9edϕ: Difference between revisions

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{{Infobox ET}}
{{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 equivalence]] is 1203.353 cents.
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.


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Revision as of 00:28, 22 April 2026

← 8edϕ 9edϕ 10edϕ →
Prime factorization 32
Step size 92.5656 ¢ 
Octave 13\9edϕ (1203.35 ¢)
(convergent)
Twelfth 21\9edϕ (1943.88 ¢) (→ 7\3edϕ)
Consistency limit 4
Distinct consistency limit 4

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.

! 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.