The 924 Hexatonics
How a computer enumerated every possible six-note scale — and what musicians found when they looked at the results.
The previous article established that there are exactly 50 distinct hexatonic prime forms — 50 families of six-note scales. But those 50 families generate a much larger number of actual scales when you account for all 12 transpositions of each family. The total comes to 924 distinct six-note scales.
For centuries, musicians navigated this territory by intuition, tradition, and accident. The major scale, the natural minor, the whole-tone scale — these were discovered, named, taught, and handed down. But 924 is a large number. How many were left unexplored?
Enumerating the Full Space
The systematic enumeration of all 924 hexatonic scales is a computational problem — generating every combination of 6 pitch classes from 12, classifying each by its prime form, and organizing the results into a navigable structure. Bruce Arnold worked with AI-assisted tools over several years to generate the complete catalog — 924 scales, each analyzed for trichord structure, interval vector, Forte number, and relationship to other scales.
The generation process follows a simple algorithm:
From Algorithm to Music
Generating 924 scales is the easy part. The harder question is: which of the 924 are musically useful? Not all interval combinations produce scales the ear can navigate. Some are too chromatic to sustain a tonal center. Others duplicate the internal logic of familiar scales without adding new harmonic color.
The trichord pair framework solves this elegantly. By building each hexatonic scale from two trichords, the system encodes musical logic directly into the structure. Each trichord has known harmonic character — its stability, its tension, its relationship to standard chord types. A scale built from two 027 trichords (quartal pairs) inherits their open, suspended quality. A scale built from two 016 trichords inherits double tension. The 924 hexatonics become the full matrix of trichord pair combinations — 78 unique pairings, one per Sound Cells volume.
What the Computer Found
When the full 924 were enumerated and analyzed, several things emerged that pure musical intuition had missed.
The Z-relation is more common than expected. Among the 50 prime forms, a surprising number occur in Z-related pairs — distinct set classes with identical interval vectors. Musically, two hexatonic scales can have the same “colour” while having completely different note content. Composers can use Z-related pairs to create smooth harmonic transitions that feel like the same scale while actually shifting to different pitch material.
Many scales have high degrees of symmetry. Some hexatonic scales map onto themselves under certain transpositions — the whole-tone scale has only 2 distinct transpositions instead of 12, because it divides the octave into six equal steps. These symmetrical scales create a harmonic ambiguity that composers have exploited since Debussy.
The most useful scales cluster around middle values of interval variety. Scales with very low interval variety tend to sound static and color-limited. Those with very high variety tend to sound uncentered and difficult to use. The trichord pairs that prove most musically productive sit in a middle range: enough variety to create interest, enough consistency to project a clear sound.
Six Scales Worth Exploring
Six hexatonic scales from across the 924 — from familiar to genuinely unexplored:
The Role of Computation in Music
The enumeration of the 924 hexatonics is a concrete example of what computation makes possible that was previously out of reach. A musician could spend a lifetime exploring scales and never systematically cover the full space. A computer covers it in seconds.
But computation doesn’t replace musicianship — it changes its scope. The 924 hexatonics don’t tell you which scales to play. They tell you what scales exist to play. The gap between that map and actual music is still filled by the ear, by practice, by taste, by the thousands of hours that constitute musical understanding. The computation ensures you know what territory you’re in.
This is the model for the Sound Cells series: computational enumeration of the full harmonic space, followed by human curation of what is musically productive, followed by years of compositional practice to demonstrate what each corner of the space actually sounds like. 924 scales. 78 volumes. 30 years. Still ongoing.
Hear Less-Explored Corners of the 924
These two pieces come from trichord pairs off the beaten path — the kind of harmonic territory that only becomes accessible once the full 924 have been mapped.
What computation makes possible
Full enumeration
C(12,6) = 924 combinations, all generated, classified, and organized. A task that would take a human years takes a computer seconds — changing what questions a musician can even ask.
Symmetry detection
The computer identifies which scales have special symmetry — fewer than 12 distinct transpositions. These are the scales with extra ambiguity, beloved by impressionist composers and jazz musicians alike.
Z-pair discovery
The Z-relation — same interval content, different pitch content — was only discoverable by systematic analysis. No musician playing by ear would notice that two very different-sounding scales share identical interval distributions.
Practice material generation
Once the 924 are catalogued, exercises, études, and drone tracks for all 924 across all 12 keys can be generated algorithmically — more practice material than any human could compose by hand in a lifetime.
The Map Is Not the Territory
Having the complete map of 924 hexatonic scales is not the same as knowing how to play them. The map is a starting point — an orientation in a space that was previously uncharted. What comes next is the musical work: hearing each scale, learning its sound, understanding its relationship to chord types, building the ear training that makes it available in real time during improvisation.
That is the work of the Sound Cells series — and of the Ear Training system built around it. The next article in this series looks at how the brain processes these scales: the neuroscience of key center perception, and why hearing everything from one tonal center is neurologically more efficient than chord-by-chord interval calculation.
The computational enumeration of hexatonic scales using Forte’s set theory framework is described in Forte, The Structure of Atonal Music (1973). The Sound Cells series covering all 78 trichord pair combinations is available at muse-eek.com. Python libraries for pitch class set analysis include music21 (MIT).