31ed8

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31ed8

32ed8 →

Prime factorization	31 (prime)
Step size	116.129 ¢
Octave	10\31ed8 (1161.29 ¢)
Twelfth	16\31ed8 (1858.06 ¢)
Consistency limit	3
Distinct consistency limit	3

31 equal divisions of the 8th harmonic (abbreviated 31ed8) is a nonoctave tuning system that divides the interval of 8/1 into 31 equal parts of about 116 ¢ each. Each step represents a frequency ratio of 8^1/31, or the 31st root of 8. It is identical to every third step of 31edo.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	116.1	15/14
2	232.3
3	348.4
4	464.5	17/13
5	580.6	7/5
6	696.8	3/2
7	812.9
8	929
9	1045.2
10	1161.3
11	1277.4	21/10, 23/11
12	1393.5
13	1509.7
14	1625.8
15	1741.9
16	1858.1
17	1974.2	22/7
18	2090.3	10/3
19	2206.5	25/7
20	2322.6
21	2438.7
22	2554.8
23	2671	14/3
24	2787.1	5/1
25	2903.2
26	3019.4
27	3135.5
28	3251.6
29	3367.7	7/1
30	3483.9	15/2
31	3600

Harmonics

Approximation of harmonics in 31ed8

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-38.7	-43.9	+38.7	+0.8	+33.5	-1.1	+0.0	+28.3	-37.9	+29.3	-5.2
	Relative (%)	-33.3	-37.8	+33.3	+0.7	+28.9	-0.9	+0.0	+24.4	-32.7	+25.3	-4.5
Steps (reduced)		10 (10)	16 (16)	21 (21)	24 (24)	27 (27)	29 (29)	31 (0)	33 (2)	34 (3)	36 (5)	37 (6)

Approximation of harmonics in 31ed8

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-27.6	-39.8	-43.1	-38.7	-27.5	-10.4	+12.2	+39.5	-45.0	-9.4	+29.8
	Relative (%)	-23.8	-34.3	-37.1	-33.3	-23.7	-8.9	+10.5	+34.0	-38.7	-8.1	+25.7
Steps (reduced)		38 (7)	39 (8)	40 (9)	41 (10)	42 (11)	43 (12)	44 (13)	45 (14)	45 (14)	46 (15)	47 (16)

Music

• A Call in Summer by Jake Freivald

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