19edo: Difference between revisions

Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-[[just]]. For example, if we are using [[49ed6]] or [[30ed3]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57{{c}}, respectively), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-limit [[tonality diamond]]. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well.

* Step size: 63.158{{c}}, octave size: 1200.0{{c}}  

{{Harmonics in equal|19|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo}}

{{Harmonics in equal|19|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo (continued)}}

; [[WE|19et, 2.3.5.11 WE tuning]]  

* Step size: 63.192{{c}}, octave size: 1200.6{{c}}

Stretching the octave of 19edo by around half a cent results in [[JND|just noticeably]] improved primes 3, 5, 7 and 13, but a just noticeably worse prime 11. This approximates all harmonics up to 16 within 19.6{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.

{{Harmonics in cet|63.192|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 2.3.5.11 WE tuning}}

{{Harmonics in cet|63.192|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 2.3.5.11 WE tuning (continued)}}

; [[WE|19et, 13-limit WE tuning]]  

* Step size: 63.291{{c}}, octave size: 1202.5{{c}}

Stretching the octave of 19edo by around 2.5{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 within 25.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|63.291|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning}}

{{Harmonics in cet|63.291|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning (continued)}}

; [[49ed6]]  

* Step size: 63.305{{c}}, octave size: 1202.8{{c}}

Stretching the octave of 19edo by just under 3{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 within 26.8{{c}}. The tuning 49ed6 does this.

{{Harmonics in equal|49|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6}}

{{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6 (continued)}}

; [[zpi|65zpi]]  

* Step size: 63.331{{c}}, octave size: 1203.3{{c}}

Stretching the octave of 19edo by around 3.5{{c}} results in greatly improved primes 3, 5, 7 and 13, but a greatly worse prime 11. This approximates all harmonics up to 16 within 28.5{{c}}. The tuning 65zpi does this.

{{Harmonics in cet|63.331|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi}}

{{Harmonics in cet|63.331|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi (continued)}}

* Step size: 63.399{{c}}, octave size: 1204.6{{c}}

Stretching the octave of 19edo by around 4.5{{c}} has similar results to 65zpi, but it "overshoot", meaning the improvements are less and the drawbacks are greater compared to 65zpi. The damage to the octave has also started to become [[JND|noticeable]] when it is stretched this far. This approximates all harmonics up to 16 within 30.4{{c}}. The tuning 30edt does this.

{{Harmonics in equal|30|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt}}

{{Harmonics in equal|30|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt (continued)}}  

One can stretch the octave even further - 12.5 cents - to get the tuning [[11edf]], but its approximations of most harmonics are worse than pure-octaves 19. So it is hard to see a use case for 11edf.