The palindrome on the Tonnetz torus
In twelve-tone equal temperament Euler's lattice closes up on itself: three major thirds make an octave, and so do four minor thirds. Roll the flat Tonnetz one way and it becomes a tube; bend the tube and it becomes a torus, a ring on which all twelve notes and all 24 major and minor triads sit side by side. Here the palindrome, its chord roots and the Pentametric areas walk on that ring.
Notes in Lluis's mirror colours (each interval and its inversion share a colour; C yellow). Warm triangles are major triads, cool ones minor. Lines: along the tube major thirds, around the ring minor thirds, diagonally fifths.
1 · Building the torus
Start flat (slide fold to the left, or press Fold from flat). The flat Tonnetz is a grid: one step along the short side is a major third, one step along the long side a minor third, and a diagonal step a fifth. In twelve tones, three major thirds come back to the same note (C, E, A♭, C), so the top edge is the bottom edge again: the sheet rolls into a tube. Then four minor thirds come back too (C, E♭, F♯, A, C), so the two ends of the tube are the same circle: the tube bends into a ring. Every note now appears exactly once, with its six neighbours around it: the fifth above and below, the major thirds above and below, the minor thirds above and below. proved
In just intonation (Euler's lattice, The palindrome on Euler's lattice) none of this closes: the comma keeps every place distinct. The torus is the picture of equal temperament.
2 · The mirror on the torus
Press Mirror through C. The mirror that turns every interval into its inversion (C → C, E ↔ A♭, G ↔ F, …) is, on the torus, a half turn about an axis through C and F♯, the two notes that stay where they are (the tritone is its own inversion). Every note lands on a note of its own colour, and every major triangle lands on a minor one: the major and minor triads are mirror images of each other, the old idea of harmonic dualism (Oettingen, Riemann). proved It is the same mirror as on the lattice, the fretboard and the circle, and as in the palindrome, which reads the same backwards.
3 · The cycle on the torus
Use the panel Walk the cycle beside the torus (below it on a narrow screen). Choose what walks: the melody (each step the interval H : (H − 1) over C), the chord roots of Chord progressions, or the Pentametric areas modulo 12. Go step by step with ◀ and ▶|, or press play and let the camera follow the note. At every step the panel shows the note, the step from the last one and its mirror, the six notes around it on the torus, the six triads it belongs to, and what lies across: its mirror through C and its tritone. With the silence switch on 5 the major thirds join, and the chord roots wind around the tube.
4 · The Pentametric areas as twelve notes: the hexatonic scale
Read the Pentametric areas 𝔓(n) modulo 12, as notes (0 = C, 1 = C♯, …). Only six notes ever appear: C, C♯, E, F, A♭, A, two augmented triads a semitone apart, the hexatonic (augmented) scale. proved
Why. 5𝔓(n) = n² + 4·(n | 5), and 5 leaves remainder 1 when divided by 4, so 𝔓(n) and n² leave the same remainder: 𝔓(n) ≡ n² (mod 4). A square leaves 0 or 1, never 2 or 3. So 𝔓(n) mod 12 is one of 0, 1, 4, 5, 8, 9, and all six occur.
Like the cycle. The twelve-note sequence repeats every 30 steps, because 𝔓(n + 30) − 𝔓(n) = 12n + 180 is a multiple of 12, and it reads the same backwards about n = 0 and n = 15, because 𝔓(−n) = 𝔓(n). proved Every step is an odd number of semitones (𝔓(n) and 𝔓(n + 1) differ in parity, as n² and (n + 1)² do) proved, and in one period of 30 each odd interval, 1, 3, 5, 7, 9, 11 semitones, occurs exactly 5 times. computed
On the torus (tick light the six notes): the six notes are two neighbouring rings around the tube, the augmented triads C–E–A♭ and C♯–F–A. Between them lie exactly six triads, F, f, D♭, c♯, A, a: the hexatonic cycle that Richard Cohn described (1996), in which each chord moves to the next by changing a single note by a semitone. proved The augmented scale is also the sound many musicians link with Coltrane's major-third cycles. interpretation
Choose the Pentametric areas in section 3 to hear the 30-step palindrome and to walk it step by step on the torus.
5 · Notes and credits
The lattice is Leonhard Euler's (1739); its twelve-tone, triangular form is the Tonnetz of Arthur von Oettingen and Hugo Riemann, and its picture as a torus is classical in music theory. The torus on this page is drawn from scratch for the site. The hexatonic reading of the Pentametric areas is new here and exploratory: it has not been checked against the literature yet.