|
|
| (8 intermediate revisions by the same user not shown) |
| Line 5: |
Line 5: |
| = Title1 = | | = Title1 = |
| == Octave stretch or compression == | | == Octave stretch or compression == |
| 64edo's approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by [[180ed7]], a [[Octave shrinking|compressed-octave]] version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
| | 38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]]. |
|
| |
|
| [[149ed5]] can also be used: it is similar to 180ed7 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as 180ed7's.
| | What follows is a comparison of stretched-octave 38edo tunings. |
|
| |
|
| If one prefers a ''[[Octave stretch|stretched-octave]]'', 64edo's approximations of 3/1, 5/1, 11/1 and 17/1 are improved by [[221ed11]], a stretched version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
| | ; 38edo |
| | * Step size: 31.579{{c}}, octave size: 1200.00{{c}} |
| | Pure-octaves 38edo approximates all harmonics up to 16 within NNN{{c}}. |
| | {{Harmonics in equal|38|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38edo}} |
| | {{Harmonics in equal|38|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38edo (continued)}} |
|
| |
|
| [[47ed5/3]] can also be used: it is similar to 221ed11 but both the improvements and shortcomings are amplified. Most notably its 2/1 is not as accurate as 221ed11's. | | ; [[WE|38et, 13-limit WE tuning]] |
| | * Step size: 31.599{{c}}, octave size: 1200.77{{c}} |
| | Stretching the octave of 38edo by around 1{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. |
| | {{Harmonics in cet|31.599|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38et, 13-limit WE tuning}} |
| | {{Harmonics in cet|31.599|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38et, 13-limit WE tuning (continued)}} |
|
| |
|
| What follows is a comparison of stretched- and compressed-octave 64edo tunings.
| | ; [[ed5|88ed5]] |
| | * Step size: 31.663{{c}}, octave size: 1203.18{{c}} |
| | Stretching the octave of 38edo 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 88ed5 does this. |
| | {{Harmonics in equal|88|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 88ed5}} |
| | {{Harmonics in equal|88|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 88ed5 (continued)}} |
|
| |
|
| ; [[ed7|179ed7]] | | ; [[zpi|166zpi]] |
| * Octave size: 1204.50{{c}} | | * Step size: 31.671{{c}}, octave size: 1203.48{{c}} |
| Stretching the octave of 64edo by around 4.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 8.99{{c}}. The tuning 179ed7 does this. So does the tuning 326zpi whose octave is identical within 0.3{{c}}. | | Stretching the octave of 38edo by around 3.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 166zpi does this. |
| {{Harmonics in equal|179|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 179ed7}} | | {{Harmonics in cet|31.671|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 166zpi}} |
| {{Harmonics in equal|179|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 179ed7 (continued)}} | | {{Harmonics in cet|31.671|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 166zpi (continued)}} |
|
| |
|
| ; [[ed6|165ed6]] | | ; [[60edt]] |
| * Octave size: 1203.18{{c}}
| | * Step size: 31.699{{c}}, octave size: 1204.57{{c}} |
| Stretching the octave of 64edo by around 3{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 9.25{{c}}. The tuning 165ed6 does this.
| | Stretching the octave of 38edo by around 4.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 60edt does this. |
| {{Harmonics in equal|165|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 165ed6}}
| | {{Harmonics in equal|60|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edt}} |
| {{Harmonics in equal|165|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 165ed6 (continued)}}
| | {{Harmonics in equal|60|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edt (continued)}} |
| | |
| ; [[ed12|229ed12]]
| |
| * Octave size: 1202.29{{c}}
| |
| Stretching the octave of 64edo by around 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.17{{c}}. The tuning 229ed12 does this. So does the tuning [[equal tuning|221ed11]] whose octave is identical within 0.1{{c}}.
| |
| {{Harmonics in equal|229|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 229ed12}}
| |
| {{Harmonics in equal|229|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 229ed12 (continued)}}
| |
| | |
| ; [[zpi|327zpi]]
| |
| * Step size: 18.767{{c}}, octave size: 1201.09{{c}} | |
| Stretching the octave of 64edo by around 1{{c}} results in improved primes 3 and 11, but worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 9.23{{c}}. The tuning 327zpi does this. | |
| {{Harmonics in cet|18.767|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 327zpi}}
| |
| {{Harmonics in cet|18.767|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 327zpi (continued)}}
| |
| | |
| ; [[WE|64et, 11-limit WE tuning]]
| |
| * Step size: 18.755{{c}}, octave size: 1200.32{{c}}
| |
| Stretching the octave of 64edo by around a third of a cent results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 8.50{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
| |
| {{Harmonics in cet|18.755|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 64et, 11-limit WE tuning}}
| |
| {{Harmonics in cet|18.755|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 64et, 11-limit WE tuning (continued)}}
| |
| | |
| ; 64edo
| |
| * Step size: 18.750{{c}}, octave size: 1200.00{{c}}
| |
| Pure-octaves 64edo approximates all harmonics up to 16 within 8.21{{c}}. The octave of 64edo's 13-limit [[WE]] tuning differs by only 0.13{{c}} from pure.
