684edo: Difference between revisions
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→Regular temperament properties: comma basis; style |
Subsets and supersets |
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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|684|columns=11}} | {{Harmonics in equal|684|columns=11}} | ||
=== Subsets and supersets === | |||
Since 684 factors as 2<sup>2</sup> × 3<sup>2</sup> × 19, 684edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342 }}. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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| 4.56 | | 4.56 | ||
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* 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]]. | * 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 === | === Rank-2 temperaments === | ||
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| [[Semihemienneadecal]] | | [[Semihemienneadecal]] | ||
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Revision as of 11:55, 11 February 2023
| ← 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) | |
Subsets and supersets
Since 684 factors as 22 × 32 × 19, 684edo has subset edos 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342.
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 |