User:BudjarnLambeth/Sandbox2: Difference between revisions

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What follows is a comparison of stretched- and compressed-octave 41edo tunings.

; [[184zpi]] / [[WE|41et, 11-limit WE tuning]]

; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]

* Step size: 29.277{{c}}, octave size: NNN{{c}}

Stretching 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 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, ~~which~~ is identical to WE within ~~1/1000 of a cent~~.

Stretching 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 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, whose octave is identical to WE within 0.02{{c}}.

{{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning}}

{{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning (continued)}}

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; 41edo

* Step size: 29.268{{c}}, octave size: 1200.0{{c}}

Pure-octaves 41edo approximates all harmonics up to 16 within NNN{{c}}.

Pure-octaves 41edo approximates all harmonics up to 16 within NNN{{c}}. The octaves of its compressed tuning [[147ed12]] differ by only 0.1{{c}} from pure. The octaves of its 13-limit [[WE]] and [[TE]] tuning differ by less than 0.1{{c}} from pure.

{{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}

{{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}

; [[~~WE|41et, 13-limit WE tuning~~]]

; [[147ed12]] / [[106ed6]] / [[65edt]]

* ~~Step~~ size: 29.~~267~~{{c}}, octave size: ~~NNN{{c}}~~

* 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}}

~~Compressing the octave of 41edo 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.~~

* 106ed6 — step size: 29.264{{c}}, octave size: 1199.69{{c}}

~~{{Harmonics in cet|29.267|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 13-limit WE tuning}}~~

* 65edt — step size: 29.261{{c}}, octave size: 1199.81{{c}}

~~{{Harmonics in cet|29.267|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 13-limit WE tuning (continued)~~}}

Compressing the octave of 41edo by around 0.2{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tunings 147ed12, 106ed6 and 65edt do this.

~~; [[147ed12]]~~

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

~~Compressing the octave of 41edo 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 147ed12 does this.~~

~~{{Harmonics in equal|147|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 147ed12~~}}

~~{{Harmonics in equal|147|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 147ed12 (continued)}}~~

~~; [[106ed6]]~~

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

Compressing the octave of 41edo 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~~ 106ed6 ~~does~~ this.

{{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}}

{{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}}

~~; [[65edt]]~~

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

~~Compressing the octave of 41edo 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 65edt does this.~~

~~{{Harmonics in equal|65|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65edt}}~~

~~{{Harmonics in equal|65|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65edt (continued)}}~~

@@ Line 9: / Line 9: @@
 What follows is a comparison of stretched- and compressed-octave 41edo tunings.
-; [[184zpi]] / [[WE|41et, 11-limit WE tuning]]
+; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]
 * Step size: 29.277{{c}}, octave size: NNN{{c}}
-Stretching 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 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, which is identical to WE within 1/1000 of a cent.
+Stretching 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 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, whose octave is identical to WE within 0.02{{c}}.
 {{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning}}
 {{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning (continued)}}
@@ Line 17: / Line 17: @@
 ; 41edo
 * Step size: 29.268{{c}}, octave size: 1200.0{{c}}
-Pure-octaves 41edo approximates all harmonics up to 16 within NNN{{c}}.
+Pure-octaves 41edo approximates all harmonics up to 16 within NNN{{c}}. The octaves of its compressed tuning [[147ed12]] differ by only 0.1{{c}} from pure. The octaves of its 13-limit [[WE]] and [[TE]] tuning differ by less than 0.1{{c}} from pure.
 {{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}
 {{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}
-; [[WE|41et, 13-limit WE tuning]]
+; [[147ed12]] / [[106ed6]] / [[65edt]]
-* Step size: 29.267{{c}}, octave size: NNN{{c}}
+* 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}}
-Compressing the octave of 41edo 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.
+* 106ed6 — step size: 29.264{{c}}, octave size: 1199.69{{c}}
-{{Harmonics in cet|29.267|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 13-limit WE tuning}}
+* 65edt — step size: 29.261{{c}}, octave size: 1199.81{{c}}
-{{Harmonics in cet|29.267|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 13-limit WE tuning (continued)}}
+Compressing the octave of 41edo by around 0.2{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tunings 147ed12, 106ed6 and 65edt do this.
-; [[147ed12]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
-Compressing the octave of 41edo 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 147ed12 does this.
-{{Harmonics in equal|147|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 147ed12}}
-{{Harmonics in equal|147|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 147ed12 (continued)}}
-; [[106ed6]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
-Compressing the octave of 41edo 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 106ed6 does this.
 {{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}}
 {{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}}
-; [[65edt]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
-Compressing the octave of 41edo 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 65edt does this.
-{{Harmonics in equal|65|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65edt}}
-{{Harmonics in equal|65|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65edt (continued)}}