23edo: Difference between revisions

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== Octave stretch or compression ==

23edo is not typically taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention.

However, when using a slightly [[stretched tuning|stretched octave]] of around 1206 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well.

Stretched-23edo is one of the best tunings to use for exploring the [[antidiatonic]] scale since its fifth is more [[consonant]] and less "[[Wolf interval|wolfish]]" than fifths in other [[pelogic family]] temperaments.

What follows is a comparison of compressed- and stretched-octave 23edo tunings.

; [[zpi|86zpi]]

* Step size: 51.653{{c}}, octave size: 1188.0{{c}}

* Approximates all harmonics <9 within 24.0{{c}}

Compressing the octave of 23edo by around 12{{c}} results in improved primes 5, 11 and 13, but worse primes 2, 3 and 7. The tuning 86zpi does this.

{{Harmonics in cet|51.653|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 86zpi}}

{{Harmonics in cet|51.653|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 86zpi (continued)}}

; [[60ed6]]

* Step size: 51.700{{c}}, octave size: 1189.1{{c}}

* Approximates all harmonics <9 within 21.8{{c}}

Compressing the octave of 23edo by around 11{{c}} results in improved primes 3, 5, 7 and 11, but a worse prime 2. The tuning 60ed6 does this. So does the tuning [[equal tuning|105ed23]] whose octave is identical within 0.01{{c}}.

{{Harmonics in equal|60|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60ed6}}

{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60ed6 (continued)}}

; [[zpi|85zpi]]

* Step size: 52.114{{c}}, octave size: 1198.6{{c}}

* Approximates all harmonics <9 within 24.9{{c}}

Compressing the octave of 23edo by around 1.5{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. The tuning 85zpi does this. So does the tuning [[ed9|73ed9]] whose octave is identical within 0.02{{c}}.

{{Harmonics in cet|52.114|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 85zpi}}

{{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 85zpi (continued)}}

; 23edo

* Step size: 52.174{{c}}, octave size: 1200.0{{c}}

* Approximates all harmonics <9 within 23.7{{c}}

Pure-octaves 23edo.

{{Harmonics in equal|23|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23edo}}

{{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23edo (continued)}}

; [[WE|23et, 13-limit WE tuning]]

* Step size: 52.237{{c}}, octave size: 1201.5{{c}}

* Approximates all harmonics <9 within 25.7{{c}}

Stretching the octave of 23edo by around 1.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning [[equal tuning|85ed13]] whose octave is identical within 0.1{{c}}.

{{Harmonics in cet|52.237|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23et, 13-limit WE tuning}}

{{Harmonics in cet|52.237|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23et, 13-limit WE tuning (continued)}}

; [[WE|23et, 2.3.5.13 WE tuning]]

* Step size: 52.447{{c}}, octave size: 1206.3{{c}}

* Approximates all harmonics <9 within 18.8{{c}}

Stretching the octave of 23edo by around 6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. Its 2.3.5.13 WE tuning and 2.3.5.13 [[TE]] tuning both do this. So does the tuning [[ed10|76ed10]] whose octave is identical within 0.01{{c}}.

{{Harmonics in cet|52.447|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23et, 2.3.5.13 WE tuning}}

{{Harmonics in cet|52.447|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23et, 2.3.5.13 WE tuning (continued)}}

; [[59ed6]]

* Step size: 52.575{{c}}, octave size: 1209.2{{c}}

* Approximates all harmonics <9 within 24.9{{c}}

Stretching the octave of 23edo by around 9{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. The tuning 59ed6 does this. So does the tuning [[ed5|53ed5]] whose octave is identical within 0.01{{c}}.

{{Harmonics in equal|59|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 59ed6}}

{{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59ed6 (continued)}}

; [[zpi|84zpi]]

* Step size: 52.615{{c}}, octave size: 1210.1{{c}}

* Approximates all harmonics <9 within 22.2{{c}}

Stretching the octave of 23edo by around 10{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. The tuning 84zpi does this.

{{Harmonics in cet|52.615|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 84zpi}}

{{Harmonics in cet|52.615|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 84zpi (continued)}}

; [[36edt]]

* Step size: 52.832{{c}}, octave size: 1215.1{{c}}

* Approximates all harmonics <9 within 22.6{{c}}

Stretching the octave of 23edo by around 15{{c}} results in improved primes 3, 5, 7 and 13, but a worse prime 2. The tuning 36edt does this.

{{Harmonics in equal|36|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 36edt}}

{{Harmonics in equal|36|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edt (continued)}}

; [[equal tuning|84ed13]]

* Step size: 52.863{{c}}, octave size: 1215.9{{c}}

* Approximates all harmonics <9 within 21.1{{c}}

Stretching the octave of 23edo by around 16{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. The tuning 84ed13 does this.