| |
| {{Harmonics in equal|64|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 64edo}}
| |
| {{Harmonics in equal|64|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 64edo (continued)}}
| |
| | |
| ; [[zpi|328zpi]]
| |
| * Step size: 18.721{{c}}, octave size: 1198.14{{c}}
| |
| Compressing the octave of 64edo by just under 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.02{{c}}. The tuning 328zpi does this.
| |
| {{Harmonics in cet|18.721|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 328zpi}}
| |
| {{Harmonics in cet|18.721|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 328zpi (continued)}}
| |
| | |
| ; [[ed7|180ed7]]
| |
| * Octave size: 1197.80{{c}}
| |
| Compressing the octave of 64edo by just over 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.34{{c}}. The tuning 180ed7 does this.
| |
| {{Harmonics in equal|180|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 180ed7}}
| |
| {{Harmonics in equal|180|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 180ed7 (continued)}}
| |
| | |
| ; [[ed12|230ed12]]
| |
| * Octave size: 1197.07{{c}}
| |
| Compressing the octave of 64edo by around 3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.80{{c}}. The tuning 230ed12 does this.
| |
| {{Harmonics in equal|230|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 230ed12}}
| |
| {{Harmonics in equal|230|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 230ed12 (continued)}}
| |
| | |
| ; [[ed5|149ed5]]
| |
| * Step size: Octave size: NNN{{c}}
| |
| Compressing the octave of 64edo 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 149ed5 does this.
| |
| {{Harmonics in equal|149|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 149ed5}} | |
| {{Harmonics in equal|149|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 149ed5 (continued)}} | |
|
| |
|
| = Title2 = | | = Title2 = |
| Line 95: |
Line 59: |
| ; High-priority | | ; High-priority |
|
| |
|
| 64edo
| | 118edo (choose ZPIS) |
| * 179ed7 (octave is identical to 326zpi within 0.3{{c}}) | | {{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}} |
| * 165ed6 | | * 187edt |
| * 229ed12 (octave is identical to 221ed11 within 0.1{{c}}) | | * 69edf |
| * 327zpi (18.767c) | | * 13-limit WE (10.171c) |
| * 11-limit WE (18.755c) | | * Best nearby ZPI(s) |
| ''pure octaves 64edo (octave is identical to 13-limit WE within 0.13{{c}}''
| | |
| * 328zpi (18.721c) | | 103edo (narrow down edonoi, choose ZPIS) |
| * 180ed7 | | {{harmonics in equal | 103 | 2 | 1 | intervals=integer | columns=12}} |
| * 230ed12 | | * 163edt |
| * 149ed5 | | * 239ed5 |
| | * 266ed6 |
| | * 289ed7 |
| | * 356ed11 |
| | * 369ed12 |
| | * 381ed13 |
| | * 421ed17 |
| | * 466ed23 |
| | * 13-limit WE (11.658c) |
| | * Best nearby ZPI(s) |
| | |
| | 111edo (choose ZPIS) |
| | {{harmonics in equal | 111 | 2 | 1 | intervals=integer | columns=12}} |
| | * Nearby edt, ed6, ed12 and/or edf |
| | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| | * 1-2 WE tunings |
| | * Best nearby ZPI(s) |
| | |
| | 13edo |
| | {{harmonics in equal | 13 | 2 | 1 | intervals=integer | columns=12}} |
| | * Main: "13edo and optimal octave stretching" |
| | * 2.5.11.13 WE (92.483c) |
| | * 2.5.7.13 WE (92.804c) |
| | * 2.3 WE (91.405c) (good for opposite 7 mapping) |
| | * 38zpi (92.531c) |
| | |
| | 104edo |
| | * Nearby edt, ed6, ed12 and/or edf |
| | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| | * 1-2 WE tunings |
| | * Best nearby ZPI(s) |
|
| |
|
| ; Medium priority | | ; Medium-priority |
|
| |
|
| 25edo | | 25edo |
| Line 132: |
Line 126: |
| 30edo | | 30edo |
| {{harmonics in equal | 30 | 2 | 1 | intervals=integer | columns=12}} | | {{harmonics in equal | 30 | 2 | 1 | intervals=integer | columns=12}} |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
|
| |
| 34edo
| |
| {{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}}
| |
| * Nearby edt, ed6, ed12 and/or edf | | * Nearby edt, ed6, ed12 and/or edf |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional) | | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| Line 165: |
Line 152: |
| * Best nearby ZPI(s) | | * Best nearby ZPI(s) |
|
| |
|
| 38edo
| | 15edo |
| {{harmonics in equal | 38 | 2 | 1 | intervals=integer | columns=12}} | | {{harmonics in equal | 15 | 2 | 1 | intervals=integer | columns=12}} |
