23ed5/4

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23ed5/4

24ed5/4 →

Prime factorization	23 (prime)
Step size	16.7962¢
Octave	71\23ed5/4 (1192.53¢)
Twelfth	113\23ed5/4 (1897.98¢)
Consistency limit	3
Distinct consistency limit	2

23 equal divisions of 5/4 (abbreviated 23ed5/4) is a nonoctave tuning system that divides the interval of 5/4 into 23 equal parts of about 16.8 ¢ each. Each step represents a frequency ratio of (5/4)^1/23, or the 23rd root of 5/4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	16.8
2	33.6
3	50.4
4	67.2	22/21
5	84	20/19, 23/22
6	100.8	19/18
7	117.6	15/14
8	134.4	14/13
9	151.2	13/12, 23/21, 25/23
10	168	11/10
11	184.8	10/9, 19/17, 21/19
12	201.6
13	218.4	17/15
14	235.1
15	251.9	15/13, 22/19, 23/20
16	268.7	20/17
17	285.5	13/11
18	302.3	6/5, 25/21
19	319.1
20	335.9	17/14, 23/19
21	352.7	11/9
22	369.5	21/17
23	386.3

Harmonics

Approximation of harmonics in 23ed5/4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-7.47	-3.98	+1.86	+1.86	+5.35	+7.22	-5.60	-7.96	-5.60	-2.64	-2.12
	Relative (%)	-44.5	-23.7	+11.1	+11.1	+31.9	+43.0	-33.4	-47.4	-33.4	-15.7	-12.6
Steps (reduced)		71 (2)	113 (21)	143 (5)	166 (5)	185 (1)	201 (17)	214 (7)	226 (19)	237 (7)	247 (17)	256 (3)

Approximation of harmonics in 23ed5/4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-6.32	-0.25	-2.12	+3.73	-0.45	+1.37	-8.25	+3.73	+3.24	+6.69	-3.09
	Relative (%)	-37.6	-1.5	-12.6	+22.2	-2.7	+8.2	-49.1	+22.2	+19.3	+39.8	-18.4
Steps (reduced)		264 (11)	272 (19)	279 (3)	286 (10)	292 (16)	298 (22)	303 (4)	309 (10)	314 (15)	319 (20)	323 (1)