684edo: Difference between revisions
ArrowHead294 (talk | contribs) mNo edit summary |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" Tags: Mobile edit Mobile web edit |
||
| (13 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 10: | Line 10: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 684 factors into {{factorization|684}}, 684edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342 }}. | Since 684 factors into {{factorization|684}}, 684edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342 }}. | ||
== Approximation to JI == | |||
=== Zeta peak index === | |||
{{ZPI | |||
| zpi = 5818 | |||
| steps = 683.938934890938 | |||
| step size = 1.75454260429165 | |||
| tempered height = 14.267321 | |||
| pure height = 7.268914 | |||
| integral = 1.773752 | |||
| gap = 20.109967 | |||
| octave = 1200.10714133549 | |||
| consistent = 18 | |||
| distinct = 18 | |||
}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| Line 27: | Line 51: | ||
| 0.0800 | | 0.0800 | ||
| 4.56 | | 4.56 | ||
|} | |||
* 684et is the first equal temperament past [[494edo|494]] with a lower 13-limit absolute error. The next equal temperament that is better tuned is [[764edo|764]]. | * 684et is the first equal temperament past [[494edo|494]] with a lower 13-limit absolute error. The next equal temperament that is better tuned is [[764edo|764]]. | ||
| Line 33: | Line 57: | ||
Note: 11-limit temperaments supported by [[342edo|342et]] are not shown. | Note: 11-limit temperaments supported by [[342edo|342et]] are not shown. | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br>per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br>ratio* | |||
! Temperaments | |||
|- | |- | ||
| 18 | | 18 | ||
| 271\684<br | | 271\684<br>(5\684) | ||
| 475.44<br | | 475.44<br>(8.77) | ||
| 1053/800<br | | 1053/800<br>(1287/1280) | ||
| [[Semihemiennealimmal]] | | [[Semihemiennealimmal]] | ||
|- | |- | ||
| 38 | | 38 | ||
| 151\684<br | | 151\684<br>(7\684) | ||
| 264.91<br | | 264.91<br>(12.28) | ||
| 500/429<br | | 500/429<br>(144/143) | ||
| [[Semihemienneadecal]] | | [[Semihemienneadecal]] | ||
|} | |||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
== Scales == | |||
* [[15-odd-limit|Diamond15]]: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64 | |||