Torus of harmony
All 24 major and minor chords fit on the surface of a torus. Each one appears exactly once, and each sits right next to the chords it moves to most smoothly. Here is how to build it, one step at a time.
Drag to turn it.Tap it to grab it, then drag to turn it. Tap a triangle to hear its chord.
Start with a grid of notes
Put C in the middle. Each step to the right goes up a perfect fifth (C, G, D), and each step to the left goes down one (C, F, B♭). Each step up goes up a major third (C, E, G♯). Now cut every square along its diagonal. Each half is a chord: the lower-left half is major (C–E–G), and the upper-right half is minor (E–G–B).
The grid is the lattice ℤ². The point (x, y) gets the note 7x + 4y mod 12, counting half steps up from C, since a fifth is 7 half steps and a major third is 4. Euler drew the first version of this grid in 1739.
The notes repeat, so roll it into a tube
On a piano, three major thirds up from C (C, E, G♯, B♯) land right back on C. So the grid repeats every three rows, and its top row holds the same notes as its bottom row. Roll the sheet until those two rows meet, and the grid becomes a tube.
Three thirds make 4 · 3 = 12 ≡ 0 (mod 12) half steps, so moving by (0, 3) never changes the note. Gluing each row y to row y + 3 turns the plane into a cylinder.
The ends match too, with a twist
Going right repeats as well, but not straight across. Four fifths up from C (C, G, D, A, E) lands on E, one row above where C started. So each note on the tube's right end matches a note on its left end one row higher. Bend the tube around, turn one end a third of a turn so the notes line up, and glue. The result is a torus.
Four fifths make 7 · 4 = 28 ≡ 4 half steps, the same as one third, so moving by (4, −1) never changes the note either. The vectors (4, −1) and (0, 3) span a lattice K, and gluing by all of K gives the torus ℝ²/K. The −1 in (4, −1) is the twist: the ends meet one row apart.
Twelve notes, twenty-four chords
The finished torus has just 12 corners, one for each note, and 24 triangles, one for every major and minor chord. The circle of fifths is a single line on its surface. It winds three times around the ring and once around the tube, passing through all 12 notes before it closes.
K has index |det| = 4 · 3 = 12 in ℤ², so the torus has 12 corners. It also has 36 edges and 24 triangles, and its Euler characteristic is 12 − 36 + 24 = 0, as every torus's is. The fifths line closes at (12, 0) = 3 · (4, −1) + (0, 3). Those coefficients are its three turns around the ring and one around the tube.
Neighbors sound smooth
Each chord shares an edge with three others, and crossing an edge changes just one note, by a step or a half step. From C major, the three flips give C minor (P), A minor (R) and E minor (L). That single moving note is why these chord changes sound so smooth. Tap any chord, then one of its neighbors, and listen for the note that moves.
P, R and L each reflect a triangle across one of its edges, so doing any of them twice gets you back where you started. Together they generate a dihedral group of order 24. It reaches each of the 24 chords from any starting chord in exactly one way.