User:BudjarnLambeth/Sandbox2: Difference between revisions

Line 9:

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.

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

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

; [[zpi|~~ZPINAME~~]]

; EDONAME

* Step size: ~~NNN~~{{c}}, octave size: NNN{{c}}

* Step size: NNN{{c}}, octave size: NNN{{c}}

Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.

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

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

; [[WE|72et, 11-limit WE tuning]]

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

_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.

{{Harmonics in cet|16.677|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}

{{Harmonics in cet|16.677|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}

; [[zpi|380zpi]]

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

_ing the octave of EDONAME 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 ZPINAME does this.

{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}

{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}

~~; [[EDONOI]]~~

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

* Step size: NNN{{c}}, octave size: NNN{{c}}

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

~~_ing the octave of EDONAME 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 EDONOI does this.~~

~~{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}~~

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

; [[WE|~~ETNAME~~, ~~SUBGROUP~~ WE tuning]]

* Step size: ~~NNN~~{{c}}, octave size: NNN{{c}}

_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.

{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}

{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}

~~; EDONAME~~

; [[Ned12]]

* Step size: NNN{{c}}, octave size: NNN{{c}}

~~Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.~~

~~{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}}~~

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

; [[~~WE|ETNAME, SUBGROUP WE tuning~~]]

* Step size: NNN{{c}}, octave size: NNN{{c}}

_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. ~~Its SUBGROUP WE~~ tuning ~~and SUBGROUP [[TE]] tuning both do~~ this.

_ing the octave of EDONAME 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 EDONOI does this.

{{Harmonics in ~~cet~~|~~100~~|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ~~ETNAME, SUBGROUP WE tuning~~}}

{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}

{{Harmonics in ~~cet~~|~~100~~|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ~~ETNAME, SUBGROUP WE tuning~~ (continued)}}

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

; [[~~EDONOI~~]]

; [[186ed6]]

* Step size: NNN{{c}}, octave size: NNN{{c}}

_ing the octave of EDONAME 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 EDONOI does this.

{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}

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

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

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

; [[~~zpi|ZPINAME~~]]

; [[144edt]]

* Step size: NNN{{c}}, octave size: NNN{{c}}

_ing the octave of EDONAME 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 ~~ZPINAME~~ does this.

_ing the octave of EDONAME 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 EDONOI does this.

{{Harmonics in ~~cet~~|~~100~~|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ~~ZPINAME~~}}

{{Harmonics in equal|144|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}

{{Harmonics in ~~cet~~|~~100~~|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ~~ZPINAME~~ (continued)}}

{{Harmonics in equal|144|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}

@@ Line 9: / Line 9: @@
 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.
-What follows is a comparison of stretched- and compressed-octave 72edo tunings.
+What follows is a comparison of stretched-octave 72edo tunings.
-; [[zpi|ZPINAME]]
+; EDONAME
 * Step size: NNN{{c}}, octave size: NNN{{c}}
+Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.
+{{Harmonics in equal|72|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}}
+{{Harmonics in equal|72|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}}
+; [[WE|72et, 11-limit WE tuning]]
+* Step size: 16.677{{c}}, octave size: NNN{{c}}
+_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
+{{Harmonics in cet|16.677|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
+{{Harmonics in cet|16.677|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}
+; [[zpi|380zpi]]
+* Step size: 16.678{{c}}, octave size: NNN{{c}}
 _ing the octave of EDONAME 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 ZPINAME does this.
 {{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
 {{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
-; [[EDONOI]]
+; [[WE|72et, 13-limit WE tuning]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
+* Step size: 16.680{{c}}, octave size: NNN{{c}}
-_ing the octave of EDONAME 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 EDONOI does this.
-{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
-{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
-; [[WE|ETNAME, SUBGROUP WE tuning]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
 _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
 {{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
 {{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}
-; EDONAME
+; [[Ned12]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
-Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.
-{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}}
-{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}}
-; [[WE|ETNAME, SUBGROUP WE tuning]]
 * Step size: NNN{{c}}, octave size: NNN{{c}}
-_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
+_ing the octave of EDONAME 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 EDONOI does this.
-{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
+{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
-{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}
+{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
-; [[EDONOI]]
+; [[186ed6]]
 * Step size: NNN{{c}}, octave size: NNN{{c}}
 _ing the octave of EDONAME 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 EDONOI does this.
-{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
+{{Harmonics in equal|186|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
-{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
+{{Harmonics in equal|186|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
-; [[zpi|ZPINAME]]
+; [[144edt]]
 * Step size: NNN{{c}}, octave size: NNN{{c}}
-_ing the octave of EDONAME 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 ZPINAME does this.
+_ing the octave of EDONAME 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 EDONOI does this.
-{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
+{{Harmonics in equal|144|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
-{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
+{{Harmonics in equal|144|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}