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| = [[7edo]] = | | == Approximations of odd harmonics == |
| == Octave stretch or compression == | | {{harmonics in equal|1|intervals=odd|columns=7}} |
| What follows is a comparison of stretched- and compressed-octave 7edo tunings.
| | {{harmonics in equal|2|intervals=odd|columns=7}} |
| | | {{harmonics in equal|3|intervals=odd|columns=7}} |
| ; 7edo
| | {{harmonics in equal|4|intervals=odd|columns=7}} |
| * Step size: 171.429{{c}}, octave size: 1200.0{{c}}
| | {{harmonics in equal|5|intervals=odd|columns=7}} |
| Pure-octaves 7edo approximates all harmonics up to 16 within NNN{{c}}.
| | {{harmonics in equal|6|intervals=odd|columns=7}} |
| {{Harmonics in equal|7|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo}} | | {{harmonics in equal|7|intervals=odd|columns=7}} |
| {{Harmonics in equal|7|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo (continued)}} | | {{harmonics in equal|8|intervals=odd|columns=7}} |
| | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| ; [[WE|7et, 2.3.11.13 WE]]
| | {{harmonics in equal|10|intervals=odd|columns=7}} |
| * Step size: 171.993{{c}}, octave size: 1204.0{{c}}
| | {{harmonics in equal|11|intervals=odd|columns=7}} |
| Stretching the octave of 7edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| {{Harmonics in cet|171.993|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE}} | | {{harmonics in equal|13|intervals=odd|columns=7}} |
| {{Harmonics in cet|171.993|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}} | | {{harmonics in equal|14|intervals=odd|columns=7}} |
| | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| ; [[18ed6]]
| | {{harmonics in equal|16|intervals=odd|columns=7}} |
| * Step size: 172.331{{c}}, octave size: 1206.3{{c}}
| | {{harmonics in equal|17|intervals=odd|columns=7}} |
| Stretching the octave of 7edo 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 18ed6 does this.
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| {{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6}} | | {{harmonics in equal|19|intervals=odd|columns=7}} |
| {{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6 (continued)}} | | {{harmonics in equal|20|intervals=odd|columns=7}} |
| | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| ; [[WE|7et, 2.3.5.11.13 WE]]
| | {{harmonics in equal|22|intervals=odd|columns=7}} |
| * Step size: 172.390{{c}}, octave size: 1206.7{{c}}
| | {{harmonics in equal|23|intervals=odd|columns=7}} |
| Stretching the octave of 7edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| {{Harmonics in cet|172.390|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}} | | {{harmonics in equal|25|intervals=odd|columns=7}} |
| {{Harmonics in cet|172.390|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}} | | {{harmonics in equal|26|intervals=odd|columns=7}} |
| | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| ; [[zpi|15zpi]]
| | {{harmonics in equal|28|intervals=odd|columns=7}} |
| * Step size: 172.495{{c}}, octave size: 1207.5{{c}}
| | {{harmonics in equal|29|intervals=odd|columns=7}} |
| Stretching the octave of 7edo 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 15zpi does this.
| | {{harmonics in equal|30|intervals=odd|columns=7}} |
| {{Harmonics in cet|172.495|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi}} | | {{harmonics in equal|31|intervals=odd|columns=7}} |
| {{Harmonics in cet|172.495|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi (continued)}} | | {{harmonics in equal|32|intervals=odd|columns=7}} |
| | | {{harmonics in equal|33|intervals=odd|columns=7}} |
| ; [[11edt]]
| | {{harmonics in equal|34|intervals=odd|columns=7}} |
| * Step size: 172.905{{c}}, octave size: 1210.3{{c}}
| | {{harmonics in equal|35|intervals=odd|columns=7}} |
| Stretching the octave of 7edo 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 11edt does this.
| | {{harmonics in equal|36|intervals=odd|columns=7}} |
| {{Harmonics in equal|11|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt}} | | {{harmonics in equal|37|intervals=odd|columns=7}} |
| {{Harmonics in equal|11|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt (continued)}} | | {{harmonics in equal|38|intervals=odd|columns=7}} |
| | {{harmonics in equal|39|intervals=odd|columns=7}} |
| | {{harmonics in equal|40|intervals=odd|columns=7}} |
| | {{harmonics in equal|41|intervals=odd|columns=7}} |
| | {{harmonics in equal|42|intervals=odd|columns=7}} |
| | {{harmonics in equal|43|intervals=odd|columns=7}} |
| | {{harmonics in equal|44|intervals=odd|columns=7}} |
| | {{harmonics in equal|45|intervals=odd|columns=7}} |
| | {{harmonics in equal|46|intervals=odd|columns=7}} |
| | {{harmonics in equal|47|intervals=odd|columns=7}} |
| | {{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}} |