The per-voice Fourier signature: the bass is the spectrally purest voice

Note 4 of the Mathematics of Music series (companion to Notes 1–3: DOIs 10. 5281/zenodo. 20820962, 10. 5281/zenodo. 20826774, 10. 5281/zenodo. 20862822). English + Spanish. For three centuries music theory has held that the bass is special — Rameau's basse fondamentale, the figured-bass tradition, Riemann's harmonic functions: the lowest voice carries the harmonic roots. That has always been a structural claim. The discrete Fourier transform of pitch-class distributions (Lewin, Quinn, Amiot, Yust) turns chord qualities into numbers — the magnitude |aₖ| of the k-th coefficient measures a character (|a5| diatonicity, |a1| chromaticity, |a6| whole-toneness). That program reads a piece by time; it has never asked whether each voice carries a different spectral fingerprint. Here it does. Over 334 four-part Bach chorales the naive guess — that the fifth-walking bass is the most diatonic voice — is false (the soprano is). But the data shows something sharper: the bass is the spectrally purest voice, lowest in chromaticity (alpha₁, paired t = -20) and whole-toneness (alpha₆, t = -15), despite using the widest pitch-class vocabulary. The cause is root realization through two channels: functional roots move by fifths/fourths (odd intervals), balancing even/odd pitch-class weight and driving alpha₆ to 0 (the parity channel), while the bass's wide tessitura disperses its weight around the chromatic circle, lowering alpha₁ (the dispersion channel). The effect is texture-bounded: strong in functional-bass homophony (Bach chorales; Palestrina), it fades in imitative (Monteverdi madrigals) and pre-functional (Trecento) textures — a diagnostic of functional-bass writing, whose boundary confirms the mechanism. The parity channel's kernel — that a6 equals the even-minus-odd pitch-class parity — is machine-checked in Lean 4 (ParityA6. lean, sorry-free, axiom-clean, general even universe). Every ingredient is classical (Rameau/Riemann's functional bass; a6 = pc-parity, Amiot; the single-note-move rule, Hoffman) ; the contribution is the per-voice magnitude law — a measured, explained, certified, and honestly bounded empirical fact — not new mathematics. Prepared with the assistance of an AI system (Claude, Anthropic) ; the Lean 4 kernel, not the assistant, certifies the formalized part.