Logarithmic phi

Interval information
Expression	[math]\displaystyle{ 2^{\varphi} = 2^{\frac{1+\sqrt{5} } {2} } }[/math]
Size in cents	1941.641¢
Name	logarithmic phi

Logarithmic phi, or [math]\displaystyle{ \varphi }[/math] octaves = 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]\displaystyle{ 2^{\varphi} }[/math], or [math]\displaystyle{ 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.

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

The phith root of phi is another interval with interesting properties, that divides acoustic phi logarithmically by phi (in the same way that logarithmic phi divides the octave by logarithmically by phi), which creates self similar, fractal-like scales.

Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: 8edo, 13edo, 21edo, 34edo, 55edo, etc.

Approximation

Edo approximations for Logarithmic phi (741.64 ¢)
*≤ 80edo, relative error ≤ 10%*
Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
5	3\5	720.00	-21.64	-9.02
8	5\8	750.00	+8.36	+5.57
13	8\13	738.46	-3.18	-3.44
21	13\21	742.86	+1.22	+2.13
26	16\26	738.46	-3.18	-6.89
29	18\29	744.83	+3.19	+7.70
34	21\34	741.18	-0.46	-1.32
42	26\42	742.86	+1.22	+4.26
47	29\47	740.43	-1.22	-4.76
50	31\50	744.00	+2.36	+9.83
55	34\55	741.82	+0.18	+0.81
60	37\60	740.00	-1.64	-8.21
63	39\63	742.86	+1.22	+6.38
68	42\68	741.18	-0.46	-2.63
76	47\76	742.11	+0.46	+2.94

Logarithmic phi

Approximation

See also

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Logarithmic phi

Approximation

See also

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