83ed4

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83ed4

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Prime factorization	83 (prime)
Step size	28.9157¢
Octave	42\83ed4 (1214.46¢)
Twelfth	66\83ed4 (1908.43¢)
Consistency limit	1
Distinct consistency limit	1

83 equal divisions of the 4th harmonic (abbreviated 83ed4) is a nonoctave tuning system that divides the interval of 4/1 into 83 equal parts of about 28.9 ¢ each. Each step represents a frequency ratio of 4^1/83, or the 83rd root of 4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	28.9
2	57.8	30/29, 31/30
3	86.7	21/20
4	115.7	31/29
5	144.6	38/35
6	173.5
7	202.4
8	231.3
9	260.2
10	289.2	13/11
11	318.1
12	347	11/9
13	375.9
14	404.8
15	433.7	9/7
16	462.7	17/13, 30/23
17	491.6
18	520.5	23/17, 27/20
19	549.4
20	578.3
21	607.2
22	636.1	13/9
23	665.1
24	694
25	722.9
26	751.8	17/11
27	780.7	11/7
28	809.6
29	838.6
30	867.5	33/20, 38/23
31	896.4
32	925.3	29/17
33	954.2
34	983.1	30/17
35	1012
36	1041	31/17
37	1069.9	13/7
38	1098.8	17/9
39	1127.7
40	1156.6	37/19, 39/20
41	1185.5
42	1214.5
43	1243.4
44	1272.3
45	1301.2
46	1330.1
47	1359
48	1388	29/13
49	1416.9
50	1445.8	30/13
51	1474.7
52	1503.6	31/13
53	1532.5
54	1561.4	37/15
55	1590.4
56	1619.3
57	1648.2
58	1677.1	29/11
59	1706
60	1734.9	30/11
61	1763.9
62	1792.8	31/11
63	1821.7
64	1850.6
65	1879.5
66	1908.4
67	1937.3
68	1966.3
69	1995.2
70	2024.1	29/9
71	2053
72	2081.9	10/3
73	2110.8
74	2139.8	31/9
75	2168.7	7/2
76	2197.6
77	2226.5
78	2255.4
79	2284.3
80	2313.3	19/5
81	2342.2
82	2371.1
83	2400

Harmonics

Approximation of harmonics in 83ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+14.5	+6.5	+0.0	-10.4	-8.0	+14.3	+14.5	+13.0	+4.0	+12.5	+6.5
	Relative (%)	+50.0	+22.4	+0.0	-36.0	-27.6	+49.5	+50.0	+44.8	+14.0	+43.4	+22.4
Steps (reduced)		42 (42)	66 (66)	83 (0)	96 (13)	107 (24)	117 (34)	125 (42)	132 (49)	138 (55)	144 (61)	149 (66)

Approximation of harmonics in 83ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+12.5	-0.2	-3.9	+0.0	+10.7	-1.5	-8.4	-10.4	-8.1	-1.9	+7.9
	Relative (%)	+43.2	-0.5	-13.6	+0.0	+37.0	-5.2	-28.9	-36.0	-28.1	-6.6	+27.2
Steps (reduced)		154 (71)	158 (75)	162 (79)	166 (0)	170 (4)	173 (7)	176 (10)	179 (13)	182 (16)	185 (19)	188 (22)