from Gestalt-psychology principles to art-perception.

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VII. Twentieth-Century Analytic Aesthetics

Beardsley, Monroe C. Aesthetics: Problems in the Philosophy of Criticism (Harcourt Brace, 1958; 2nd ed. with new preface, Hackett, 1981). The systematic mid-twentieth-century analytic- aesthetic treatment. Beardsley's five criteria for aesthetic excellence — unity, complexity, intensity, magnitude, regional quality — are an explicit analytic-aesthetic restatement of the Aquinas / Hutcheson / Kant formalist tradition. Cited in V2 §5.1 and §5.2 as the closest twentieth-century analytic-aesthetic parallel to the Gradus three parent principles and their sub- categories.

Goodman, Nelson. Languages of Art: An Approach to a Theory of Symbols (Bobbs-Merrill, 1968; 2nd ed., Hackett, 1976). The analytic-philosophical apparatus for treating art works as symbol systems: a musical score is a notation, a performance is a compliant of the score, the work exemplifies certain properties through its structure. Cited in V2 §8.3 as the philosophical pedigree for the present chapter's careful separation between perceived-intention (a property of structure) and biographical- intent (a fact about the artist's psychology).

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VIII. Music-Cognitive Aesthetics

Meyer, Leonard B. Emotion and Meaning in Music (U Chicago Press, 1956). The founding work of the cognitive-music-psychology tradition. Meyer's central thesis: musical meaning is grounded in expectation; specific musical events confirm or violate the listener's internalized stylistic norms, and the experience of musical meaning is the experience of expectation, fulfillment, and surprise. Cited in V2 §5.3 as the music-specific empirical apparatus for the convergence-of-harmonic-and-aesthetic claim. See also Meyer's Music, the Arts, and Ideas (U Chicago Press, 1967) and Style and Music (U Pennsylvania Press, 1989) for the broader framework.

Narmour, Eugene. The Analysis and Cognition of Basic Melodic Structures: The Implication-Realization Model (U Chicago Press, 1990); The Analysis and Cognition of Melodic Complexity: The Implication-Realization Model (U Chicago Press, 1992). The rigorous formalization of Meyer's expectation theory for melodic perception: a melodic interval implies a continuation; the actual continuation either realizes or violates the implication. Cited in V2 §5.3.

Huron, David. Sweet Anticipation: Music and the Psychology of Expectation (MIT Press, 2006). The comprehensive synthesis of the Meyer-Narmour tradition with contemporary cognitive- psychological and evolutionary apparatus. The ITPRA model (Imagination, Tension, Prediction, Reaction, Appraisal) names five distinct phases of the listener's response to a musical event. Cited in V2 §5.3.

Bregman, Albert S. Auditory Scene Analysis: The Perceptual Organization of Sound (MIT Press, 1990). The canonical translation of Gestalt-psychology principles to auditory perception. Auditory streaming, auditory closure, auditory good continuation, similarity, common fate organize the listener's perception of sound into structured events, recognizable motives, perceived voices, and integrated streams. Cited in V2 §8.3 as the empirical apparatus for the Order claim's perceptual mechanism.

Krumhansl, Carol L. See V1 Appendix D, Section II for the probe-tone work (Krumhansl-Kessler 1982; Krumhansl 1990) cited in V2 §5.1 (as the falsifiability test for the formalist position) and in V2 §8.6 (with Bharucha 1984/1987 and Lerdahl 2001 as the empirical record on long-range tonal hearing).

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IX. Perceptual Psychology — The Gestalt Tradition

Wertheimer, Max. "Untersuchungen zur Lehre von der Gestalt II" (Psychologische Forschung 4, 1923: 301–350). English trans. in W.D. Ellis, ed., A Source Book of Gestalt Psychology (London: Routledge & Kegan Paul, 1938). The classic paper establishing the Gestalt principles of perceptual organization (Prägnanz, proximity, similarity, good continuation, closure, common fate). Cited in V2 §8.3 as the empirical apparatus for the perceptual mechanism by which the listener detects non-random structure.

Köhler, Wolfgang. Gestalt Psychology (Liveright, 1929; revised ed. 1947). The most accessible introductory treatment of Gestalt psychology in English. Köhler's experimental work on chimpanzee problem-solving and his theoretical treatment of the isomorphism between perceptual structure and the underlying neural processes both bear on the Order chapter's claim that order is detected by built-in perceptual apparatus rather than by conscious inference.

Koffka, Kurt. Principles of Gestalt Psychology (Harcourt Brace, 1935). The comprehensive Gestalt-psychology textbook. The most thorough treatment of the perceptual-organization principles that the present volume's §8.3 cites in their auditory analogues via Bregman 1990.

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Editorial note on V1/V2 bibliography overlap

Several primary sources cited in this volume are also cited in Volume 1 and have their full bibliographic entries there: Schenker (V1 Appendix D, Section II); Hindemith and Boulanger (via Lasser; V1 Appendix D, Section VII); Krumhansl, Bharucha, Bigand-Pineau (V1 Appendix D, Section II added in this session); Fux, Mozart-Attwood, Vidal, Insanguine, Handel, Kellner (V1 Appendix D, Section IV); Mozart letters and Padre Martini (V1 Appendix D, Section VIII). The volumes share a bibliography for these cross-cited sources; each lists them only once. The present volume cross-references V1 Appendix D for the full entries where applicable.

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Editor's Notes — For Iteration

These are working notes for the author, not part of the treatise itself.

1. Volume 2 is genuinely a separate book. The author should consider whether Volume 2 has its own readership (teachers, conservatory admission committees, parents of students at the Gradus method) that does not overlap fully with Volume 1's readership (theorists, composers, musicology graduate students). If the readerships differ, each volume should be self-contained enough to stand alone, with V2 citing V1 only where the citation does work that paraphrase cannot.

2. Length estimate. Volume 2 fully written would be approximately 25,000–40,000 words, somewhat shorter than Volume 1.

3. Citations. Citation apparatus should match Volume 1's conventions when fully written.

4. The Lasser attribution. Lasser appears in the bibliography (Appendix V2.C) as a specific source the method draws on, not in the canonical lineage of the main acknowledgments. The author should confirm the exact form of the citation (which of Lasser's published works, with what page references) when the bibliography is written in full prose.