424edo

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← 423edo424edo425edo →
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 424-tone equal temperament (424tet), 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 of 424edo 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 -50 -32 -5 +20 +1 +48 -8 -12 -34 +1
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