Pythagorean Tuning: The Geometric Comma of Pure Ratios — E8 Intelligence Research

Abstract

FINDING: Pythagorean tuning is generated entirely by stacking perfect fifths (3:2) and octaves (2:1), producing a 12-tone cycle that never closes perfectly — the "Pythagorean comma" — and this pure-ratio system is geometrically encoded in the tetractys and Platonic solids. | MATH: Perfect fifth = 3:2; perfect fourth = 4:3 (inverse of 3:2, since 2/3 × 2 = 4/3); whole tone = 9:8 (=(3/2)²/2); Pythagorean comma = (3/2)¹² / 2⁷ = 531441/524288 ≈ 1.01364 (≈ 23.46 cents); 12-tone cycle: 12 fifths ≈ 7 octaves + comma; tetractys: 1+2+3+4=10, with ratios 2:1, 3:2, 4:3 derived from first four integers; all tones generated by 2^a × 3^b (no other primes). | CONNECTION: The 3:2 ratio is the geometric mean of 1 and 2 (√2 ≈ 1.414, but 3/2 = 1.5 is the arithmetic mean of 1 and 2 — a different harmonic mean); the 9:8 whole tone is the ratio of the fifth to the fourth: (3/2)/(4/3) = 9/8; the comma 531441/524288 = 3¹²/2¹⁹ — a pure power-of-3-to-power-of-2 ratio, linking to base-60 (60 = 2²×3×5) and to the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com