The Symmetrical Difference
How the closest possible relationship between two scales — major and pentatonic a tritone away — becomes a compositional structure covering all 12 pitch classes.
In set theory, the symmetrical difference of two sets is the collection of elements that belong to one set or the other, but not both — what is unique to each after removing what they share. Applied to pitch class sets, it reveals a structural relationship most musicians have never thought to look for, even though they encounter it constantly.
The most striking example involves two of the most familiar scales in all of Western music.
The Major Scale and the Pentatonic a Tritone Away
Take the C major scale: C, D, E, F, G, A, B — seven notes. Now take the major pentatonic starting on F# (a tritone away from C): F#, G#, A#, C#, F — five notes. Look at what they share:
Zero notes in common. Between them, all 12 pitch classes. The C major scale is the white keys on a piano. The F# major pentatonic is five of the black keys. Together they cover every note available. No relationship in Western music gets closer than this — two of the most commonly used scales, perfectly interlocking with no overlap.
This is not a curiosity. It is the fundamental relationship that the A/B compositional structure is built on.
“The major scale and the major pentatonic a tritone away share no notes and together cover all 12 pitch classes. The relationship between familiar and exotic is mathematically exact.”
— The symmetrical difference in practiceThe Mathematics
For any hexatonic scale (six notes), its symmetrical difference with the remaining six pitch classes produces another hexatonic — its complement. Six plus six equals twelve: the complete chromatic aggregate. The formula is clean:
Here is how the chromatic scale divides when a 026-026 hexatonic and its symmetrical difference are placed side by side — red for the A section hexatonic, green for the B section complement:
The A/B Compositional Structure
Many of Bruce Arnold’s compositions are built on exactly this structure. The A section uses one hexatonic scale — two trichords alternating, pivoting, and combining across the phrase. The B section uses the symmetrical difference of that hexatonic — also two trichords, drawn from the complementary six pitch classes. Together the two sections cover all 12 pitch classes while maintaining trichord pair logic throughout.
This structure achieves something neither traditional tonal composition nor twelve-tone serialism does in quite the same way. Traditional tonal music uses all 12 pitch classes but privileges some over others. Twelve-tone serialism uses all 12 equally but abandons the trichord pair logic that makes each section harmonically coherent. The A/B symmetrical difference structure uses all 12 while maintaining clear harmonic identity in each section — and the relationship between the sections is mathematically precise.
The A section and B section are not random contrasts. They are each other’s exact complement. Every note present in A is absent from B. Every note absent from A appears in B. The two sections are as far apart harmonically as two hexatonics can be, while remaining bound together by the logic of the chromatic aggregate.
Specific Examples from the Catalog
Several compositions in the Sound Cells series use this structure explicitly:
In each case the B section is mathematically determined — not a random harmonic departure but the exact complement of the A section. A listener with trained ears hears a complete change of harmonic color. A listener without technical training hears a contrast, a B section that belongs to a different world while still being part of the same piece. Both perceptions are correct.
Why the Tritone Is Central — Again
The tritone relationship appears throughout this framework because the tritone is the only interval that divides the octave exactly in half. This means a hexatonic scale and the hexatonic scale a tritone away are always potential symmetrical difference pairs — they live on opposite sides of the exact midpoint of the pitch class circle.
The C major scale and the F# major pentatonic are the most familiar example. But every hexatonic trichord pair in the Sound Cells system has a tritone-related complement. Learning one hexatonic automatically gives you a structural relationship with its partner — two halves of the same chromatic whole. And learning to hear the shift from A section to B section — from one hexatonic world to its complement — is one of the most advanced exercises in the ear training system built around this music.
Hear the A/B Structure
Both pieces use the A/B symmetrical difference structure. Listen for the moment the B section arrives — a complete change of pitch-class territory while the trichord pair logic continues.
Set theory and composition, the same structure
Symmetrical difference
A △ B = (A ∪ B) − (A ∩ B). For hexatonics this always produces another hexatonic covering the remaining six pitch classes — the exact complement, mathematically determined.
Chromatic aggregate
A hexatonic and its symmetrical difference together cover all 12 pitch classes. The A/B structure achieves twelve-tone completeness through form — each section coherent, together exhaustive.
Tritone axis
The tritone divides the octave exactly in half. The most familiar example: C major (7 notes, white keys) and F# major pentatonic (5 notes, five black keys) share zero pitch classes and together cover all 12.
Compositional economy
One hexatonic pair generates an entire piece. A section from the hexatonic, B section from its complement. No arbitrary modulations — the harmonic structure of the whole follows from a single mathematical relationship.
Completeness Without Serialism
Schoenberg invented twelve-tone serialism to achieve chromatic completeness — equal weight for all 12 pitch classes. The symmetrical difference structure arrives at a similar completeness through a different path: not a row that enforces equal treatment, but a trichord pair logic that organizes the 12 notes into two coherent harmonic worlds that together cover the full chromatic spectrum.
The listener hears both organization and totality. This is the structural achievement that makes the Sound Cells system more than a scale exercise — it is a compositional architecture that solves the problem of chromatic completeness while preserving harmonic coherence in each individual section.
The next article moves into the neuroscience of how the brain processes these scales and why hearing everything from one tonal center is neurologically more efficient than chord-by-chord interval calculation.
The symmetrical difference in music theory is discussed in Forte, The Structure of Atonal Music (1973). Complementary hexachords and twelve-tone technique: Babbitt, “Some Aspects of Twelve-Tone Composition,” The Score (1955). The A/B symmetrical difference compositional structure is documented in the Sound Cells Pitch Class Set Improvisation series at muse-eek.com.