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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 | |||