Coltrane's Geometry

Nicolas Slonimsky's Thesaurus of Scales and Melodic Patterns (1947) is organized by equal divisions of the octave. He picks out principal tones that split the octave evenly, then decorates them with notes placed between them (interpolation), below them (infrapolation) or above them (ultrapolation). Coltrane is widely reported to have practiced from the book.

Put the twelve pitch classes on a clock face and each division becomes a regular polygon: a diameter, a triangle, a square, a hexagon, a dodecagon. In group theory these are the subgroups of the twelve-tone cycle.

Switch to circle-of-fifths order and every one of these shapes survives, at most rotated. A major scale rearranged the same way changes shape completely. That is how symmetrical these sets are.

“Giant Steps” (recorded 1959) moves among three key centers, B, G and E♭, a major third apart. Each is reached through its own dominant seventh, often with its ii chord first, at two chords to the bar and about 290 beats per minute.

Plot the roots and they form three augmented triangles, one for each harmonic function. Only four augmented triads exist in the twelve-tone system. The tune uses three of them and leaves the fourth, C–E–G♯, empty.

Coltrane made the move into a general substitution, now called Coltrane changes, and laid it over standards: “Countdown” over “Tune Up,” “26-2” over “Confirmation,” “Satellite” over “How High the Moon.”

The chord chart is the standard lead sheet. Pick any chord to hear it, or play the whole chorus at a slower tempo first.

Each tonic triad of “Giant Steps” sits one half step from the same augmented triad, B–D♯–G. Raise one note of B major, G major or E♭ major by a half step and you land on it.

Richard Cohn calls a family like this a hexatonic system. The symmetrical chord at its center never sounds as a tonic in the tune. The three keys circle it.

Reading this as an unseen center the music keeps circling is a metaphor, not a finding. It does fit where Coltrane went next.

Lay out pitch classes so that one direction adds minor thirds and the other adds major thirds. Four minor thirds make an octave, and so do three major thirds, so both directions wrap around. A grid that wraps in two directions is a torus. This is the Tonnetz of Euler and Riemann, folded by octave and enharmonic equivalence.

The twelve pitch classes fit exactly: four positions around the ring, three around the tube. A perfect fifth is one minor third plus one major third, so it is a single diagonal step.

The tonics of “Giant Steps” take three steps around the tube and close. The circle of fifths takes twelve diagonal steps, winding three times one way and four times the other, and ties a knot.

Follow the shapes. A triangle in 1959. A near-even seven-note mode held over a bass pedal in the classic quartet years. All twelve points in use by Ascension. Then sounds between the points: overtones, multiphonics, bends and screams above the written range.

As a polygon gains vertices it approaches a circle, the way Archimedes closed in on π. Time follows the same path. The barline grid gives way to layered cycles, then to Rashied Ali's free pulse.

The modal stage shows D Dorian, the mode of “Impressions.” The rhythm lanes are schematic, not transcriptions.

A polyrhythm is two cycles that share a downbeat. Send one voice around the ring of a torus and the other around the tube, and the combined path closes into a torus knot. Three against two draws the trefoil.

The ring voice sounds each time the path crosses the blue loop. The tube voice sounds each time it crosses the red loop. They meet at the downbeat, where the loops cross.

This is a picture of layered time like Elvin Jones's, not a claim about how he thought about it.