The measure beneath the music

The LCM Model

If tonal music is a story of distance from home, the LCM is the ruler that measures the distance. LCM — least common multiple — takes the simplest whole-number ratio between a note and the tonic and reports how large the numbers have to grow. A perfect fifth (2:3) is a small number; the tritone is a very large one. Small means close — consonant, at rest. Large means far — dissonant, leaning home. It is Euler’s old gradus suavitatis, the “degree of sweetness,” put to work as the spine of the curriculum.

Below is a real Bach chorale — Aus tiefer Not schrei ich zu dir (BWV 38.6) — plotted as a path through that space. The center is home; each sounding note is a point at its measured distance; each phrase is a journey out and back. Press play and watch the music move.

Phrase 1 of 5 · mm. 1–3

audio

Tempo60 bpm (1.00×)

Position

m1 · b1.0

event 1 / 57 · dur 2♩

Roman numeral

I⁴⁄₂

Local key

E major (0.73)

Pitch LCM (max / mean)

144 / 42.8

Chord LCM

root 1 · internal 420

Region LCM (vs A)

Voices

S: B4A: E4T: G#3B: D3

Tendency tones

none

jump:compare:

How to read it

Center is the local tonic — home. It shifts phrase by phrase as the key re-estimates.
Radial distance is each pitch’s LCM measure against that tonic. The dashed rings mark the canonical distances; closer in is more consonant.
Angle follows the circle of fifths — twelve o’clock is the tonic, one o’clock the dominant — so notes that are tonally close sit visually close.
Color names the pull: chord root, leading tone, the Phrygian pull (♭2̂ / ♭6̂), the chordal seventh, a secondary-function tone. The same physical note can sit at different distances depending on the harmony carrying it.

Watch the leading-tone cadences (the half-step from below) against the Phrygian moments (the half-step from above): the Phrygian pull lands acoustically closer to home — exactly what the theory predicts.

Go deeper

The Theory of Tonal MovementThe full theory the LCM measures — pulls, resolutions, and the pedagogy that follows.Distance from HomeThe static diagram — every diatonic pitch placed by its distance from the tonic, in one glance.Distance PlaygroundPlay any pitch against any tonic and hear — and see — how far from home it sits.Progression ViewerTrace a chord progression as a journey through the gradient, chord by chord.

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The LCM Model

How to read it

Center is the local tonic — home. It shifts phrase by phrase as the key re-estimates.

Radial distance is each pitch’s LCM measure against that tonic. The dashed rings mark the canonical distances; closer in is more consonant.

Angle follows the circle of fifths — twelve o’clock is the tonic, one o’clock the dominant — so notes that are tonally close sit visually close.

Color names the pull: chord root, leading tone, the Phrygian pull (♭2̂ / ♭6̂), the chordal seventh, a secondary-function tone. The same physical note can sit at different distances depending on the harmony carrying it.