414edo
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 (%) | +0.0 | -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) | |
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
Template:Comma basis begin |- | 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 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf
Music
- DISconnectioN (2024)