The phase taxonomy of pitch-class set invariants

Working over the chromatic universe Z₁2, represent a pitch-class set A by its 0/1 indicator and take its discrete Fourier transform = F (ind A), with = |Â| e^iφ. We record an exact organization: every classical pitch-class-set invariant we examine sorts cleanly by how much of the phase φ it retains. The interval vector, Babbitt's hexachord theorem, homometry / the Z-relation, deep scales, the rhythmic-oddity property, and common tones under transposition are all functions of the phase-blind power spectrum |Â|² alone — equivalently the autocorrelation, equivalently (in crystallography) the Patterson function. The sum / index vector — common tones under inversion — is a function of ², which retains only partial phase and, as an invariant of the set class, is weak. Then comes the twist: the third-order triple correlation (whose transform is the bispectrum ··conj (Â) ) is also phase-blind, yet over Z₁2 it resolves the Z-relation completely — separating homometric chords needs not more phase but higher order. The full (magnitude and phase) is, last, a complete invariant determining the set exactly. The boundaries of this ladder of increasing resolving power are machine-checked in Lean 4 (Fourier. lean): the phase-blind collapse, the third-order resolution, and the complete-invariant capstone, in one file with a clean axiom audit. The crystallographic crossing is rigorous, not analogy: autocorrelation = Patterson function, |Â|² = X-ray diffraction intensity, the Z-relation = the phase problem. The contribution is the unified taxonomy as one organization with its boundaries machine-checked — a first formalization, not new mathematics. Note #2 of the music-math series; continues the 6-30 note. This record bundles the English and Spanish versions of the note.