# 416edo

 ← 415edo 416edo 417edo →
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 416ed2), also called 416-tone equal temperament (416tet) or 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 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 (%) -34.4 +7.8 +14.0 +31.1 -12.4 -38.3 -26.6 -38.5 -13.8 -20.4 +19.8
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

## Music

birdshite stalactite