684edo: Difference between revisions
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m Infobox ET now computes most parameters automatically |
→Regular temperament properties: comma basis; style |
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! rowspan="2" | [[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 27: | Line 27: | ||
|- | |- | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 1089/1088, 1701/1700, 2025/2023 | | 1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005 | ||
| [{{val| 684 1084 1588 1920 2366 2531 2796 }}] | | [{{val| 684 1084 1588 1920 2366 2531 2796 }}] | ||
| +0.0744 | | +0.0744 | ||
| Line 33: | Line 33: | ||
| 4.56 | | 4.56 | ||
|} | |} | ||
* 684et is the first equal temperament with a lower 13-limit absolute error | * 684et is the first equal temperament past 494 with a lower 13-limit absolute error. The next equal temperament that is better tuned is [[764edo|764]]. | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per 8ve | ||
! Generator<br>(Reduced) | ! Generator<br>(Reduced) | ||
! Cents<br>(Reduced) | ! Cents<br>(Reduced) | ||
Revision as of 09:59, 6 October 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, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005 | [⟨684 1084 1588 1920 2366 2531 2796]] | +0.0744 | 0.0800 | 4.56 |
- 684et is the first equal temperament past 494 with a lower 13-limit absolute error. The next equal temperament that is better tuned is 764.
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 18 | 271\684 (5\684) |
475.44 (8.77) |
1053/800 (1287/1280) |
Semihemiennealimmal |
| 38 | 151\684 (7\684) |
264.91 (12.28) |
500/429 (144/143) |
Semihemienneadecal |