524edo
524 equal division divides the octave into steps of 2.29 cents each.
Theory
| 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) | |
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.
524 years is the length of a calendar leap week cycle with 93 leap weeks, creating a 93 out of 524 maximum evenness scale, 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, 97 & 400, and 52 & 293 already have their identities and names.
In the 13-limit, 524edo tempers out 1001/1000 and 6664/6655.
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 |