412edo

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← 411edo412edo413edo →
Prime factorization 22 × 103
Step size 2.91262¢
Fifth 241\412 (701.942¢)
Semitones (A1:m2) 39:31 (113.6¢ : 90.29¢)
Consistency limit 9
Distinct consistency limit 9

412 equal divisions of the octave (abbreviated 412edo), or 412-tone equal temperament (412tet), 412 equal temperament (412et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 412 equal parts of about 2.91 ¢ each. Each step of 412edo represents a frequency ratio of 21/412, or the 412th root of 2.

Theory

412edo has a very accurate perfect fifth, but it is not quite accurate beyond that. The equal temperament tempers out [32 -7 -9 (escapade comma) and [-69 45 -1 (counterschisma) in the 5-limit; 6144/6125, 118098/117649, 2460375/2458624, 49009212/48828125, and notably the nanisma in the 7-limit. It supports nanic and counterschismic.

Prime harmonics

Approximation of prime harmonics in 412edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.00 -0.01 +1.06 +1.08 -0.83 +1.22 -0.10 -0.43 +0.85 -1.42 -0.38
relative (%) +0 -0 +37 +37 -29 +42 -3 -15 +29 -49 -13
Steps
(reduced)
412
(0)
653
(241)
957
(133)
1157
(333)
1425
(189)
1525
(289)
1684
(36)
1750
(102)
1864
(216)
2001
(353)
2041
(393)

Subsets and supersets

412 factors into 22 × 103, with subset edos 2, 4, 103, and 206. 1236edo, which triples it, gives a good correction to harmonics 5, 7, and 11.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-653 412 [412 653]] +0.0042 0.0042 0.14
2.3.5 [32 -7 -9, [-5 31 -19 [412 653 957]] -0.1501 0.2182 7.49
2.3.5.7 6144/6125, 2460375/2458624, 49009212/48828125 [412 653 957 1157]] -0.2085 0.2143 7.36

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 9\412 26.21 49/48 Sfourth (5-limit)
1 19\412 55.34 16875/16384 Escapade (5-limit)
1 171\412 498.06 4/3 Counterschismic
Nanic
2 19\412 55.34 16875/16384 Septisuperfourth (7-limit)

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct