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	28.916
2	57.831	30/29, 31/30
3	86.747	21/20
4	115.663	31/29
5	144.578	38/35
6	173.494
7	202.41
8	231.325
9	260.241
10	289.157	13/11
11	318.072
12	346.988	11/9
13	375.904
14	404.819
15	433.735	9/7
16	462.651	17/13, 30/23
17	491.566
18	520.482	23/17, 27/20
19	549.398
20	578.313
21	607.229
22	636.145	13/9
23	665.06
24	693.976
25	722.892
26	751.807	17/11
27	780.723	11/7
28	809.639
29	838.554
30	867.47	33/20, 38/23
31	896.386
32	925.301	29/17
33	954.217
34	983.133	30/17
35	1012.048
36	1040.964	31/17
37	1069.88	13/7
38	1098.795	17/9
39	1127.711
40	1156.627	37/19, 39/20
41	1185.542
42	1214.458
43	1243.373
44	1272.289
45	1301.205
46	1330.12
47	1359.036
48	1387.952	29/13
49	1416.867
50	1445.783	30/13
51	1474.699
52	1503.614	31/13
53	1532.53
54	1561.446	37/15
55	1590.361
56	1619.277
57	1648.193
58	1677.108	29/11
59	1706.024
60	1734.94	30/11
61	1763.855
62	1792.771	31/11
63	1821.687
64	1850.602
65	1879.518
66	1908.434
67	1937.349
68	1966.265
69	1995.181
70	2024.096	29/9
71	2053.012
72	2081.928	10/3
73	2110.843
74	2139.759	31/9
75	2168.675	7/2
76	2197.59
77	2226.506
78	2255.422
79	2284.337
80	2313.253	19/5
81	2342.169
82	2371.084
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)