684edo: Difference between revisions
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+infobox; +RTT table and rank-2 temperaments |
→Regular temperament properties: +semihemienneadecal |
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| Line 55: | Line 55: | ||
| 1053/800<br>(1287/1280) | | 1053/800<br>(1287/1280) | ||
| [[Semihemiennealimmal]] | | [[Semihemiennealimmal]] | ||
|- | |||
| 38 | |||
| 151\684<br>(7\684) | |||
| 264.91<br>(12.28) | |||
| 500/429<br>(144/143) | |||
| [[Semihemienneadecal]] | |||
|} | |} | ||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
Revision as of 00:06, 21 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 |
| 38 | 151\684 (7\684) |
264.91 (12.28) |
500/429 (144/143) |
Semihemienneadecal |