23ed7/4

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23ed7/4

24ed7/4 →

Prime factorization	23 (prime)
Step size	42.1229¢
Octave	28\23ed7/4 (1179.44¢)
Twelfth	45\23ed7/4 (1895.53¢) (semiconvergent)
Consistency limit	3
Distinct consistency limit	3

23 equal divisions of 7/4 (abbreviated 23ed7/4) is a nonoctave tuning system that divides the interval of 7/4 into 23 equal parts of about 42.1 ¢ each. Each step represents a frequency ratio of (7/4)^1/23, or the 23rd root of 7/4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	42.123
2	84.246	22/21, 23/22
3	126.369	14/13, 15/14
4	168.491	21/19, 23/21
5	210.614	17/15, 25/22
6	252.737	15/13, 22/19
7	294.86	25/21
8	336.983	17/14, 23/19
9	379.106	21/17
10	421.229	19/15
11	463.352	13/10, 17/13, 25/19
12	505.474
13	547.597
14	589.72	7/5
15	631.843	13/9
16	673.966	22/15, 25/17
17	716.089
18	758.212	14/9
19	800.334
20	842.457	21/13
21	884.58	5/3
22	926.703	17/10
23	968.826

Harmonics

Approximation of harmonics in 23ed7/4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-20.6	-6.4	+1.0	-6.2	+15.1	+1.0	-19.6	-12.9	+15.4	+18.8	-5.4
	Relative (%)	-48.8	-15.3	+2.4	-14.7	+35.9	+2.4	-46.4	-30.5	+36.5	+44.7	-12.9
Steps (reduced)		28 (5)	45 (22)	57 (11)	66 (20)	74 (5)	80 (11)	85 (16)	90 (21)	95 (3)	99 (7)	102 (10)

Approximation of harmonics in 23ed7/4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-17.6	-19.6	-12.6	+2.0	-18.7	+8.7	-0.6	-5.2	-5.4	-1.7	+5.6
	Relative (%)	-41.8	-46.4	-30.0	+4.8	-44.4	+20.7	-1.5	-12.3	-12.9	-4.1	+13.2
Steps (reduced)		105 (13)	108 (16)	111 (19)	114 (22)	116 (1)	119 (4)	121 (6)	123 (8)	125 (10)	127 (12)	129 (14)