The Groovy Commutator in Music

Why funk sounds different from EDM, mathematically.

Setup for Rhythm

For binary rhythm patterns r ∈ {0,1}^n:

• D: accent-mask (where changes occur)
• I: overlay (union of onsets)
• C: XOR (difference pattern)

The Key Insight

• Grid music (quantized EDM): G(r) = 0 often (distributes)
• Funk / Groove: G(r) ≠ 0 (the non-distribution IS the groove)

"Combine patterns then find accents" ≠ "Find accents then combine"

The groovy commutator literally formalizes groove.

Harmonic Extension

For pitch-class sets P ⊆ Z₁₂:

• D: interval vector
• I: union
• C: symmetric difference

G(P) ≠ ∅ detects chords with harmonic complexity—where combining-then-analyzing differs from analyzing-then-combining.

Related: Polyrhythm, Syncopation, Music Theory