User:BudjarnLambeth/Sandbox2: Difference between revisions

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== Octave stretch or compression ==

What follows is a comparison of stretched- and compressed-octave 22edo tunings.

; [[51ed5]]

* Step size: NNN{{c}}, octave size: nnn{{c}}

Stretching 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]] temperament). The tuning 57ed6 does this.

{{Harmonics in equal|51|5|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6}}

{{Harmonics in equal|51|5|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6 (continued)}}

; 22edo

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

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

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

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

Compressing the octave of 22edo by around 1{{c}} results in improved primes 3 and 7, but worse primes 5, 11 and 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.

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.

{{Harmonics in cet|54.494|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning}}

{{Harmonics in cet|54.494|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}

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; [[zpi|80zpi]]

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

Compressing the octave of 22edo by around 1{{c}} results in improved primes 3 and 7, but worse primes 5, 11 and 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this.

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.

{{Harmonics in cet|54.483|columns=11|collapsed=true|intervals=integer|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)}}

; [[57ed6]]

* Step size: NNN{{c}}, octave size: ~~NNN~~{{c}}

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

~~_ing~~ the octave of 22edo by around ~~NNN~~{{c}} results in improved primes ~~NNN~~, but worse primes ~~NNN~~. This approximates all harmonics up to 16 within ~~NNN~~{{c}}. The tuning 57ed6 does this.

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]] 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|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6 (continued)}}

; [[35edt]]

* Step size: NNN{{c}}, octave size: ~~NNN~~{{c}}

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

~~_ing~~ the octave of 22edo by around ~~NNN~~{{c}} results in improved primes ~~NNN~~, but worse primes ~~NNN~~. This approximates all harmonics up to 16 within ~~NNN~~{{c}}. The ~~tuning~~ 35edt ~~does~~ this.

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 [[JND|just noticeably 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|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt (continued)}}

@@ Line 8: / Line 8: @@
 == Octave stretch or compression ==
 What follows is a comparison of stretched- and compressed-octave 22edo tunings.
+; [[51ed5]]
+* Step size: NNN{{c}}, octave size: nnn{{c}}
+Stretching 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]] temperament). The tuning 57ed6 does this.
+{{Harmonics in equal|51|5|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6}}
+{{Harmonics in equal|51|5|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6 (continued)}}
 ; 22edo
 * Step size: 54.545{{c}}, octave size: 1200.0{{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.
+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|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo (continued)}}
@@ Line 17: / Line 23: @@
 ; [[WE|22et, 11-limit WE tuning]]
 * Step size: 54.494{{c}}, octave size: 1198.9{{c}}
-Compressing the octave of 22edo by around 1{{c}} results in improved primes 3 and 7, but worse primes 5, 11 and 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.
+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.
 {{Harmonics in cet|54.494|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning}}
 {{Harmonics in cet|54.494|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}
@@ Line 23: / Line 29: @@
 ; [[zpi|80zpi]]
 * Step size: 54.483{{c}}, octave size: 1198.6{{c}}
-Compressing the octave of 22edo by around 1{{c}} results in improved primes 3 and 7, but worse primes 5, 11 and 13. This approximates all harmonics up to 16 within 27.1{{c}}. The tuning 80zpi does this.
+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.
 {{Harmonics in cet|54.483|columns=11|collapsed=true|intervals=integer|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)}}
 ; [[57ed6]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
+* Step size: NNN{{c}}, octave size: 1197.2{{c}}
-_ing the octave of 22edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 57ed6 does this.
+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]] 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|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 57ed6 (continued)}}
 ; [[35edt]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
+* Step size: NNN{{c}}, octave size: 1195.5{{c}}
-_ing the octave of 22edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 35edt does this.
+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 [[JND|just noticeably 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|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 35edt (continued)}}