# 574edo

 ← 573edo 574edo 575edo →
Prime factorization 2 × 7 × 41
Step size 2.09059¢
Fifth 336\574 (702.439¢) (→24\41)
Semitones (A1:m2) 56:42 (117.1¢ : 87.8¢)
Consistency limit 5
Distinct consistency limit 5

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

Approximation of prime harmonics in 574edo
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

Table of rank-2 temperaments by generator
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