60 scales · 9 symmetric hubs · drag to rotate · scroll to zoom (dive right into the centre) · click a scale
Each of the 60 scales here is one of five seven-note scale types built on the twelve keys — and every line is a precise musical relationship. The shape is not decoration: the relationships genuinely close into a torus.
Around the big ring runs the circle of fifths. Around the tube runs the alteration square: with degrees 1 2 4 5 7 held fixed, Major, Melodic Minor, Harmonic Minor and Harmonic Major are exactly the four combinations of {3 or ♭3} × {6 or ♭6}. Each step around the tube alters a single note, and four steps bring you home — a closed cycle. A cycle crossed with a cycle is a torus: 4 types × 12 keys = the 48-scale surface. Natural Minor rides just off the inner surface, joined to Harmonic Minor by 7 ↔ ♭7.
Fifths lattice — neighbouring keys, same scale type. Alteration square — same key, one degree altered. Semitone diagonals — every pair of scales whose note collections differ by moving one note a single semitone, computed from the notes themselves. Relative strands — same notes, different seating (C Natural Minor ↔ E♭ Major).
Follow the alterations downward — flatten the 3rd, flatten the 6th, flatten the 7th — and you arrive at Natural Minor, which shares its notes with the Major three fifths flat-ward. Glue those together and the path closes: the scales resolve into three interlocked helical strands, each winding round the tube four times per lap of the ring. Hence: Harmonic DNA. (Try Helix mode in the controls.)
The three equal divisions of the octave each gather the torus by their own cycle. Split Melodic Minor's 5th both ways (5 → ♯5 + ♭5) and the diminished scale appears — three collections, one per helix strand family, in the hole. Converge Melodic Minor's 1 and 2 onto ♭2 and six wholetone notes remain — two collections, the poles. Converge the harmonic pair's 2 and 4 onto the missing third and you reach the augmented scale — four collections, the outer ring. Only the two harmonic scales — the ones with the augmented-2nd gap — can reach the augmented world by a single move.
Based on Joel Purnell's Harmony Matrix research (2011), which first mapped these scales as a doubly cyclic lattice — an object that always wanted to be a torus. Realised in 3D in 2026.