39ed4

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39ed4

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Prime factorization	3 × 13
Step size	61.5385¢
Octave	20\39ed4 (1230.77¢)
Twelfth	31\39ed4 (1907.69¢)
Consistency limit	1
Distinct consistency limit	1

39 equal divisions of the 4th harmonic (abbreviated 39ed4) is a nonoctave tuning system that divides the interval of 4/1 into 39 equal parts of about 61.5 ¢ each. Each step represents a frequency ratio of 4^1/39, or the 39th root of 4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	61.538	27/26
2	123.077	29/27
3	184.615	10/9, 29/26
4	246.154	15/13
5	307.692
6	369.231	21/17, 26/21
7	430.769	9/7
8	492.308
9	553.846	29/21
10	615.385	10/7, 27/19
11	676.923
12	738.462	23/15, 26/17, 29/19
13	800	27/17
14	861.538
15	923.077	17/10, 29/17
16	984.615	23/13
17	1046.154
18	1107.692	17/9, 19/10
19	1169.231
20	1230.769
21	1292.308	19/9
22	1353.846
23	1415.385	25/11
24	1476.923
25	1538.462	17/7
26	1600
27	1661.538	13/5
28	1723.077	19/7, 27/10
29	1784.615
30	1846.154	29/10
31	1907.692	3/1
32	1969.231
33	2030.769	29/9
34	2092.308
35	2153.846
36	2215.385
37	2276.923	26/7
38	2338.462	27/7
39	2400

Harmonics

Approximation of harmonics in 39ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+30.8	+5.7	+0.0	-17.1	-25.0	+15.8	+30.8	+11.5	+13.7	-28.2	+5.7
	Relative (%)	+50.0	+9.3	+0.0	-27.8	-40.7	+25.7	+50.0	+18.6	+22.2	-45.9	+9.3
Steps (reduced)		20 (20)	31 (31)	39 (0)	45 (6)	50 (11)	55 (16)	59 (20)	62 (23)	65 (26)	67 (28)	70 (31)

Approximation of harmonics in 39ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-9.8	-15.0	-11.3	+0.0	+18.1	-19.3	+10.2	-17.1	+21.5	+2.5	-12.9
	Relative (%)	-15.9	-24.3	-18.4	+0.0	+29.4	-31.4	+16.5	-27.8	+35.0	+4.1	-20.9
Steps (reduced)		72 (33)	74 (35)	76 (37)	78 (0)	80 (2)	81 (3)	83 (5)	84 (6)	86 (8)	87 (9)	88 (10)