4ed5/3

4 equal divisions of 5/3 (abbreviated 4ed5/3) is a nonoctave tuning system that divides the interval of 5/3 into 4 equal parts of about 221 ¢ each. Each step represents a frequency ratio of (5/3)^1/4, or the 4th root of 5/3.

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4ed5/3

5ed5/3 →

Prime factorization	22 (highly composite)
Step size	221.09 ¢
Octave	5\4ed5/3 (1105.45 ¢)
Twelfth	9\4ed5/3 (1989.81 ¢)
Consistency limit	2
Distinct consistency limit	2

Theory

This tuning tempers out 10/9 in the 5-limit; 8/7 in the 7-limit; 12/11 and 22/21 in the 11-limit; 14/13 and 16/13 in the 13-limit; 17/15 in the 17-limit; 19/18 and 20/19 in the 19-limit; 26/23 and 28/23 in the 23-limit; 29/25 in the 29-limit; 31/30 in the 31-limit; and 43/41 in the 41-limit.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1, 19/18, 20/19
1	221.1	6/5, 7/6, 11/10, 13/11, 13/12, 19/17, 20/17, 21/19
2	442.2	13/10, 17/13, 21/17
3	663.3	10/7, 17/11, 17/12, 19/13, 20/13
4	884.4	5/3, 12/7, 17/10, 18/11, 19/11, 19/12, 21/13

Harmonics

Approximation of prime harmonics in 4ed5/3

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89
Error	Absolute (¢)	-94.6	+87.9	+87.9	-52.5	+49.4	-18.7	-41.0	-12.5	+99.0	-81.2	+24.4	-60.8	-17.5	-99.9	-32.8	-19.7	+15.7	-42.0	+16.7	-83.7	+89.3	-47.5	+88.1	-32.7
	Relative (%)	-42.8	+39.7	+39.7	-23.7	+22.3	-8.5	-18.5	-5.6	+44.8	-36.7	+11.0	-27.5	-7.9	-45.2	-14.8	-8.9	+7.1	-19.0	+7.5	-37.9	+40.4	-21.5	+39.8	-14.8
Steps (reduced)		5 (1)	9 (1)	13 (1)	15 (3)	19 (3)	20 (0)	22 (2)	23 (3)	25 (1)	26 (2)	27 (3)	28 (0)	29 (1)	29 (1)	30 (2)	31 (3)	32 (0)	32 (0)	33 (1)	33 (1)	34 (2)	34 (2)	35 (3)	35 (3)

Approximation of prime harmonics in 4ed5/3

Harmonic		97	101	103	107	109	113	127	131	137	139	149	151	157	163	167	173	179	181	191	193	197	199	211	223
Error	Absolute (¢)	+39.3	-30.6	-64.6	+90.6	+58.5	-3.9	+15.0	-38.7	+104.9	+79.8	-40.5	-63.6	+90.0	+25.1	-16.9	-78.0	+84.1	+64.9	-28.2	-46.3	-81.8	-99.3	+20.4	-75.3
	Relative (%)	+17.8	-13.9	-29.2	+41.0	+26.5	-1.8	+6.8	-17.5	+47.4	+36.1	-18.3	-28.8	+40.7	+11.4	-7.6	-35.3	+38.0	+29.3	-12.8	-20.9	-37.0	-44.9	+9.2	-34.1
Steps (reduced)		36 (0)	36 (0)	36 (0)	37 (1)	37 (1)	37 (1)	38 (2)	38 (2)	39 (3)	39 (3)	39 (3)	39 (3)	40 (0)	40 (0)	40 (0)	40 (0)	41 (1)	41 (1)	41 (1)	41 (1)	41 (1)	41 (1)	42 (2)	42 (2)