15edo: Difference between revisions

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== Octave stretch or compression ==

15edo's [[prime]]s 3, 5, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].

; 15edo

* Step size: 80.000{{c}}, octave size: 1200.0{{c}}

Pure-octaves 15edo approximates all harmonics up to 12 within 36.1{{c}}.

{{Harmonics in equal|15|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15edo}}

{{Harmonics in equal|15|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15edo (continued)}}

; [[equal tuning|52ed11]]

* Octave size: 1197.5{{c}}

Compressing the octave of 15edo by around 2.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 12 within 28.1{{c}}. The tuning 52ed11 does this.

{{Harmonics in equal|52|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 52ed11}}

{{Harmonics in equal|52|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 52ed11 (continued)}}

; [[WE|15et, 11-limit WE tuning]]

* Step size: 79.770{{c}}, octave size: 1196.5{{c}}

Compressing the octave of 15edo by around 3.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 12 within 25.0{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.

{{Harmonics in cet|79.770|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15et, 11-limit WE tuning}}

{{Harmonics in cet|79.770|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15et, 11-limit WE tuning (continued)}}

; [[ed10|50ed10]]

* Octave size: 1195.1{{c}}

Compressing the octave of 15edo by around 4{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 12 within 23.0{{c}}. The tuning 50ed10 does this.

{{Harmonics in equal|50|10|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 50ed10}}

{{Harmonics in equal|50|10|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 50ed10 (continued)}}

; [[zpi|47zpi]]

* Step size: 79.715{{c}}, octave size: 1195.7{{c}}

Compressing the octave of 15edo by around 4.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 12 within 22.4{{c}}. The tuning 47zpi does this.

{{Harmonics in cet|79.715|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 47zpi}}

{{Harmonics in cet|79.715|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 47zpi (continued)}}

; [[ed12|54ed12]]

* Octave size: 1195.0{{c}}

Compressing the octave of 15edo by around 5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 12 within 20.0{{c}}. The tuning 54ed12 does this.

{{Harmonics in equal|54|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54ed12}}

{{Harmonics in equal|54|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54ed12 (continued)}}

== Scales==

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 <references />
+== Octave stretch or compression ==
+edo's [[prime]]s 3, 5, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].
+; 15edo
+* Step size: 80.000{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 15edo approximates all harmonics up to 12 within 36.1{{c}}.
+{{Harmonics in equal|15|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15edo}}
+{{Harmonics in equal|15|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15edo (continued)}}
+; [[equal tuning|52ed11]]
+* Octave size: 1197.5{{c}}
+Compressing the octave of 15edo by around 2.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 12 within 28.1{{c}}. The tuning 52ed11 does this.
+{{Harmonics in equal|52|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 52ed11}}
+{{Harmonics in equal|52|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 52ed11 (continued)}}
+; [[WE|15et, 11-limit WE tuning]]
+* Step size: 79.770{{c}}, octave size: 1196.5{{c}}
+Compressing the octave of 15edo by around 3.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 12 within 25.0{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
+{{Harmonics in cet|79.770|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15et, 11-limit WE tuning}}
+{{Harmonics in cet|79.770|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15et, 11-limit WE tuning (continued)}}
+; [[ed10|50ed10]]
+* Octave size: 1195.1{{c}}
+Compressing the octave of 15edo by around 4{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 12 within 23.0{{c}}. The tuning 50ed10 does this.
+{{Harmonics in equal|50|10|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 50ed10}}
+{{Harmonics in equal|50|10|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 50ed10 (continued)}}
+; [[zpi|47zpi]]
+* Step size: 79.715{{c}}, octave size: 1195.7{{c}}
+Compressing the octave of 15edo by around 4.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 12 within 22.4{{c}}. The tuning 47zpi does this.
+{{Harmonics in cet|79.715|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 47zpi}}
+{{Harmonics in cet|79.715|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 47zpi (continued)}}
+; [[ed12|54ed12]]
+* Octave size: 1195.0{{c}}
+Compressing the octave of 15edo by around 5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 12 within 20.0{{c}}. The tuning 54ed12 does this.
+{{Harmonics in equal|54|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54ed12}}
+{{Harmonics in equal|54|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54ed12 (continued)}}
 == Scales==