31ed7/3

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31ed7/3

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Prime factorization	31 (prime)
Step size	47.3184¢
Octave	25\31ed7/3 (1182.96¢)
Twelfth	40\31ed7/3 (1892.74¢)
Consistency limit	3
Distinct consistency limit	3

31 equal divisions of 7/3 (abbreviated 31ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 31 equal parts of about 47.3 ¢ each. Each step represents a frequency ratio of (7/3)^1/31, or the 31st root of 7/3.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	47.318
2	94.637
3	141.955	25/23
4	189.274	10/9, 19/17
5	236.592
6	283.91	13/11
7	331.229	23/19
8	378.547	26/21
9	425.866	9/7
10	473.184	17/13, 25/19
11	520.503	23/17, 27/20
12	567.821
13	615.139	10/7
14	662.458	19/13, 22/15, 25/17
15	709.776	3/2
16	757.095	14/9, 17/11
17	804.413
18	851.731
19	899.05
20	946.368	19/11, 26/15
21	993.687	23/13
22	1041.005
23	1088.324
24	1135.642	25/13, 27/14
25	1182.96
26	1230.279
27	1277.597	21/10, 23/11
28	1324.916	15/7
29	1372.234	11/5
30	1419.552	25/11
31	1466.871	7/3

Harmonics

Approximation of harmonics in 31ed7/3

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-17.0	-9.2	+13.2	+5.5	+21.1	-9.2	-3.8	-18.4	-11.6	+12.7	+4.0
	Relative (%)	-36.0	-19.5	+28.0	+11.6	+44.5	-19.5	-8.0	-39.0	-24.4	+26.8	+8.5
Steps (reduced)		25 (25)	40 (9)	51 (20)	59 (28)	66 (4)	71 (9)	76 (14)	80 (18)	84 (22)	88 (26)	91 (29)

Approximation of harmonics in 31ed7/3

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+7.4	+21.1	-3.7	-20.8	+16.2	+11.8	+12.9	+18.7	-18.4	-4.3	+13.3
	Relative (%)	+15.6	+44.5	-7.9	-44.0	+34.2	+25.0	+27.2	+39.5	-39.0	-9.2	+28.2
Steps (reduced)		94 (1)	97 (4)	99 (6)	101 (8)	104 (11)	106 (13)	108 (15)	110 (17)	111 (18)	113 (20)	115 (22)