72edo: Difference between revisions

Line 1,472:

== Octave stretch or compression ==

72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.

72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]], [[zpi|380zpi]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 380zpi and 186ed6 are milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.

~~What follows is a comparison of stretched-octave 72edo tunings.~~

~~; 72edo~~

* Step size: 16.667{{c}}, octave size: 1200.00{{c}}

~~Pure-octaves 72edo approximates all harmonics up to 16 within 7.19{{c}}.~~

~~{{Harmonics in equal|72|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 72edo}}~~

~~{{Harmonics in equal|72|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 72edo (continued)}}~~

~~; [[ed11|249ed11]]~~

* Step size: 16.672{{c}}, octave size: 1200.38{{c}}

Stretching the octave of 72edo by around 0.4{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within 5.79{{c}}. The tuning 249ed11 does this.

~~{{Harmonics in equal|249|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 249ed11}}~~

~~{{Harmonics in equal|249|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 249ed11 (continued)}}~~

~~; [[258ed12]]~~

* Step size: 16.674{{c}}, octave size: 1200.55{{c}}

Stretching the octave of 72edo by around 0.5{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within 5.18{{c}}. The tuning 258ed12 does this.

~~{{Harmonics in equal|258|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 258ed12}}~~

~~{{Harmonics in equal|258|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 258ed12 (continued)}}~~

~~; [[186ed6]] / [[WE|72et, 11-limit WE tuning]] / [[ed7|202ed7]]~~

* Step size: 16.677{{c}}, octave size: 1200.76{{c}}

Stretching the octave of 72edo by around 0.75{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within 4.40{{c}}. The tuning 186ed6 does this. 72et's 11-limit WE tuning and 11-limit [[TE]] tuning both do this, their octave differing from 186ed6's by only 0.02{{c}}. The tuning 202ed7 does this also, it's octave differing from 186ed6 by less than a hundredth of a cent.

~~{{Harmonics in equal|186|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 186ed6}}~~

~~{{Harmonics in equal|186|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 186ed6 (continued)}}~~

~~; [[zpi|380zpi]]~~

* Step size: 16.678{{c}}, octave size: 1200.82{{c}}

Stretching the octave of 72edo by around 0.8{{c}} results in a [[JND|just-noticeably]] better primes 13 and unnoticeable better primes 3, 5, and 7, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within 4.18{{c}}. The tuning 380zpi does this.

~~{{Harmonics in cet|16.678|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 380zpi}}~~

~~{{Harmonics in cet|16.678|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 380zpi (continued)}}~~

~~; [[WE|72et, 13-limit WE tuning]]~~

* Step size: 16.680{{c}}, octave size: 1200.96{{c}}

Stretching the octave of 72edo by around 1{{c}} results in a [[JND|just-noticeably]] better primes 13 and unnoticeable better primes 3, 5, and 7, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within 3.84{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

~~{{Harmonics in cet|16.680|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 72et, 13-limit WE tuning}}~~

~~{{Harmonics in cet|16.680|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 72et, 13-limit WE tuning (continued)}}~~

~~; [[114edt]] / [[ed5|167ed5]]~~

* Step size: 16.684{{c}}, octave size: 1201.23{{c}}

Stretching the octave of 72edo by around 1.25{{c}} results in a [[JND|just-noticeably]] better primes 13 and unnoticeably better primes 3, 5, and 7, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within 4.94{{c}}. The tuning 144edt does this. The tuning 167ed5 does this also, its octave differing from 114edt by only 0.05{{c}}.

~~{{Harmonics in equal|114|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 114edt}}~~

~~{{Harmonics in equal|114|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 114edt (continued)}}~~

~~; [[42edf]]~~

* Step size: 16.713{{c}}, octave size: 1203.35{{c}}

Stretching the octave of 72edo by around 3{{c}} results in better primes 13, 19 and 23, but worse primes 2, 3, 5, 7, 11 and 17. This approximates all harmonics up to 16 within 8.14{{c}}. The tuning 42edf does this.

