31ed4

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31ed4

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Prime factorization	31 (prime)
Step size	77.4194¢
Octave	16\31ed4 (1238.71¢)
Twelfth	25\31ed4 (1935.48¢)
Consistency limit	1
Distinct consistency limit	1

31 equal divisions of the 4th harmonic (abbreviated 31ed4) is a nonoctave tuning system that divides the interval of 4/1 into 31 equal parts of about 77.4 ¢ each. Each step represents a frequency ratio of 4^1/31, or the 31st root of 4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	77.4	21/20, 22/21, 26/25
2	154.8
3	232.3
4	309.7
5	387.1
6	464.5	17/13
7	541.9	15/11, 26/19
8	619.4	10/7
9	696.8	3/2
10	774.2	11/7, 14/9
11	851.6
12	929
13	1006.5
14	1083.9
15	1161.3
16	1238.7
17	1316.1	15/7
18	1393.5
19	1471	7/3
20	1548.4	22/9
21	1625.8
22	1703.2
23	1780.6
24	1858.1
25	1935.5
26	2012.9
27	2090.3	10/3
28	2167.7	7/2
29	2245.2	11/3
30	2322.6
31	2400

Harmonics

Approximation of harmonics in 31ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+38.7	+33.5	+0.0	+0.8	-5.2	+37.6	+38.7	-10.4	-37.9	+29.3	+33.5
	Relative (%)	+50.0	+43.3	+0.0	+1.0	-6.7	+48.6	+50.0	-13.4	-49.0	+37.9	+43.3
Steps (reduced)		16 (16)	25 (25)	31 (0)	36 (5)	40 (9)	44 (13)	47 (16)	49 (18)	51 (20)	54 (23)	56 (25)

Approximation of harmonics in 31ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-27.6	-1.1	+34.3	+0.0	-27.5	+28.3	+12.2	+0.8	-6.3	-9.4	-8.9
	Relative (%)	-35.7	-1.4	+44.3	+0.0	-35.6	+36.6	+15.7	+1.0	-8.1	-12.1	-11.5
Steps (reduced)		57 (26)	59 (28)	61 (30)	62 (0)	63 (1)	65 (3)	66 (4)	67 (5)	68 (6)	69 (7)	70 (8)