624edo
| ← 623edo | 624edo | 625edo → |
624 equal divisions of the octave (abbreviated 624edo or 624ed2), also called 624-tone equal temperament (624tet) or 624 equal temperament (624et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 624 equal parts of about 1.92 ¢ each. Each step represents a frequency ratio of 21/624, or the 624th root of 2.
Theory
624edo is consistent to the 27-odd-limit. As an equal temperament, it tempers out [23 6 -14⟩ (vishnuzma) and [-69 45 -1⟩ (counterschisma) in the 5-limit; 250047/250000, 2460375/2458624, and 134217728/133984375 in the 7-limit; 9801/9800, 46656/46585, 131072/130977, and 151263/151250 in the 11-limit; 1716/1715, 2080/2079, 4096/4095, 34398/34375, and 39366/39325 in the 13-limit; 936/935, 1701/1700, 2025/2023, and 2058/2057 in the 17-limit; 1521/1520, 2376/2375, 2432/2431, and 3328/3325 in the 19-limit; 2024/2023, 2025/2024, 2646/2645, 3520/3519, and 3888/3887 in the 23-limit.
It provides an excellent optimal patent val for the rank-6 temperament tempering out 936/935, as well as the rank-5 2.3.5.11.13.17-subgroup restriction thereof.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.032 | +0.225 | +0.405 | +0.605 | -0.143 | +0.814 | +0.564 | +0.572 | -0.731 | -0.805 |
| Relative (%) | +0.0 | -1.7 | +11.7 | +21.1 | +31.5 | -7.4 | +42.3 | +29.3 | +29.7 | -38.0 | -41.8 | |
| Steps (reduced) |
624 (0) |
989 (365) |
1449 (201) |
1752 (504) |
2159 (287) |
2309 (437) |
2551 (55) |
2651 (155) |
2823 (327) |
3031 (535) |
3091 (595) | |
Subsets and supersets
Since 624 factors into 24 × 3 × 13, 624edo has subset edos 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-989 624⟩ | [⟨624 989]] | +0.0101 | 0.0101 | 0.52 |
| 2.3.5 | [23 6 -14⟩, [-69 45 -1⟩ | [⟨624 989 1449]] | −0.0256 | 0.0510 | 2.65 |
| 2.3.5.7 | 250047/250000, 2460375/2458624, [27 0 -8 -3⟩ | [⟨624 989 1449 1752]] | −0.0552 | 0.0678 | 3.52 |
| 2.3.5.7.11 | 9801/9800, 46656/46585, 131072/130977, 151263/151250 | [⟨624 989 1449 1752 2159]] | −0.0792 | 0.0772 | 4.02 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325 | [⟨624 989 1449 1752 2159 2309]] | −0.0595 | 0.0831 | 4.32 |
| 2.3.5.7.11.13.17 | 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011 | [⟨624 989 1449 1752 2159 2309 2551]] | −0.0795 | 0.0911 | 4.74 |
| 2.3.5.7.11.13.17.19 | 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011 | [⟨624 989 1449 1752 2159 2309 2551 2651]] | −0.0861 | 0.0870 | 4.53 |
| 2.3.5.7.11.13.17.19.23 | 936/935, 1521/1520, 1701/1700, 1716/1715, 2024/2023, 2025/2023, 2376/2375, 2646/2645 | [⟨624 989 1449 1752 2159 2309 2551 2651 2823]] | −0.0906 | 0.0830 | 4.32 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 259\624 | 498.08 | 4/3 | Counterschismic |
| 1 | 311\624 | 598.08 | 847/600 | Vydubychi |
| 2 | 37\624 | 71.15 | 25/24 | Vishnu (5-limit) |
| 3 | 73\624 | 140.38 | 243/224 | Septichrome |
| 6 | 177\624 (31\624) |
340.38 (59.62) |
162/133 (88/85) |
Semiseptichrome |
| 12 | 259\624 (1\624) |
498.08 (1.92) |
4/3 (32805/32768) |
Atomic |
| 13 | 259\624 (19\624) |
498.08 (36.54) |
4/3 (?) |
Aluminium (5-limit) |
| 16 | 259\624 (14\624) |
498.08 (48.077) |
4/3 (?) |
Sulfur |
| 24 | 303\624 (17\624) |
582.692 (32.692) |
7/5 (?) |
Chromium |
| 26 | 259\624 (19\624) |
498.08 (36.54) |
4/3 (?) |
Iron |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Music
- Etude in Iron (2024)