Subsets of MMA
The 10 hexatonics, 12 pentatonics, and 16 arpeggios hidden inside the MMA scale — and why the power set of a 7-note scale is a complete harmonic toolkit.
The Melodic Minor Ascending scale contains 7 pitch classes. Its power set — the collection of all possible subsets — has 2⁷ = 128 members. Among these 128 subsets are 7 single notes, 21 intervals (dyads), 35 trichords, 35 tetrads (arpeggios), 21 pentatonics, 7 hexatonics, and the full scale itself. Every possible chord, interval, and melodic fragment available within MMA is an element of this power set.
The SuperScales book specifically documents the most harmonically useful subset sizes: the 10 hexatonics (6-note subsets that form complete hexatonic scales), the 12 pentatonics (5-note subsets that form pentatonic scales), and the 16 arpeggios (4-note subsets that form standard seventh chord arpeggios). These subsets are independent harmonic resources — each can be used as a melodic or harmonic palette in its own right, drawn from the MMA sound world.
The 10 Hexatonics — Not 7
The power set of a 7-note scale contains 7 hexatonic subsets (removing one note at a time from all 7 positions). But the SuperScales book documents 10 hexatonics from MMA, not 7. Why the discrepancy? Because the 10 includes hexatonics from the modes — not just the hexatonics formed by removing one note from the root-position MMA scale, but also the hexatonics formed by removing one note from each MMA mode. Some of these are identical (hence 10 rather than 49), but they represent a broader set of hexatonic resources available within the MMA sound world.
The 7 direct hexatonics (one note removed from C MMA) are: the most immediately available — the scale with one of its seven notes removed. Each produces a hexatonic with a specific Forte classification that connects it to the larger hexatonic trichord pair system documented in the Sound Cells series.
The 16 Arpeggios — Highest Harmonic Richness
The SuperScales book specifically highlights the 16 arpeggios (4-note subsets) as the subset size with the highest harmonic richness per note. A 4-note arpeggio from MMA contains a root, third, fifth, and seventh — the complete jazz chord — drawn from the MMA pitch palette. The 16 arpeggios represent every jazz chord type available within the MMA sound world.
Why 16 rather than 35? Of the 35 possible 4-note subsets of C MMA, 16 correspond to recognizable seventh chord arpeggios with clear root-quality-bass functions. The other 19 are either inversions already counted, or 4-note sets without a clear chord-root relationship. The 16 include: major 7th, dominant 7th, minor 7th, minor-major 7th, half-diminished 7th, augmented major 7th, and diminished 7th arpeggios, plus their modal variants.
Why Subsets Matter for Improvisation
A musician who knows only the full MMA scale has one tool: run up and down 7 notes. A musician who knows MMA’s subsets has a palette: choose a trichord for a tense moment, a pentatonic for a lyrical phrase, a hexatonic for a longer line, an arpeggio to outline the chord clearly. The power set is the full vocabulary; the scale itself is just one word in it.
The connection to the hexatonic trichord pair system (Sound Cells) is direct: many of the hexatonics hidden inside MMA are the same hexatonics covered in the Sound Cells series. This means that a musician studying both systems is not learning two separate things — they are learning different facets of the same underlying harmonic structure. The trichord pairs of Sound Cells are the atoms; MMA is the molecule that contains many of them.
Subsets and harmonic richness
Power set = harmonic universe
The 128 subsets of MMA are the complete harmonic universe available within the scale. Every chord, interval, and melodic fragment is an element of this power set. The scale is the vocabulary; the power set is the grammar.
Pascal’s triangle again
The subset counts 1,7,21,35,35,21,7,1 are the 7th row of Pascal’s triangle — binomial coefficients C(7,k). The same mathematical structure appears in probability, algebra, and combinatorics. Music subsets follow the same counting principle.
Hexatonics inside MMA
The hexatonics inside MMA connect the SuperScales system to the Sound Cells hexatonic trichord pair system. The two series are not separate — MMA contains many Sound Cells hexatonics as subsets, and Sound Cells hexatonics are often fragments of MMA transpositions.
16 arpeggios — highest density
Among all subset sizes, the 4-note arpeggios have the highest ratio of harmonic information to notes — a complete seventh chord in four pitches. The 16 MMA arpeggios give the improviser chord-tone clarity within the MMA sound world.
The next article examines MMA as a generating system — how one scale family, through its 12 transpositions, 7 modes, and complete subset structure, generates more harmonic material than any other 7-note scale in Western music theory.
Power set and subset theory: Forte, The Structure of Atonal Music (1973). The 10 hexatonics, 12 pentatonics, and 16 arpeggios of MMA are documented with exercises in all 12 keys in the SuperScales book at muse-eek.com.