From the Theory of Tonal Movement, §3.4
Distance from Home
Tonal music is a story of pulls and resolutions. The pulls are felt — the leading tone urgently close-and-unstable, the dominant near home, the chromatic tritone far out — and the felt distance has a measurable structure. This diagram is that structure, rendered in one glance.
| Pitch | Ratio | LCM | Diatonic | Context |
|---|---|---|---|---|
| C (tonic) | 1:1 | 1 | yes | — |
| G | 2:3 | 6 | yes | — |
| F | 3:4 | 12 | yes | — |
| A | 3:5 | 15 | yes | — |
| E | 4:5 | 20 | yes | — |
| B♭ | 4:7 | 28 | no | V⁷ context (partial 7 of G) |
| E♭ | 5:6 | 30 | no | — |
| F♯ | 5:7 | 35 | no | just (V⁷ partial 7 supports) |
| A♭ | 5:8 | 40 | no | — |
| D | 8:9 | 72 | yes | — |
| B | 8:15 | 120 | yes | — |
| B♭ | 9:16 | 144 | no | diatonic ♭7 |
| D♭ | 15:16 | 240 | no | ♭2 |
| C♯ | 16:17 | 272 | no | ♯1 |
| F♯ | 32:45 | 1440 | no | Pythagorean (strict diatonic ♯4) |
What you are looking at
Each pitch sits at a cycle-of-fifths angle around the tonic — G one fifth up at one o'clock, F one fifth down at eleven, D and B♭ at two and ten, and so on. The radial distance from the center is the LCM measure: the least common multiple of the integers in the simplest just-intonation ratio between the pitch and the tonic. Lower LCM means closer to home.
The apparatus is Euler's gradus suavitatis of 1739, refined in Stolzenburg (2015) and cross-checked against the empirical probe-tone data of Krumhansl and Kessler. The two come into agreement on the structural shape of the gradient. The reader does not compute LCMs; she hears the pitch and feels its position. The number is the theorist's label on what the listener already perceives.
Three things to notice
The cycle of fifths is not a uniform spiral. A (three fifths up, LCM 15) sits much closer to the tonic than D (two fifths up, LCM 72). A reaches the tonic by an intervallic shortcut — the 3:5 ratio between partials 3 and 5 — that fifth-stacking alone does not see. F (one fifth down, LCM 12) is slightly farther than G (one fifth up, LCM 6), because the inverse route to the tonic has bigger integers than the direct route. The picture tilts where the harmonic series tilts.
Harmonic context literally moves a pitch. The two B♭ readings — one short segment connecting the V⁷-context dot (LCM 28, pulled inward by the dominant's partial-7 support) to the strict-diatonic reading (LCM 144, much farther out) — are the same physical note placed at different distances by the context the listener is bringing to it. The F♯ pair (just 5:7 versus Pythagorean 32:45) shows the same effect at much greater amplitude.
The strict-diatonic Pythagorean tritone sits in a category of its own. Out at log₂(1,440) ≈ 10.5, F♯ heard as the diminished fifth of common-practice harmony is roughly twice the radial distance of every other diatonic-mode pitch. The visualization makes plain what the table only states.
What this is not
This is not a claim about musical form, climax placement, or proportions of sections. The gradient operates at the local scales the listener's auditory short-term memory tracks — pitch-to-pitch, beat-to-beat, phrase-to-phrase, and across short sections in which a single tonal center remains the perceived home. It is not extended to whole-movement structure, because the empirical record on long-range tonal hearing does not support a perceptual claim at that scale.