559edo

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559edo

560edo →

Prime factorization	13 × 43
Step size	2.14669¢
Fifth	327\559 (701.968¢)
Semitones (A1:m2)	53:42 (113.8¢ : 90.16¢)
Consistency limit	11
Distinct consistency limit	11

559 equal divisions of the octave (559edo), or 559-tone equal temperament (559tet), 559 equal temperament (559et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 559 equal parts of about 2.15 ¢ each.

Theory

559edo tempers out the luna comma, [38 -2 -15⟩ and the minortone comma, [-16 35 -17⟩ in the 5-limit, as well as the monzisma, [54 -37 2⟩; 4375/4374, 2100875/2097152, and 282475249/281250000 in the 7-limit; 12005/11979, 41503/41472, 160083/160000, and 172032/171875 in the 11-limit. Rank-2 temperaments it supports include mitonic, lunatic, acrokleismic, monzism, and meridic.

Prime harmonics

Approximation of prime harmonics in 559edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	absolute (¢)	+0.000	+0.013	+0.091	-0.668	+0.382	+0.975	+0.232	+0.877	+0.706	+0.834	-0.850
	relative (%)	+0	+1	+4	-31	+18	+45	+11	+41	+33	+39	-40
Steps (reduced)		559 (0)	886 (327)	1298 (180)	1569 (451)	1934 (257)	2069 (392)	2285 (49)	2375 (139)	2529 (293)	2716 (480)	2769 (533)

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[886 -559⟩	[⟨559 886]]	-0.0040	0.0040	0.19
2.3.5	[38 -2 -15⟩, [-16 35 -17⟩	[⟨559 886 1298]]	-0.0157	0.0168	0.78
2.3.5.7	4375/4374, 2100875/2097152, [-4 -2 -9 10⟩	[⟨559 886 1298 1569]]	+0.0478	0.1109	5.16
2.3.5.7.11	4375/4374, 12005/11979, 41503/41472, 172032/171875	[⟨559 886 1298 1569 1934]]	0.0161	0.1175	5.48

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per Octave	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	90\559	182.47	10/9	Mitonic
1	90\559	193.20	352/315	Lunatic
1	116\559	249.02	[-27 11 3 1⟩	Monzismic
1	147\559	315.56	6/5	Acrokleismic
43	232\559 (2\559)	498.03 (4.29)	4/3 (385/384)	Meridic