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| = Title1 = | | == Approximations of odd harmonics == |
| {{Harmonics in equal|40|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|1|intervals=odd|columns=7}} |
| {{Harmonics in equal|7|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|2|intervals=odd|columns=7}} |
| {{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|3|intervals=odd|columns=7}} |
| {{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|4|intervals=odd|columns=7}} |
| | | {{harmonics in equal|5|intervals=odd|columns=7}} |
| = Title2 = | | {{harmonics in equal|6|intervals=odd|columns=7}} |
| == Octave stretch or compression == | | {{harmonics in equal|7|intervals=odd|columns=7}} |
| What follows is a comparison of stretched- and compressed-octave 22edo tunings.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| ; 22edo
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| * Step size: 54.545{{c}}, octave size: 1200.0{{c}}
| | {{harmonics in equal|11|intervals=odd|columns=7}} |
| Pure-octaves 22edo approximates all harmonics up to 16 within 22.3{{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.
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| {{Harmonics in equal|22|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo}} | | {{harmonics in equal|13|intervals=odd|columns=7}} |
| {{Harmonics in equal|22|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo (continued)}} | | {{harmonics in equal|14|intervals=odd|columns=7}} |
| | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| ; [[WE|22et, 11-limit WE tuning]]
| | {{harmonics in equal|16|intervals=odd|columns=7}} |
| * Step size: 54.494{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|17|intervals=odd|columns=7}} |
| 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.
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| {{Harmonics in cet|54.494|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning}} | | {{harmonics in equal|19|intervals=odd|columns=7}} |
| {{Harmonics in cet|54.494|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22et, 11-limit WE tuning (continued)}} | | {{harmonics in equal|20|intervals=odd|columns=7}} |
| | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| ; [[zpi|80zpi]]
| | {{harmonics in equal|22|intervals=odd|columns=7}} |
| * Step size: 54.483{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|23|intervals=odd|columns=7}} |
| 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.
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| {{Harmonics in cet|54.483|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi}} | | {{harmonics in equal|25|intervals=odd|columns=7}} |
| {{Harmonics in cet|54.483|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi (continued)}} | | {{harmonics in equal|26|intervals=odd|columns=7}} |
| | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| ; [[35edt]]
| | {{harmonics in equal|28|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|29|intervals=odd|columns=7}} |
| _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|30|intervals=odd|columns=7}} |
| {{Harmonics in equal|57|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}} | | {{harmonics in equal|31|intervals=odd|columns=7}} |
| {{Harmonics in equal|57|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}} | | {{harmonics in equal|32|intervals=odd|columns=7}} |
| | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| ; [[13edf]]
| | {{harmonics in equal|34|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|35|intervals=odd|columns=7}} |
| _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|36|intervals=odd|columns=7}} |
| {{Harmonics in equal|13|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}} | | {{harmonics in equal|37|intervals=odd|columns=7}} |
| {{Harmonics in equal|13|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}} | | {{harmonics in equal|38|intervals=odd|columns=7}} |
| | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| ; [[35edt]]
| | {{harmonics in equal|40|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|41|intervals=odd|columns=7}} |
| _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|42|intervals=odd|columns=7}} |
| {{Harmonics in equal|35|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}} | | {{harmonics in equal|43|intervals=odd|columns=7}} |
| {{Harmonics in equal|35|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}} | | {{harmonics in equal|44|intervals=odd|columns=7}} |
| | | {{harmonics in equal|45|intervals=odd|columns=7}} |
| | | {{harmonics in equal|46|intervals=odd|columns=7}} |
| 13edf
| | {{harmonics in equal|47|intervals=odd|columns=7}} |
| 35edt
| | {{harmonics in equal|48|intervals=odd|columns=7}} |
| | {{harmonics in equal|49|intervals=odd|columns=7}} |
| | {{harmonics in equal|50|intervals=odd|columns=7}} |
| | {{harmonics in equal|51|intervals=odd|columns=7}} |
| | {{harmonics in equal|52|intervals=odd|columns=7}} |
| | {{harmonics in equal|53|intervals=odd|columns=7}} |