83ed11

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← 82ed11 83ed11 84ed11 →
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 111/83, 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