Logarithmic phi: Difference between revisions
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'''Logarithmic phi''', or [[phi|<math>\varphi</math>]] [[ | '''Logarithmic phi''', or [[phi|<math>\varphi</math>]] [[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 not to be confused with [[acoustic phi]], which is 833.1{{c}}. | ||
Latest revision as of 14:15, 14 July 2026
| 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