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= Title1 =
= Title1 =
== Octave stretch or compression ==
== Octave stretch or compression ==
64edo's approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by [[180ed7]], a [[Octave shrinking|compressed-octave]] version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.


; [[ed6|152ed6]]  
[[149ed5]] can also be used: it is similar to 180ed7 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as 180ed7's.
 
If one prefers a ''[[Octave stretch|stretched-octave]]'', 64edo's approximations of 3/1, 5/1, 11/1 and 17/1 are improved by [[221ed11]], a stretched version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
 
[[47ed5/3]] can also be used: it is similar to 221ed11 but both the improvements and shortcomings are amplified. Most notably its 2/1 is not as accurate as 221ed11's.
 
What follows is a comparison of stretched- and compressed-octave 64edo tunings.
 
; [[ed7|179ed7]]
* Octave size: NNN{{c}}
* Octave size: NNN{{c}}
_ing the octave of 59edo 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 152ed6 does this.
Stretching the octave of 64edo 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 179ed7 does this. So does the tuning 326zpi whose octave is identical within 0.3{{c}}.
{{Harmonics in equal|152|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 152ed6}}
{{Harmonics in equal|179|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 179ed7}}
{{Harmonics in equal|152|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 152ed6 (continued)}}
{{Harmonics in equal|179|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 179ed7 (continued)}}


; [[zpi|294zpi]]  
; [[ed6|165ed6]]
* Step size: 20.399{{c}}, octave size: NNN{{c}}
* Octave size: NNN{{c}}
_ing the octave of 59edo 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 294zpi does this.
Stretching the octave of 64edo 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 165ed6 does this.
{{Harmonics in cet|20.399|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 294zpi}}
{{Harmonics in equal|165|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in cet|20.399|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 294zpi (continued)}}
{{Harmonics in equal|165|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 165ed6 (continued)}}


; [[211ed12]]  
; [[ed12|229ed12]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Octave size: NNN{{c}}
_ing the octave of 59edo 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 211ed12 does this.
Stretching the octave of 64edo 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 229ed12 does this. So does the tuning [[equal tuning|221ed11]] whose octave is identical within 0.1{{c}}.
{{Harmonics in equal|211|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 211ed12}}
{{Harmonics in equal|229|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 229ed12}}
{{Harmonics in equal|211|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 211ed12 (continued)}}
{{Harmonics in equal|229|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 229ed12 (continued)}}


; [[zpi|ZPINAME]]  
; [[zpi|327zpi]]
* Step size: 20.342{{c}}, octave size: NNN{{c}}
* Step size: 18.767{{c}}, octave size: NNN{{c}}
_ing the octave of 59edo 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 295zpi does this.
Stretching the octave of 64edo 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 327zpi does this.
{{Harmonics in cet|20.342|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 295zpi}}
{{Harmonics in cet|18.767|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 327zpi}}
{{Harmonics in cet|20.342|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 295zpi (continued)}}
{{Harmonics in cet|18.767|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 327zpi (continued)}}


; 59edo
; [[WE|64et, 11-limit WE tuning]]
* Step size: 20.339{{c}}, octave size: 1200.00{{c}}  
* Step size: 18.755{{c}}, octave size: NNN{{c}}
Pure-octaves 59edo approximates all harmonics up to 16 within NNN{{c}}. So does the tuning [[ed|137ed5]] whose octave is identical within 0.05{{c}}.
Stretching the octave of 64edo 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.
{{Harmonics in equal|59|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 59edo}}
{{Harmonics in cet|18.755|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 64et, 11-limit WE tuning}}
{{Harmonics in equal|59|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59edo (continued)}}
{{Harmonics in cet|18.755|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 64et, 11-limit WE tuning (continued)}}


; [[WE|59et, 13-limit WE tuning]]
; 64edo
* Step size: 20.320{{c}}, octave size: NNN{{c}}
* Step size: 18.750{{c}}, octave size: 1200.00{{c}}  
_ing the octave of 59edo 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.
Pure-octaves 64edo approximates all harmonics up to 16 within NNN{{c}}. The octave of 64edo's 13-limit [[WE]] tuning differs by only 0.13{{c}} from pure.
{{Harmonics in cet|20.320|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 59et, 13-limit WE tuning}}
{{Harmonics in equal|64|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 64edo}}
{{Harmonics in cet|20.320|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59et, 13-limit WE tuning (continued)}}
{{Harmonics in equal|64|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 64edo (continued)}}


; [[WE|59et, 7-limit WE tuning]]  
; [[zpi|328zpi]]
* Step size: 20.301{{c}}, octave size: NNN{{c}}
* Step size: 18.721{{c}}, octave size: NNN{{c}}
_ing the octave of 59edo by around NNN{{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.
Compressing the octave of 64edo 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 328zpi does this.
{{Harmonics in cet|20.301|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, 59et, 7-limit WE tuning}}
{{Harmonics in cet|18.721|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 328zpi}}
{{Harmonics in cet|20.301|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59et, 7-limit WE tuning (continued)}}
{{Harmonics in cet|18.721|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 328zpi (continued)}}


; [[166ed7]]  
; [[ed7|180ed7]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Octave size: NNN{{c}}
_ing the octave of 59edo 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 166ed7 does this.
Compressing the octave of 64edo 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 180ed7 does this.
{{Harmonics in equal|166|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 166ed7}}
{{Harmonics in equal|180|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 180ed7}}
{{Harmonics in equal|166|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 166ed7 (continued)}}
{{Harmonics in equal|180|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 180ed7 (continued)}}


; [[212ed12]]  
; [[ed12|230ed12]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Octave size: NNN{{c}}
_ing the octave of 59edo 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 212ed12 does this.
Compressing the octave of 64edo 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 230ed12 does this.
{{Harmonics in equal|212|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 212ed12}}
{{Harmonics in equal|230|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 230ed12}}
{{Harmonics in equal|212|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 212ed12 (continued)}}
{{Harmonics in equal|230|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 230ed12 (continued)}}
 
; [[zpi|296zpi]]
* Step size: 20.282{{c}}, octave size: NNN{{c}}
_ing the octave of 59edo 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 296zpi does this.
{{Harmonics in cet|20.282|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 296zpi}}
{{Harmonics in cet|20.282|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 296zpi (continued)}}


; [[153ed6]]  
; [[ed5|149ed5]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: Octave size: NNN{{c}}
_ing the octave of 59edo 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 153ed6 does this.
Compressing the octave of 64edo 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 149ed5 does this.
{{Harmonics in equal|153|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 153ed6}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 149ed5}}
{{Harmonics in equal|153|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 153ed6 (continued)}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 149ed5 (continued)}}


= Title2 =
= Title2 =
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