2024edo
| ← 2023edo | 2024edo | 2025edo → |
2024 equal divisions of the octave (abbreviated 2024edo or 2024ed2), also called 2024-tone equal temperament (2024tet) or 2024 equal temperament (2024et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2024 equal parts of about 0.593 ¢ each. Each step represents a frequency ratio of 21/2024, or the 2024th root of 2.
Theory
2024edo is enfactored in the 13-limit, with the same tuning as 1012edo, which is also a zeta edo. It corrects 1012edo's mapping for 17, being a strong 2.3.7.11.17.19 subgroup temperament. A comma basis for the said subgroup is {23409/23408, 117649/117612, 323456/323433, 131072/131043, 26042368/26040609}.
In the 2.3.7.11.17.19, it tunes the 629 & 2024 temperament, which has 19/17 as a generator and reaches 17/16 in 13 steps. The comma basis in the 2.3.7.11.17.19 subgroup is {23409/23408, 131072/131043, 323456/323433, 818107136/817887699}.
It has two suitable mappings for 5th harmonic, one which derives from 1012edo, and other in the 2024c val. In the 2024c val, it tempers out the wizma, 420175/419904 in the 7-limit, as well as 3025/3024, 4225/4224 and 10648/10647 in the 13-limit. If the sharp and flat mappings of 5/4 are combined, then 2024edo is a good 2.3.25 subgroup tuning. In the 2.3.25.7.11 subgroup, it tempers out 4375/4374 and tunes a messed-up version of the heimdall temperament, which reaches 7th harmonic in 2 second generators instead of 4, and 11th harmonic in 6 second generators instead of 12, taking half as much.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | +0.021 | +0.248 | -0.051 | +0.065 | +0.184 | -0.015 | +0.115 | +0.184 | +0.265 | -0.174 |
| relative (%) | +0 | +4 | +42 | -9 | +11 | +31 | -2 | +19 | +31 | +45 | -29 | |
| Steps (reduced) |
2024 (0) |
3208 (1184) |
4700 (652) |
5682 (1634) |
7002 (930) |
7490 (1418) |
8273 (177) |
8598 (502) |
9156 (1060) |
9833 (1737) |
10027 (1931) | |
Subsets and supersets
Since 2024 factors into 23 × 11 × 23, 2024edo has subset edos 2, 4, 8, 11, 22, 23, 44, 46, 88, 92, 184, 253, 506, and 1012.
Music
- Piano Juice (2024) - palladium[92] (from 506edo), ruthenium[88] (from 1012edo), 629 & 2024[25], major-arcana[88] (from 506edo)