432edo

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432edo

433edo →

Prime factorization	24 × 33
Step size	2.77778¢
Fifth	253\432 (702.778¢)
Semitones (A1:m2)	43:31 (119.4¢ : 86.11¢)
Consistency limit	7
Distinct consistency limit	7

432 equal divisions of the octave (abbreviated 432edo or 432ed2), also called 432-tone equal temperament (432tet) or 432 equal temperament (432et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 432 equal parts of about 2.78 ¢ each. Each step represents a frequency ratio of 2^1/432, or the 432nd root of 2.

Theory

432edo has a reasonable approximation to the 7-limit, where the equal temperament tempers out 5120/5103, 703125/702464, 40353607/40310784, 102760448/102515625, and 283115520/282475249. It supports and provides a good tuning for the 5-limit maja temperament.

Odd harmonics

Approximation of odd harmonics in 432edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+0.82	-0.20	+0.62	-1.13	-1.32	+1.14	+0.62	+0.60	-0.29	-1.34	-0.50
	relative (%)	+30	-7	+22	-41	-47	+41	+22	+22	-10	-48	-18
Steps (reduced)		685 (253)	1003 (139)	1213 (349)	1369 (73)	1494 (198)	1599 (303)	1688 (392)	1766 (38)	1835 (107)	1897 (169)	1954 (226)

Subsets and supersets

432 is a highly factorable number, factoring into 2⁴ × 3³, so 432edo has subset edos 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, and 216.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[685 -432⟩	[⟨432 685]]	-0.2596	0.2595	9.34
2.3.5	[41 -20 -4⟩, [-3 -23 17⟩	[⟨432 685 1003]]	-0.1440	0.2676	9.63
2.3.5.7	5120/5103, 703125/702464, 6565234375/6530347008	[⟨432 685 1003 1213]]	-0.1631	0.2341	8.43

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	163\432	452.78	125/96	Maja
4	179\432 (37\432)	497.22 (102.78)	4/3	Undim

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