# 524edo

 ← 523edo 524edo 525edo →
Prime factorization 22 × 131
Step size 2.29008¢
Fifth 307\524 (703.053¢)
Semitones (A1:m2) 53:37 (121.4¢ : 84.73¢)
Dual sharp fifth 307\524 (703.053¢)
Dual flat fifth 306\524 (700.763¢) (→153\262)
Dual major 2nd 89\524 (203.817¢)
Consistency limit 5
Distinct consistency limit 5

524 equal divisions of the octave (abbreviated 524edo or 524ed2), also called 524-tone equal temperament (524tet) or 524 equal temperament (524et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 524 equal parts of about 2.29 ¢ each. Each step represents a frequency ratio of 21/524, or the 524th root of 2.

## Theory

524edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise it has a good approximation to harmonics 7, 9, 11, 13, 15, 17, and 19, making it suitable for a 2.9.15.7.11.13.17.19 subgroup interpretation. The 2.9.7.13.19 subgroup is particularly good, where it supports Eliora's ostara, the 93 & 524 temperament. The generator 293\524 represents the 28/19 interval in the 2.7.19 subgroup and it serves as ostara's generator in the no-threes 19-limit. Being around 671 cents, can also be used to as a generator for mavila or pelog.

In the 13-limit, 524edo tempers out 1001/1000 and 6664/6655.

### Odd harmonics

Approximation of odd harmonics in 524edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.10 +0.71 -0.12 -0.09 +0.59 -0.07 -0.48 +0.39 +0.20 +0.97 -0.79
Relative (%) +48.0 +31.0 -5.4 -4.1 +25.8 -3.0 -21.1 +16.9 +8.6 +42.6 -34.6
Steps
(reduced)
831
(307)
1217
(169)
1471
(423)
1661
(89)
1813
(241)
1939
(367)
2047
(475)
2142
(46)
2226
(130)
2302
(206)
2370
(274)

### Subsets and supersets

Since 524 factors into 22 × 131, 524edo has subset edos 2, 4, 131, and 262.

## Regular temperament properties

Based on treating 524edo as a no-threes system:

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.5 [1217 -524 [524 1217]] -0.152 0.153 6.67
2.5.7 [33 -13 -1, [-4 -43 37 [524 1217 1471]] -0.087 0.155 6.79
2.5.7.11 1835008/1830125, [3 7 3 -8, [-13 -5 10 -1 [524 1217 1471 1813]] -0.108 0.139 6.09
2.5.7.11.13 1001/1000, 742586/741125, 2097152/2093663, 14201915/14172488 [524 1217 1471 1813 1939]] -0.082 0.135 5.88
2.5.7.11.13.17 1001/1000, 6664/6655, 54080/54043, 147968/147875, 285719/285610 [524 1217 1471 1813 1939 2142]] -0.084 0.122 5.37

## Scales

• Ostara[7]: 62 62 62 107 62 62 107 - 2L 5s