Logarithmic phi: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Ratio = 2^{\varphi} = 2^{\frac{1+\sqrt{5 | | Ratio = 2^{\varphi} = 2^{\frac{1+\sqrt{5} } {2} } | ||
| Cents = 1941.640786499874 | | Cents = 1941.640786499874 | ||
| Name = logarithmic phi | | Name = logarithmic phi | ||
}} | }} | ||
'''Logarithmic phi''', or | '''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. | Logarithmic phi is not to be confused with [[acoustic phi]], which is 833.1{{c}}. | ||
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 == | |||
{{Interval edo approximation|interval = 353/230}} | |||
;Music | == See also == | ||
*[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]] | * [[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 family|Father temperament]] | |||
* [[Keegic temperaments #Aurora|Aurora temperament]] | |||
* [[Triforce]] divides an 1/3 octave period into logarithmic-phi-sized fractions. | |||
; 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]] | |||
[[Category:Golden ratio]] | [[Category:Golden ratio]] | ||
Latest revision as of 22:09, 26 November 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 ¢.
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 | 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 |
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