524edo
| ← 523edo | 524edo | 525edo → |
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
| 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