The Missing Zigzag: Cycles of Semitone Trichords and a Conservation Law in Equal Temperament

Abstract

A cycle of nine pitch classes in twelve-tone equal temperament can be arranged so that its nine consecutive three-note windows realize, exactly once each, every trichord type containing a semitone -- types taken up to transposition but not inversion. Such cycles exist in abundance, and generalize to n-tone equal temperament, where the windows realize the n-3 semitone-containing types. In every one of the 1764 such cycles in twelve-tone equal temperament, the chromatic trichord 012 appears scalarly, as a direct chromatic run, and never in a broken "zigzag" contour such as 1-0-2. Why is the zigzag missing? We show that the zigzag presentation behaves as a conserved topological charge of an underlying interval-flow network: it occurs if and only if 3 divides n and n is neither 6 nor 12. The forward obstruction is a Z/3-valued conservation law, proved for all n by a weight-function argument; the two exceptional temperaments -- one of them the common Western tuning -- are excluded by finite certificates; and existence for every other multiple of 3 follows from an explicit construction. We also quantify rarity, and give exact censuses well beyond the reach of direct search. The classification is machine verified in Lean 4 in both directions. The paper is written for two audiences. Musical sections require no more than pitch-class set vocabulary; the mathematical development is self-contained and elementary, yet fully rigorous.