364edo
| ← 363edo | 364edo | 365edo → |
364 equal divisions of the octave (abbreviated 364edo or 364ed2), also called 364-tone equal temperament (364tet) or 364 equal temperament (364et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 364 equal parts of about 3.3 ¢ each. Each step represents a frequency ratio of 21/364, or the 364th root of 2.
Theory
364edo is consistent through the 21-odd-limit. The equal temperament tempers out 1600000/1594323 (amity comma) and [-65 0 28⟩ (oquatonic comma) in the 5-limit; 65625/65536 (horwell comma), 390625/388962 (dimcomp comma), and 420175/419904 (wizma) in the 7-limit (supporting fifthplus and oquatonic); 1375/1372, 6250/6237, 19712/19683, and 41503/41472 in the 11-limit (as well as 9801/9800); 625/624, 1716/1715, 2080/2079, 2200/2197, and 14641/14625 in the 13-limit (as well as 4096/4095, 4225/4224, and 10985/10976); 715/714, 1089/1088, 1225/1224, 1275/1274, 2025/2023, and 8624/8619 in the 17-limit (as well as 2431/2430, 4914/4913, and 5832/5831); 1216/1215, 1331/1330, 1540/1539, and 1729/1728 in the 19-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.24 | -0.60 | +0.40 | -0.77 | +0.13 | +0.54 | -0.81 | +1.40 | -1.01 | -1.08 |
| Relative (%) | +0.0 | +7.4 | -18.2 | +12.3 | -23.3 | +4.0 | +16.4 | -24.6 | +42.3 | -30.5 | -32.7 | |
| Steps (reduced) |
364 (0) |
577 (213) |
845 (117) |
1022 (294) |
1259 (167) |
1347 (255) |
1488 (32) |
1546 (90) |
1647 (191) |
1768 (312) |
1803 (347) | |
Subsets and supersets
Since 364 factors into 22 × 7 × 13, 364edo has subset edos 2, 4, 7, 13, 14, 26, 28, 52, 91, 182.
Miscellaneous properties
364edo can act as "pseudo-24024edo" in a sense that it can replicate being a multiple of 11edo, 12edo, 13edo and 14edo. It has 13 and 14 as its divisors, while at the same time supporting the Supermajor[11] scale from 91edo, which is a very precise temperament, and WorldCalendar[12] scale, which mimics 12edo. While it does not exactly replicate 11edo and 12edo, it comes close enough in harmonic parameters these edos are sought after.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [577 -364⟩ | [⟨364 577]] | −0.0766 | 0.0766 | 2.32 |
| 2.3.5 | 1600000/1594323, [-65 0 28⟩ | [⟨364 577 845]] | +0.0350 | 0.1698 | 5.15 |
| 2.3.5.7 | 65625/65536, 390625/388962, 420125/419904 | [⟨364 577 845 1022]] | −0.0098 | 0.1662 | 5.04 |
| 2.3.5.7.11 | 1375/1372, 6250/6237, 19712/19683, 41503/41472 | [⟨364 577 845 1022 1259]] | +0.0366 | 0.1753 | 5.32 |
| 2.3.5.7.11.13 | 625/624, 1375/1372, 2080/2079, 2200/2197, 14641/14625 | [⟨364 577 845 1022 1259 1347]] | +0.0245 | 0.1622 | 4.92 |
| 2.3.5.7.11.13.17 | 625/624, 715/714, 1089/1088, 1225/1224, 2025/2023, 2200/2197 | [⟨364 577 845 1022 1259 1347 1488]] | +0.0022 | 0.1599 | 4.85 |
| 2.3.5.7.11.13.17.19 | 625/624, 715/714, 1089/1088, 1216/1215, 1225/1224, 1331/1330, 1729/1728 | [⟨364 577 845 1022 1259 1347 1488 1546]] | +0.0257 | 0.1620 | 4.91 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 103\364 | 339.56 | 243/200 | Amity / paramity |
| 1 | 125\364 | 412.09 | 80/63 | Witch |
| 1 | 149\364 | 491.21 | 3645/2744 | Fifthplus |
| 1 | 151\364 | 497.80 | 4/3 | Gary |
| 2 | 57\364 | 187.91 | 49/44 | Semiwitch |
| 4 | 30\364 | 98.90 | 18/17 | World calendar |
| 13 | 151\364 (11\364) |
497.80 (36.26) |
4/3 (?) |
Aluminium |
| 26 | 151\364 (11\364) |
497.80 (36.26) |
4/3 (?) |
Iron |
| 28 | 151\364 (5\364) |
497.80 (16.48) |
4/3 (105/104) |
Oquatonic |
| 91 | 151\364 (3\364) |
497.80 (3.30) |
4/3 (176/175) |
Protactinium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
- WorldCalendar[12]: 31 30 30 31 30 30 31 30 30 31 30 30