Tonnetz Space

Physics: Tonnetz: triangular lattice where each triangle = triad. Edges = shared intervals (P5=7st, M3=4st, m3=3st). P (parallel): share root+5th, toggle 3rd (C major ↔ C minor). L (leading-tone): share M3, move by semitone (C major ↔ E minor). R (relative): share P5, swap root/3rd (C major ↔ A minor). Cohn (1996): PLR group is the dihedral group D24. Hexatonic poles: C major ↔ A♭ minor (share no notes). Beethoven's 9th slow movement traces Riemannian region cycles. Film composers (Williams, Elfman) use chromatic mediants.