User:BudjarnLambeth/Sandbox2: Difference between revisions

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What follows is a comparison of stretched- and compressed-octave 39edo tunings.

~~171zpi~~

; [[zpi|171zpi]]

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

* Step size: 30.973{{c}}, octave size: 107.9{{c}}

~~_ing~~ the octave of 39edo 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 171zpi does this. Because it shares error evenly between 39edo's fifths, it is suited for use as a [[dual-fifths]] ~~tuning~~ of 39edo.

Stretching the octave of 39edo by around 8{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 171zpi does this. Because it shares error evenly between 39edo's fifths, it is suited for use as a [[dual-fifths tuning]] of 39edo.

{{Harmonics in cet|30.973|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 171zpi}}

{{Harmonics in cet|30.973|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 171zpi (continued)}}

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{{Harmonics in equal|39|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39edo (continued)}}

~~13-limit WE~~

; [[WE|39et, 13-limit WE tuning]]

* Step size: 30.757{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 30.757{{c}}, octave size: 1199.5{{c}}

~~_ing~~ the octave of 39edo by ~~around NNN{{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.

Compressing the octave of 39edo by about half a cent 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|30.757|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning}}

{{Harmonics in cet|30.757|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning (continued)}}

~~101ed6~~

; [[ed6|101ed6]]

; [[101ed6]]

* Octave size: 1197.8{{c}}

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

Compressing the octave of 101ed6 by around 2{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 101ed6 does this. So does [[zpi|172zpi]] whose octave differs by only 0.4{{c}}.

~~_ing~~ the octave of 101ed6 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 101ed6 does this. So does [[zpi|172zpi]] whose octave differs by only 0.4{{c}}.

{{Harmonics in equal|101|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 101ed6}}

{{Harmonics in equal|101|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 101ed6 (continued)}}

~~2.3.5.11 WE~~

; [[WE|39et, 2.3.5.11 WE tuning]]

* Step size: 30.703{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 30.703{{c}}, octave size: 1197.4{{c}}

~~_ing~~ the octave of 39edo by around ~~NNN~~{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.

Compressing the octave of 39edo by around 2.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.

{{Harmonics in cet|30.703|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning}}

{{Harmonics in cet|30.703|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning (continued)}}

~~173zpi~~

; [[zpi|173zpi]]

* Step size: 30.672{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 30.672{{c}}, octave size: 1196.2{{c}}

~~_ing~~ the octave of 39edo 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 173zpi does this. So does [[62edt]] whose octave differs by only 0.2{{c}}.

Compressing the octave of 39edo by around 4{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 173zpi does this. So does [[62edt]] whose octave differs by only 0.2{{c}}.

{{Harmonics in cet|30.672|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 173zpi}}

{{Harmonics in cet|30.672|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 173zpi (continued)}}

; [[ed7|110ed7]]

~~110ed7~~

* Octave size: 1194.4{{c}}

; [[110ed7]]

Compressing the octave of 39edo by around 5.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 110ed7 does this. So does [[equal tuning|145ed13]] whose octave differs by only 0.1{{c}}.

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

~~_ing~~ the octave of 39edo 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 110ed7 does this. So does [[equal tuning|145ed13]] whose octave differs by only 0.1{{c}}.

{{Harmonics in equal|110|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 110ed7}}

{{Harmonics in equal|110|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 110ed7 (continued)}}

; [[ed5|91ed5]]

~~91ed5~~

* Octave size: 1194.1{{c}}

; [[91ed5]]

Compressing the octave of 39edo by around 6{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 91ed5 does this.

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

~~_ing~~ the octave of 39edo 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 91ed5 does this.

