374edo
| ← 373edo | 374edo | 375edo → |
374 equal divisions of the octave (abbreviated 374edo or 374ed2), also called 374-tone equal temperament (374tet) or 374 equal temperament (374et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 374 equal parts of about 3.21 ¢ each. Each step represents a frequency ratio of 21/374, or the 374th root of 2.
Theory
374et is inconsistent to the 5-odd-limit since harmonic 5 is about halfway between its steps. Omitting the harmonic 5, it is consistent to the 31-odd-limit.
Using the patent val, the equal temperament tempers out 5120/5103, 1071875/1062882, 1500625/1492992, 2100875/2097152, and 9765625/9680832 in the 7-limit; 1375/1372, 4375/4356, 12005/11979, and 41503/41472 in the 11-limit. It supports quintakwai and quartemka.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.72 | -1.29 | +0.16 | +0.55 | +0.11 | +0.93 | +0.88 | +0.60 | +0.37 | +0.42 |
| Relative (%) | +0.0 | +22.4 | -40.1 | +4.9 | +17.3 | +3.6 | +28.9 | +27.5 | +18.8 | +11.5 | +13.1 | |
| Steps (reduced) |
374 (0) |
593 (219) |
868 (120) |
1050 (302) |
1294 (172) |
1384 (262) |
1529 (33) |
1589 (93) |
1692 (196) |
1817 (321) |
1853 (357) | |
Subsets and supersets
Since 374 factors into 2 × 11 × 17, 374edo has subset edos 2, 11, 17, 22, 34, and 187. 748edo, which doubles it, gives a good correction to the harmonic 5, but its approximation of harmonic 3 has drifted too far to render it inconsistent in the 9-odd-limit.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [593 -374⟩ | [⟨374 593]] | -0.2268 | 0.2267 | 7.07 |
| 2.3.7 | [4 -22 11⟩, [51 -18 -8⟩ | [⟨374 593 1050]] | -0.1699 | 0.2018 | 6.29 |
| 2.3.7.11 | 41503/41472, 1362944/1361367, 70493667328/70027449129 | [⟨374 593 1050 1294]] | -0.1675 | 0.1748 | 5.45 |
| 2.3.7.11.13 | 10648/10647, 20449/20412, 41503/41472, 652288/649539 | [⟨374 593 1050 1294 138 4]] | -0.1401 | 0.1656 | 5.16 |
| 2.3.7.11.13.17 | 2058/2057, 8281/8262, 8624/8619, 22528/22491, 34816/34749 | [⟨374 593 1050 1294 1384 1529]] | -0.1546 | 0.1546 | 4.82 |
| 2.3.7.11.13.17.19 | 1729/1728, 2058/2057, 2912/2907, 5929/5928, 22528/22491, 34816/34749 | [⟨374 593 1050 1294 1384 1529 1589]] | -0.1622 | 0.1444 | 4.50 |