523edo

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← 522edo523edo524edo →
Prime factorization 523 (prime)
Step size 2.29446¢ 
Fifth 306\523 (702.103¢)
Semitones (A1:m2) 50:39 (114.7¢ : 89.48¢)
Consistency limit 15
Distinct consistency limit 15

523 equal divisions of the octave (abbreviated 523edo or 523ed2), also called 523-tone equal temperament (523tet) or 523 equal temperament (523et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 523 equal parts of about 2.29 ¢ each. Each step represents a frequency ratio of 21/523, or the 523rd root of 2.

Theory

523et tempers out 2401/2400 in the 7-limit; 56723625/56689952, 26214400/26198073, 78675968/78594219, 759375/758912, 496125/495616, 6250/6237, 42875/42768, 19712/19683, 1479016/1476225, 16808715/16777216, 3294225/3294172, 43923/43904, 102487/102400 and 322102/321489 in the 11-limit.

Prime harmonics

Approximation of prime harmonics in 523edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.15 -0.85 -0.57 -0.65 -0.76 +0.59 +0.77 +0.41 +0.63 -0.10
Relative (%) +0.0 +6.5 -36.8 -24.7 -28.3 -33.0 +25.7 +33.4 +17.7 +27.6 -4.5
Steps
(reduced)
523
(0)
829
(306)
1214
(168)
1468
(422)
1809
(240)
1935
(366)
2138
(46)
2222
(130)
2366
(274)
2541
(449)
2591
(499)

Subsets and supersets

523edo is the 99th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [829 -523 [523 829]] -0.0468 0.0468 2.04
2.3.5 1600000/1594323, [-101 11 36 [523 829 1214]] +0.0901 0.1974 8.60
2.3.5.7 2401/2400, 1600000/1594323, 341796875/339738624 [523 829 1214 1468]] +0.1180 0.1776 7.74
2.3.5.7.11 6250/6237, 2401/2400, 19712/19683, 166375/165888 [523 829 1214 1468 1809]] +0.1319 0.1613 7.03
2.3.5.7.11.13 2080/2079, 625/624, 4459/4455, 19712/19683, 50193/50176 [523 829 1214 1468 1809 1935]] +0.1440 0.1497 6.52

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 37\523 84.89 21/20 Amicable
1 52\523 119.31 15/14 Septidiasemi
1 148\523 339.58 243/200 Amity

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Francium