431edo

431edo

Prime factorization	431 (prime)
Step size	2.78422¢
Fifth	252\431 (701.62¢)
Semitones (A1:m2)	40:33 (111.37¢ : 91.88¢)
Consistency limit	15

The 431 equal divisions of the octave (431edo), or the 431(-tone) equal temperament (431tet, 431et) when viewed from a regular temperament perspective, is the equal division of the octave into 431 parts of about 2.78 cents each.

Theory

431edo is consistent to the 15-odd-limit, tempering out the schisma in the 5-limit; 2401/2400 in the 7-limit; 5632/5625 and 8019/8000 in the 11-limit; 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655 and 10648/10647 in the 13-limit. It supports the sesquiquartififths temperament.

431edo is the 83rd prime edo.

Prime harmonics

Approximation of prime harmonics in 431edo

Harmonic		2	3	5	7	11	13	17	19	23	29
Error	absolute (¢)	+0.00	-0.33	+0.69	+0.08	-0.04	+0.31	+0.85	+0.40	+0.96	+0.59
	relative (%)	+0	-12	+25	+3	-2	+11	+30	+14	+34	+21
Steps (reduced)		431 (0)	683 (252)	1001 (139)	1210 (348)	1491 (198)	1595 (302)	1762 (38)	1831 (107)	1950 (226)	2094 (370)

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-683 431⟩	[⟨431 683]]	+0.1044	0.1044	3.75
2.3.5	32805/32768, [7 63 -46⟩	[⟨431 683 1001]]	-0.0230	0.2082	7.48
2.3.5.7	2401/2400, 32805/32768, [3 16 -11 -1⟩	[⟨431 683 1001 1210]]	-0.0299	0.1803	6.48
2.3.5.7.11	2401/2400, 5632/5625, 8019/8000, 43923/43904	[⟨431 683 1001 1210 1491]]	-0.0215	0.1621	5.82
2.3.5.7.11.13	729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655	[⟨431 683 1001 1210 1491 1595]]	-0.0318	0.1498	5.38

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	63\431	175.41	448/405	Sesquiquartififths
1	172\431	498.55	4/3	Helmholtz