User:BudjarnLambeth/Sandbox2: Difference between revisions

What follows is a comparison of stretched- and compressed-octave 54edo tunings.

Stretching the octave of 54edo 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 139ed6 does this. So does the tuning 262zpi whose octave is identical to 139ed6 within 0.2{{c}}.

; [[ed7|151ed7]]

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

Stretching the octave of 54edo 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 151ed7 does this.

{{Harmonics in equal|151|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed7}}

{{Harmonics in equal|151|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed7 (continued)}}

; [[ed12|193ed12]]

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

Stretching the octave of 54edo 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 193ed12 does this.

{{Harmonics in equal|193|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 193ed12}}

{{Harmonics in equal|193|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 193ed12 (continued)}}

; [[zpi|263zpi]]  

* Step size: 22.243{{c}}, octave size: NNN{{c}}

Stretching the octave of 54edo 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 263zpi does this.

{{Harmonics in cet|22.243|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 263zpi}}

{{Harmonics in cet|22.243|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 263zpi (continued)}}

; 54edo

* Step size: 22.222{{c}}, octave size: NNN{{c}}  

Pure-octaves 54edo approximates all harmonics up to 16 within NNN{{c}}.

{{Harmonics in equal|54|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54edo}}

{{Harmonics in equal|54|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54edo (continued)}}

; [[WE|54et, 13-limit WE tuning]]  

* Step size: 22.198{{c}}, octave size: NNN{{c}}

Compressing the octave of 54edo 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. So does the tuning [[equal tuning|187ed11]] whose octave is identical to 13lim WE within 0.1{{c}}.

 

* Step size: 22.175{{c}}, octave size: NNN{{c}}

Compressing the octave of 54edo 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 264zpi does this. So does the tuning 194ed12 whose octave is identical to 264zpi within 0.01{{c}}.

 

Compressing the octave of 54edo 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 152ed7 does this.

{{Harmonics in equal|152|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 152ed7}}

{{Harmonics in equal|152|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 152ed7 (continued)}}

 

 

{{Harmonics in equal|126|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 126ed5 (continued)}}

* 139ed6 (octave is identical to 262zpi within 0.2{{c}})

* 193ed12

* 263zpi (22.243c)

* 13-limit WE (22.198c)  (octave is identical to 187ed11 within 0.1{{c}})

* 264zpi (22.175c) (octave is identical to 194ed12 within 0.01{{c}})

* 152ed7

* 126ed5 (octave is identical to 86edt within 0.1{{c}})

* 179ed7 (octave is identical to 326zpi within 0.3{{c}})

* 165ed6

* 229ed12 (octave is identical to 221ed11 within 0.1{{c}})

* 327zpi (18.767c)

* 11-limit WE (18.755c)

* 328zpi (18.721c)

59edo (reduce # of edonoi or zpi)

* 294zpi (20.399c)

''pure octaves 59edo octave is identical to 137ed5 within 0.05{{c}}''

* 13-limit WE (20.320c)

* 7-limit WE (20.301c)

* 166ed7

* 296zpi (20.282c)

; Medium priority

{{harmonics in equal | 38 | 2 | 1 | intervals=integer | columns=12}}

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

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

 

 

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

* Best nearby ZPI(s)

* Best nearby ZPI(s)

{{harmonics in equal | 13 | 2 | 1 | intervals=integer | columns=12}}

* Main: "13edo and optimal octave stretching"

* 2.5.11.13 WE (92.483c)

* 2.5.7.13 WE (92.804c)

* 2.3 WE (91.405c) (good for opposite 7 mapping)

 

{{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}}

* 13-limit WE (10.171c)

{{harmonics in equal | 103 | 2 | 1 | intervals=integer | columns=12}}

* 13-limit WE (11.658c)

 

{{harmonics in equal | 111 | 2 | 1 | intervals=integer | columns=12}}