User:BudjarnLambeth/Sandbox2: Difference between revisions

31edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an [[11-limit]] equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13.

What follows is a comparison of stretched-octave 31edo tunings.

; 31edo

* Step size: 38.710{{c}}, octave size: 1200.0{{c}}  

Pure-octaves 31edo approximates all harmonics up to 16 within NNN{{c}}.

{{Harmonics in equal|31|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo}}

{{Harmonics in equal|31|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo (continued)}}

; [[WE|31et, 13-limit WE tuning]]  

* Step size: 38.725{{c}}, octave size: 1200.5{{c}}

Stretching the octave of 31edo by around 0.5{{c}} results in slightly improved primes 3, 7 and 11, but slightly worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 12.8{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|38.725|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 13-limit WE tuning}}

{{Harmonics in cet|38.725|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)}}

; [[zpi|127zpi]]  

* Step size: 38.737{{c}}, octave size: 1200.8{{c}}

Stretching the octave of 31edo by slightly less than 1{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 127zpi does this.

{{Harmonics in cet|38.737|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi}}

{{Harmonics in cet|38.737|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi (continued)}}

; [[WE|31et, 11-limit WE tuning]]  

* Step size: 38.748{{c}}, octave size: 1201.2{{c}}

Stretching the octave of 31edo by slightly more than 1{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5 and 7, and much worse 13. This approximates all harmonics up to 16 within 15.5{{c}} Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this, so does the tuning [[111ed12]].

{{Harmonics in cet|38.748|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning}}

{{Harmonics in cet|38.748|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning (continued)}}

; [[80ed6]]  

* Step size: 38.774{{c}}, octave size: 1202.0{{c}}

Stretching the octave of 31edo by about 2{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5 and 7, and much worse 13. This is approaching 2.239{{c}} - the most octave stretch 31edo can tolerate before the mapping of the 13th harmonic changes. This approximates all harmonics up to 16 within 18.5{{c}}. The tuning 80ed6 does this.

{{Harmonics in equal|80|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6}}

{{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (continued)}

; [[49edt]]  

* Step size: 38.815{{c}}, octave size: 1203.3{{c}}

Stretching the octave of 31edo by about 3.5{{c}} results in improved primes 3 and 11, especially 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 15.6{{c}}. The tuning 49edt does this.

{{Harmonics in equal|49|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49edt}}

{{Harmonics in equal|49|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49edt (continued)}}