684edo: Difference between revisions
Created page with "'''684EDO''' is the equal division of the octave into 684 parts of 1.75439 cents each (dividing the steps of 171EDO into four). It is consistent to the..." Tags: Mobile edit Mobile web edit |
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 |
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{{Infobox ET}} | |||
{{ED intro}} | |||
[[ | == Theory == | ||
684edo divides the steps of [[171edo]] into four. It is [[consistent]] to the [[17-odd-limit]], [[tempering out]] [[2401/2400]], [[3025/3024]], [[4225/4224]], [[4375/4374]], and [[32805/32768]] in the 13-limit; [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], [[2500/2499]], 8624/8619, and 14875/14872 in the 17-limit. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|684|columns=11}} | |||
=== 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 }}. | |||
== 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 == | |||
{| 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 | |||
| 2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768 | |||
| {{mapping| 684 1084 1588 1920 2366 2531 }} | |||
| +0.0994 | |||
| 0.0558 | |||
| 3.18 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005 | |||
| {{mapping| 684 1084 1588 1920 2366 2531 2796 }} | |||
| +0.0744 | |||
| 0.0800 | |||
| 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]]. | |||
=== Rank-2 temperaments === | |||
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 | |||
| 271\684<br>(5\684) | |||
| 475.44<br>(8.77) | |||
| 1053/800<br>(1287/1280) | |||
| [[Semihemiennealimmal]] | |||
|- | |||
| 38 | |||
| 151\684<br>(7\684) | |||
| 264.91<br>(12.28) | |||
| 500/429<br>(144/143) | |||
| [[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 | |||