Fibonacci Frequency Ratios: Golden-Ratio Spacing in a Non-Octave Spectrum — E8 Intelligence Research

FINDING: Fibonacci numbers as frequency ratios produce a non-octave, inharmonic spectrum that diverges from the harmonic series, yet exhibits golden-ratio spacing between successive partials. MATH: - Fibonacci sequence: F_n = F_{n-1} + F_{n-2}, with F_1=1, F_2=1. - Frequency ratios in Hertz: 89, 144, 233, 377, 610, 987, 1600, ... - Ratio between successive Fibonacci frequencies: F_{n+1}/F_n → φ = (1+√5)/2 ≈ 1.6180339887. - Octave ratio = 2:1. Fibonacci ratios are not powers of 2; they are powers of φ (e.g., 144/89 ≈ 1.61798, 233/144 ≈ 1.61806). - Harmonic series: frequencies are integer multiples of a fundamental f0: f_n = n·f0. Ratio between successive harmonics: (n+1)/n → 1 as n→∞. - Fibonacci spacing is constant (≈ φ) across all partials, unlike harmonic series which compresses. CONNECTION: - Golden ratio φ = 1.618... appears directly as the frequency ratio between successive Fibonacci tones. - Reciprocal 1/φ = 0.618... and φ² = 2.618... are implicit in the sequenc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com