User:BudjarnLambeth/Draft related tunings section: Difference between revisions
→Octave stretch or compression: mention 93ed6 |
→Octave stretch or compression: primes -> harmonics. Italic seems unnecessary here |
||
| Line 72: | Line 72: | ||
; [[21edf]] | ; [[21edf]] | ||
* Step size: 33.426{{c}}, octave size: 1203. | * Step size: 33.426{{c}}, octave size: 1203.351{{c}} | ||
{{Harmonics in equal|21|3|2|columns=11|collapsed=true}} | {{Harmonics in equal|21|3|2|columns=11|collapsed=true}} | ||
{{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 | 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 harmonics up to 16 within 13.4{{c}}. The tuning 21edf does this. | ||
; [[57edt]] | ; [[57edt]] | ||
* Step size: 33.368{{c}}, octave size: 1201. | * Step size: 33.368{{c}}, octave size: 1201.235{{c}} | ||
{{Harmonics in equal|57|3|1|columns=11|collapsed=true}} | {{Harmonics in equal|57|3|1|columns=11|collapsed=true}} | ||
{{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 | 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 harmonics up to 16 within 16.6{{c}}. Several almost-identical tunings do this: 57edt, [[93ed6]], [[101ed7]], [[zpi|155zpi]], and the 2.3.7.13-subgroup [[TE]] and [[WE]] tunings of 36et. | ||
; 36edo | ; 36edo | ||
| Line 90: | Line 90: | ||
{{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|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edo (continued)}} | ||
Pure-octaves 36edo approximates all | Pure-octaves 36edo approximates all harmonics up to 16 within 15.3{{c}}. | ||
; [[TE|36et, 13-limit TE tuning]] | ; [[TE|36et, 13-limit TE tuning]] | ||
| Line 97: | Line 97: | ||
{{Harmonics in cet|33.303596|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 13-limit TE tuning of 36et (continued)}} | {{Harmonics in cet|33.303596|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 13-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 | 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 harmonics up to 16 within 11.6{{c}}. The 11- and 13-limit TE tunings of 36et both do this, as do their respective WE tunings. | ||
{| class="wikitable sortable center-all mw-collapsible mw-collapsed" | {| class="wikitable sortable center-all mw-collapsible mw-collapsed" | ||