User:BudjarnLambeth/Draft related tunings section: Difference between revisions
→Octave stretch or compression: cleanup. Try to make the section more compact |
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What follows is a comparison of stretched- and compressed-octave 36edo tunings. | What follows is a comparison of stretched- and compressed-octave 36edo tunings. | ||
; [[21edf]] | |||
; [[21edf]] | * Step size: 33.426{{c}}, octave size: 1203.3{{c}} | ||
* Step size: 33.426{{c}} | {{Harmonics in equal|21|3|2|columns=11|collapsed=true}} | ||
{{Harmonics in equal|21|3|2|columns= | |||
{{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}} | {{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. | 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]] | ; [[57edt]] | ||
* Step size: 33.368{{c}} | * Step size: 33.368{{c}}, octave size: 1201.2{{c}} | ||
{{Harmonics in equal|57|3|1|columns=11|collapsed=true}} | |||
{{Harmonics in equal|57|3|1|columns= | |||
{{Harmonics in equal|57|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}} | {{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]] | 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 2.3.7.13-subgroup [[TE]] and [[WE]] tunings of 36et. | ||
; 36edo | |||
; | * Step size: 33.333{{c}}, octave size: 1200.0{{c}} | ||
* Step size: 33.333{{c}} | {{Harmonics in equal|36|2|1|intervals=integer|columns=11|collapsed=true}} | ||
{{Harmonics in equal|36|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edo (continued)}} | |||
{{Harmonics in equal|36|2|1|columns= | |||
{{Harmonics in equal|36|2|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edo (continued)}} | |||
Pure-octaves 36edo approximates all primes up to 11 within ''15.3{{c}}''. | Pure-octaves 36edo approximates all primes up to 11 within ''15.3{{c}}''. | ||
; [[TE|36et, 11-limit TE tuning]] | |||
* Step size: 33.287{{c}}, octave size: 1198.3{{c}} | |||
{{Harmonics in cet|33.287|columns=11|collapsed=true|title=Approximation of harmonics in 11-limit TE tuning of 36et}} | |||
{{Harmonics in cet|33.287|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11-limit TE tuning of 36et (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 36et both do this, as do their respective WE tunings. | |||
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 | |||
{| class="wikitable sortable center-all mw-collapsible mw-collapsed" | {| class="wikitable sortable center-all mw-collapsible mw-collapsed" | ||
|+ | |+ style="white-space: nowrap;" | Comparison of stretched and compressed tunings | ||
|- | |- | ||
! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(cents) !! colspan="6" | Prime error (cents) | ! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(cents) !! colspan="6" | Prime error (cents) | ||