Commas of Tuning: Syntonic vs. Pythagorean in Just Intonation and 12-TET — E8 Intelligence Research

FINDING: The syntonic comma (81/80) and Pythagorean comma (531441/524288) define the two fundamental tuning inconsistencies separating just intonation, Pythagorean tuning, and 12-TET, with the syntonic comma being the kernel of the 5-limit just intonation lattice. | MATH: Pythagorean comma = (3/2)^12 / 2^7 = 531441/524288 ≈ 23.460 cents; Syntonic comma = 81/80 = (3^4)/(2^4·5) ≈ 21.506 cents; Difference (schisma) = 32805/32768 ≈ 1.954 cents; 12-TET semitone = 100 cents exactly, so 12-TET fifth = 700 cents vs pure fifth 701.955 cents (error ≈ 1.955 cents ≈ schisma); Pythagorean major third = 81/64 = 407.820 cents vs just major third 5/4 = 386.314 cents (difference = syntonic comma). | CONNECTION: The syntonic comma 81/80 = (3/2)^4 / (5/4) — a 4:1 ratio in the 3-direction vs one step in the 5-direction — defines the rank-2 lattice of 5-limit just intonation (basis vectors: 2, 3, 5). The Pythagorean comma is the 12-step closure failure in the 3-lattice (Z_12 subgroup of the 3-adic torus). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com