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	77.419	21/20, 22/21, 26/25
2	154.839
3	232.258
4	309.677
5	387.097
6	464.516	17/13
7	541.935	15/11, 26/19
8	619.355	10/7
9	696.774	3/2
10	774.194	11/7, 14/9
11	851.613
12	929.032
13	1006.452
14	1083.871
15	1161.29
16	1238.71
17	1316.129	15/7
18	1393.548
19	1470.968	7/3
20	1548.387	22/9
21	1625.806
22	1703.226
23	1780.645
24	1858.065
25	1935.484
26	2012.903
27	2090.323	10/3
28	2167.742	7/2
29	2245.161	11/3
30	2322.581
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)