# 412edo

 ← 411edo 412edo 413edo →
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 412ed2), also called 412-tone equal temperament (412tet) or 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 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 -0.5 +36.6 +37.0 -28.6 +41.9 -3.5 -14.6 +29.2 -48.8 -12.9
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