31ed7/4

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31ed7/4

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Prime factorization	31 (prime)
Step size	31.2524¢
Octave	38\31ed7/4 (1187.59¢)
Twelfth	61\31ed7/4 (1906.4¢)
Consistency limit	2
Distinct consistency limit	2

31 equal divisions of 7/4 (abbreviated 31ed7/4) is a nonoctave tuning system that divides the interval of 7/4 into 31 equal parts of about 31.3 ¢ each. Each step represents a frequency ratio of (7/4)^1/31, or the 31st root of 7/4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	31.252
2	62.505	26/25
3	93.757	18/17, 19/18
4	125.01	14/13, 15/14
5	156.262	23/21
6	187.515	19/17
7	218.767	17/15, 25/22
8	250.02	15/13, 22/19
9	281.272	13/11, 27/23
10	312.524	6/5
11	343.777	11/9, 17/14
12	375.029	21/17
13	406.282	19/15
14	437.534	9/7, 22/17
15	468.787	17/13, 25/19
16	500.039
17	531.292	15/11, 19/14, 23/17
18	562.544	18/13, 25/18
19	593.797
20	625.049
21	656.301	19/13, 22/15
22	687.554
23	718.806
24	750.059	17/11
25	781.311	11/7
26	812.564
27	843.816	18/11
28	875.069	5/3
29	906.321	22/13
30	937.573	19/11
31	968.826

Harmonics

Approximation of harmonics in 31ed7/4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-12.4	+4.4	+6.4	-4.8	-8.0	+6.4	-6.0	+8.9	+14.0	+5.3	+10.9
	Relative (%)	-39.7	+14.2	+20.6	-15.5	-25.5	+20.6	-19.1	+28.4	+44.8	+16.8	+34.8
Steps (reduced)		38 (7)	61 (30)	77 (15)	89 (27)	99 (6)	108 (15)	115 (22)	122 (29)	128 (4)	133 (9)	138 (14)

Approximation of harmonics in 31ed7/4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-2.7	-6.0	-0.4	+12.9	+1.7	-3.5	-3.4	+1.6	+10.9	-7.1	+9.7
	Relative (%)	-8.6	-19.1	-1.3	+41.2	+5.4	-11.3	-10.8	+5.1	+34.8	-22.9	+30.9
Steps (reduced)		142 (18)	146 (22)	150 (26)	154 (30)	157 (2)	160 (5)	163 (8)	166 (11)	169 (14)	171 (16)	174 (19)