~~{{Harmonics in equal|42|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42edf}}~~

~~{{Harmonics in equal|42|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42edf (continued)}}~~

== Scales ==

@@ Line 1,472: / Line 1,472: @@
 == Octave stretch or compression ==
-edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.
+edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]], [[zpi|380zpi]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 380zpi and 186ed6 are milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.
-What follows is a comparison of stretched-octave 72edo tunings.
-; 72edo
-* Step size: 16.667{{c}}, octave size: 1200.00{{c}}
-Pure-octaves 72edo approximates all harmonics up to 16 within 7.19{{c}}.
-{{Harmonics in equal|72|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 72edo}}
-{{Harmonics in equal|72|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 72edo (continued)}}
-; [[ed11|249ed11]]
-* Step size: 16.672{{c}}, octave size: 1200.38{{c}}
-Stretching the octave of 72edo by around 0.4{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within 5.79{{c}}. The tuning 249ed11 does this.
-{{Harmonics in equal|249|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 249ed11}}
-{{Harmonics in equal|249|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 249ed11 (continued)}}
-; [[258ed12]]
-* Step size: 16.674{{c}}, octave size: 1200.55{{c}}
-Stretching the octave of 72edo by around 0.5{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within 5.18{{c}}. The tuning 258ed12 does this.
-{{Harmonics in equal|258|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 258ed12}}
-{{Harmonics in equal|258|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 258ed12 (continued)}}
-; [[186ed6]] / [[WE|72et, 11-limit WE tuning]] / [[ed7|202ed7]]
-* Step size: 16.677{{c}}, octave size: 1200.76{{c}}
-Stretching the octave of 72edo by around 0.75{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within 4.40{{c}}. The tuning 186ed6 does this. 72et's 11-limit WE tuning and 11-limit [[TE]] tuning both do this, their octave differing from 186ed6's by only 0.02{{c}}. The tuning 202ed7 does this also, it's octave differing from 186ed6 by less than a hundredth of a cent.
-{{Harmonics in equal|186|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 186ed6}}
-{{Harmonics in equal|186|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 186ed6 (continued)}}
-; [[zpi|380zpi]]
-* Step size: 16.678{{c}}, octave size: 1200.82{{c}}
-Stretching the octave of 72edo by around 0.8{{c}} results in a [[JND|just-noticeably]] better primes 13 and unnoticeable better primes 3, 5, and 7, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within 4.18{{c}}. The tuning 380zpi does this.
-{{Harmonics in cet|16.678|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 380zpi}}
-{{Harmonics in cet|16.678|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 380zpi (continued)}}
-; [[WE|72et, 13-limit WE tuning]]
-* Step size: 16.680{{c}}, octave size: 1200.96{{c}}
-Stretching the octave of 72edo by around 1{{c}} results in a [[JND|just-noticeably]] better primes 13 and unnoticeable better primes 3, 5, and 7, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within 3.84{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
-{{Harmonics in cet|16.680|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 72et, 13-limit WE tuning}}
-{{Harmonics in cet|16.680|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 72et, 13-limit WE tuning (continued)}}
-; [[114edt]] / [[ed5|167ed5]]
-* Step size: 16.684{{c}}, octave size: 1201.23{{c}}
-Stretching the octave of 72edo by around 1.25{{c}} results in a [[JND|just-noticeably]] better primes 13 and unnoticeably better primes 3, 5, and 7, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within 4.94{{c}}. The tuning 144edt does this. The tuning 167ed5 does this also, its octave differing from 114edt by only 0.05{{c}}.
-{{Harmonics in equal|114|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 114edt}}
-{{Harmonics in equal|114|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 114edt (continued)}}
-; [[42edf]]
-* Step size: 16.713{{c}}, octave size: 1203.35{{c}}
-Stretching the octave of 72edo by around 3{{c}} results in better primes 13, 19 and 23, but worse primes 2, 3, 5, 7, 11 and 17. This approximates all harmonics up to 16 within 8.14{{c}}. The tuning 42edf does this.
-{{Harmonics in equal|42|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42edf}}
-{{Harmonics in equal|42|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42edf (continued)}}
 == Scales ==