22edo: Difference between revisions

Line 1,451:

; 22edo

* Step size: 54.545{{c}}, octave size: 1200.0{{c}}

* Step size: 54.545{{c}}, octave size: 1200.000{{c}}

Pure-octaves 22edo approximates all harmonics up to 16 within 22.3{{c}}. The optimal 13-limit [[WE]] tuning has octaves only 0.01{{c}} different from pure-octaves 22edo, and the 13-limit [[TE]] tuning has octaves only 0.08{{c}} different, so by those metrics pure-octaves 22edo might be considered already optimal. It is a good 13-limit tuning for its size.

{{Harmonics in equal|22|2|1|columns=11|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 22edo}}

{{Harmonics in equal|22|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22edo}}

{{Harmonics in equal|22|2|1|columns=12|start=12|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 22edo (continued)}}

{{Harmonics in equal|22|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edo (continued)}}

; [[WE|22et, 11-limit WE tuning]]

* Step size: 54.494{{c}}, octave size: 1198.9{{c}}

* Step size: 54.494{{c}}, octave size: 1198.859{{c}}

Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 26.5{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. It is a good 11-limit tuning for its size.

Compressing the octave of 22edo by around 1.1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 26.5{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. It is a good 11-limit tuning for its size.

{{Harmonics in cet|54.~~494~~|columns=11|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 22et, 11-limit WE tuning}}

{{Harmonics in cet|54.493592|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning}}

{{Harmonics in cet|54.~~494~~|columns=12|start=12|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}

{{Harmonics in cet|54.493592|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}

; [[~~zpi~~|80zpi]]

; [[ZPI|80zpi]]

* Step size: 54.483{{c}}, octave size: 1198.6{{c}}

* Step size: 54.483{{c}}, octave size: 1198.630{{c}}

Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this. It is a good 11-limit tuning for its size.

Compressing the octave of 22edo by around 1.4{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this. It is a good 11-limit tuning for its size.

{{Harmonics in cet|54.483|columns=11|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 80zpi}}

{{Harmonics in cet|54.483|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 80zpi}}

{{Harmonics in cet|54.483|columns=12|start=12|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 80zpi (continued)}}

{{Harmonics in cet|54.483|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 80zpi (continued)}}

; [[57ed6]]

* Step size: 54.420{{c}}, octave size: 1197.2{{c}}

* Step size: 54.420{{c}}, octave size: 1197.246{{c}}

Compressing the octave of 22edo by around 3{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 within 21.9{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7]] tuning, eg for [[archy]] (2.3.7 superpyth) temperament. The tuning 57ed6 does this.

Compressing the octave of 22edo by around 2.8{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 within 21.9{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7 subgroup|2.3.7-subgroup]] tuning, e.g. for [[archy]] (2.3.7-subgroup superpyth) temperament. The tuning 57ed6 does this.

{{Harmonics in equal|57|6|1|columns=11|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 57ed6}}

{{Harmonics in equal|57|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed6}}

{{Harmonics in equal|57|6|1|columns=12|start=12|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 57ed6 (continued)}}

{{Harmonics in equal|57|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed6 (continued)}}

; [[35edt]]

* Step size: 54.342{{c}}, octave size: 1195.5{{c}}

* Step size: 54.342{{c}}, octave size: 1195.515{{c}}

Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a moderately worse 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[~~equal tuning|~~62ed7]] both do this. This extends 57ed6's 2.3.7 tuning into a 2.3.7.13 [[subgroup]] tuning.

Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a moderately worse 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[62ed7]] both do this. This extends 57ed6's 2.3.7 tuning into a 2.3.7.13 [[subgroup]] tuning.

