The Pitch Class Set as a Power Set
How every scale contains every chord hidden within it — and why the power set is the complete map of Western harmony.
In set theory, the power set of a set S is the collection of all subsets of S — including the empty set and S itself. If S has n elements, its power set has 2ⁿ elements. The power set of {a, b, c} is: {}, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c} — eight subsets from three elements.
Applied to a musical scale, the power set gives you every possible chord, interval, note, and combination that can be drawn from that scale. The power set of the C major scale contains every chord in the key of C — every triad, seventh chord, ninth chord, cluster, and single note. It is the complete harmonic universe of C major, derived from first principles.
The Power Set of a Hexatonic Scale
A hexatonic scale has 6 pitch classes. Its power set has 2⁶ = 64 subsets. These 64 subsets include:
E A B
CA, CB, DG…
037, 027, 027…
and extensions
of the hexatonic
The hexatonic itself
Complete harmonic universe
The 20 three-note subsets (trichords) are highlighted because they represent the harmonic atoms of the hexatonic — the smallest units that carry a distinct harmonic color. The trichord pair system is built on exactly these: by choosing which trichords you want in your hexatonic, you determine which 20 trichords are available as harmonic building blocks.
Why This Matters for Improvisation
The power set framework makes explicit something that experienced improvisers know intuitively: you don’t just play the scale. You play subsets of the scale — dyads, trichords, arpeggios, pentatonic fragments, full runs. The scale is the vocabulary; the power set is the grammar that tells you all the sentences you can construct from that vocabulary.
Understanding the power set of your hexatonic tells you, precisely, what harmonic material you have available at any moment. The 15 dyads are the interval relationships within the scale. The 20 trichords are the harmonic cells. The 15 tetrads are the arpeggios and seventh chords. The 6 pentatonics are the five-note subsets that can be played as complete melodic units. All 64 — minus the empty set — are legitimate harmonic choices.
Subsets by Size — The Complete Table
| Size | Count in 6-note scale | Musical name and significance |
|---|---|---|
| 0 | 1 | Empty set — silence, rest |
| 1 | 6 | Single notes — the six pitch classes of the scale |
| 2 | 15 | Dyads / intervals — all interval relationships within the scale |
| 3 | 20 | Trichords — harmonic atoms, the building blocks of the system |
| 4 | 15 | Tetrads — arpeggios, seventh chords, upper-structure voicings |
| 5 | 6 | Pentatonics — five-note subsets, melodically complete units |
| 6 | 1 | The full hexatonic scale — the complete harmonic palette |
Notice the symmetry: the counts 1, 6, 15, 20, 15, 6, 1 form Pascal’s triangle’s 6th row — a consequence of the combinatorial formula C(6,k) for choosing k items from 6. This symmetry is not a musical observation but a mathematical fact about binomial coefficients, appearing in music because the musical structure is genuinely combinatorial.
The Trichord as the Fundamental Unit
The 20 trichords in the power set of a hexatonic are not all distinct trichord types. Many will be transpositions or inversions of each other — members of the same set class. In the 027-027 hexatonic, the 20 trichords include multiple instances of the 027 trichord (the primary building block), several instances of the 025, and instances of 024 and 037 — all the trichords that can be formed from combinations of C, D, G, E, A, B.
This redundancy is not a problem — it is information. The trichords that appear most frequently in a hexatonic’s power set are the ones most naturally generated by that hexatonic’s interval structure. They are the “preferred atoms” of that harmonic world. For the 027-027 hexatonic, the dominant trichord is — unsurprisingly — the 027 itself, appearing in multiple transpositions and inversions.
Scale Analysis and the Complete Subset Index
Bruce Arnold’s Scale Analysis course (700 pages, 109 scales, 36 videos) systematically documents the complete power set structure for each of its scales — listing every 2-note, 3-note, 4-note, 5-note, and 6-note subset with its interval content, its Forte number, its relationship to standard chord types, and whether each note in each subset is a chord tone, a tension, or an avoid note over each of the chord types the scale can be used with.
This is the full map. Not just “what scale to play over this chord” but “what are all the harmonic units available within that scale, and how does each one relate to the chord underneath?” The power set framework is what makes this complete description possible — without it, scale analysis remains a set of rules of thumb rather than a systematic account of harmonic possibility.
The 013 Trichord — The Universal Subset
The 013 trichord appears as a subset in more hexatonics than any other trichord type — it is the most “universal” element of the power set across the full system. Both pieces are built exclusively from 013-013 hexatonics.
Set theory and music theory, the same mathematics
Power set
The set of all subsets of S. For a 6-note scale, 2⁶ = 64 subsets. Every chord, interval, and melodic fragment available within the scale is an element of its power set — the complete harmonic universe of that scale.
Pascal’s triangle
The subset counts 1, 6, 15, 20, 15, 6, 1 are the 6th row of Pascal’s triangle — binomial coefficients C(6,k). The same numbers appear in probability theory, combinatorics, and the expansion of (a+b)⁶. Music is combinatorics.
Trichord as atom
The 20 three-note subsets occupy the center of the power set table — the most harmonically rich size class. Not too small to carry harmonic color, not too large to be manipulated easily in real time. The fundamental unit of the hexatonic system.
Systematic enumeration
Scale Analysis documents the full power set structure for 109 scales — a project that is only tractable with computational tools. The power set framework transforms “what can I play?” from an open question into a finite, enumerable answer.
From Map to Music
The power set is a map, not a prescription. Knowing that a hexatonic contains 20 trichords, 15 dyads, and 6 pentatonics does not tell you which ones to play or when. It tells you what is available — the full vocabulary from which any phrase must be drawn. The musical judgment of what to play and when is still the musician’s work, and it is still the work that takes decades to develop.
But the map changes what is possible. A musician who knows the power set of their hexatonic knows that they are never “stuck” — there are always 63 other harmonic choices available within the scale. And a musician who can hear which power-set element they are using — which trichord, which dyad, which pentatonic — is operating at a level of harmonic clarity that transforms both composition and improvisation.
The next article moves into technology: how Python and SuperCollider were used to generate and validate all 9,240 trichord permutations — the computational backbone of the Sound Cells series.
The power set and combinatorics of musical scales are discussed in Forte, The Structure of Atonal Music (1973) and Rahn, Basic Atonal Theory (1980). Bruce Arnold’s Scale Analysis course, documenting the complete subset index for 109 scales, is available at muse-eek.com.