Music Theory Lattice

5-limit tuning · just intonation grid · Riemann Tonnetz

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Physics: 5-limit just intonation: pitch = 2^a · 3^b · 5^c. Lattice axes: perfect fifths (3/2) horizontal, major thirds (5/4) vertical. Riemann's Tonnetz (1880s) predates neo-Riemannian theory. Major triad = 1:5/4:3/2. The syntonic comma (81/80 ≈ 21.5¢) is the discrepancy from traversing the comma loop: 4 fifths (81/16) vs 5/4×2 = 5/4 up two octaves (80/16). Euler's Gradus function measures consonance. In 12-TET, the lattice wraps into a torus (finite = equal temperament).