{{Harmonics in equal|84|13|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 84ed13}}

{{Harmonics in equal|84|13|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 84ed13 (continued)}}

== Scales ==

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 | Werckisma
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+== Octave stretch or compression ==
+{{main|23edo and octave stretching}}
+edo is not typically taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention.
+However, when using a slightly [[stretched tuning|stretched octave]] of around 1206 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well.
+Stretched-23edo is one of the best tunings to use for exploring the [[antidiatonic]] scale since its fifth is more [[consonant]] and less "[[Wolf interval|wolfish]]" than fifths in other [[pelogic family]] temperaments.
+What follows is a comparison of compressed- and stretched-octave 23edo tunings.
+; [[zpi|86zpi]]
+* Step size: 51.653{{c}}, octave size: 1188.0{{c}}
+* Approximates all harmonics <9 within 24.0{{c}}
+Compressing the octave of 23edo by around 12{{c}} results in improved primes 5, 11 and 13, but worse primes 2, 3 and 7. The tuning 86zpi does this.
+{{Harmonics in cet|51.653|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 86zpi}}
+{{Harmonics in cet|51.653|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 86zpi (continued)}}
+; [[60ed6]]
+* Step size: 51.700{{c}}, octave size: 1189.1{{c}}
+* Approximates all harmonics <9 within 21.8{{c}}
+Compressing the octave of 23edo by around 11{{c}} results in improved primes 3, 5, 7 and 11, but a worse prime 2. The tuning 60ed6 does this. So does the tuning [[equal tuning|105ed23]] whose octave is identical within 0.01{{c}}.
+{{Harmonics in equal|60|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60ed6}}
+{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60ed6 (continued)}}
+; [[zpi|85zpi]]
+* Step size: 52.114{{c}}, octave size: 1198.6{{c}}
+* Approximates all harmonics <9 within 24.9{{c}}
+Compressing the octave of 23edo by around 1.5{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. The tuning 85zpi does this. So does the tuning [[ed9|73ed9]] whose octave is identical within 0.02{{c}}.
+{{Harmonics in cet|52.114|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 85zpi}}
+{{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 85zpi (continued)}}
+; 23edo
+* Step size: 52.174{{c}}, octave size: 1200.0{{c}}
+* Approximates all harmonics <9 within 23.7{{c}}
+Pure-octaves 23edo.
+{{Harmonics in equal|23|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23edo}}
+{{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23edo (continued)}}
+; [[WE|23et, 13-limit WE tuning]]
+* Step size: 52.237{{c}}, octave size: 1201.5{{c}}
+* Approximates all harmonics <9 within 25.7{{c}}
+Stretching the octave of 23edo by around 1.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning [[equal tuning|85ed13]] whose octave is identical within 0.1{{c}}.
+{{Harmonics in cet|52.237|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23et, 13-limit WE tuning}}
+{{Harmonics in cet|52.237|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23et, 13-limit WE tuning (continued)}}
+; [[WE|23et, 2.3.5.13 WE tuning]]
+* Step size: 52.447{{c}}, octave size: 1206.3{{c}}
+* Approximates all harmonics <9 within 18.8{{c}}
+Stretching the octave of 23edo by around 6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. Its 2.3.5.13 WE tuning and 2.3.5.13 [[TE]] tuning both do this. So does the tuning [[ed10|76ed10]] whose octave is identical within 0.01{{c}}.
+{{Harmonics in cet|52.447|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23et, 2.3.5.13 WE tuning}}
+{{Harmonics in cet|52.447|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23et, 2.3.5.13 WE tuning (continued)}}
+; [[59ed6]]
+* Step size: 52.575{{c}}, octave size: 1209.2{{c}}
+* Approximates all harmonics <9 within 24.9{{c}}
+Stretching the octave of 23edo by around 9{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. The tuning 59ed6 does this. So does the tuning [[ed5|53ed5]] whose octave is identical within 0.01{{c}}.
+{{Harmonics in equal|59|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 59ed6}}
+{{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59ed6 (continued)}}
+; [[zpi|84zpi]]
+* Step size: 52.615{{c}}, octave size: 1210.1{{c}}
+* Approximates all harmonics <9 within 22.2{{c}}
+Stretching the octave of 23edo by around 10{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. The tuning 84zpi does this.
+{{Harmonics in cet|52.615|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 84zpi}}
+{{Harmonics in cet|52.615|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 84zpi (continued)}}
+; [[36edt]]
+* Step size: 52.832{{c}}, octave size: 1215.1{{c}}
+* Approximates all harmonics <9 within 22.6{{c}}
+Stretching the octave of 23edo by around 15{{c}} results in improved primes 3, 5, 7 and 13, but a worse prime 2. The tuning 36edt does this.
+{{Harmonics in equal|36|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 36edt}}
+{{Harmonics in equal|36|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edt (continued)}}
+; [[equal tuning|84ed13]]
+* Step size: 52.863{{c}}, octave size: 1215.9{{c}}
+* Approximates all harmonics <9 within 21.1{{c}}
+Stretching the octave of 23edo by around 16{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. The tuning 84ed13 does this.
+{{Harmonics in equal|84|13|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 84ed13}}
+{{Harmonics in equal|84|13|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 84ed13 (continued)}}
 == Scales ==