User:BudjarnLambeth/Sandbox2: Difference between revisions

Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and [[gamelan]], with [[37ed5]] and [[57ed12]] being good options.

What follows is a comparison of stretched- and compressed-octave 16edo tunings.

; 16edo

* Step size: 75.000{{c}}, octave size: 1200.0{{c}}  

Pure-octaves 16edo approximates all harmonics up to 16 within 36.7{{c}}.

{{Harmonics in equal|16|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16edo}}

{{Harmonics in equal|16|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16edo (continued)}}

; [[WE|16et, 2.5.7.13 WE tuning]]  

* Step size: 75.105{{c}}, octave size: 1201.7{{c}}

Stretching the octave of 16edo by around 2{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 31.8{{c}}. Its 2.5.7.13 WE tuning and 2.5.7.13 [[TE]] tuning both do this.

{{Harmonics in cet|75.105|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning}}

{{Harmonics in cet|75.105|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning (continued)}}

; [[zpi|15zpi]] / [[equal tuning|59ed13]]

* Step size: 75.262{{c}}, octave size: 1204.2{{c}}

Stretching the octave of 16edo by around 4{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 34.5{{c}}. The tunings 15zpi and 59ed13 do this.

{{Harmonics in cet|75.262|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15zpi}}

{{Harmonics in cet|75.262|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15zpi (continued)}}

; [[WE|16et, 13-limit WE tuning]] / [[37ed5]]

* Step size (WE 16et): 75.315{{c}}, octave size (WE 16et): 1205.0{{c}}

Stretching the octave of 16edo by around 5{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 37.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this, so does the tuning 37ed5.

{{Harmonics in cet|75.315|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning}}

{{Harmonics in cet|75.315|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning (continued)}}

{{Harmonics in equal|37|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 37ed5 (continued)}}

; [[57ed12]] / [[equal tuning|55ed11]]

* Step size (57ed12): 75.473{{c}}, octave size (57ed12): 1207.6{{c}}

Stretching the octave of 16edo by around 7.5{{c}} results in especially improved primes 3, 5 and 11, but far worse primes 2 and 7. This approximates all harmonics up to 16 within NNN{{c}}. The tunings 57ed12 and 55ed11 do this.

{{Harmonics in equal|57|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed12}}

{{Harmonics in equal|57|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed12 (continued)}}

{{Harmonics in equal|55|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 55ed11}}

{{Harmonics in equal|55|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 55ed11 (continued)}}