User:BudjarnLambeth/Sandbox2: Difference between revisions

58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[92edt]] or [[150ed6]].

What follows is a comparison of stretched- and compressed-octave 58edo tunings.

; [[zpi|288zpi]]

* Step size: 20.736{{c}}, octave size: 1202.69{{c}}

Stretching the octave of 58edo 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 288zpi does this.

{{Harmonics in cet|20.736|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 288zpi}}

{{Harmonics in cet|20.736|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 288zpi (continued)}}

; 58edo

* Step size: 20.690{{c}}, octave size: 1200.00{{c}}  

Pure-octaves 58edo approximates all harmonics up to 16 within NNN{{c}}.

{{Harmonics in equal|58|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edo}}

{{Harmonics in equal|58|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edo (continued)}}

; [[150ed6]]  

* Step size: 20.680{{c}}, octave size: 1199.42{{c}}

Compressing the octave of 58edo by around half a cent results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 150ed6 does this.

{{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed6}}

{{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed6 (continued)}}

; [[92edt]]  

* Step size: 20.673{{c}}, octave size: 1199.06{{c}}

Compressing the octave of 58edo 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 92edt does this.

{{Harmonics in equal|92|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 92edt}}

{{Harmonics in equal|92|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 92edt (continued)}}

; [[zpi|289zpi]] / [[WE|58et, 7-limit WE tuning]]  

* Step size: 20.666{{c}}, octave size: 1198.63{{c}}

Compressing the octave of 58edo by just under 1.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. The tuning 289zpi also does this, its octave differing from 7-limit WE by only 0.06{{c}}.  

{{Harmonics in cet|20.666|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 289zpi}}

{{Harmonics in cet|20.666|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 289zpi (continued)}}

; [[WE|58et, 13-limit WE tuning]]  

* Step size: 20.663{{c}}, octave size: 1198.45{{c}}

Compressing the octave of 58edo by just over 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|20.663|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning}}

{{Harmonics in cet|20.663|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning (continued)}}