416edo

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← 415edo416edo417edo →
Prime factorization 25 × 13
Step size 2.88462¢
Fifth 243\416 (700.962¢)
Semitones (A1:m2) 37:33 (106.7¢ : 95.19¢)
Dual sharp fifth 244\416 (703.846¢) (→61\104)
Dual flat fifth 243\416 (700.962¢)
Dual major 2nd 71\416 (204.808¢)
Consistency limit 7
Distinct consistency limit 7

416 equal divisions of the octave (abbreviated 416edo), or 416-tone equal temperament (416tet), 416 equal temperament (416et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 416 equal parts of about 2.88 ¢ each. Each step of 416edo represents a frequency ratio of 21/416, or the 416th root of 2.

Theory

416et is consistent to the 7-odd-limit, but the error of harmonic 3 is quite large. Nonetheless, it gives reasonable approximations to harmonics 5, 7, 9, 11, 19, and 23, making it suitable for a 2.9.5.7.11.19.23 subgroup interpretation, where it notably tempers out 1331/1330, 1540/1539, 5632/5625, 9801/9800, and 10241/10240.

Using the patent val regardless, it tempers out 321489/320000, 589824/588245, and 703125/702464, supporting tridecatonic and fermionic.

Odd harmonics

Approximation of odd harmonics in 416edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) -0.99 +0.22 +0.40 +0.90 -0.36 -1.10 -0.77 -1.11 -0.40 -0.59 +0.57
relative (%) -34 +8 +14 +31 -12 -38 -27 -38 -14 -20 +20
Steps
(reduced)
659
(243)
966
(134)
1168
(336)
1319
(71)
1439
(191)
1539
(291)
1625
(377)
1700
(36)
1767
(103)
1827
(163)
1882
(218)

Subsets and supersets

Since 416 factors into 25 × 13, 416edo subset edos 2, 4, 8, 13, 16, 26, 32, 52, 104, and 208. 832edo, which doubles it, gives a good correction to the harmonic 3.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [1319 -416 [416 1319]] -0.1416 0.1416 4.91
2.9.5 [56 -14 -5, [-5 -16 24 [416 1319 966]] -0.1267 0.1175 4.07
2.9.5.7 420175/419904, 102760448/102515625, [14 -6 7 -4 [416 1319 966 1168]] -0.1310 0.1021 3.54
2.9.5.7.11 5632/5625, 9801/9800, 41503/41472, 774400000/771895089 [416 1319 966 1168 1439]] -0.0842 0.1308 4.53