684edo: Difference between revisions
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Improve intro; +prime error table |
+infobox; +RTT table and rank-2 temperaments |
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{{Infobox ET | |||
| Prime factorization = 2<sup>2</sup> × 3<sup>2</sup> × 19 | |||
| Step size = 1.75439¢ | |||
| Fifth = 400\684 (701.75¢) (→ [[171edo|100\171]]) | |||
| Semitones = 64:52 (112.28¢ : 91.23¢) | |||
| Consistency = 17 | |||
}} | |||
{{EDO intro|684}} | {{EDO intro|684}} | ||
== 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. | 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}} | {{Harmonics in equal|684|columns=11}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list|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 | |||
| [{{val| 684 1084 1588 1920 2366 2531 }}] | |||
| +0.0994 | |||
| 0.0558 | |||
| 3.18 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 1089/1088, 1701/1700, 2025/2023, 2058/2057, 4225/4224, 13013/13005 | |||
| [{{val| 684 1084 1588 1920 2366 2531 2796 }}] | |||
| +0.0744 | |||
| 0.0800 | |||
| 4.56 | |||
|} | |||
* 684et is the first equal temperament with a lower 13-limit absolute error after 494. The next equal temperament that is better tuned is 764. | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per Octave | |||
! Generator<br>(Reduced) | |||
! Cents<br>(Reduced) | |||
! Associated<br>Ratio | |||
! Temperaments | |||
|- | |||
| 18 | |||
| 271\684<br>(5\684) | |||
| 475.44<br>(8.77) | |||
| 1053/800<br>(1287/1280) | |||
| [[Semihemiennealimmal]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
Revision as of 09:01, 20 September 2022
| ← 683edo | 684edo | 685edo → |
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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.201 | -0.349 | -0.405 | -0.441 | -0.177 | +0.308 | +0.733 | -0.204 | +0.247 | +0.578 |
| Relative (%) | +0.0 | -11.4 | -19.9 | -23.1 | -25.1 | -10.1 | +17.5 | +41.8 | -11.6 | +14.1 | +33.0 | |
| Steps (reduced) |
684 (0) |
1084 (400) |
1588 (220) |
1920 (552) |
2366 (314) |
2531 (479) |
2796 (60) |
2906 (170) |
3094 (358) |
3323 (587) |
3389 (653) | |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7.11.13 | 2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768 | [⟨684 1084 1588 1920 2366 2531]] | +0.0994 | 0.0558 | 3.18 |
| 2.3.5.7.11.13.17 | 1089/1088, 1701/1700, 2025/2023, 2058/2057, 4225/4224, 13013/13005 | [⟨684 1084 1588 1920 2366 2531 2796]] | +0.0744 | 0.0800 | 4.56 |
- 684et is the first equal temperament with a lower 13-limit absolute error after 494. The next equal temperament that is better tuned is 764.
Rank-2 temperaments
| Periods per Octave |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 18 | 271\684 (5\684) |
475.44 (8.77) |
1053/800 (1287/1280) |
Semihemiennealimmal |