83ed11

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83ed11

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Prime factorization	83 (prime)
Step size	50.0159 ¢
Octave	24\83ed11 (1200.38 ¢) (convergent)
Twelfth	38\83ed11 (1900.6 ¢) (semiconvergent)
Consistency limit	6
Distinct consistency limit	6

83 equal divisions of the 11th harmonic (abbreviated 83ed11) is a nonoctave tuning system that divides the interval of 11/1 into 83 equal parts of about 50 ¢ each. Each step represents a frequency ratio of 11^1/83, or the 83rd root of 11.

Theory

83ed11 is closely related to 24edo (quarter-tone tuning), but with the 11th harmonic rather than the octave being just, which stretches the octave by about 0.381 cents. Like 24edo, 83ed11 is consistent to the 6-integer-limit.

Harmonics

Approximation of harmonics in 83ed11

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.4	-1.4	+0.8	+14.6	-1.0	-17.8	+1.1	-2.7	+15.0	+0.0	-0.6
	Relative (%)	+0.8	-2.7	+1.5	+29.1	-1.9	-35.5	+2.3	-5.4	+29.9	+0.0	-1.2
Steps (reduced)		24 (24)	38 (38)	48 (48)	56 (56)	62 (62)	67 (67)	72 (72)	76 (76)	80 (80)	83 (0)	86 (3)

Approximation of harmonics in 83ed11 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+10.9	-17.4	+13.2	+1.5	-3.4	-2.3	+4.1	+15.3	-19.1	+0.4	+23.5	-0.2
	Relative (%)	+21.8	-34.8	+26.4	+3.0	-6.8	-4.6	+8.2	+30.7	-38.2	+0.8	+46.9	-0.4
Steps (reduced)		89 (6)	91 (8)	94 (11)	96 (13)	98 (15)	100 (17)	102 (19)	104 (21)	105 (22)	107 (24)	109 (26)	110 (27)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	50	34/33, 35/34, 36/35, 37/36
2	100	18/17, 35/33
3	150	12/11
4	200.1	37/33
5	250.1	15/13, 37/32
6	300.1	19/16
7	350.1	11/9, 38/31
8	400.1	29/23, 34/27, 39/31
9	450.1	35/27
10	500.2	4/3
11	550.2	11/8
12	600.2	17/12
13	650.2	16/11, 35/24
14	700.2	3/2
15	750.2	37/24
16	800.3	27/17
17	850.3	18/11, 31/19
18	900.3	32/19, 37/22
19	950.3	26/15
20	1000.3
21	1050.3	11/6
22	1100.3	17/9
23	1150.4	33/17, 35/18
24	1200.4	2/1
25	1250.4	33/16, 35/17, 37/18
26	1300.4	36/17
27	1350.4	24/11
28	1400.4
29	1450.5	30/13, 37/16
30	1500.5	19/8
31	1550.5	22/9
32	1600.5
33	1650.5
34	1700.5	8/3
35	1750.6	11/4
36	1800.6	17/6
37	1850.6	32/11, 35/12
38	1900.6	3/1
39	1950.6	34/11, 37/12
40	2000.6
41	2050.7	36/11
42	2100.7	37/11
43	2150.7
44	2200.7
45	2250.7	11/3
46	2300.7	34/9
47	2350.7	35/9
48	2400.8	4/1
49	2450.8	33/8
50	2500.8
51	2550.8
52	2600.8	9/2
53	2650.8	37/8
54	2700.9
55	2750.9
56	2800.9
57	2850.9
58	2900.9	16/3
59	2950.9	11/2
60	3001	17/3
61	3051	35/6
62	3101	6/1
63	3151	37/6
64	3201
65	3251
66	3301
67	3351.1
68	3401.1
69	3451.1	22/3
70	3501.1
71	3551.1
72	3601.1	8/1
73	3651.2	33/4
74	3701.2
75	3751.2
76	3801.2	9/1
77	3851.2	37/4
78	3901.2
79	3951.3
80	4001.3
81	4051.3
82	4101.3
83	4151.3	11/1

See also

• 14edf – relative edf
• 24edo – relative edo
• 38edt – relative edt
• 56ed5 – relative ed5
• 62ed6 – relative ed6
• 86ed12 – relative ed12
• 198ed304 – close to the zeta-optimized tuning for 24edo