524edo

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← 523edo524edo525edo →
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 524-tone equal temperament (524tet), 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 of 524edo 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 +31 -5 -4 +26 -3 -21 +17 +9 +43 -35
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