36edo: Difference between revisions
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<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Octave stretch or compression == | |||
What follows is a comparison of stretched- and compressed-octave 36edo tunings. | |||
; [[21edf]] | |||
* Step size: 33.426{{c}} | |||
* Octave size: 1203.3{{c}} | |||
{{Harmonics in equal|21|3|2|columns=12|collapsed=true}} | |||
{{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}} | |||
Stretching the octave of 36edo by a little over 3{{c}} results in improved primes 5, 11, and 13, but worse primes 2, 3, and 7. This approximates all primes up to 11 within ''12.0{{c}}''. The tuning 21edf does this. | |||
; [[57edt]] | |||
* Step size: 33.368{{c}} | |||
* Octave size: 1201.2{{c}} | |||
{{Harmonics in equal|57|3|1|columns=12|collapsed=true}} | |||
{{Harmonics in equal|57|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}} | |||
If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1{{c}} optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11, and 13. This approximates all primes up to 11 within ''16.6{{c}}''. Five almost-identical tunings do this: 57edt, [[101ed7]], [[zpi|155zpi]], and the [[TE]] and [[WE]] 2.3.7.13 subgroup WE tunings of 36edo. | |||
; Pure-octaves 36edo | |||
* Step size: 33.333{{c}} | |||
* Octave size: 1200.0{{c}} | |||
Pure-octaves 36edo approximates all primes up to 11 within ''15.3{{c}}''. | |||
; [[TE|11-limit TE 36edo]] | |||
* Step size: 33.287{{c}} | |||
* Octave size: 1198.3{{c}} | |||
{{Harmonics in cet|33.287|columns=12|collapsed=true|title=Approximation of harmonics in 11lim WE-tuned 36edo}} | |||
{{Harmonics in cet|33.287|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11lim WE-tuned 36edo (continued)}} | |||
Compressing the octave of 36edo by about 2{{c}} results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all primes up to 11 within ''9.7{{c}}''. The 11- and 13-limit TE tunings of 36edo both do this, as do their respective WE tunings. | |||
{| class="wikitable sortable center-all mw-collapsible mw-collapsed" | |||
|+ Stretched/compressed tunings comparison table | |||
|- | |||
! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(cents) !! colspan="6" | Prime error (cents) | |||
! rowspan="2" | Mapping of primes 2–13 (steps) | |||
! rowspan="2" | Octave stretch | |||
|- | |||
! 2 !! 3 !! 5 !! 7 !! 11 | |||
! 13 | |||
|- | |||
! 21edf | |||
| 33.426 | |||
| +3.3 || +3.3 || −12.0 || +7.2 || −6.5 || +5.1 | |||
| 36, 57, 83, 101, 124, 133 | |||
| +0.275% | |||
|- | |||
! 57edt | |||
| 33.368 || +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6 | |||
| 36, 57, 84, 101, 124, 133 | |||
| +0.001% | |||
|- | |||
! 155zpi | |||
| 33.346 | |||
| +0.6 || −1.0 || +15.1 || −0.5 || −16.0|| −5.0 | |||
| 36, 57, 83, 101, 124, 133 | |||
| +0.0005% | |||
|- | |||
! 36edo | |||
| '''33.333''' | |||
| '''0.0''' || '''−2.0''' || '''+13.7''' || '''−2.2''' || '''+15.3''' || '''−7.2''' | |||
| '''36, 57, 84, 101, 125, 133''' | |||
| '''0%''' | |||
|- | |||
! 13-limit WE | |||
| 33.302 | |||
| −1.1 || −3.7 || +11.1 || −5.3 || +11.4 || −11.4 | |||
| 36, 57, 84, 101, 125, 133 | |||
| -0.0009% | |||
|- | |||
! 11-limit WE | |||
| 33.286 | |||
| −1.7 || −4.7 || +9.7 || −6.9 || +9.4 || −13.5 | |||
| 36, 57, 84, 101, 125, 133 | |||
| -0.00142% | |||
|} | |||
== Scales == | == Scales == | ||