{{Harmonics in equal|35|3|1|columns=11|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 35edt}}

{{Harmonics in equal|35|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 35edt}}

{{Harmonics in equal|35|3|1|columns=12|start=12|collapsed=true~~|intervals=integer~~|title=Approximation of harmonics in 35edt (continued)}}

{{Harmonics in equal|35|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edt (continued)}}

== Scales ==

@@ Line 1,451: / Line 1,451: @@
 ; 22edo
-* Step size: 54.545{{c}}, octave size: 1200.0{{c}}
+* Step size: 54.545{{c}}, octave size: 1200.000{{c}}
 Pure-octaves 22edo approximates all harmonics up to 16 within 22.3{{c}}. The optimal 13-limit [[WE]] tuning has octaves only 0.01{{c}} different from pure-octaves 22edo, and the 13-limit [[TE]] tuning has octaves only 0.08{{c}} different, so by those metrics pure-octaves 22edo might be considered already optimal. It is a good 13-limit tuning for its size.
-{{Harmonics in equal|22|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo}}
+{{Harmonics in equal|22|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22edo}}
-{{Harmonics in equal|22|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo (continued)}}
+{{Harmonics in equal|22|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edo (continued)}}
 ; [[WE|22et, 11-limit WE tuning]]
-* Step size: 54.494{{c}}, octave size: 1198.9{{c}}
+* Step size: 54.494{{c}}, octave size: 1198.859{{c}}
-Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 26.5{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. It is a good 11-limit tuning for its size.
+Compressing the octave of 22edo by around 1.1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 26.5{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. It is a good 11-limit tuning for its size.
-{{Harmonics in cet|54.494|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning}}
+{{Harmonics in cet|54.493592|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning}}
-{{Harmonics in cet|54.494|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}
+{{Harmonics in cet|54.493592|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}
-; [[zpi|80zpi]]
+; [[ZPI|80zpi]]
-* Step size: 54.483{{c}}, octave size: 1198.6{{c}}
+* Step size: 54.483{{c}}, octave size: 1198.630{{c}}
-Compressing the octave of 22edo by around 1{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this. It is a good 11-limit tuning for its size.
+Compressing the octave of 22edo by around 1.4{{c}} results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this. It is a good 11-limit tuning for its size.
-{{Harmonics in cet|54.483|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi}}
+{{Harmonics in cet|54.483|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 80zpi}}
-{{Harmonics in cet|54.483|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi (continued)}}
+{{Harmonics in cet|54.483|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 80zpi (continued)}}
 ; [[57ed6]]
-* Step size: 54.420{{c}}, octave size: 1197.2{{c}}
+* Step size: 54.420{{c}}, octave size: 1197.246{{c}}
-Compressing the octave of 22edo by around 3{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 within 21.9{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7]] tuning, eg for [[archy]] (2.3.7 superpyth) temperament. The tuning 57ed6 does this.
+Compressing the octave of 22edo by around 2.8{{c}} results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a [[JND|just noticeably worse]] 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 within 21.9{{c}}. With its worse 5 and 11, it only really makes sense as a [[2.3.7 subgroup|2.3.7-subgroup]] tuning, e.g. for [[archy]] (2.3.7-subgroup superpyth) temperament. The tuning 57ed6 does this.
-{{Harmonics in equal|57|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6}}
+{{Harmonics in equal|57|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed6}}
-{{Harmonics in equal|57|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6 (continued)}}
+{{Harmonics in equal|57|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed6 (continued)}}
 ; [[35edt]]
-* Step size: 54.342{{c}}, octave size: 1195.5{{c}}
+* Step size: 54.342{{c}}, octave size: 1195.515{{c}}
-Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a moderately worse 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[equal tuning|62ed7]] both do this. This extends 57ed6's 2.3.7 tuning into a 2.3.7.13 [[subgroup]] tuning.
+Compressing the octave of 22edo by around 4.5{{c}} results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a moderately worse 2. This approximates all harmonics up to 16 within 21.4{{c}}. The tunings 35edt and [[62ed7]] both do this. This extends 57ed6's 2.3.7 tuning into a 2.3.7.13 [[subgroup]] tuning.
-{{Harmonics in equal|35|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt}}
+{{Harmonics in equal|35|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 35edt}}
-{{Harmonics in equal|35|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt (continued)}}
+{{Harmonics in equal|35|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edt (continued)}}
 == Scales ==