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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 12edo tunings.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| ; [[40ed10]]
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| * Step size: 99.658{{c}}, octave size: 1195.9{{c}}
| | {{harmonics in equal|11|intervals=odd|columns=7}} |
| Compressing the octave of EDONAME 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 40ed10 does this.
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| {{Harmonics in equal|40|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 40ed10}} | | {{harmonics in equal|13|intervals=odd|columns=7}} |
| {{Harmonics in equal|40|10|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 40ed10 (continued)}} | | {{harmonics in equal|14|intervals=odd|columns=7}} |
| | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| ; [[WE|12et, 7-limit WE tuning]]
| | {{harmonics in equal|16|intervals=odd|columns=7}} |
| * Step size: 99.664{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|17|intervals=odd|columns=7}} |
| Compressing 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 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| {{Harmonics in cet|99.664|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning}} | | {{harmonics in equal|19|intervals=odd|columns=7}} |
| {{Harmonics in cet|99.664|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}} | | {{harmonics in equal|20|intervals=odd|columns=7}} |
| | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| ; [[zpi|34zpi]]
| | {{harmonics in equal|22|intervals=odd|columns=7}} |
| * Step size: 99.807{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|23|intervals=odd|columns=7}} |
| Compressing the octave of 12edo 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 34zpi does this.
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| {{Harmonics in cet|99.807|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi}} | | {{harmonics in equal|25|intervals=odd|columns=7}} |
| {{Harmonics in cet|99.807|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi (continued)}} | | {{harmonics in equal|26|intervals=odd|columns=7}} |
| | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| ; [[WE|12et, 5-limit WE tuning]]
| | {{harmonics in equal|28|intervals=odd|columns=7}} |
| * Step size: 99.868{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|29|intervals=odd|columns=7}} |
| Compressing 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 5-limit WE tuning and 5-limit [[TE]] tuning both do this.
| | {{harmonics in equal|30|intervals=odd|columns=7}} |
| {{Harmonics in cet|99.868|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning}} | | {{harmonics in equal|31|intervals=odd|columns=7}} |
| {{Harmonics in cet|99.868|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}} | | {{harmonics in equal|32|intervals=odd|columns=7}} |
| | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| ; [[WE|12et, 2.3.5.17.19 WE tuning]]
| | {{harmonics in equal|34|intervals=odd|columns=7}} |
| * Step size: 99.930{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|35|intervals=odd|columns=7}} |
| Compressing the octave of 12edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The 2.3.5.17.19 WE tuning and 2.3.5.17.19 [[TE]] tuning both do this.
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| {{Harmonics in cet|99.930|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 2.3.5.17.19 WE tuning}} | | {{harmonics in equal|37|intervals=odd|columns=7}} |
| {{Harmonics in cet|99.930|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 2.3.5.17.19 WE tuning (continued)}} | | {{harmonics in equal|38|intervals=odd|columns=7}} |
| | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| ; 12edo
| | {{harmonics in equal|40|intervals=odd|columns=7}} |
| * Step size: 100.000{{c}}, octave size: 1200.0{{c}}
| | {{harmonics in equal|41|intervals=odd|columns=7}} |
| Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.
| | {{harmonics in equal|42|intervals=odd|columns=7}} |
| {{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONAME}} | | {{harmonics in equal|43|intervals=odd|columns=7}} |
| {{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONAME (continued)}} | | {{harmonics in equal|44|intervals=odd|columns=7}} |
| | | {{harmonics in equal|45|intervals=odd|columns=7}} |
| ; [[31ed6]]
| | {{harmonics in equal|46|intervals=odd|columns=7}} |
| * Step size: 100.063{{c}}, octave size: 1200.8{{c}}
| | {{harmonics in equal|47|intervals=odd|columns=7}} |
| Stretching the octave of 12edo by a little less than 1{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 31ed6 does this.
| | {{harmonics in equal|48|intervals=odd|columns=7}} |
| {{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6}} | | {{harmonics in equal|49|intervals=odd|columns=7}} |
| {{Harmonics in equal|31|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6 (continued)}} | | {{harmonics in equal|50|intervals=odd|columns=7}} |
| | | {{harmonics in equal|51|intervals=odd|columns=7}} |
| ; [[19edt]]
| | {{harmonics in equal|52|intervals=odd|columns=7}} |
| * Step size: 101.103{{c}}, octave size: 1201.2{{c}}
| | {{harmonics in equal|53|intervals=odd|columns=7}} |
| Stretching the octave of 12edo by a little more than 1{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 19edt does this.
| |
| {{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt}} | |
| {{Harmonics in equal|19|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt (continued)}} | |
| | |
| ; [[7edf]]
| |
| * Step size: 100.3{{c}}, octave size: 1203.35{{c}}
| |
| Stretching the octave of 12edo by around 3{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 7edf does this.
| |
| {{Harmonics in equal|7|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf}} | |
| {{Harmonics in equal|7|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf (continued)}} | |