# 424edo

 ← 423edo 424edo 425edo →
Prime factorization 23 × 53
Step size 2.83019¢
Fifth 248\424 (701.887¢) (→31\53)
Semitones (A1:m2) 40:32 (113.2¢ : 90.57¢)
Consistency limit 9
Distinct consistency limit 9

424 equal divisions of the octave (abbreviated 424edo or 424ed2), also called 424-tone equal temperament (424tet) or 424 equal temperament (424et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 424 equal parts of about 2.83 ¢ each. Each step represents a frequency ratio of 21/424, or the 424th root of 2.

## Theory

424edo is consistent to the 9-odd-limit, but the harmonic 5 is about halfway between its steps. It is enfactored in the 7-limit, with the same tuning as 212edo. The approximation to 11, although closer to just than 212edo's, tends sharp, so its improvement is debatable. All things considered, a 2.3.13.17.19.23 subgroup interpretation with optional additions of 7, 11, or both, seems most reasonable.

### Odd harmonics

Approximation of odd harmonics in 424edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.07 -1.41 -0.90 -0.14 +0.57 +0.04 +1.35 -0.24 -0.34 -0.97 +0.03
Relative (%) -2.4 -49.8 -31.8 -4.8 +20.1 +1.4 +47.8 -8.4 -12.1 -34.3 +1.0
Steps
(reduced)
672
(248)
984
(136)
1190
(342)
1344
(72)
1467
(195)
1569
(297)
1657
(385)
1733
(37)
1801
(105)
1862
(166)
1918
(222)

### Subsets and supersets

Since 424 factors into 23 × 53, 424edo has subset edos 2, 4, 8, 53, 106, and 212. 848edo, which doubles it, gives a good correction to the harmonic 5.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.7.11 41503/41472, 117649/117128, [-26 19 1 -2 [424 672 1190 1467]] +0.0499 0.1747 6.17