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= Title2 =
= Title2 =
== Octave stretch or compression ==
== Octave stretch or compression ==
What follows is a comparison of stretched- and compressed-octave 22edo tunings.
58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[92edt]] or [[150ed6]].


; [[zpi|ZPINAME]]
What follows is a comparison of stretched- and compressed-octave 58edo tunings.
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of EDONAME 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 ZPINAME does this.
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME (continued)}}


; [[EDONOI]]  
; [[zpi|288zpi]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: 20.736{{c}}, octave size: 1202.69{{c}}
_ing the octave of EDONAME 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 EDONOI does this.
Stretching the octave of 58edo by around 2.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 288zpi does this.
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}}
{{Harmonics in cet|20.736|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 288zpi}}
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}
{{Harmonics in cet|20.736|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 288zpi (continued)}}


; 22edo
; 58edo
* Step size: 54.545{{c}}, octave size: 1200.0{{c}}  
* Step size: 20.690{{c}}, octave size: 1200.00{{c}}  
Pure-octaves 22edo approximates all harmonics up to 16 within NNN{{c}}. The optimal 13-limit [[WE]] tuning has octaves only 0.01{{c}} different from pure-octaves 22edo, and the 13-limit [[TE]] tuning has octaves only 0.08{{c}} different, so by those metrics pure-octaves 22edo might be considered already optimal.
Pure-octaves 58edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|22|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo}}
{{Harmonics in equal|58|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edo}}
{{Harmonics in equal|22|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo (continued)}}
{{Harmonics in equal|58|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edo (continued)}}


; [[WE|22et, 11-limit WE tuning]]  
; [[150ed6]]  
* Step size: 54.494{{c}}, octave size: NNN{{c}}
* Step size: 20.680{{c}}, octave size: 1199.42{{c}}
Compressing the octave of 22edo by around half a cent 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.
Compressing the octave of 58edo by around half a cent results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 150ed6 does this.
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed6}}
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}
{{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed6 (continued)}}


; [[zpi|80zpi]]  
; [[92edt]]  
* Step size: 54.483{{c}}, octave size: NNN{{c}}
* Step size: 20.673{{c}}, octave size: 1199.06{{c}}
Compressing the octave of 22edo 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 80zpi does this.
Compressing the octave of 58edo by around 1{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 92edt does this.
{{Harmonics in cet|54.483|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi}}
{{Harmonics in equal|92|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 92edt}}
{{Harmonics in cet|54.483|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi (continued)}}
{{Harmonics in equal|92|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 92edt (continued)}}


; [[13edf]]  
; [[zpi|289zpi]] / [[WE|58et, 7-limit WE tuning]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: 20.666{{c}}, octave size: 1198.63{{c}}
_ing the octave of EDONAME 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 EDONOI does this.
Compressing the octave of 58edo by just under 1.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. The tuning 289zpi also does this, its octave differing from 7-limit WE by only 0.06{{c}}.  
{{Harmonics in equal|13|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}}
{{Harmonics in cet|20.666|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 289zpi}}
{{Harmonics in equal|13|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}}
{{Harmonics in cet|20.666|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 289zpi (continued)}}


; [[35edt]]  
; [[WE|58et, 13-limit WE tuning]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: 20.663{{c}}, octave size: 1198.45{{c}}
_ing the octave of EDONAME 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 EDONOI does this.
Compressing the octave of 58edo by just over 1.5{{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|35|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}}
{{Harmonics in cet|20.663|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning}}
{{Harmonics in equal|35|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}}
{{Harmonics in cet|20.663|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning (continued)}}
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