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 1200*[math]\displaystyle{ \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]\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¢.

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

See also

• Generating a scale through successive divisions of the octave by the Golden Ratio
• Golden sequences and tuning
• Golden meantone
• Metallic MOS

The MOS patterns generated by logarithmic phi

• 3L 2s
• 5L 3s
• 8L 5s
• 13L 8s
• 21L 13s
• ...

Related regular temperaments

• Father temperament
• Aurora temperament
• Triforce divides an 1/3 octave period into logarithmic-phi-sized fractions.

Music

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