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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}} |
| 72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| What follows is a comparison of stretched-octave 72edo tunings.
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| | | {{harmonics in equal|11|intervals=odd|columns=7}} |
| ; EDONAME
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|13|intervals=odd|columns=7}} |
| Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.
| | {{harmonics in equal|14|intervals=odd|columns=7}} |
| {{Harmonics in equal|72|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}} | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| {{Harmonics in equal|72|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}} | | {{harmonics in equal|16|intervals=odd|columns=7}} |
| | | {{harmonics in equal|17|intervals=odd|columns=7}} |
| ; [[WE|72et, 11-limit WE tuning]]
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| * Step size: 16.677{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|19|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|20|intervals=odd|columns=7}} |
| {{Harmonics in cet|16.677|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}} | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| {{Harmonics in cet|16.677|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}} | | {{harmonics in equal|22|intervals=odd|columns=7}} |
| | | {{harmonics in equal|23|intervals=odd|columns=7}} |
| ; [[zpi|380zpi]]
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| * Step size: 16.678{{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}}. The tuning ZPINAME does this.
| | {{harmonics in equal|26|intervals=odd|columns=7}} |
| {{Harmonics in cet|16.678|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| {{Harmonics in cet|16.678|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} | | {{harmonics in equal|28|intervals=odd|columns=7}} |
| | | {{harmonics in equal|29|intervals=odd|columns=7}} |
| ; [[WE|72et, 13-limit WE tuning]]
| | {{harmonics in equal|30|intervals=odd|columns=7}} |
| * Step size: 16.680{{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}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
| | {{harmonics in equal|32|intervals=odd|columns=7}} |
| {{Harmonics in cet|16.680|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}} | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| {{Harmonics in cet|16.680|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}} | | {{harmonics in equal|34|intervals=odd|columns=7}} |
| | | {{harmonics in equal|35|intervals=odd|columns=7}} |
| ; [[258ed12]]
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| * Step size: NNN{{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}}. The tuning EDONOI does this.
| | {{harmonics in equal|38|intervals=odd|columns=7}} |
| {{Harmonics in equal|258|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| {{Harmonics in equal|258|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} | | {{harmonics in equal|40|intervals=odd|columns=7}} |
| | | {{harmonics in equal|41|intervals=odd|columns=7}} |
| ; [[186ed6]]
| | {{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|186|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|186|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}} |
| ; [[144edt]]
| | {{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|144|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | | {{harmonics in equal|51|intervals=odd|columns=7}} |
| {{Harmonics in equal|144|3|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}} |