Logarithmic phi

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Interval information
Expression [math]2^{\varphi} = 2^{\frac{1+\sqrt{5)} {2} }[/math]
Size in cents 1941.6408¢
Name logarithmic phi
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~4.65399 bits

Logarithmic phi, or 1200*[math]\varphi[/math] cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is useful as a generator, for example in Erv Wilson's "Golden Horagrams". As a frequency relation it is [math]2^{\varphi}[/math], or [math]2^{\varphi - 1} = 2^{1/\varphi}[/math] when octave-reduced. Logarithmic phi is notable for being the most difficult interval to approximate by EDOs, and as such a "small equal division of logarithmic phi" nonoctave tuning would minimize pseudo-octaves. With or without pseudo-octaves, an "equal division of logarithmic phi" nonoctave tuning forms an Intense Phrygian-Subpental Aeolian mode.

Logarithmic phi is not to be confused with acoustic phi, which is 833.1¢.

See also

The MOS patterns generated by logarithmic phi
Related regular temperaments
Music