# 422edo

 ← 421edo 422edo 423edo →
Prime factorization 2 × 211
Step size 2.8436¢
Fifth 247\422 (702.37¢)
Semitones (A1:m2) 41:31 (116.6¢ : 88.15¢)
Consistency limit 27
Distinct consistency limit 27
Special properties

422 equal divisions of the octave (abbreviated 422edo or 422ed2), also called 422-tone equal temperament (422tet) or 422 equal temperament (422et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 422 equal parts of about 2.84 ¢ each. Each step represents a frequency ratio of 21/422, or the 422nd root of 2.

## Theory

422edo is a zeta peak edo, though not zeta integral nor zeta gap. It is distinctly consistent through the 27-odd-limit, with harmonics of 3 through 23 all tuned sharp. The equal temperament tempers out the vishnuzma, [23 6 -14 and the countritonic comma, [33 -34 9, in the 5-limit; 4375/4374 and 589824/588245 in the 7-limit; 3025/3024, 5632/5625, and 9801/9800 in the 11-limit; 1716/1715, 2080/2079, and 2200/2197 in the 13-limit; 1156/1155, 1275/1274, and 2431/2430 in the 17-limit; 1216/1215, 1331/1330, 1445/1444, and 2432/2431 in the 19-limit. It supports provides the optimal patent vals for gamera in the 7-limit and hemigamera in the 13-limit. Other notable temperaments it supports are vishnu, semisupermajor, and countritonic.

### Prime harmonics

Approximation of prime harmonics in 422edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.41 +0.42 +0.84 +0.34 +1.18 +0.26 +1.07 +0.16 -0.19 +0.94
Relative (%) +0.0 +14.6 +14.6 +29.6 +12.0 +41.4 +9.1 +37.5 +5.7 -6.8 +32.9
Steps
(reduced)
422
(0)
669
(247)
980
(136)
1185
(341)
1460
(194)
1562
(296)
1725
(37)
1793
(105)
1909
(221)
2050
(362)
2091
(403)

### Subsets and supersets

Since 422 factors into 2 × 211, 422edo has subset edos 2edo and 211edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [669 -422 [422 669]] -0.1308 0.1308 4.60
2.3.5 [23 6 -14, [33 -34 9 [422 669 980]] -0.1469 0.1092 3.84
2.3.5.7 4375/4374, 589824/588245, 29360128/29296875 [422 669 980 1185]] -0.1852 0.1155 4.06
2.3.5.7.11 3025/3024, 4375/4374, 5632/5625, 589824/588245 [422 669 980 1185 1460]] -0.1679 0.1090 3.83
2.3.5.7.11.13 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625 [422 669 980 1185 1460 1562]] -0.1930 0.1142 4.02
2.3.5.7.11.13.17 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430 [422 669 980 1185 1460 1562 1725]] -0.1744 0.1151 4.05
2.3.5.7.11.13.17.19 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197 [422 669 980 1185 1460 1562 1725 1793]] -0.1839 0.1106 3.89
• 422et has lower absolute errors than any previous equal temperaments in the 17-, 19- and 23-limit. In the 17- and 19-limit it beats 400 and is bettered by 460. In the 23-limit it beats 373g and is bettered by 525.

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 81\422 230.33 8/7 Gamera