82ed7/3

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82ed7/3

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Prime factorization	2 × 41
Step size	17.8887¢
Octave	67\82ed7/3 (1198.54¢) (semiconvergent)
Twelfth	106\82ed7/3 (1896.2¢) (→53\41ed7/3)
Consistency limit	4
Distinct consistency limit	4

82 equal divisions of 7/3 (abbreviated 82ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 82 equal parts of about 17.9⁠ ⁠¢ each. Each step represents a frequency ratio of (7/3)^1/82, or the 82nd root of 7/3.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	17.9
2	35.8
3	53.7	31/30, 32/31, 33/32, 34/33
4	71.6	24/23
5	89.4	20/19, 39/37
6	107.3	17/16, 33/31
7	125.2	14/13
8	143.1	38/35
9	161	34/31
10	178.9	31/28
11	196.8	37/33
12	214.7	17/15, 26/23
13	232.6	8/7
14	250.4	15/13, 37/32
15	268.3	7/6
16	286.2	13/11, 33/28
17	304.1	31/26, 37/31
18	322
19	339.9	28/23, 39/32
20	357.8	16/13
21	375.7
22	393.6
23	411.4	19/15, 33/26
24	429.3
25	447.2	22/17
26	465.1	17/13, 38/29
27	483	37/28
28	500.9	4/3
29	518.8	31/23
30	536.7	15/11
31	554.5	40/29
32	572.4	32/23, 39/28
33	590.3
34	608.2	37/26
35	626.1	23/16, 33/23
36	644	29/20
37	661.9	22/15
38	679.8	34/23
39	697.7
40	715.5
41	733.4	26/17, 29/19
42	751.3	17/11, 37/24
43	769.2
44	787.1
45	805	35/22
46	822.9	37/23
47	840.8	13/8
48	858.7	23/14
49	876.5
50	894.4
51	912.3	22/13, 39/23
52	930.2	12/7
53	948.1	19/11
54	966	7/4
55	983.9	30/17
56	1001.8
57	1019.7
58	1037.5	20/11, 31/17
59	1055.4	35/19
60	1073.3	13/7
61	1091.2	15/8
62	1109.1	19/10
63	1127	23/12
64	1144.9	31/16
65	1162.8
66	1180.7
67	1198.5	2/1
68	1216.4
69	1234.3
70	1252.2	33/16, 35/17
71	1270.1
72	1288	40/19
73	1305.9	17/8
74	1323.8
75	1341.7	13/6
76	1359.5
77	1377.4	31/14
78	1395.3	38/17
79	1413.2
80	1431.1	16/7
81	1449	30/13, 37/16
82	1466.9	7/3

Harmonics

Approximation of harmonics in 82ed7/3

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-1.46	-5.76	-2.92	+4.32	-7.22	-5.76	-4.38	+6.38	+2.86	-1.15	-8.67
	Relative (%)	-8.2	-32.2	-16.3	+24.1	-40.3	-32.2	-24.5	+35.6	+16.0	-6.4	-48.5
Steps (reduced)		67 (67)	106 (24)	134 (52)	156 (74)	173 (9)	188 (24)	201 (37)	213 (49)	223 (59)	232 (68)	240 (76)

Approximation of harmonics in 82ed7/3

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-4.14	-7.22	-1.44	-5.84	-3.46	+4.92	+0.76	+1.40	+6.38	-2.61	-8.01
	Relative (%)	-23.1	-40.3	-8.0	-32.6	-19.3	+27.5	+4.2	+7.8	+35.6	-14.6	-44.8
Steps (reduced)		248 (2)	255 (9)	262 (16)	268 (22)	274 (28)	280 (34)	285 (39)	290 (44)	295 (49)	299 (53)	303 (57)