Golden Ratio

Introduction

The "golden ratio" or "phi" (Greek letter Φ / φ / ϕ ) may be defined by a/b such that a/b = (a+b)/a. It follows that ϕ-1 = 1/ϕ, and also that ϕ = (1+sqrt(5))/2, or approximately 1.6180339887... ϕ is an irrational number that appears in many branches of mathematics.

Wikipedia article on phi

Musical applications

Phi taken as a musical ratio (ϕ*f where f=1/1) is about 833.1 cents. This is sometimes called "acoustical phi".

As the ratios of successive terms of the Fibonacci sequence converge on phi, the just intonation intervals 3/2, 5/3, 8/5, 13/8, 21/13, ... converge on ~833.1 cents.

"Logarithmic phi", or 1200*ϕ cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is also useful as a generator, for example in Erv Wilson's "Golden Horagrams".

Additional reading

Generating a scale through successive divisions of the octave by the Golden Ratio

Phi as a Generator

sqrtphi, a temperament based on the square root of phi (~416.5 cents) as a generator

Golden Meantone

833 Cent Golden Scale (Bohlen)

The Noble Mediant: Complex ratios and metastable musical intervals, by Margo Schulter and David Keenan

5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree, by David Finnamore