22edo: Difference between revisions

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== Octave stretch or compression ==

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

22edo can benefit from slightly compressing the octave, especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 7th harmonic at the expense of somewhat less accurate approximations of 5 and 11.

; 22edo

* 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.

Pure-octaves 22edo approximates all harmonics up to 16 but 13 within 14.3{{c}}.

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

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

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

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

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~~.

Compressing the octave of 22edo by around 1.1{{c}} results in slightly improved primes 3, 7, and 17, but slightly worse primes 5 and 11. This approximates all harmonics up to 16 but 13 within 10.6{{c}}. Both 11-limit TE and WE tunings do this.

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

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

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; [[ZPI|80zpi]]

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

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~~.

Compressing the octave of 22edo by around 1.4{{c}} results in slightly improved primes 3, 7 and 17, but slightly worse primes 5 and 11. This approximates all harmonics up to 16 but 13 within 10.6{{c}}. The tuning 80zpi does this.

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

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

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; [[57ed6]]

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

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.

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 but 13 within 15.4{{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|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed6}}

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

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; [[35edt]]

* 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 [[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-subgroup tuning into a [[2.3.7.13 subgroup|2.3.7.13-subgroup]] tuning.

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

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

@@ Line 1,448: / Line 1,448: @@
 == Octave stretch or compression ==
-What follows is a comparison of compressed-octave 22edo tunings.
+edo can benefit from slightly compressing the octave, especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 7th harmonic at the expense of somewhat less accurate approximations of 5 and 11.
 ; 22edo
 * 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.
+Pure-octaves 22edo approximates all harmonics up to 16 but 13 within 14.3{{c}}.
 {{Harmonics in equal|22|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22edo}}
 {{Harmonics in equal|22|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edo (continued)}}
@@ Line 1,458: / Line 1,458: @@
 ; [[WE|22et, 11-limit WE tuning]]
 * Step size: 54.494{{c}}, octave size: 1198.859{{c}}
-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.
+Compressing the octave of 22edo by around 1.1{{c}} results in slightly improved primes 3, 7, and 17, but slightly worse primes 5 and 11. This approximates all harmonics up to 16 but 13 within 10.6{{c}}. Both 11-limit TE and WE tunings do this.
 {{Harmonics in cet|54.493592|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning}}
 {{Harmonics in cet|54.493592|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}}
@@ Line 1,464: / Line 1,464: @@
 ; [[ZPI|80zpi]]
 * Step size: 54.483{{c}}, octave size: 1198.630{{c}}
-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.
+Compressing the octave of 22edo by around 1.4{{c}} results in slightly improved primes 3, 7 and 17, but slightly worse primes 5 and 11. This approximates all harmonics up to 16 but 13 within 10.6{{c}}. The tuning 80zpi does this.
 {{Harmonics in cet|54.483|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 80zpi}}
 {{Harmonics in cet|54.483|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 80zpi (continued)}}
@@ Line 1,470: / Line 1,470: @@
 ; [[57ed6]]
 * Step size: 54.420{{c}}, octave size: 1197.246{{c}}
-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.
+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 but 13 within 15.4{{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|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed6}}
 {{Harmonics in equal|57|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed6 (continued)}}
@@ Line 1,476: / Line 1,476: @@
 ; [[35edt]]
 * 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 [[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-subgroup tuning into a [[2.3.7.13 subgroup|2.3.7.13-subgroup]] tuning.
 {{Harmonics in equal|35|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 35edt}}
 {{Harmonics in equal|35|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edt (continued)}}