# 2024edo

 ← 2023edo 2024edo 2025edo →
Prime factorization 23 × 11 × 23
Step size 0.592885¢
Fifth 1184\2024 (701.976¢) (→148\253)
Semitones (A1:m2) 192:152 (113.8¢ : 90.12¢)
Consistency limit 5
Distinct consistency limit 5

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

Approximation of prime harmonics in 2024edo
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 (%) +0.0 +3.6 +41.8 -8.6 +11.0 +31.0 -2.5 +19.5 +31.1 +44.6 -29.3
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.

Eliora