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= Title1 =
= Title1 =
== Octave stretch or compression ==
== 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.
Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and gamelans, with [[25edt]], [[41ed6]], and [[57ed12]] being good options.


What follows is a comparison of stretched- and compressed-octave 14edo tunings.
What follows is a comparison of stretched- and compressed-octave 16edo tunings.


; 14edo
; 16edo
* Step size: 85.714{{c}}, octave size: 1200.0{{c}}  
* Step size: 75.000{{c}}, octave size: 1200.0{{c}}  
Pure-octaves 14edo approximates all harmonics up to 16 within NNN{{c}}.
Pure-octaves 16edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|14|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14edo}}
{{Harmonics in equal|16|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16edo}}
{{Harmonics in equal|14|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14edo (continued)}}
{{Harmonics in equal|16|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16edo (continued)}}


; [[WE|14et, 13-limit WE tuning]]  
; [[WE|16et, 2.5.7.13 WE tuning]]  
* Step size: 85.759{{c}}, octave size: NNN{{c}}
* Step size: 75.105{{c}}, octave size: NNN{{c}}
Stretching the octave of 14edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.5.7.13 WE tuning and 2.5.7.13 [[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|75.105|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 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)}}
{{Harmonics in cet|75.105|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning (continued)}}


; [[WE|14et, 11-limit WE tuning]]  
; [[zpi|15zpi]]  
* Step size: 85.842{{c}}, octave size: NNN{{c}}
* Step size: 75.262{{c}}, octave size: NNN{{c}}
Stretching the octave of 14edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 15zpi does 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|75.262|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15zpi}}
{{Harmonics in cet|85.842|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning (continued)}}
{{Harmonics in cet|75.262|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15zpi (continued)}}


; [[36ed6]]  
; [[WE|16et, 13-limit WE tuning]]  
* Step size: 86.165{{c}}, octave size: 1206.3{{c}}
* Step size: 75.315{{c}}, octave size: NNN{{c}}
Stretching the octave of 14edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 36ed6 does this.
Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in equal|36|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 36ed6}}
{{Harmonics in cet|75.315|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning}}
{{Harmonics in equal|36|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36ed6 (continued)}}
{{Harmonics in cet|75.315|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning (continued)}}


; [[zpi|42zpi]]  
; [[57ed12]]  
* Step size: 86.329{{c}}, octave size: NNN{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Stretching the octave of 14edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 42zpi does this.
Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 57ed12 does this.
{{Harmonics in cet|86.329|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42zpi}}
{{Harmonics in equal|57|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed12}}
{{Harmonics in cet|86.329|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42zpi (continued)}}
{{Harmonics in equal|57|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed12 (continued)}}


; [[22edt]]  
; [[41ed6]]  
* Step size: 86.453{{c}}, octave size: 1210.3{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Stretching the octave of 14edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 22edt does this.
Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 41ed6 does this.
{{Harmonics in equal|22|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|41|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41ed6}}
{{Harmonics in equal|22|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edt (continued)}}
{{Harmonics in equal|41|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41ed6 (continued)}}
 
; [[25edt]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
Stretching the octave of 16edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 25edt does this.
{{Harmonics in equal|25|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 25edt}}
{{Harmonics in equal|25|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 25edt (continued)}}


= Title2 =
= Title2 =
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