414edo
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| ← 413edo | 414edo | 415edo → |
The 414 equal divisions of the octave (414edo), or the 414(-tone) equal temperament (414tet, 414et) when viewed from a regular temperament perspective, is the equal division of the octave into 414 parts of about 2.90 cents each.
Theory
414edo is closely related to 207edo, but the patent vals differ on the mapping for 5. It is consistent to the 17-odd-limit, tempering out [-36 11 8⟩ (submajor comma) and [1 -27 18⟩ (ennealimma) in the 5-limit; 2401/2400, 4375/4374, and [-37 4 12 1⟩ in the 7-limit; 3025/3024, 9801/9800, 41503/41472, and 1265625/1261568 in the 11-limit; 625/624, 729/728, 1575/1573, 2200/2197, and 26411/26364 in the 13-limit; 833/832, 1089/1088, 1225/1224, 1275/1274, and 1701/1700 in the 17-limit. It supports the 11-limit hemiennealimmal and the 13-limit quatracot.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.51 | -0.81 | -0.71 | -0.59 | +0.05 | -0.61 | +1.04 | +0.71 | -0.59 | -0.11 |
| Relative (%) | +0.0 | -17.4 | -27.8 | -24.5 | -20.5 | +1.8 | -21.0 | +35.8 | +24.5 | -20.4 | -3.7 | |
| Steps (reduced) |
414 (0) |
656 (242) |
961 (133) |
1162 (334) |
1432 (190) |
1532 (290) |
1692 (36) |
1759 (103) |
1873 (217) |
2011 (355) |
2051 (395) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [-36 11 8⟩, [1 -27 18⟩ | [⟨414 656 961]] | +0.2222 | 0.1575 | 5.43 |
| 2.3.5.7 | 2401/2400, 4375/4374, [-36 11 8⟩ | [⟨414 656 961 1162]] | +0.2299 | 0.1371 | 4.73 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 4375/4374, 1366875/1362944 | [⟨414 656 961 1162 1432]] | +0.2182 | 0.1248 | 4.30 |
| 2.3.5.7.11.13 | 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400 | [⟨414 656 961 1162 1432 1532]] | +0.1795 | 0.1431 | 4.94 |
| 2.3.5.7.11.13.17 | 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197 | [⟨414 656 961 1162 1432 1532 1692]] | +0.1751 | 0.1329 | 4.58 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 125\414 | 362.31 | 10125/8192 | Submajor (5-limit) |
| 2 | 61\414 | 176.81 | 195/176 | Quatracot |
| 9 | 109\414 (17\414) |
315.94 (49.28) |
6/5 (36/35) |
Ennealimmal |
| 18 | 86\414 (6\414) |
249.28 (17.39) |
231/200 (99/98) |
Hemiennealimmal |
| 18 | 164\414 (3\414) |
475.36 (8.70) |
1053/800 (1287/1280) |
Semihemiennealimmal |