The Tonnetz spectrum is the generating triad's Fourier balance profile

The neo-Riemannian Tonnetz — the graph on the 24 major and minor triads joined by the parsimonious P, L, R moves — has an adjacency spectrum, and the main result identifies it exactly: the eigenvalues of the Tonnetz are ± the Fourier balances of the single triad that generates it, the signed multiset ±{3, sqrt(5), sqrt(3), 1, 2cos(pi/12), 2cos(5pi/12)} = {±|hat 1_T(k)|} for k = 0..6. The augmented balance sqrt(5) = |a_3|, the diminished sqrt(3) = |a_4|, the whole-tone 1 = |a_6| — the chord's own periodicity weights — are literally the graph's vibration frequencies. The proof builds one tool: the discrete Fourier transform is the shared eigenbasis of pitch-class symmetry. (i) every translation-invariant operator on Z_N (the chromatic cycle C_12, every abelian Cayley graph) is diagonalized by the DFT characters, eigenvalue the Fourier coefficient of its connection set (Babai). (ii) adjoining inversion makes the T/I group D_2N block-diagonal, pairing a_k <-> a_{-k} into conjugate-pair irreducibles. (iii) the Tonnetz is the Cayley graph of the non-abelian PLR = D_12 group, and reading it through (i)-(ii) yields the balance-profile identity, complete with multiplicities. Every step is machine-checked in Lean 4, sorry-free and axiom-clean, including the unconditional completeness of the spectrum (an explicit basis of eigenvectors; no Sage, no irrep classification). The spectral formula itself is standard — the dihedral-Cayley computation of Gao-Luo, of which the Tonnetz is the S = {P,L,R} instance; the Tonnetz Laplacian was studied by Lostanlen. What is added is the reading of that adjacency spectrum as the generating triad's balance profile, the identification of its spectral fibres with the music-theoretic Z-relation (homometric chords give cospectral graphs — Babai's 1979 criterion, read musically), and the Lean certification. An appendix instantiates the N-generic saturation rule at N = 24 (quarter tones): the limited-transposition catalogue strictly extends the chromatic one, with one symmetric mode (H_8) landing off the 12-tone grid; the Z_24 cosets are verified by exact cyclotomic computation in Sage. Not new mathematics — a unified, certified account. Note #3 of the music-math series; continues the phase-taxonomy note. This record bundles the English and Spanish versions of the note.Revision 1.1 (October 2026): the reference-table fifth-frequency magnitude of the seven-note diatonic collection is corrected to 2 + sqrt(3); 2 cos(pi/12) belongs to the generating triad. The unit-frequency fixed-cardinality Huddling maximizer is stated as an inverse-unit image of a consecutive arc, rather than as a maximally even set at every unit frequency. English and Spanish sources/PDFs are synchronized; Spanish layout is reviewed with Babel. The Tonnetz spectral theorem is unchanged. Corrections prepared with the assistance of artificial intelligence tools.