Logarithmic phi
| 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.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 5 | 3\5 | 720.00 | -21.60 | -9.00 |
| 8 | 5\8 | 750.00 | +8.40 | +5.60 |
| 13 | 8\13 | 738.46 | -3.14 | -3.40 |
| 21 | 13\21 | 742.86 | +1.25 | +2.19 |
| 26 | 16\26 | 738.46 | -3.14 | -6.81 |
| 29 | 18\29 | 744.83 | +3.22 | +7.79 |
| 34 | 21\34 | 741.18 | -0.43 | -1.21 |
| 42 | 26\42 | 742.86 | +1.25 | +4.38 |
| 47 | 29\47 | 740.43 | -1.18 | -4.62 |
| 50 | 31\50 | 744.00 | +2.40 | +9.98 |
| 55 | 34\55 | 741.82 | +0.21 | +0.98 |
| 60 | 37\60 | 740.00 | -1.60 | -8.02 |
| 63 | 39\63 | 742.86 | +1.25 | +6.58 |
| 68 | 42\68 | 741.18 | -0.43 | -2.42 |
| 76 | 47\76 | 742.11 | +0.50 | +3.17 |
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