| * Nearby edt, ed6, ed12 and/or edf | | * Nearby edt, ed6, ed12 and/or edf |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional) | | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| Line 179: |
Line 166: |
| * Best nearby ZPI(s) | | * Best nearby ZPI(s) |
|
| |
|
| 10edo
| | 18edo |
| {{harmonics in equal | 10 | 2 | 1 | intervals=integer | columns=12}} | | {{harmonics in equal | 18 | 2 | 1 | intervals=integer | columns=12}} |
| * Nearby edt, ed6, ed12 and/or edf | | * Nearby edt, ed6, ed12 and/or edf |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional) | | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| Line 186: |
Line 173: |
| * Best nearby ZPI(s) | | * Best nearby ZPI(s) |
|
| |
|
| 11edo
| | 24edo |
| {{harmonics in equal | 11 | 2 | 1 | intervals=integer | columns=12}} | | {{harmonics in equal | 24 | 2 | 1 | intervals=integer | columns=12}} |
| * Nearby edt, ed6, ed12 and/or edf | | * Nearby edt, ed6, ed12 and/or edf |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional) | | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| * 1-2 WE tunings | | * 1-2 WE tunings |
| * Best nearby ZPI(s) | | * Best nearby ZPI( |
| | |
| 15edo
| |
| {{harmonics in equal | 15 | 2 | 1 | intervals=integer | columns=12}}
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
| | |
| 18edo
| |
| {{harmonics in equal | 18 | 2 | 1 | intervals=integer | columns=12}}
| |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
|
| |
|
| 48edo | | 48edo |
| {{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}} | | {{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}} |
| * Nearby edt, ed6, ed12 and/or edf
| |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional)
| |
| * 1-2 WE tunings
| |
| * Best nearby ZPI(s)
| |
|
| |
| 24edo
| |
| {{harmonics in equal | 24 | 2 | 1 | intervals=integer | columns=12}}
| |
| * Nearby edt, ed6, ed12 and/or edf | | * Nearby edt, ed6, ed12 and/or edf |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional) | | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| Line 233: |
Line 199: |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional) | | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| * 1-2 WE tunings | | * 1-2 WE tunings |
| * Best nearby ZPI(s) | | * Best nearby ZPI(s)s) |
|
| |
|
| 13edo
| | 10edo |
| {{harmonics in equal | 13 | 2 | 1 | intervals=integer | columns=12}} | | {{harmonics in equal | 10 | 2 | 1 | intervals=integer | columns=12}} |
| * Main: "13edo and optimal octave stretching" | | * Nearby edt, ed6, ed12 and/or edf |
| * 2.5.11.13 WE (92.483c) | | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| * 2.5.7.13 WE (92.804c)
| | * 1-2 WE tunings |
| * 2.3 WE (91.405c) (good for opposite 7 mapping) | |
| * 38zpi (92.531c)
| |
| | |
| 118edo (choose ZPIS)
| |
| {{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}}
| |
| * 187edt
| |
| * 69edf
| |
| * 13-limit WE (10.171c)
| |
| * Best nearby ZPI(s) | | * Best nearby ZPI(s) |
|
| |
|
| 103edo (narrow down edonoi, choose ZPIS)
| | 11edo |
| {{harmonics in equal | 103 | 2 | 1 | intervals=integer | columns=12}} | | {{harmonics in equal | 11 | 2 | 1 | intervals=integer | columns=12}} |
| * 163edt
| |
| * 239ed5
| |
| * 266ed6
| |
| * 289ed7
| |
| * 356ed11
| |
| * 369ed12
| |
| * 381ed13
| |
| * 421ed17
| |
| * 466ed23
| |
| * 13-limit WE (11.658c)
| |
| * Best nearby ZPI(s)
| |
| | |
| 111edo (choose ZPIS)
| |
| {{harmonics in equal | 111 | 2 | 1 | intervals=integer | columns=12}}
| |
| * Nearby edt, ed6, ed12 and/or edf | | * Nearby edt, ed6, ed12 and/or edf |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional) | | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| Line 271: |
Line 215: |
| * Best nearby ZPI(s) | | * Best nearby ZPI(s) |
|
| |
|
| ; Low priority
| | 34edo |
| | | {{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}} |
| 104edo
| |
| * Nearby edt, ed6, ed12 and/or edf | | * Nearby edt, ed6, ed12 and/or edf |
| * Nearby ed5, ed10, ed7 and/or ed11 (optional) | | * Nearby ed5, ed10, ed7 and/or ed11 (optional) |
| * 1-2 WE tunings | | * 1-2 WE tunings |
| * Best nearby ZPI(s) | | * Best nearby ZPI(s) |
| | |
| | ; Low priority |
|
| |
|
| 125edo | | 125edo |