31ed5/4

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31ed5/4

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Prime factorization	31 (prime)
Step size	12.4617¢
Octave	96\31ed5/4 (1196.33¢)
Twelfth	153\31ed5/4 (1906.65¢)
Consistency limit	2
Distinct consistency limit	2

31 equal divisions of 5/4 (abbreviated 31ed5/4) is a nonoctave tuning system that divides the interval of 5/4 into 31 equal parts of about 12.5⁠ ⁠¢ each. Each step represents a frequency ratio of (5/4)^1/31, or the 31st root of 5/4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	12.5
2	24.9
3	37.4
4	49.8
5	62.3
6	74.8	22/21
7	87.2	19/18, 20/19, 21/20
8	99.7	18/17
9	112.2
10	124.6	14/13
11	137.1	13/12, 27/25
12	149.5	12/11, 25/23
13	162	11/10, 23/21
14	174.5	10/9, 21/19
15	186.9	19/17
16	199.4
17	211.8	17/15
18	224.3	8/7
19	236.8
20	249.2	22/19, 23/20
21	261.7	7/6
22	274.2	20/17
23	286.6	13/11, 27/23
24	299.1
25	311.5	6/5, 25/21
26	324
27	336.5	23/19
28	348.9
29	361.4	21/17, 26/21
30	373.9
31	386.3

Harmonics

Approximation of harmonics in 31ed5/4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-3.67	+4.69	+5.11	+5.11	+1.02	-4.16	+1.44	-3.08	+1.44	-1.56	-2.66
	Relative (%)	-29.5	+37.6	+41.0	+41.0	+8.2	-33.4	+11.6	-24.7	+11.6	-12.5	-21.3
Steps (reduced)		96 (3)	153 (29)	193 (7)	224 (7)	249 (1)	270 (22)	289 (10)	305 (26)	320 (10)	333 (23)	345 (4)

Approximation of harmonics in 31ed5/4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-4.15	+4.63	-2.66	-2.23	+4.97	+5.71	-0.66	-2.23	+0.53	-5.23	+5.04
	Relative (%)	-33.3	+37.2	-21.3	-17.9	+39.9	+45.8	-5.3	-17.9	+4.3	-42.0	+40.5
Steps (reduced)		356 (15)	367 (26)	376 (4)	385 (13)	394 (22)	402 (30)	409 (6)	416 (13)	423 (20)	429 (26)	436 (2)