48 scales · 9 symmetric hubs · drag to rotate · scroll to zoom (dive right into the centre) · click a scale
Each of the 48 scales here is one of the four asymmetric seven-note scale types built on the twelve keys — and every line is a precise musical relationship. With the three symmetric collections as hubs, the diagram shows exactly the seven parent types at the top of the scale taxonomy, and nothing else. 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 is not a node: Aeolian is a mode of Major, not a scale type — flatten Harmonic Minor's 7th and you land directly on the relative Major. The "Relative minor seating" toggle re-roots every Major on its 6th degree to show that reading — same notes, same number, different nature.
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 (these include Harmonic Minor to its relative Major: the old Natural Minor edge, now an honest voice-leading move).
Follow the alterations downward — flatten the 3rd, flatten the 6th, flatten the 7th — and you land directly on the Major three fifths flat-ward (in its relative-minor seating). The path closes: the scales resolve into three interlocked helical strands of twelve, each winding round the tube four times per lap of the ring. Hence: Harmonic DNA. (Try Helix mode in the controls.) In integers, each strand's twelve steps use every power of two exactly once — +1, +2, +4, … +1024 — repaid by a single −2047 crossing of the B↔C seam: 1+2+…+1024 = 2047. The helix is the binary expansion of the seam.
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.
Every structure here is a 12-bit binary word — one bit per semitone, C leftmost — and therefore a number: C Major is 101011010101 = 2773. Every semitone edge on this torus changes that number by ± a power of two (binary carries and borrows are voice leading), a re-seated Major keeps its number exactly (nature lives in the spelling, not the notes), and on the C column the descent is literal counting: C Harmonic Minor 2905 + 1 = 2906 = E♭ Major — which is C Natural Minor in its relative-minor seating. Every node now carries a distinct integer. Toggle "Integer badges" in the controls, or click any scale and count ±1 from it.
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.