23edo: Difference between revisions

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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 ==

@@ Line 373: / Line 373: @@
 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 ==