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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}} |
| 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]].
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| What follows is a comparison of stretched- and compressed-octave 58edo tunings.
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| | | {{harmonics in equal|11|intervals=odd|columns=7}} |
| ; [[zpi|288zpi]]
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| * Step size: 20.736{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|13|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 ZPINAME does this.
| | {{harmonics in equal|14|intervals=odd|columns=7}} |
| {{Harmonics in cet|20.736|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| {{Harmonics in cet|20.736|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} | | {{harmonics in equal|16|intervals=odd|columns=7}} |
| | | {{harmonics in equal|17|intervals=odd|columns=7}} |
| ; 58edo
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| * Step size: 20.690{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|19|intervals=odd|columns=7}} |
| Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.
| | {{harmonics in equal|20|intervals=odd|columns=7}} |
| {{Harmonics in equal|58|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}} | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| {{Harmonics in equal|58|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}} | | {{harmonics in equal|22|intervals=odd|columns=7}} |
| | | {{harmonics in equal|23|intervals=odd|columns=7}} |
| ; [[WE|58et, 7-limit WE tuning]]
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| * Step size: 20.667{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|25|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}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
| | {{harmonics in equal|26|intervals=odd|columns=7}} |
| {{Harmonics in cet|20.667|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}} | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| {{Harmonics in cet|20.667|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}} | | {{harmonics in equal|28|intervals=odd|columns=7}} |
| | | {{harmonics in equal|29|intervals=odd|columns=7}} |
| ; [[zpi|289zpi]]
| | {{harmonics in equal|30|intervals=odd|columns=7}} |
| * Step size: 20.666{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|31|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 ZPINAME does this.
| | {{harmonics in equal|32|intervals=odd|columns=7}} |
| {{Harmonics in cet|20.666|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| {{Harmonics in cet|20.666|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} | | {{harmonics in equal|34|intervals=odd|columns=7}} |
| | | {{harmonics in equal|35|intervals=odd|columns=7}} |
| ; [[WE|58et, 13-limit WE tuning]]
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| * Step size: 20.663{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|37|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}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
| | {{harmonics in equal|38|intervals=odd|columns=7}} |
| {{Harmonics in cet|20.663|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}} | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| {{Harmonics in cet|20.663|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}} | | {{harmonics in equal|40|intervals=odd|columns=7}} |
| | | {{harmonics in equal|41|intervals=odd|columns=7}} |
| ; [[Ned12]]
| | {{harmonics in equal|42|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|43|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|44|intervals=odd|columns=7}} |
| {{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | | {{harmonics in equal|45|intervals=odd|columns=7}} |
| {{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} | | {{harmonics in equal|46|intervals=odd|columns=7}} |
| | | {{harmonics in equal|47|intervals=odd|columns=7}} |
| ; [[150ed6]]
| | {{harmonics in equal|48|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|49|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|50|intervals=odd|columns=7}} |
| {{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | | {{harmonics in equal|51|intervals=odd|columns=7}} |
| {{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} | | {{harmonics in equal|52|intervals=odd|columns=7}} |
| | | {{harmonics in equal|53|intervals=odd|columns=7}} |
| ; [[92edt]]
| |
| * 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 EDONOI does this.
| |
| {{Harmonics in equal|92|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | |
| {{Harmonics in equal|92|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} | |