232edo

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← 231edo232edo233edo →
Prime factorization 23 × 29
Step size 5.17241¢ 
Fifth 136\232 (703.448¢) (→17\29)
Semitones (A1:m2) 24:16 (124.1¢ : 82.76¢)
Consistency limit 5
Distinct consistency limit 5

232 equal divisions of the octave (abbreviated 232edo or 232ed2), also called 232-tone equal temperament (232tet) or 232 equal temperament (232et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 232 equal parts of about 5.17 ¢ each. Each step represents a frequency ratio of 21/232, or the 232nd root of 2.

Theory

232 = 8 × 29, and 232edo shares its fifth with 29edo. The equal temperament supports and provides the optimal patent val for the 13-limit mystery temperament, the rank-3 pele temperament and the rank-3 trimyna temperament and other temperaments tempering out 196/195, for which it gives the optimal patent val for the corresponding rank-5 temperament.

Aside from its patent val, the 232d val 232 368 539 652 803 859] is worth considering. Both temper out the würschmidt comma, 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, 5120/5103 and the trimyna comma, 50421/50000; and 232d 4375/4374 and 16875/16807, supporting octoid. In the 11-limit, the patent val tempers out 441/440 and 896/891, and 232d 540/539, 1375/1372 and 4000/3993. In the 13-limit, the patent val tempers out 196/195, 352/351, 364/363, 676/675, and 847/845, which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.

Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows werckismic chords, because it tempers out 196/195 it allows mynucumic chords, because it tempers out 352/351 it allows major minthmic chords, and because it tempers out 364/363 it allows minor minthmic chords, and because it tempers out 847/845 it allows the cuthbert chords, making it a very flexible harmonic system.

Odd harmonics

Approximation of odd harmonics in 232edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.49 +1.62 -1.58 -2.19 +2.13 +2.58 -2.06 -1.51 +2.49 -0.09 -2.41
Relative (%) +28.9 +31.3 -30.6 -42.3 +41.2 +49.8 -39.9 -29.1 +48.1 -1.8 -46.6
Steps
(reduced)
368
(136)
539
(75)
651
(187)
735
(39)
803
(107)
859
(163)
906
(210)
948
(20)
986
(58)
1019
(91)
1049
(121)

Subsets and supersets

Since 232 factors into 23 × 29, 232edo has subset edos 2, 4, 8, 29, 58, and 116.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 393216/390625, [46 -29 0 [232 368 539]] -0.5461 0.3989 7.71

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 61\232 315.52 6/5 Acrokleismic (7-limit, 232d)
1 75\232 387.93 5/4 Würschmidt (5-limit)
8 113\232
(3\232)
584.48
(15.52)
7/5
(100/99)
Octoid (232d)
29 96\232
(3\232)
496.55
(15.52)
4/3
(105/104)
Mystery (232)