|
|
| (173 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| = Title1 = | | == Approximations of odd harmonics == |
| == Octave stretch or compression == | | {{harmonics in equal|1|intervals=odd|columns=7}} |
| Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and [[gamelan]], with [[37ed5]] and [[57ed12]] being good options.
| | {{harmonics in equal|2|intervals=odd|columns=7}} |
| | | {{harmonics in equal|3|intervals=odd|columns=7}} |
| What follows is a comparison of stretched- and compressed-octave 16edo tunings.
| | {{harmonics in equal|4|intervals=odd|columns=7}} |
| | | {{harmonics in equal|5|intervals=odd|columns=7}} |
| ; 16edo
| | {{harmonics in equal|6|intervals=odd|columns=7}} |
| * Step size: 75.000{{c}}, octave size: 1200.0{{c}}
| | {{harmonics in equal|7|intervals=odd|columns=7}} |
| Pure-octaves 16edo approximates all harmonics up to 16 within 36.7{{c}}.
| | {{harmonics in equal|8|intervals=odd|columns=7}} |
| {{Harmonics in equal|16|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16edo}} | | {{harmonics in equal|9|intervals=odd|columns=7}} |
| {{Harmonics in equal|16|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16edo (continued)}} | | {{harmonics in equal|10|intervals=odd|columns=7}} |
| | | {{harmonics in equal|11|intervals=odd|columns=7}} |
| ; [[WE|16et, 2.5.7.13 WE tuning]]
| | {{harmonics in equal|12|intervals=odd|columns=7}} |
| * Step size: 75.105{{c}}, octave size: 1201.7{{c}}
| | {{harmonics in equal|13|intervals=odd|columns=7}} |
| Stretching the octave of 16edo by around 2{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 31.8{{c}}. Its 2.5.7.13 WE tuning and 2.5.7.13 [[TE]] tuning both do this.
| | {{harmonics in equal|14|intervals=odd|columns=7}} |
| {{Harmonics in cet|75.105|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning}} | | {{harmonics in equal|15|intervals=odd|columns=7}} |
| {{Harmonics in cet|75.105|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning (continued)}} | | {{harmonics in equal|16|intervals=odd|columns=7}} |
| | | {{harmonics in equal|17|intervals=odd|columns=7}} |
| ; [[zpi|15zpi]] / [[equal tuning|59ed13]]
| | {{harmonics in equal|18|intervals=odd|columns=7}} |
| * Step size: 75.262{{c}}, octave size: 1204.2{{c}}
| | {{harmonics in equal|19|intervals=odd|columns=7}} |
| Stretching the octave of 16edo by around 4{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 34.5{{c}}. The tunings 15zpi and 59ed13 do this.
| | {{harmonics in equal|20|intervals=odd|columns=7}} |
| {{Harmonics in cet|75.262|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15zpi}} | | {{harmonics in equal|21|intervals=odd|columns=7}} |
| {{Harmonics in cet|75.262|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15zpi (continued)}} | | {{harmonics in equal|22|intervals=odd|columns=7}} |
| | | {{harmonics in equal|23|intervals=odd|columns=7}} |
| ; [[WE|16et, 13-limit WE tuning]] / [[37ed5]]
| | {{harmonics in equal|24|intervals=odd|columns=7}} |
| * Step size (WE 16et): 75.315{{c}}, octave size (WE 16et): 1205.0{{c}}
| | {{harmonics in equal|25|intervals=odd|columns=7}} |
| Stretching the octave of 16edo by around 5{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 37.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this, so does the tuning 37ed5.
| | {{harmonics in equal|26|intervals=odd|columns=7}} |
| {{Harmonics in cet|75.315|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning}} | | {{harmonics in equal|27|intervals=odd|columns=7}} |
| {{Harmonics in cet|75.315|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning (continued)}} | | {{harmonics in equal|28|intervals=odd|columns=7}} |
| {{Harmonics in equal|37|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 37ed5}} | | {{harmonics in equal|29|intervals=odd|columns=7}} |
| {{Harmonics in equal|37|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 37ed5 (continued)}} | | {{harmonics in equal|30|intervals=odd|columns=7}} |
| | | {{harmonics in equal|31|intervals=odd|columns=7}} |
| ; [[57ed12]] / [[equal tuning|55ed11]]
| | {{harmonics in equal|32|intervals=odd|columns=7}} |
| * Step size (57ed12): 75.473{{c}}, octave size (57ed12): 1207.6{{c}}
| | {{harmonics in equal|33|intervals=odd|columns=7}} |
| Stretching the octave of 16edo by around 7.5{{c}} results in especially improved primes 3, 5 and 11, but far worse primes 2 and 7. This approximates all harmonics up to 16 within NNN{{c}}. The tunings 57ed12 and 55ed11 do this.
| | {{harmonics in equal|34|intervals=odd|columns=7}} |
| {{Harmonics in equal|57|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed12}} | | {{harmonics in equal|35|intervals=odd|columns=7}} |
| {{Harmonics in equal|57|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed12 (continued)}} | | {{harmonics in equal|36|intervals=odd|columns=7}} |
| {{Harmonics in equal|55|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 55ed11}} | | {{harmonics in equal|37|intervals=odd|columns=7}} |
| {{Harmonics in equal|55|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 55ed11 (continued)}} | | {{harmonics in equal|38|intervals=odd|columns=7}} |
| | | {{harmonics in equal|39|intervals=odd|columns=7}} |
| = Title2 = | | {{harmonics in equal|40|intervals=odd|columns=7}} |
| === Placeholder === | | {{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}} |