Logarithmic phi: Difference between revisions
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'''Logarithmic phi''', or | {{Infobox Interval | ||
| Ratio = 2^{\varphi} = 2^{\frac{1+\sqrt{5} } {2} } | |||
| Cents = 1941.640786499874 | |||
| Name = logarithmic phi | |||
}} | |||
'''Logarithmic phi''', or [[phi|<math>\varphi</math>]] [[2/1|octave]]s = 1941.6 [[cent]]s (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 [[edo]]s, 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{{c}}. | |||
Logarithmic phi is | Logarithmic phi is well-approximated in equal divisions of the octave corresponding to the Fibonacci sequence: [[8edo]], [[13edo]], [[21edo]], [[34edo]], [[55edo]], etc. | ||
== See also == | == See also == | ||
* [[Generating a scale through successive divisions of the octave by the Golden Ratio]] | * [[Generating a scale through successive divisions of the octave by the Golden Ratio]] | ||
* [[Golden sequences and tuning]] | |||
* [[Golden meantone]] | * [[Golden meantone]] | ||
* [[Metallic MOS]] | * [[Metallic MOS]] | ||
; The MOS patterns generated by logarithmic phi | ; The MOS patterns generated by logarithmic phi | ||
* [[3L 2s]] | * [[3L 2s]] | ||
* [[5L 3s]] | * [[5L 3s]] | ||
| Line 14: | Line 22: | ||
* [[13L 8s]] | * [[13L 8s]] | ||
* [[21L 13s]] | * [[21L 13s]] | ||
* | * … | ||
; Related regular temperaments | ; Related regular temperaments | ||
* [[Father family|Father temperament]] | * [[Father family|Father temperament]] | ||
* [[Keegic temperaments #Aurora|Aurora temperament]] | * [[Keegic temperaments #Aurora|Aurora temperament]] | ||
* [[Triforce]] divides an 1/3 octave period into logarithmic-phi-sized fractions. | |||
; Music | ; Music | ||
* [http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm 5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree], by [[David Finnamore]] | * [http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm 5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree], by [[David Finnamore]] | ||
[[Category:Golden ratio]] | [[Category:Golden ratio]] | ||
Latest revision as of 13:47, 31 July 2025
| Interval information |
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 ¢.
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