Set Theory and the Hexatonic Scale
How mathematics organizes every possible six-note scale into 50 families — and why composers have barely scratched the surface.
In 1960, music theorist Allen Forte published a systematic classification of every possible collection of pitch classes using the mathematics of set theory. What emerged was a catalog of 208 distinct set classes, organized by their internal interval structure.
Hidden inside that catalog, at exactly six notes per set, are 50 hexatonic prime forms. Fifty distinct six-note scales, each with its own interval fingerprint. Western music has been using perhaps five or six of them for centuries. The hexatonic system built on trichord pairs explores all 50.
What Set Theory Actually Does
Set theory provided a framework for music by focusing on interval content rather than note names. Two collections of notes are members of the same set class if one can be transformed into the other by transposition or inversion. C major and D major belong to the same set class — identical interval structures, different starting pitches.
A prime form is the most compact, left-packed version of a set class — the canonical representative. Forte gave each prime form a number: 6-1 through 6-50 for hexachords. These are called Forte numbers.
“Set theory doesn’t tell you what music to write. It tells you what territory exists to explore.”
— The organizing principleFrom 924 to 50 — The Math
The number of ways to choose 6 notes from 12 is C(12,6) = 924. But this includes many sets that are transpositions or inversions of each other. When we collapse those equivalences — using Burnside’s lemma from group theory — we get exactly 50 prime forms.
The 12 Trichords
Every hexatonic scale can be split into two trichords. There are exactly 12 distinct trichord types. The trichord pair system organizes the 50 prime forms by which two trichords combine to create them:
| Intervals | Example | Character |
|---|---|---|
| 012 | C, C♯, D | Chromatic cluster |
| 013 | C, C♯, E♭ | Universal — works over any chord |
| 014 | C, C♯, E | Monk-ish — diminished flavour |
| 015 | C, C♯, F | Modern major/minor substitute |
| 016 | C, C♯, F♯ | Dominant powerhouse — tritone-bearing |
| 024 | C, D, E | Lyrical — whole-tone adjacent |
| 025 | C, D, F | Pentatonic DNA — blues foundation |
| 026 | C, D, F♯ | Whole-tone dominant — hovering |
| 027 | C, D, G | Quartal — open, modern jazz |
| 036 | C, E♭, F♯ | Diminished triad |
| 037 | C, E♭, G | Minor/major triad — traditional |
| 048 | C, E, A♭ | Augmented triad |
A Sample of the 50 Prime Forms
Here is a selection of the 50 hexatonic prime forms. The highlighted ones (amber background) are core starting points in the Sound Cells series — chosen for their musical accessibility and harmonic richness.
The “Z” in some Forte numbers (like 6-Z23) indicates a Z-related pair — two different set classes with identical interval content that cannot be transformed into each other. These were one of the surprises in Forte’s original analysis, with analogues in abstract algebra and topology.
Why Avoid Notes Matter
One of the most practical insights from applying set theory to improvisation is the concept of avoid notes — scale degrees that create dissonance with a given chord. In a major scale over a major chord, the fourth degree clashes with the major third and typically needs to resolve.
Hexatonic scales built from trichord pairs are deliberately constructed to minimize avoid notes. By choosing trichords compatible with a wide range of chord types, the resulting six-note scale can be played freely over chord changes without note-by-note navigation. The mathematical selection of trichords does harmonic work that used to require conscious decision-making on every beat — turning what felt like musical intuition into a system with clear, auditable logic.
Hear the 027-027 Hexatonic
The 027-027 hexatonic (quartal pairs, Forte 6-34) is the most open-sounding of the 50 prime forms — stable enough to hear as a coherent scale, modern enough to avoid cliché.
Mathematics and music, the same structure
Equivalence classes
Set theory collapses 924 six-note combinations into 50 families by treating transpositions and inversions as equivalent — the same operation that underlies the mathematical concept of a quotient group.
Prime form
The most compact, left-packed version of a set class. Every set class has exactly one prime form, just as every fraction has one reduced form — the canonical representative of its equivalence class.
Z-relation
Some pairs of hexachords have identical interval content but cannot be transformed into each other. Mathematically they are distinct objects with the same invariant — a phenomenon with analogues in algebra and topology.
Burnside’s lemma
The reduction from 924 to 50 uses Burnside’s lemma from group theory — counting orbits of a set under the action of a symmetry group. Abstract algebra and music theory arrive at the same 50.
50 Families, 30 Years of Exploration
Bruce Arnold has been composing in the hexatonic trichord pair system since 1990. The Sound Cells series — 78 volumes covering the most musically productive of the 50 prime forms — represents the most thorough practical exploration of this territory ever assembled. Each volume is a set of compositions and exercises built on one specific hexatonic family.
Western music has used perhaps a handful of the 50 prime forms throughout its entire history. The mathematics says there are 45 more waiting to be explored. The next article looks at the 924 hexatonics in full, and the computational methods used to generate and categorize all of them.
Allen Forte’s classification is documented in The Structure of Atonal Music (Yale University Press, 1973). The trichord pair system is covered in the Sound Cells Pitch Class Set Improvisation series at muse-eek.com. Burnside’s lemma and music theory: Fripertinger, “Enumeration in Music Theory” (1992).