2190edo

← 2189edo

2190edo

2191edo →

Prime factorization	2 × 3 × 5 × 73
Step size	0.547945 ¢
Fifth	1281\2190 (701.918 ¢) (→ 427\730)
Semitones (A1:m2)	207:165 (113.4 ¢ : 90.41 ¢)
Consistency limit	15
Distinct consistency limit	15

2190 equal divisions of the octave (abbreviated 2190edo or 2190ed2), also called 2190-tone equal temperament (2190tet) or 2190 equal temperament (2190et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2190 equal parts of about 0.548 ¢ each. Each step represents a frequency ratio of 2^1/2190, or the 2190th root of 2.

2190edo is a very strong 13-limit system; no smaller division has a smaller 13-limit relative error, and nothing beats it until 2684. It is closely related to 730edo, the Woolhouse unit system, with which it shares the same tuning in the 5-limit, but the harmonics 7, 11, and 13 are all mapped differently. A basis for the 13-limit commas is {9801/9800, 10648/10647, 105644/105625, 140625/140608, 196625/196608}; also tempered out are 123201/123200, 151263/151250, and 250047/250000.

It is not as impressive beyond the 13-limit, though it does well in the 2.3.5.7.11.13.19 subgroup, where it holds the record of lowest relative error until 6079, and the 2.3.5.7.11.13.19.29 subgroup, where it holds the record of lowest relative error until 14618.

Prime harmonics

Approximation of prime harmonics in 2190edo

Harmonic		2	3	5	7	11	13	17	19	23
Error	Absolute (¢)	+0.000	-0.037	-0.012	-0.059	-0.085	+0.020	+0.250	+0.021	+0.219
	Relative (%)	+0.0	-6.8	-2.3	-10.7	-15.5	+3.7	+45.6	+3.9	+39.9
Steps (reduced)		2190 (0)	3471 (1281)	5085 (705)	6148 (1768)	7576 (1006)	8104 (1534)	8952 (192)	9303 (543)	9907 (1147)

Approximation of prime harmonics in 2190edo (continued)

Harmonic		29	31	37	41	43	47	53	59	61
Error	Absolute (¢)	+0.012	+0.170	+0.163	-0.021	+0.263	+0.247	-0.080	+0.006	-0.172
	Relative (%)	+2.2	+31.0	+29.7	-3.9	+48.0	+45.0	-14.6	+1.2	-31.5
Steps (reduced)		10639 (1879)	10850 (2090)	11409 (459)	11733 (783)	11884 (934)	12165 (1215)	12544 (1594)	12883 (1933)	12988 (2038)

Subsets and supersets

Since 2190 factors into primes as 2 × 3 × 5 × 73, 2190edo has subset edos 2, 3, 5, 6, 10, 12, 15, 30, 73, 146, 219, 365, 438, 730, and 1095. A step of 2190edo is exactly 1⁄3 Woolhouse unit.

4380edo, which doubles 2190edo, provides a good correction to the harmonics 17 and 23.