14edo: Difference between revisions

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== Octave stretch or compression ==

14edo benefits from [[octave stretch]] as harmonics 3, 7, and 11 are all tuned flat. [[22edt]], [[ed6|36ed6]] and [[42zpi]] are among the possible choices.

What follows is a comparison of stretched-octave 14edo tunings.

; 14edo

* Step size: 85.714{{c}}, octave size: 1200.0{{c}}

Pure-octaves 14edo approximates all no-5s harmonics up to 16 within 37.0{{c}}.

{{Harmonics in equal|14|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14edo}}

{{Harmonics in equal|14|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14edo (continued)}}

; [[WE|14et, 13-limit WE tuning]]

* Step size: 85.759{{c}}, octave size: 1200.6{{c}}

Stretching the octave of 14edo by around half a cent results in improved primes 3, 7 and 11, but a worse prime 13. This approximates all no-5s harmonics up to 16 within 34.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|85.759|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14et, 13-limit WE tuning}}

{{Harmonics in cet|85.759|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14et, 13-limit WE tuning (continued)}}

; [[WE|14et, 11-limit WE tuning]]

* Step size: 85.842{{c}}, octave size: 1201.8{{c}}

Stretching the octave of 14edo by around 2{{c}} results in yet more improved primes 3, 7 and 11, but worse primes 2 and especially 13. This approximates all no-5s harmonics up to 16 within 30.9{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.

{{Harmonics in cet|85.842|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning}}

{{Harmonics in cet|85.842|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning (continued)}}

; [[ed6|36ed6]]

* Step size: 86.165{{c}}, octave size: 1206.3{{c}}

Stretching the octave of 14edo by around NNN{{c}} results in improved primes 3, 7 and 11, but a worse prime 2 and completely unusable prime 13. This approximates all no-13s harmonics up to 16 within 35.3{{c}}. The tuning 36ed6 does this.

{{Harmonics in equal|36|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 36ed6}}

{{Harmonics in equal|36|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36ed6 (continued)}}

; [[zpi|42zpi]]

* Step size: 86.329{{c}}, octave size: 1208.6{{c}}

Stretching the octave of 14edo by around 8.5{{c}} results in dramatically improved primes 3, 5, 7 and 11, but a much worse prime 2 and unusable 13. This approximates all no-13s harmonics up to 16 within 34.4{{c}}. The tuning 42zpi does this.

{{Harmonics in cet|86.329|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42zpi}}

{{Harmonics in cet|86.329|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42zpi (continued)}}

; [[22edt]]

* Step size: 86.453{{c}}, octave size: 1210.3{{c}}

Stretching the octave of 14edo by around 10{{c}} results in dramatically improved primes 3, 5, 7 and 11, but a much worse prime 2 and unusable 13. It could be argued this is the only one of the 14edo tunings discussed here that has a truly use able 5th harmonic, allowing for full 7-limit harmony. The price for that, however, are quite wobbly octaves with 10 cents of error. This approximates all no-13s harmonics up to 16 within 41.3{{c}}. The tuning 22edt does this.

{{Harmonics in equal|22|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}

{{Harmonics in equal|22|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edt (continued)}}

== Scales ==

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 <references group="note" />
+== Octave stretch or compression ==
+edo benefits from [[octave stretch]] as harmonics 3, 7, and 11 are all tuned flat. [[22edt]], [[ed6|36ed6]] and [[42zpi]] are among the possible choices.
+What follows is a comparison of stretched-octave 14edo tunings.
+; 14edo
+* Step size: 85.714{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 14edo approximates all no-5s harmonics up to 16 within 37.0{{c}}.
+{{Harmonics in equal|14|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14edo}}
+{{Harmonics in equal|14|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14edo (continued)}}
+; [[WE|14et, 13-limit WE tuning]]
+* Step size: 85.759{{c}}, octave size: 1200.6{{c}}
+Stretching the octave of 14edo by around half a cent results in improved primes 3, 7 and 11, but a worse prime 13. This approximates all no-5s harmonics up to 16 within 34.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|85.759|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14et, 13-limit WE tuning}}
+{{Harmonics in cet|85.759|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14et, 13-limit WE tuning (continued)}}
+; [[WE|14et, 11-limit WE tuning]]
+* Step size: 85.842{{c}}, octave size: 1201.8{{c}}
+Stretching the octave of 14edo by around 2{{c}} results in yet more improved primes 3, 7 and 11, but worse primes 2 and especially 13. This approximates all no-5s harmonics up to 16 within 30.9{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
+{{Harmonics in cet|85.842|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning}}
+{{Harmonics in cet|85.842|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning (continued)}}
+; [[ed6|36ed6]]
+* Step size: 86.165{{c}}, octave size: 1206.3{{c}}
+Stretching the octave of 14edo by around NNN{{c}} results in improved primes 3, 7 and 11, but a worse prime 2 and completely unusable prime 13. This approximates all no-13s harmonics up to 16 within 35.3{{c}}. The tuning 36ed6 does this.
+{{Harmonics in equal|36|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 36ed6}}
+{{Harmonics in equal|36|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36ed6 (continued)}}
+; [[zpi|42zpi]]
+* Step size: 86.329{{c}}, octave size: 1208.6{{c}}
+Stretching the octave of 14edo by around 8.5{{c}} results in dramatically improved primes 3, 5, 7 and 11, but a much worse prime 2 and unusable 13. This approximates all no-13s harmonics up to 16 within 34.4{{c}}. The tuning 42zpi does this.
+{{Harmonics in cet|86.329|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42zpi}}
+{{Harmonics in cet|86.329|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42zpi (continued)}}
+; [[22edt]]
+* Step size: 86.453{{c}}, octave size: 1210.3{{c}}
+Stretching the octave of 14edo by around 10{{c}} results in dramatically improved primes 3, 5, 7 and 11, but a much worse prime 2 and unusable 13. It could be argued this is the only one of the 14edo tunings discussed here that has a truly use able 5th harmonic, allowing for full 7-limit harmony. The price for that, however, are quite wobbly octaves with 10 cents of error. This approximates all no-13s harmonics up to 16 within 41.3{{c}}. The tuning 22edt does this.
+{{Harmonics in equal|22|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
+{{Harmonics in equal|22|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edt (continued)}}
 == Scales ==