9edϕ: Difference between revisions
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9 equal divisions of [[acoustic phi]] (9edϕ) is a [[tuning system]] that [[Edφ|evenly divides ϕ]] into 9 parts of approximately 92.566 cents each. This scale is closely related to [[13edo]], but slightly stretched. The result is an equal 13-tone-per-octave scale where acoustic phi is [[Just intonation|justly intonated]], and the | {{Infobox ET}} | ||
9 equal divisions of [[acoustic phi]] (9edϕ) is a [[tuning system]] that [[Edφ|evenly divides ϕ]] into 9 parts of approximately 92.566 cents each. This scale is closely related to [[13edo]], but slightly stretched. The result is an equal 13-tone-per-octave scale where acoustic phi is [[Just intonation|justly intonated]], and the octave is stretched by 3.353 cents. | |||
{{Harmonics in cet|92.5656|columns=15}} | |||
<pre> | <pre> | ||
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== Logarithmic phi == | == Logarithmic phi == | ||
9edϕ has a very close approximation of [[logarithmic phi]] on its 21st step, with only +2.2 cents of error. This is because 9edϕ is related to 13edo, which is a Fibonacci edo. | |||
Latest revision as of 00:47, 22 April 2026
| ← 8edϕ | 9edϕ | 10edϕ → |
(convergent)
9 equal divisions of acoustic phi (9edϕ) is a tuning system that evenly divides ϕ into 9 parts of approximately 92.566 cents each. This scale is closely related to 13edo, but slightly stretched. The result is an equal 13-tone-per-octave scale where acoustic phi is justly intonated, and the octave is stretched by 3.353 cents.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.4 | +41.9 | +6.7 | -9.3 | +45.3 | -36.5 | +10.1 | -8.7 | -6.0 | +14.1 | -43.9 | +2.6 | -33.1 | +32.6 | +13.4 |
| Relative (%) | +3.6 | +45.3 | +7.2 | -10.1 | +48.9 | -39.4 | +10.9 | -9.4 | -6.5 | +15.3 | -47.5 | +2.8 | -35.8 | +35.2 | +14.5 | |
| Step | 13 | 21 | 26 | 30 | 34 | 36 | 39 | 41 | 43 | 45 | 46 | 48 | 49 | 51 | 52 | |
! 9edphi.scl ! 9 equal divisions of acoustic phi 9 ! 92.565588 185.131177 277.696765 370.262354 462.827942 555.393531 647.959119 740.524708 833.090296
! 13tet_phi-stretched.scl ! 13-tone-per-octave with acoustic phi justly intonated 13 ! 92.565588 185.131177 277.696765 370.262354 462.827942 555.393531 647.959119 740.524708 833.090296 925.655885 1018.221473 1110.787062 1203.35265
Logarithmic phi
9edϕ has a very close approximation of logarithmic phi on its 21st step, with only +2.2 cents of error. This is because 9edϕ is related to 13edo, which is a Fibonacci edo.