65ed4

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65 equal divisions of the 4th harmonic (abbreviated 65ed4) is a nonoctave tuning system that divides the interval of 4/1 into 65 equal parts of about 36.9 ¢ each. Each step represents a frequency ratio of 4^1/65, or the 65th root of 4.

← 63ed4

65ed4

67ed4 →

Prime factorization	5 × 13
Step size	36.9231 ¢
Octave	33\65ed4 (1218.46 ¢)
Twelfth	52\65ed4 (1920 ¢) (→ 4\5ed4)
Consistency limit	1
Distinct consistency limit	1

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	36.9
2	73.8	23/22
3	110.8	33/31
4	147.7
5	184.6
6	221.5	33/29
7	258.5
8	295.4
9	332.3	17/14, 23/19
10	369.2	21/17, 26/21
11	406.2	19/15
12	443.1	22/17
13	480	29/22
14	516.9	31/23
15	553.8
16	590.8	31/22
17	627.7	33/23
18	664.6	22/15
19	701.5	3/2
20	738.5	23/15, 26/17
21	775.4
22	812.3
23	849.2	31/19
24	886.2
25	923.1	29/17
26	960
27	996.9
28	1033.8
29	1070.8	13/7
30	1107.7	19/10
31	1144.6	29/15, 33/17
32	1181.5
33	1218.5
34	1255.4	31/15
35	1292.3
36	1329.2
37	1366.2	11/5
38	1403.1	9/4
39	1440	23/10
40	1476.9
41	1513.8
42	1550.8
43	1587.7
44	1624.6
45	1661.5
46	1698.5
47	1735.4
48	1772.3
49	1809.2
50	1846.2	29/10
51	1883.1
52	1920
53	1956.9	31/10
54	1993.8
55	2030.8
56	2067.7	33/10
57	2104.6	27/8
58	2141.5
59	2178.5
60	2215.4
61	2252.3
62	2289.2
63	2326.2
64	2363.1
65	2400

Harmonics

Approximation of harmonics in 65ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+18.5	+18.0	+0.0	-17.1	-0.4	-8.8	+18.5	-0.8	+1.4	-15.9	+18.0
	Relative (%)	+50.0	+48.9	+0.0	-46.3	-1.1	-23.9	+50.0	-2.3	+3.7	-43.2	+48.9
Steps (reduced)		33 (33)	52 (52)	65 (0)	75 (10)	84 (19)	91 (26)	98 (33)	103 (38)	108 (43)	112 (47)	117 (52)

Approximation of harmonics in 65ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-9.8	+9.6	+1.0	+0.0	+5.8	+17.6	-2.1	-17.1	+9.2	+2.5	-0.6
	Relative (%)	-26.4	+26.1	+2.6	+0.0	+15.7	+47.7	-5.8	-46.3	+25.0	+6.8	-1.6
Steps (reduced)		120 (55)	124 (59)	127 (62)	130 (0)	133 (3)	136 (6)	138 (8)	140 (10)	143 (13)	145 (15)	147 (17)