22edo: Difference between revisions

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=== Subsets and supersets ===

As 22 is divisible by 11, a 22edo instrument can play any music in 11edo, in the same way that [[12edo]] can play [[6edo]] (the whole tone scale). 11edo is interesting for sounding melodically very similar to 12edo (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.

~~=== Stretched and compressed tunings ===~~

The [[The Riemann zeta function and tuning|local zeta peak]] around 22, '''80zpi''', is located at 22.025147, which has the octave [[Stretched and compressed tuning|compressed]] by 1.37{{c}}. The step size of [[APS|1ed54.5c]] differs from 22edo by only 0.05{{c}}. It improves the tuning of primes 3 and 7, but worsens that of primes 5 and 11, so it may be considered when treating 22edo as a tuning of [[archy]] (2.3.7 superpyth).

~~{{Harmonics in cet|54.5|intervals=prime|columns=11|collapsed=true}}~~

== Defining features ==

Line 1,450:

Line 1,446:

| [[Undeka]]<br>[[Hendecatonic]]

|}

== Octave stretch or compression ==

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 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)}}

; [[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, 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)}}

; [[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, 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)}}

; [[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 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)}}

; [[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-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)}}

== Scales ==

@@ Line 27: / Line 27: @@
 === Subsets and supersets ===
 As 22 is divisible by 11, a 22edo instrument can play any music in 11edo, in the same way that [[12edo]] can play [[6edo]] (the whole tone scale). 11edo is interesting for sounding melodically very similar to 12edo (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22.
-=== Stretched and compressed tunings ===
-The [[The Riemann zeta function and tuning|local zeta peak]] around 22, '''80zpi''', is located at 22.025147, which has the octave [[Stretched and compressed tuning|compressed]] by 1.37{{c}}. The step size of [[APS|1ed54.5c]] differs from 22edo by only 0.05{{c}}. It improves the tuning of primes 3 and 7, but worsens that of primes 5 and 11, so it may be considered when treating 22edo as a tuning of [[archy]] (2.3.7 superpyth).
-{{Harmonics in cet|54.5|intervals=prime|columns=11|collapsed=true}}
 == Defining features ==
@@ Line 1,450: / Line 1,446: @@
 | [[Undeka]]<br>[[Hendecatonic]]
 |}
+== Octave stretch or compression ==
+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 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)}}
+; [[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, 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)}}
+; [[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, 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)}}
+; [[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 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)}}
+; [[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-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)}}
 == Scales ==