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= Title1 =
= Title1 =
== Octave stretch or compression ==
== Octave stretch or compression ==
What follows is a comparison of stretched- and compressed-octave 39edo tunings.
What follows is a comparison of stretched- and compressed-octave EDONAME tunings.


; [[zpi|171zpi]]  
; [[zpi|ZPINAME]]  
* Step size: 30.973{{c}}, octave size: 107.9{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Stretching the octave of 39edo by around 8{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 171zpi does this. Because it shares error evenly between 39edo's fifths, it is suited for use as a [[dual-fifths tuning]] of 39edo.
_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|30.973|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 171zpi}}
{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|30.973|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 171zpi (continued)}}
{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}


; 39edo
; [[EDONOI]]
* Step size: 30.769{{c}}, octave size: 1200.00{{c}}  
* Step size: NNN{{c}}, octave size: NNN{{c}}
Pure-octaves 39edo approximates all harmonics up to 16 within 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 EDONOI does this.
{{Harmonics in equal|39|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39edo}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|39|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39edo (continued)}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}


; [[WE|39et, 13-limit WE tuning]]  
; [[WE|ETNAME, SUBGROUP WE tuning]]  
* Step size: 30.757{{c}}, octave size: 1199.5{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 39edo by about half a cent 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.
_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}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
{{Harmonics in cet|30.757|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning}}
{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|30.757|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning (continued)}}
{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}


; [[ed6|101ed6]]
; EDONAME
* Octave size: 1197.8{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}  
Compressing the octave of 101ed6 by around 2{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 101ed6 does this. So does [[zpi|172zpi]] whose octave differs by only 0.4{{c}}.
Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|101|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 101ed6}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}}
{{Harmonics in equal|101|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 101ed6 (continued)}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}}


; [[WE|39et, 2.3.5.11 WE tuning]]  
; [[WE|ETNAME, SUBGROUP WE tuning]]  
* Step size: 30.703{{c}}, octave size: 1197.4{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 39edo by around 2.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.
_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}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
{{Harmonics in cet|30.703|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning}}
{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|30.703|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning (continued)}}
{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}


; [[zpi|173zpi]]  
; [[EDONOI]]  
* Step size: 30.672{{c}}, octave size: 1196.2{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 39edo by around 4{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 173zpi does this. So does [[62edt]] whose octave differs by only 0.2{{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.
{{Harmonics in cet|30.672|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 173zpi}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in cet|30.672|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 173zpi (continued)}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}


; [[ed7|110ed7]]  
; [[zpi|ZPINAME]]  
* Octave size: 1194.4{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 39edo by around 5.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 110ed7 does this. So does [[equal tuning|145ed13]] whose octave differs by only 0.1{{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 equal|110|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 110ed7}}
{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in equal|110|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 110ed7 (continued)}}
{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
 
; [[ed5|91ed5]]
* Octave size: 1194.1{{c}}
Compressing the octave of 39edo by around 6{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 91ed5 does this.
{{Harmonics in equal|91|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 91ed5}}
{{Harmonics in equal|91|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 91ed5 (continued)}}


= Title2 =
= Title2 =
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