20203edo

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20203edo

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Prime factorization	89 × 227
Step size	0.0593971 ¢
Fifth	11818\20203 (701.955 ¢)
Semitones (A1:m2)	1914:1519 (113.7 ¢ : 90.22 ¢)
Consistency limit	45
Distinct consistency limit	45

20203 equal divisions of the octave (abbreviated 20203edo or 20203ed2), also called 20203-tone equal temperament (20203tet) or 20203 equal temperament (20203et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 20203 equal parts of about 0.0594 ¢ each. Each step represents a frequency ratio of 2^1/20203, or the 20203rd root of 2.

20203edo is a very strong high-limit system, and specializes in the 17- and 19-limit, with lower 17- and 19-limit relative errors than any smaller edo until 102557 and 128215, respectively. It is also distinctly consistent through the 45-odd-limit, and has a lower 43-limit relative error than any smaller edo except for 7361, 14348 and 17461.

A 43-limit comma basis for this temperament is {29792/29791, 32799/32798, 43264/43263, 45696/45695, 47151/47150, 52326/52325, 53361/53360, 69875/69874, 81796/81795, 83521/83520, 87465/87464, 96876/96875, 111112/111111}. In the 13-limit it tempers out 123201/123200 and 1990656/1990625; in the 17-limit 194481/194480 and 336141/336140; in the 19-limit 89376/89375, 104976/104975, and 165376/165375; in the 23-limit 43264/43263 and 52326/52325 among others.

Prime harmonics

Approximation of prime harmonics in 20203edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0000	+0.0002	+0.0051	+0.0005	+0.0061	+0.0010	-0.0007	+0.0072	+0.0284	+0.0125	+0.0221
	Relative (%)	+0.0	+0.3	+8.7	+0.9	+10.3	+1.6	-1.2	+12.0	+47.8	+21.0	+37.2
Steps (reduced)		20203 (0)	32021 (11818)	46910 (6504)	56717 (16311)	69891 (9282)	74760 (14151)	82579 (1767)	85821 (5009)	91390 (10578)	98146 (17334)	100090 (19278)

Approximation of prime harmonics in 20203edo (continued)

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	+0.0246	+0.0224	+0.0103	-0.0213	-0.0105	-0.0022	+0.0186	-0.0119	-0.0203	-0.0015	-0.0168
	Relative (%)	+41.4	+37.7	+17.3	-35.9	-17.7	-3.8	+31.4	-20.0	-34.1	-2.6	-28.2
Steps (reduced)		105247 (4232)	108239 (7224)	109627 (8612)	112219 (11204)	115721 (14706)	118847 (17832)	119819 (18804)	122553 (1335)	124243 (3025)	125053 (3835)	127355 (6137)