{{Harmonics in equal|91|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 91ed5}}

{{Harmonics in equal|91|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 91ed5 (continued)}}

@@ Line 7: / Line 7: @@
 What follows is a comparison of stretched- and compressed-octave 39edo tunings.
-zpi
 ; [[zpi|171zpi]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
+* Step size: 30.973{{c}}, octave size: 107.9{{c}}
-_ing the octave of 39edo 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 171zpi does this. Because it shares error evenly between 39edo's fifths, it is suited for use as a [[dual-fifths]] tuning of 39edo.
+Stretching the octave of 39edo by around 8{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 171zpi does this. Because it shares error evenly between 39edo's fifths, it is suited for use as a [[dual-fifths tuning]] of 39edo.
 {{Harmonics in cet|30.973|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 171zpi}}
 {{Harmonics in cet|30.973|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 171zpi (continued)}}
@@ Line 20: / Line 19: @@
 {{Harmonics in equal|39|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39edo (continued)}}
--limit WE
 ; [[WE|39et, 13-limit WE tuning]]
-* Step size: 30.757{{c}}, octave size: NNN{{c}}
+* Step size: 30.757{{c}}, octave size: 1199.5{{c}}
-_ing the octave of 39edo by around NNN{{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.
+Compressing the octave of 39edo by about half a cent 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|30.757|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning}}
 {{Harmonics in cet|30.757|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning (continued)}}
-ed6
+; [[ed6|101ed6]]
-; [[101ed6]]
+* Octave size: 1197.8{{c}}
-* Step size: NNN{{c}}, octave size: NNN{{c}}
+Compressing the octave of 101ed6 by around 2{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 101ed6 does this. So does [[zpi|172zpi]] whose octave differs by only 0.4{{c}}.
-_ing the octave of 101ed6 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 101ed6 does this. So does [[zpi|172zpi]] whose octave differs by only 0.4{{c}}.
 {{Harmonics in equal|101|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 101ed6}}
 {{Harmonics in equal|101|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 101ed6 (continued)}}
-.3.5.11 WE
 ; [[WE|39et, 2.3.5.11 WE tuning]]
-* Step size: 30.703{{c}}, octave size: NNN{{c}}
+* Step size: 30.703{{c}}, octave size: 1197.4{{c}}
-_ing the octave of 39edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.
+Compressing the octave of 39edo by around 2.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.
 {{Harmonics in cet|30.703|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning}}
 {{Harmonics in cet|30.703|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning (continued)}}
-zpi
 ; [[zpi|173zpi]]
-* Step size: 30.672{{c}}, octave size: NNN{{c}}
+* Step size: 30.672{{c}}, octave size: 1196.2{{c}}
-_ing the octave of 39edo 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 173zpi does this. So does [[62edt]] whose octave differs by only 0.2{{c}}.
+Compressing the octave of 39edo by around 4{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 173zpi does this. So does [[62edt]] whose octave differs by only 0.2{{c}}.
 {{Harmonics in cet|30.672|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 173zpi}}
 {{Harmonics in cet|30.672|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 173zpi (continued)}}
+; [[ed7|110ed7]]
-ed7
+* Octave size: 1194.4{{c}}
-; [[110ed7]]
+Compressing the octave of 39edo by around 5.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 110ed7 does this. So does [[equal tuning|145ed13]] whose octave differs by only 0.1{{c}}.
-* Step size: NNN{{c}}, octave size: NNN{{c}}
-_ing the octave of 39edo 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 110ed7 does this. So does [[equal tuning|145ed13]] whose octave differs by only 0.1{{c}}.
 {{Harmonics in equal|110|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 110ed7}}
 {{Harmonics in equal|110|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 110ed7 (continued)}}
+; [[ed5|91ed5]]
-ed5
+* Octave size: 1194.1{{c}}
-; [[91ed5]]
+Compressing the octave of 39edo by around 6{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 91ed5 does this.
-* Step size: NNN{{c}}, octave size: NNN{{c}}
-_ing the octave of 39edo 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 91ed5 does this.
 {{Harmonics in equal|91|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 91ed5}}
 {{Harmonics in equal|91|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 91ed5 (continued)}}