|
|
| Line 410: |
Line 410: |
| == Octave stretch or compression == | | == Octave stretch or compression == |
| 14edo benefits from [[octave stretch]] as harmonics 3, 7, and 11 are all tuned flat. [[22edt]], [[ed6|36ed6]] and [[42zpi]] are among the possible choices. | | 14edo benefits from [[octave stretch]] as harmonics 3, 7, and 11 are all tuned flat. [[22edt]], [[ed6|36ed6]] and [[42zpi]] are among the possible choices. |
|
| |
| What follows is a comparison of stretched-octave 14edo tunings.
| |
|
| |
| ; 14edo
| |
| * Step size: 85.714{{c}}, octave size: 1200.0{{c}}
| |
| Pure-octaves 14edo approximates all no-5s harmonics up to 16 within 37.0{{c}}.
| |
| {{Harmonics in equal|14|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14edo}}
| |
| {{Harmonics in equal|14|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14edo (continued)}}
| |
|
| |
| ; [[WE|14et, 13-limit WE tuning]]
| |
| * Step size: 85.759{{c}}, octave size: 1200.6{{c}}
| |
| Stretching the octave of 14edo by around half a cent results in improved primes 3, 7 and 11, but a worse prime 13. This approximates all no-5s harmonics up to 16 within 34.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
| |
| {{Harmonics in cet|85.759|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14et, 13-limit WE tuning}}
| |
| {{Harmonics in cet|85.759|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14et, 13-limit WE tuning (continued)}}
| |
|
| |
| ; [[WE|14et, 11-limit WE tuning]]
| |
| * Step size: 85.842{{c}}, octave size: 1201.8{{c}}
| |
| Stretching the octave of 14edo by around 2{{c}} results in yet more improved primes 3, 7 and 11, but worse primes 2 and especially 13. This approximates all no-5s harmonics up to 16 within 30.9{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
| |
| {{Harmonics in cet|85.842|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning}}
| |
| {{Harmonics in cet|85.842|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning (continued)}}
| |
|
| |
| ; [[ed6|36ed6]]
| |
| * Step size: 86.165{{c}}, octave size: 1206.3{{c}}
| |
| Stretching the octave of 14edo by around NNN{{c}} results in improved primes 3, 7 and 11, but a worse prime 2 and completely unusable prime 13. This approximates all no-13s harmonics up to 16 within 35.3{{c}}. The tuning 36ed6 does this.
| |
| {{Harmonics in equal|36|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 36ed6}}
| |
| {{Harmonics in equal|36|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36ed6 (continued)}}
| |
|
| |
| ; [[zpi|42zpi]]
| |
| * Step size: 86.329{{c}}, octave size: 1208.6{{c}}
| |
| Stretching the octave of 14edo by around 8.5{{c}} results in dramatically improved primes 3, 5, 7 and 11, but a much worse prime 2 and unusable 13. This approximates all no-13s harmonics up to 16 within 34.4{{c}}. The tuning 42zpi does this.
| |
| {{Harmonics in cet|86.329|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42zpi}}
| |
| {{Harmonics in cet|86.329|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42zpi (continued)}}
| |
|
| |
| ; [[22edt]]
| |
| * Step size: 86.453{{c}}, octave size: 1210.3{{c}}
| |
| Stretching the octave of 14edo by around 10{{c}} results in dramatically improved primes 3, 5, 7 and 11, but a much worse prime 2 and unusable 13. It could be argued this is the only one of the 14edo tunings discussed here that has a truly use able 5th harmonic, allowing for full 7-limit harmony. The price for that, however, are quite wobbly octaves with 10 cents of error. This approximates all no-13s harmonics up to 16 within 41.3{{c}}. The tuning 22edt does this.
| |
| {{Harmonics in equal|22|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
| |
| {{Harmonics in equal|22|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 22edt (continued)}}
| |
|
| |
|
| == Scales == | | == Scales == |