31ed5/2

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31ed5/2

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Prime factorization	31 (prime)
Step size	51.1714¢
Octave	23\31ed5/2 (1176.94¢)
Twelfth	37\31ed5/2 (1893.34¢) (semiconvergent)
Consistency limit	3
Distinct consistency limit	3

31 equal divisions of 5/2 (abbreviated 31ed5/2) is a nonoctave tuning system that divides the interval of 5/2 into 31 equal parts of about 51.2 ¢ each. Each step represents a frequency ratio of (5/2)^1/31, or the 31st root of 5/2.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	51.171
2	102.343
3	153.514	23/21
4	204.686	26/23
5	255.857
6	307.028	6/5
7	358.2	21/17, 27/22
8	409.371	14/11
9	460.543	17/13
10	511.714
11	562.886	25/18
12	614.057
13	665.228	19/13, 22/15
14	716.4
15	767.571	14/9
16	818.743
17	869.914
18	921.085
19	972.257
20	1023.428	9/5
21	1074.6	13/7
22	1125.771	21/11
23	1176.942
24	1228.114
25	1279.285	23/11, 25/12
26	1330.457
27	1381.628
28	1432.799
29	1483.971	26/11
30	1535.142	17/7
31	1586.314	5/2

Harmonics

Approximation of harmonics in 31ed5/2

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-23.1	-8.6	+5.1	-23.1	+19.5	+8.5	-18.0	-17.2	+5.1	-6.4	-3.6
	Relative (%)	-45.1	-16.8	+9.9	-45.1	+38.1	+16.6	-35.2	-33.7	+9.9	-12.6	-7.0
Steps (reduced)		23 (23)	37 (6)	47 (16)	54 (23)	61 (30)	66 (4)	70 (8)	74 (12)	78 (16)	81 (19)	84 (22)

Approximation of harmonics in 31ed5/2

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+11.4	-14.6	+19.5	+10.1	+7.5	+10.9	+19.6	-18.0	-0.1	+21.7	-4.1
	Relative (%)	+22.2	-28.5	+38.1	+19.8	+14.7	+21.3	+38.4	-35.2	-0.2	+42.4	-8.0
Steps (reduced)		87 (25)	89 (27)	92 (30)	94 (1)	96 (3)	98 (5)	100 (7)	101 (8)	103 (10)	105 (12)	106 (13)