31edo: 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.

 

* Step size: 38.710{{c}}, octave size: 1200.000{{c}}

 

* Step size: 38.725{{c}}, octave size: 1200.481{{c}}

 

; [[ZPI|127zpi]]  

* Step size: 38.737{{c}}, octave size: 1200.837{{c}}

 

; [[WE|31et, 11-limit WE tuning]]  

* Step size: 38.748{{c}}, octave size: 1201.196{{c}}

Stretching the octave of 31edo by around 1.2{{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|38.748261|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31et, 11-limit WE tuning (continued)}}

 

Stretching the octave of 31edo by around 2.0{{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.

 

; [[49edt]]  

* Step size: 38.815{{c}}, octave size: 1203.278{{c}}

Stretching the octave of 31edo by around 3.3{{c}} results in improved primes 3 and 11, especially 11, but slightly worse primes 2, 5, 7 and 13. The 13 is now differently mapped than – and much better than – 80ed6's (but not as good as the pure octaves 13). This approximates all harmonics up to 16 within 15.6{{c}}. The tuning 49edt does this.