2022edo

 ← 2021edo 2022edo 2023edo →
Prime factorization 2 × 3 × 337
Step size 0.593472¢
Fifth 1183\2022 (702.077¢)
Semitones (A1:m2) 193:151 (114.5¢ : 89.61¢)
Consistency limit 5
Distinct consistency limit 5

2022 equal divisions of the octave (abbreviated 2022edo or 2022ed2), also called 2022-tone equal temperament (2022tet) or 2022 equal temperament (2022et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2022 equal parts of about 0.593 ¢ each. Each step represents a frequency ratio of 21/2022, or the 2022nd root of 2.

Theory

2022edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. Nonetheless, it offers good appoximations of the 2.3.5.11.17.29.41.43.53.61 subgroup. When using smaller numbers, 2.3.5.11 is a good choice, and if rougher errors are allowed, no-sevens 29-limit is a satisfactory choice.

In the 5-limit, 2022edo supports the pirate temperament, 323 & 407, and tempers out the [-90 -15 49 comma.

In the 2.3.5.11 subgroup, 2022edo supports the rank-3 temperament that eliminates the [25 -17 -23 16 comma. If the 11-limit is taken as a whole, 2022edo tempers out 3025/3024 and 4375/4374 when it is 7/4 is put on the 1633rd step (2022d val), and 41503/41472 with 250047/250000 when using the 1632nd step of the patent val.

In the 2.5.11.17.29.41.43.53.61 subgroup, 2022edo tempers out 17630/17629, 18491/18490, 21200/21199, and 22528/22525.

If the 29-limit is taken as a whole even including the 7-limit inconsistency, 2022edo tempers out 2002/2001, 3451/3450, 5104/5103, and 16445/16443.

Prime harmonics

Approximation of prime harmonics in 2022edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.122 +0.036 -0.280 +0.017 -0.172 +0.089 -0.184 +0.212 +0.096 -0.228
Relative (%) +0.0 +20.6 +6.1 -47.2 +2.9 -28.9 +15.0 -30.9 +35.8 +16.2 -38.5
Steps
(reduced)
2022
(0)
3205
(1183)
4695
(651)
5676
(1632)
6995
(929)
7482
(1416)
8265
(177)
8589
(501)
9147
(1059)
9823
(1735)
10017
(1929)

Subsets and supersets

Since 2022 factors into 2 × 3 × 337, 2022edo has subset edos 2, 3, 6, 337, 674, and 1011.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [3205 -2022 [2022 3205]] -0.038534 0.038533 6.493
2.3.5 [25 -48 22, [-90 -15 49 [2022 3205 4695]] -0.030920 0.033254 5.603
2.3.5.11.13.17.19.23.29 2431/2430, 2755/2754, 3520/3519, 142025/141984, 2582624/2581875, 9096256/9092061, 11293425/11290976, 51054848/51046875 [2022 3205 4695 6955 7482 8265 8589 9147 9283]] -0.010752 0.036910 6.219

Eliora