23ed4

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23ed4

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Prime factorization	23 (prime)
Step size	104.348¢
Octave	12\23ed4 (1252.17¢)
Twelfth	18\23ed4 (1878.26¢)
Consistency limit	1
Distinct consistency limit	1

23 equal divisions of the 4th harmonic (abbreviated 23ed4) is a nonoctave tuning system that divides the interval of 4/1 into 23 equal parts of about 104 ¢ each. Each step represents a frequency ratio of 4^1/23, or the 23rd root of 4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	104.348	18/17, 19/18
2	208.696	17/15, 25/22
3	313.043	6/5
4	417.391	14/11, 19/15, 23/18
5	521.739	19/14, 23/17
6	626.087
7	730.435	23/15
8	834.783
9	939.13	19/11
10	1043.478	11/6
11	1147.826
12	1252.174
13	1356.522	11/5
14	1460.87	7/3
15	1565.217
16	1669.565
17	1773.913	14/5
18	1878.261
19	1982.609	19/6
20	2086.957
21	2191.304
22	2295.652
23	2400

Harmonics

Approximation of harmonics in 23ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+52.2	-23.7	+0.0	+31.1	+28.5	-29.7	+52.2	-47.4	-21.1	+22.6	-23.7
	Relative (%)	+50.0	-22.7	+0.0	+29.8	+27.3	-28.5	+50.0	-45.4	-20.2	+21.7	-22.7
Steps (reduced)		12 (12)	18 (18)	23 (0)	27 (4)	30 (7)	32 (9)	35 (12)	36 (13)	38 (15)	40 (17)	41 (18)

Approximation of harmonics in 23ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+46.4	+22.5	+7.4	+0.0	-0.6	+4.8	+15.5	+31.1	+51.0	-29.6	-2.2
	Relative (%)	+44.5	+21.5	+7.1	+0.0	-0.6	+4.6	+14.9	+29.8	+48.8	-28.3	-2.1
Steps (reduced)		43 (20)	44 (21)	45 (22)	46 (0)	47 (1)	48 (2)	49 (3)	50 (4)	51 (5)	51 (5)	52 (6)