574edo
| ← 573edo | 574edo | 575edo → |
574 equal divisions of the octave (abbreviated 574edo or 574ed2), also called 574-tone equal temperament (574tet) or 574 equal temperament (574et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 574 equal parts of about 2.09 ¢ each. Each step represents a frequency ratio of 21/574, or the 574th root of 2.
Theory
574 = 14 × 41, and 574edo shares the harmonic 3 with 41edo. Unfortunately, it is only consistent to the 5-odd-limit, and the patent val and the 574d val are about equally viable.
The 574d val tempers out 4375/4374 (ragisma), 29360128/29296875 (quasiorwellisma), and 40500000/40353607 in the 7-limit; 3025/3024, 5632/5625, 9801/9800, and 19712/19683 in the 11-limit; 676/675, 4225/4224, and 10648/10647 in the 13-limit.
The patent val tempers out 2100875/2097152 (rainy comma), 6115295232/6103515625 (vishnuzma), 49009212/48828125, and 78125000/78121827 (euzenius comma) in the 7-limit; 5632/5625, 42875/42768, 117649/117128, 131072/130977, 160083/160000, 161280/161051, 166698/166375, 391314/390625, and 532400/531441 in the 11-limit; 676/675, 1001/1000, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.484 | +0.446 | -0.882 | +0.598 | -0.110 | -0.426 | -0.649 | +0.994 | -1.006 | +0.609 |
| Relative (%) | +0.0 | +23.2 | +21.3 | -42.2 | +28.6 | -5.2 | -20.4 | -31.0 | +47.5 | -48.1 | +29.1 | |
| Steps (reduced) |
574 (0) |
910 (336) |
1333 (185) |
1611 (463) |
1986 (264) |
2124 (402) |
2346 (50) |
2438 (142) |
2597 (301) |
2788 (492) |
2844 (548) | |
Subsets and supersets
Since 574 factors into 2 × 7 × 41, 574edo has subset edos 2, 7, 14, 41, 82, and 287.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [23 6 -14⟩, [42 -47 14⟩ | [⟨574 910 1333]] | −0.1658 | 0.1260 | 6.03 |
| 2.3.5.7 | 4375/4374, 29360128/29296875, 40500000/40353607 | [⟨574 910 1333 1612]] (574d) | −0.2320 | 0.1583 | 7.57 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 5632/5625, 40500000/40353607 | [⟨574 910 1333 1612 1986]] (574d) | −0.2202 | 0.1435 | 6.87 |
| 2.3.5.7.11.13 | 676/675, 3025/3024, 4225/4224, 4375/4374, 41600/41503 | [⟨574 910 1333 1612 1986 2124]] (574d) | −0.1786 | 0.1607 | 7.69 |
| 2.3.5.7 | 10976/10935, 2100875/2097152, 49009212/48828125 | [⟨574 910 1333 1611]] (574) | −0.0459 | 0.2347 | 11.23 |
| 2.3.5.7.11 | 5632/5625, 10976/10935, 131072/130977, 166698/166375 | [⟨574 910 1333 1611 1986]] (574) | −0.0713 | 0.2160 | 10.33 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 4096/4095, 10976/10935, 166698/166375 | [⟨574 910 1333 1611 1986 2124]] (574) | −0.0544 | 0.2007 | 9.60 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 2 | 34\571 | 71.454 | 25/24 | Vishnu (574d) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct