3889edo

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3889edo

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Prime factorization	3889 (prime)
Step size	0.308563¢
Fifth	2275\3889 (701.98¢)
Semitones (A1:m2)	369:292 (113.9¢ : 90.1¢)
Consistency limit	27
Distinct consistency limit	27

3889 equal divisions of the octave (abbreviated 3889edo or 3889ed2), also called 3889-tone equal temperament (3889tet) or 3889 equal temperament (3889et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3889 equal parts of about 0.309 ¢ each. Each step represents a frequency ratio of 2^1/3889, or the 3889th root of 2.

Theory

3889edo is consistent to the 27-odd-limit, tempering out 12376/12375, 14400/14399, 6175/6174, 8625/8624, 89376/89375, 123201/123200, 1549184/1549125 and 1990656/1990625 in the 23-limit. It is strong in the 2.3.5.7.13.17.19.23.31 subgroup, tempering out 6175/6174, 14365/14364, 426496/426465, 52326/52325, 52003/52000, 1549184/1549125, 1990656/1990625 and 22816/22815. The equal temperament also tempers out 19251/19250 in the 2.3.5.7.11.23.31 subgroup and 5440/5439 in the 2.3.5.7.17.37 subgroup.

Prime harmonics

Approximation of prime harmonics in 3889edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.025	+0.007	+0.061	+0.083	-0.003	-0.044	-0.059	-0.041	+0.096	+0.040
	Relative (%)	+0.0	+8.1	+2.2	+19.7	+27.0	-1.0	-14.3	-19.0	-13.2	+31.2	+13.1
Steps (reduced)		3889 (0)	6164 (2275)	9030 (1252)	10918 (3140)	13454 (1787)	14391 (2724)	15896 (340)	16520 (964)	17592 (2036)	18893 (3337)	19267 (3711)

Subsets and supersets

3889edo is the 539th prime edo.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[6164 -3889⟩	[⟨3889 6164]]	−0.0079	0.0079	2.56
2.3.5	[-90 -15 49⟩, [56 -91 38⟩	[⟨3889 6164 9030]]	−0.0062	0.0068	2.20
2.3.5.7	[-4 17 1 -9⟩, [2 -20 14 -1⟩, [46 -14 -3 -6⟩	[⟨3889 6164 9030 10918]]	−0.0101	0.0089	2.88
2.3.5.7.11	21437500/21434787, 47265625/47258883, 56953125/56942116, 104857600/104825259	[⟨3889 6164 9030 10918 13454]]	−0.0129	0.0098	3.18
2.3.5.7.11.13	123201/123200, 6656/6655, 1990656/1990625, 492128/492075, 29046875/29042496	[⟨3889 6164 9030 10918 13454 14391]]	−0.0106	0.0103	3.34
2.3.5.7.11.13.17	12376/12375, 14400/14399, 123201/123200, 37180/37179, 1990656/1990625, 361250/361179	[⟨3889 6164 9030 10918 13454 14391 15896]]	−0.0075	0.0121	3.92

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	602\3889	185.755	[24 4 -13⟩	Pirate

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