45edo: Difference between revisions

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== Octave stretch and compression ==

45edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by an [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed6|116ed6]]. The trade-off is a slightly worse 2/1.

45edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by an [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed12|161ed12]] or [[ed6|116ed6]]. The trade-off is a slightly worse 2/1.

The tuning [[equal tuning|183ed17]] may be used ~~for this purpose too~~, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1, but at the cost of a noticeably worse 2/1 than ~~116ed6.~~

The tuning [[equal tuning|183ed17]] may also be used, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1 (with different mappings for many) but at the cost of a noticeably worse 2/1 than the others.

~~What follows is a comparison of compressed- and stretched-octave 45edo tunings.~~

~~; [[zpi|209zpi]]~~

* Step size: 26.550{{c}}, octave size: 1194.8{{c}}

~~Compressing~~ the ~~octave of 45edo by around 5{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 11.1{{c}}. The tuning 209zpi does this.~~

~~{{Harmonics in cet|26.550|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 209zpi}}~~

~~{{Harmonics in cet|26~~.~~550|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 209zpi (continued)}}~~

~~; 45edo~~

* Step size: 26.667{{c}}, octave size: 1200.0{{c}}

~~Pure-octaves 45edo approximates all harmonics up to 16 within 13.0{{c}}.~~

~~{{Harmonics in equal|45|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45edo}}~~

~~{{Harmonics in equal|45|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45edo (continued)}}~~

~~; [[WE|45et, 13-limit WE tuning]]~~

* Step size: 26.695{{c}}, octave size: 1201.3{{c}}

Stretching the octave of 45edo by around 1{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

~~{{Harmonics in cet|26.695|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning}}~~

~~{{Harmonics in cet|26.695|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning (continued)}}~~

~~; [[ed12|161ed12]]~~

* Step size: Octave size: 1202.4{{c}}

Stretching the octave of 45edo by around 2.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 12.2{{c}}. The tuning 161ed12 does this.

~~{{Harmonics in equal|161|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 161ed12}}~~

~~{{Harmonics in equal|161|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 161ed12 (continued)}}~~

~~; [[ed6|116ed6]]~~

* Step size: Octave size: 1203.3{{c}}

Stretching the octave of 45edo by around 3{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 116ed6 does this. So does [[ed7|126ed7]] whose octave is identical within 0.1{{c}}.

~~{{Harmonics in equal|116|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 116ed6}}~~

~~{{Harmonics in equal|116|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 116ed6 (continued)}}~~

~~; [[WE|45et, 7-limit WE tuning]]~~

* Step size: 26.745{{c}}, octave size: 1203.5{{c}}

Stretching the octave of 45edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.6{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.

~~{{Harmonics in cet|26.745|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning}}~~

~~{{Harmonics in cet|26.745|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning (continued)}}~~

~~; [[zpi|207zpi]]~~

* Step size: 26.762{{c}}, octave size: 1204.3{{c}}

Stretching the octave of 45edo by around 4{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 207zpi does this.

~~{{Harmonics in cet|26.762|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 207zpi}}~~

~~{{Harmonics in cet|26.762|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 207zpi (continued)}}~~

~~; [[71edt]]~~

* Step size: 26.788{{c}}, octave size: 1205.5{{c}}

Stretching the octave of 45edo by around 5.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 71edt does this. So do the tunings [[ed5|104ed5]] and [[equal tuning|155ed11]] whose octave is identical within 0.3{{c}}.

~~{{Harmonics in equal|71|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 71edt}}~~

~~{{Harmonics in equal|71|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 71edt (continued)}}~~

~~; [[equal tuning|183ed17]]~~

* Octave size: 1206.1{{c}}

Stretching the octave of 45edo by around 6{{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 13.3{{c}}. The tuning 183ed17 does this.

~~{{Harmonics in equal|183|17|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 183ed17}}~~

~~{{Harmonics in equal|183|17|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 183ed17 (continued)}}~~

== Instruments ==

@@ Line 584: / Line 584: @@
 == Octave stretch and compression ==
-edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by an [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed6|116ed6]]. The trade-off is a slightly worse 2/1.
+edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by an [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed12|161ed12]] or [[ed6|116ed6]]. The trade-off is a slightly worse 2/1.
-The tuning [[equal tuning|183ed17]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1, but at the cost of a noticeably worse 2/1 than 116ed6.
+The tuning [[equal tuning|183ed17]] may also be used, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1 (with different mappings for many) but at the cost of a noticeably worse 2/1 than the others.
-What follows is a comparison of compressed- and stretched-octave 45edo tunings.
-; [[zpi|209zpi]]
-* Step size: 26.550{{c}}, octave size: 1194.8{{c}}
-Compressing the octave of 45edo by around 5{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 11.1{{c}}. The tuning 209zpi does this.
-{{Harmonics in cet|26.550|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 209zpi}}
-{{Harmonics in cet|26.550|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 209zpi (continued)}}
-; 45edo
-* Step size: 26.667{{c}}, octave size: 1200.0{{c}}
-Pure-octaves 45edo approximates all harmonics up to 16 within 13.0{{c}}.
-{{Harmonics in equal|45|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45edo}}
-{{Harmonics in equal|45|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45edo (continued)}}
-; [[WE|45et, 13-limit WE tuning]]
-* Step size: 26.695{{c}}, octave size: 1201.3{{c}}
-Stretching the octave of 45edo by around 1{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
-{{Harmonics in cet|26.695|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning}}
-{{Harmonics in cet|26.695|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning (continued)}}
-; [[ed12|161ed12]]
-* Step size: Octave size: 1202.4{{c}}
-Stretching the octave of 45edo by around 2.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 12.2{{c}}. The tuning 161ed12 does this.
-{{Harmonics in equal|161|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 161ed12}}
-{{Harmonics in equal|161|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 161ed12 (continued)}}
-; [[ed6|116ed6]]
-* Step size: Octave size: 1203.3{{c}}
-Stretching the octave of 45edo by around 3{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 116ed6 does this. So does [[ed7|126ed7]] whose octave is identical within 0.1{{c}}.
-{{Harmonics in equal|116|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 116ed6}}
-{{Harmonics in equal|116|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 116ed6 (continued)}}
-; [[WE|45et, 7-limit WE tuning]]
-* Step size: 26.745{{c}}, octave size: 1203.5{{c}}
-Stretching the octave of 45edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.6{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
-{{Harmonics in cet|26.745|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning}}
-{{Harmonics in cet|26.745|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning (continued)}}
-; [[zpi|207zpi]]
-* Step size: 26.762{{c}}, octave size: 1204.3{{c}}
-Stretching the octave of 45edo by around 4{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 207zpi does this.
-{{Harmonics in cet|26.762|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 207zpi}}
-{{Harmonics in cet|26.762|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 207zpi (continued)}}
-; [[71edt]]
-* Step size: 26.788{{c}}, octave size: 1205.5{{c}}
-Stretching the octave of 45edo by around 5.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 71edt does this. So do the tunings [[ed5|104ed5]] and [[equal tuning|155ed11]] whose octave is identical within 0.3{{c}}.
-{{Harmonics in equal|71|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 71edt}}
-{{Harmonics in equal|71|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 71edt (continued)}}
-; [[equal tuning|183ed17]]
-* Octave size: 1206.1{{c}}
-Stretching the octave of 45edo by around 6{{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 13.3{{c}}. The tuning 183ed17 does this.
-{{Harmonics in equal|183|17|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 183ed17}}
-{{Harmonics in equal|183|17|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 183ed17 (continued)}}
 == Instruments ==