User:BudjarnLambeth/Sandbox2: Difference between revisions

72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.

What follows is a comparison of stretched-octave 72edo tunings.

; 72edo

* Step size: 16.667{{c}}, octave size: 1200.00{{c}}  

Pure-octaves 72edo approximates all harmonics up to 16 within NNN{{c}}.

{{Harmonics in equal|72|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 72edo}}

{{Harmonics in equal|72|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 72edo (continued)}}

; [[249ed11]]

* Step size: NNN{{c}}, octave size: 1200.38{{c}}

Stretching the octave of 72edo by around 0.4{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 249ed11 does this.

{{Harmonics in equal|249|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 249ed11}}

{{Harmonics in equal|249|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 249ed11 (continued)}}

; [[258ed12]]  

* Step size: NNN{{c}}, octave size: 1200.55{{c}}

Stretching the octave of 72edo by around 0.5{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 258ed12 does this.

{{Harmonics in equal|258|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 258ed12}}

{{Harmonics in equal|258|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 258ed12 (continued)}}

; [[186ed6]] / [[WE|72et, 11-limit WE tuning]] / [[ed7|202ed7]]

* Step size: NNN{{c}}, octave size: 1200.76{{c}}

Stretching the octave of 72edo by around 0.75{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 186ed6 does this. 72et's 11-limit WE tuning and 11-limit [[TE]] tuning both do this, their octave differing from 186ed6's by only 0.02{{c}}. The tuning 202ed7 does this also, it's octave differing from 186ed6 by less than a hundredth of a cent.

{{Harmonics in equal|186|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 186ed6}}

{{Harmonics in equal|186|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 186ed6 (continued)}}

; [[zpi|380zpi]]  

* Step size: 16.678{{c}}, octave size: 1200.82{{c}}

Stretching the octave of 72edo by around 0.8{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7 and 13, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 380zpi does this.

{{Harmonics in cet|16.678|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 380zpi}}

{{Harmonics in cet|16.678|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 380zpi (continued)}}

; [[WE|72et, 13-limit WE tuning]]  

* Step size: 16.680{{c}}, octave size: 1200.96{{c}}

Stretching the octave of 72edo by around NNN{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7 and 13, but unnoticeably worse primes 2 and 11. 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|16.680|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 72et, 13-limit WE tuning}}

{{Harmonics in cet|16.680|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 72et, 13-limit WE tuning (continued)}}

; [[114edt]] / [[167ed5]]

* Step size: NNN{{c}}, octave size: 1201.23{{c}}

Stretching the octave of 72edo by around NNN{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7 and 13, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 144edt does this. The tuning 167ed5 does this also, its octave differing from 114edt by only 0.05{{c}}.

{{Harmonics in equal|114|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 114edt}}

{{Harmonics in equal|114|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 114edt (continued)}}