Logarithmic phi
| Interval information |
(Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])
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
- Related regular temperaments
- Father temperament
- Aurora temperament
- Triforce divides an 1/3 octave period into logarithmic-phi-sized fractions.
- Music