Acoustic phi
| Interval information |
ϕ taken as a frequency ratio (ϕ⋅f where f = 1/1) is about 833.1 cents. This metastable interval is sometimes called acoustic phi, or the phi neutral sixth. It is wider than a 12edo minor sixth (800 cents) by about a sixth-tone (33.3… cents).
ϕ is the most difficult interval to approximate by rational numbers, as its continued fraction consists entirely of 1's. The convergents (rational number approximations, obtained from the continued fractions) are the ratios of successive terms of the Fibonacci sequence converge on ϕ, the just intonation intervals 3/2, 5/3 (~884.4¢), 8/5 (~814.7¢), 13/8 (~840.5¢), 21/13 (~830.3¢), … converge on ~833.1 cents.
Erv Wilson accordingly described ϕ as "the worstest of the worst — and yet somehow with divinity imbued, Lord have mercy!", inspiring the term merciful intonation.
Acoustic phi is not to be confused with logarithmic phi, which is 1941.6¢ (741.6¢ octave-reduced).
The phith root of phi is another interval with interesting properties, that divides acoustic phi logarithmically by phi, which creates self similar, fractal-like scales.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 3 | 2\3 | 800.00 | -33.05 | -8.26 |
| 10 | 7\10 | 840.00 | +6.95 | +5.79 |
| 13 | 9\13 | 830.77 | -2.28 | -2.48 |
| 23 | 16\23 | 834.78 | +1.73 | +3.31 |
| 26 | 18\26 | 830.77 | -2.28 | -4.95 |
| 33 | 23\33 | 836.36 | +3.31 | +9.10 |
| 36 | 25\36 | 833.33 | +0.28 | +0.84 |
| 39 | 27\39 | 830.77 | -2.28 | -7.43 |
| 46 | 32\46 | 834.78 | +1.73 | +6.63 |
| 49 | 34\49 | 832.65 | -0.40 | -1.64 |
| 52 | 36\52 | 830.77 | -2.28 | -9.90 |
| 59 | 41\59 | 833.90 | +0.84 | +4.15 |
| 62 | 43\62 | 832.26 | -0.80 | -4.11 |
| 69 | 48\69 | 834.78 | +1.73 | +9.94 |
| 72 | 50\72 | 833.33 | +0.28 | +1.68 |
| 75 | 52\75 | 832.00 | -1.05 | -6.59 |
See also
- 833 Cent Golden Scale (Bohlen)
- Edφ, tunings created by dividing acoustic phi into equally sized smaller steps
- Phi as a generator
- sqrtphi, a temperament based on the square root of phi (~416.5 cents) as a generator.
- photosynthesis, a temperament using phi as a “prime” in its subgroup