414edo
| ← 413edo | 414edo | 415edo → |
414 equal divisions of the octave (abbreviated 414edo or 414ed2), also called 414-tone equal temperament (414tet) or 414 equal temperament (414et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 414 equal parts of about 2.9 ¢ each. Each step represents a frequency ratio of 21/414, or the 414th root of 2.
Theory
414edo is closely related to 207edo, but the patent vals differ on the mapping for harmonic 5. It is consistent to the 17-odd-limit with a flat tendency for most of the harmonics, making for a good full 17-limit system. The equal temperament tempers 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 | -17 | -28 | -24 | -20 | +2 | -21 | +36 | +25 | -20 | -4 | |
| Steps (reduced) |
414 (0) |
656 (242) |
961 (133) |
1162 (334) |
1432 (190) |
1532 (290) |
1692 (36) |
1759 (103) |
1873 (217) |
2011 (355) |
2051 (395) | |
Subsets and supersets
Since 414 factors into 2 × 32 × 23, 414edo has subset edos 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207.
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 8ve |
Generator* | Cents* | 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 |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct