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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-octave 31edo tunings.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| ; 31edo
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| * Step size: 38.710{{c}}, octave size: 1200.0{{c}}
| | {{harmonics in equal|11|intervals=odd|columns=7}} |
| Pure-octaves 31edo approximates all harmonics up to 16 within NNN{{c}}.
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| {{Harmonics in equal|31|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo}} | | {{harmonics in equal|13|intervals=odd|columns=7}} |
| {{Harmonics in equal|31|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo (continued)}} | | {{harmonics in equal|14|intervals=odd|columns=7}} |
| | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| ; [[WE|31et, 13-limit WE tuning]]
| | {{harmonics in equal|16|intervals=odd|columns=7}} |
| * Step size: 38.725{{c}}, octave size: 1200.5{{c}}
| | {{harmonics in equal|17|intervals=odd|columns=7}} |
| Stretching the octave of 31edo by around 0.5{{c}} results in slightly improved primes 3, 7 and 11, but slightly worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 12.8{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| {{Harmonics in cet|38.725|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 13-limit WE tuning}} | | {{harmonics in equal|19|intervals=odd|columns=7}} |
| {{Harmonics in cet|38.725|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)}} | | {{harmonics in equal|20|intervals=odd|columns=7}} |
| | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| ; [[zpi|127zpi]]
| | {{harmonics in equal|22|intervals=odd|columns=7}} |
| * Step size: 38.737{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|23|intervals=odd|columns=7}} |
| Stretching the octave of 31edo 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 127zpi does this.
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| {{Harmonics in cet|38.737|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi}} | | {{harmonics in equal|25|intervals=odd|columns=7}} |
| {{Harmonics in cet|38.737|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi (continued)}} | | {{harmonics in equal|26|intervals=odd|columns=7}} |
| | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| ; [[WE|31et, 11-limit WE tuning]]
| | {{harmonics in equal|28|intervals=odd|columns=7}} |
| * Step size: 38.748{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|29|intervals=odd|columns=7}} |
| _Stretching the octave of 31edo 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|30|intervals=odd|columns=7}} |
| {{Harmonics in cet|38.748|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning}} | | {{harmonics in equal|31|intervals=odd|columns=7}} |
| {{Harmonics in cet|38.748|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning (continued)}} | | {{harmonics in equal|32|intervals=odd|columns=7}} |
| | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| ; [[111ed12]]
| | {{harmonics in equal|34|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|35|intervals=odd|columns=7}} |
| Stretching the octave of 31edo 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 111ed12 does this.
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| {{Harmonics in equal|111|12|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 111ed12}} | | {{harmonics in equal|37|intervals=odd|columns=7}} |
| {{Harmonics in equal|111|12|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 111ed12 (continued)}} | | {{harmonics in equal|38|intervals=odd|columns=7}} |
| | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| ; [[80ed6]]
| | {{harmonics in equal|40|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|41|intervals=odd|columns=7}} |
| Stretching the octave of 31edo 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 80ed6 does this.
| | {{harmonics in equal|42|intervals=odd|columns=7}} |
| {{Harmonics in equal|80|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6}} | | {{harmonics in equal|43|intervals=odd|columns=7}} |
| {{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (continued)}} | | {{harmonics in equal|44|intervals=odd|columns=7}} |
| | | {{harmonics in equal|45|intervals=odd|columns=7}} |
| ; [[25ed7/4]]
| | {{harmonics in equal|46|intervals=odd|columns=7}} |
| * Step size: NNN{{c}}, octave size: NNN{{c}}
| | {{harmonics in equal|47|intervals=odd|columns=7}} |
| Stretching the octave of 31edo 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 25ed7/4 does this.
| | {{harmonics in equal|48|intervals=odd|columns=7}} |
| {{Harmonics in equal|25|7|4|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 25ed7/4}} | | {{harmonics in equal|49|intervals=odd|columns=7}} |
| {{Harmonics in equal|25|7|4|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 25ed7/4 (continued)}} | | {{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}} |