User:BudjarnLambeth/Sandbox2: Difference between revisions

What follows is a comparison of stretched- and compressed-octave 42edo tunings.

Stretching the octave of 42edo 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 108ed6 does this. So does the tuning [[97ed5]] whose octave differs by only 0.1{{c}}.

; [[zpi|189zpi]]

* Step size: 28.689{{c}}, octave size: 1204.9{{c}}

Stretching the octave of 42edo by around 5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 189zpi does this.

{{Harmonics in cet|28.689|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 189zpi}}

{{Harmonics in cet|28.689|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 189zpi (continued)}}

; [[ed12|150ed12]]  

* Step size: NNN{{c}}, octave size: 1204.5{{c}}

Stretcing the octave of 42edo 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 150ed12 does this.

{{Harmonics in equal|150|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed12}}

{{Harmonics in equal|150|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed12 (continued)}}

; [[equal tuning|145ed11]]  

* Step size: NNN{{c}}, octave size: 1202.5{{c}}

Stretching the octave of 42edo by around 2.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 145ed11 does this.

{{Harmonics in equal|145|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 145ed11}}

{{Harmonics in equal|145|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 145ed11 (continued)}}

; 42edo

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

Pure-octaves 42edo approximates all harmonics up to 16 within NNN{{c}}. The tuning [[zpi|190zpi]] is almost exactly the same as pure-octaves 42edo, its octave differing by less than 0.05{{c}}.

{{Harmonics in equal|42|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42edo}}

{{Harmonics in equal|42|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42edo (continued)}}

; [[ed7|118ed7]]  

* Step size: NNN{{c}}, octave size: 1199.1{{c}}

Compressing the octave of 42edo by around 1{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 118ed7 does this.

 

* Step size: 28.534{{c}}, octave size: 1198.4{{c}}

 

 

 

 

Compressing the octave of 42edo by around 7.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 67edt does this.

{{Harmonics in equal|67|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 67edt}}

{{Harmonics in equal|67|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 67edt (continued)}}

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

* 151ed7

* 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

* 140ed6

* 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)

''pure octaves 64edo (octave is identical to 13-limit WE within 0.13{{c}}''

* 328zpi (18.721c)

* 152ed6

* 294zpi (20.399c)

* 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}}

* 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}}