2024edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 2023edo2024edo2025edo →
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.

Music

Eliora