524edo

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524edo
Prime factorization 22 × 131
Step size 2.29008¢
Fifth 307\524 (703.0534¢)

The 524 equal divisions of the octave (524edo), or 524(-tone) equal temperament (524-tet, 524et) when viewed from a regular temperament perspective, divides the octave into 524 equal parts of about 2.29 cents each.

Theory

Approximation of odd harmonics in 524edo
Harmonic 3 5 7 9 11 13 15 17 19 21
Error absolute (¢) +1.10 +0.71 -0.12 -0.09 +0.59 -0.07 -0.48 +0.39 +0.20 +0.97
relative (%) +48 +31 -5 -4 +26 -3 -21 +17 +9 +43
Steps
(reduced)
831
(307)
1217
(169)
1471
(423)
1661
(89)
1813
(241)
1939
(367)
2047
(475)
2142
(46)
2226
(130)
2302
(206)

524edo is excellent in the 2.7.13.19 subgroup, and good in the no-threes 19-limit. In the 3-limit, it is wise to treat 524edo as a dual-fifth system. The minor fifth, 306\524 reduces to 153\262, as such, on this val 524edo is contorted.

524 years is the length of a calendar leap week cycle with 93 leap weeks, creating a 93 out of 524 maximum evenness scale with the generator 293\524, represented by the 93 & 524 temperament.

In addition, both 93 and 524 represent well the 13:17:19 harmonics. The corresponding comma list in the 2.7.13.17.19 subgroup is 16807/16796, 157339/157216, 47071232/47045881. Eliora proposes that this temperament be named ostara, after the feast of the spring equinox, which 93\524 leap week rule approximates well. Other spring equinoctial temperaments, such as 41 & 231 (Dee leap week), 97 & 400 (Gregorian leap day), and 52 & 293 (Sym454) already have their identities and names.

The generator 293\524, 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.

Scales

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

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