User:BudjarnLambeth/Sandbox2: Difference between revisions

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

; [[ed6|108ed6]]  

* Step size: NNN{{c}}, octave size: 1206.3{{c}}

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]]  

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]]  

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.

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.

; 42edo

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.

{{Harmonics in equal|118|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 118ed7}}

{{Harmonics in equal|118|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 118ed7 (continued)}}

; [[WE|42et, 13-limit WE tuning]]  

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

Compressing the octave of 42edo by around 1.5{{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|28.534|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning}}

{{Harmonics in cet|28.534|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning (continued)}}

; [[ed12|151ed12]]  

* Step size: NNN{{c}}, octave size: 1196.6{{c}}

Compressing the octave of 42edo 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 151ed12 does this. So do the 7-limit [[WE]] and [[TE]] tunings of 42et, whose octaves are within 0.3{{c}} of 151ed12.

{{Harmonics in equal|151|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed12}}

{{Harmonics in equal|151|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed12 (continued)}}

; [[ed6|109ed6]]  

* Step size: NNN{{c}}, octave size: 1195.2{{c}}

Compressing 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 109ed6 does this.

{{Harmonics in equal|109|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 109ed6}}

{{Harmonics in equal|109|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 109ed6 (continued)}}

; [[zpi|191zpi]]  

* Step size: 28.444{{c}}, octave size: 1194.6{{c}}

Compressing the octave of 42edo 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 191zpi does this.

{{Harmonics in cet|28.444|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 191zpi}}

{{Harmonics in cet|28.444|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 191zpi (continued)}}

; [[67edt]]  

* Step size: NNN{{c}}, octave size: 1192.3{{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

* 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}})

* 140ed6

* 126ed5 (octave is identical to 86edt 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}}''

* 180ed7

* 230ed12

* 149ed5

 

59edo (reduce # of edonoi or zpi)

* 153ed6

{{harmonics in equal | 25 | 2 | 1 | intervals=integer | columns=12}}

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

{{harmonics in equal | 15 | 2 | 1 | intervals=integer | columns=12}}

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

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

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

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

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

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

 

 

 

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

 

* 241edt

* 13-limit WE (7.894c)