User:BudjarnLambeth/Sandbox2: Difference between revisions

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= Title1 =

== Octave stretch or compression ==

38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]].

What follows is a comparison of stretched-octave 38edo tunings.

; 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)}}

; [[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)}}

; [[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)}}

; [[zpi|166zpi]]

* Step size: 31.671{{c}}, octave size: 1203.48{{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 cet|31.671|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 166zpi}}

{{Harmonics in cet|31.671|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 166zpi (continued)}}

; [[60edt]]

* Step size: 31.699{{c}}, octave size: 1204.57{{c}}

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|60|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edt}}

{{Harmonics in equal|60|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edt (continued)}}

= Title2 =

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30edo

{{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 ed5, ed10, ed7 and/or ed11 (optional)

Line 126:

Line 152:

* 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 ed5, ed10, ed7 and/or ed11 (optional)

Line 140:

Line 166:

* 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 ed5, ed10, ed7 and/or ed11 (optional)

Line 147:

Line 173:

* 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 ed5, ed10, ed7 and/or ed11 (optional)

* 1-2 WE tunings

* Best nearby ZPI(s)

* Best nearby ZPI(

~~15edo~~

48edo

{{harmonics in equal | 15 | 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)

Line 161:

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* Best nearby ZPI(s)

~~18edo~~

5edo

{{harmonics in equal | 18 | 2 | 1 | intervals=integer | columns=12}}

{{harmonics in equal | 5 | 2 | 1 | intervals=integer | columns=12}}

* Nearby edt, ed6, ed12 and/or edf

* Nearby ed5, ed10, ed7 and/or ed11 (optional)

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* Best nearby ZPI(s)

~~48edo~~

6edo

{{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}}

{{harmonics in equal | 6 | 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)

* Best nearby ZPI(s)s)

~~24edo~~

10edo

{{harmonics in equal | 24 | 2 | 1 | intervals=integer | columns=12}}

{{harmonics in equal | 10 | 2 | 1 | intervals=integer | columns=12}}

* Nearby edt, ed6, ed12 and/or edf

* Nearby ed5, ed10, ed7 and/or ed11 (optional)

Line 182:

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* Best nearby ZPI(s)

~~5edo~~

11edo

{{harmonics in equal | 5 | 2 | 1 | intervals=integer | columns=12}}

{{harmonics in equal | 11 | 2 | 1 | intervals=integer | columns=12}}

* Nearby edt, ed6, ed12 and/or edf

* Nearby ed5, ed10, ed7 and/or ed11 (optional)

Line 189:

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* Best nearby ZPI(s)

~~6edo~~

34edo

{{harmonics in equal | 6 | 2 | 1 | intervals=integer | columns=12}}

{{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}}

* Nearby edt, ed6, ed12 and/or edf

* Nearby ed5, ed10, ed7 and/or ed11 (optional)

@@ Line 5: / Line 5: @@
 = Title1 =
 == Octave stretch or compression ==
+edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]].
+What follows is a comparison of stretched-octave 38edo tunings.
+; 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)}}
+; [[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)}}
+; [[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)}}
+; [[zpi|166zpi]]
+* Step size: 31.671{{c}}, octave size: 1203.48{{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 cet|31.671|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 166zpi}}
+{{Harmonics in cet|31.671|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 166zpi (continued)}}
+; [[60edt]]
+* Step size: 31.699{{c}}, octave size: 1204.57{{c}}
+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|60|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edt}}
+{{Harmonics in equal|60|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edt (continued)}}
 = Title2 =
@@ Line 93: / Line 126: @@
 edo
 {{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)
-edo
-{{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}}
 * Nearby edt, ed6, ed12 and/or edf
 * Nearby ed5, ed10, ed7 and/or ed11 (optional)
@@ Line 126: / Line 152: @@
 * Best nearby ZPI(s)
 edo
-{{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 ed5, ed10, ed7 and/or ed11 (optional)
@@ Line 140: / Line 166: @@
 * Best nearby ZPI(s)
 edo
-{{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 ed5, ed10, ed7 and/or ed11 (optional)
@@ Line 147: / Line 173: @@
 * Best nearby ZPI(s)
 edo
-{{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 ed5, ed10, ed7 and/or ed11 (optional)
 * 1-2 WE tunings
-* Best nearby ZPI(s)
+* Best nearby ZPI(
 edo
-{{harmonics in equal | 15 | 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)
@@ Line 161: / Line 187: @@
 * Best nearby ZPI(s)
 edo
-{{harmonics in equal | 18 | 2 | 1 | intervals=integer | columns=12}}
+{{harmonics in equal | 5 | 2 | 1 | intervals=integer | columns=12}}
 * Nearby edt, ed6, ed12 and/or edf
 * Nearby ed5, ed10, ed7 and/or ed11 (optional)
@@ Line 168: / Line 194: @@
 * Best nearby ZPI(s)
 edo
-{{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}}
+{{harmonics in equal | 6 | 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)
+* Best nearby ZPI(s)s)
 edo
-{{harmonics in equal | 24 | 2 | 1 | intervals=integer | columns=12}}
+{{harmonics in equal | 10 | 2 | 1 | intervals=integer | columns=12}}
 * Nearby edt, ed6, ed12 and/or edf
 * Nearby ed5, ed10, ed7 and/or ed11 (optional)
@@ Line 182: / Line 208: @@
 * Best nearby ZPI(s)
 edo
-{{harmonics in equal | 5 | 2 | 1 | intervals=integer | columns=12}}
+{{harmonics in equal | 11 | 2 | 1 | intervals=integer | columns=12}}
 * Nearby edt, ed6, ed12 and/or edf
 * Nearby ed5, ed10, ed7 and/or ed11 (optional)
@@ Line 189: / Line 215: @@
 * Best nearby ZPI(s)
 edo
-{{harmonics in equal | 6 | 2 | 1 | intervals=integer | columns=12}}
+{{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}}
 * Nearby edt, ed6, ed12 and/or edf
 * Nearby ed5, ed10, ed7 and/or ed11 (optional)