111ed4

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← 109ed4 111ed4 113ed4 →
Prime factorization 3 × 37
Step size 21.6216¢ 
Octave 56\111ed4 (1210.81¢)
Twelfth 88\111ed4 (1902.7¢)
Consistency limit 1
Distinct consistency limit 1

111 equal divisions of the 4th harmonic (abbreviated 111ed4) is a nonoctave tuning system that divides the interval of 4/1 into 111 equal parts of about 21.6 ¢ each. Each step represents a frequency ratio of 41/111, or the 111th root of 4.

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 21.6
2 43.2
3 64.9 27/26
4 86.5 41/39
5 108.1 33/31
6 129.7
7 151.4
8 173 21/19
9 194.6 19/17
10 216.2 17/15
11 237.8 31/27, 47/41
12 259.5 43/37
13 281.1
14 302.7 25/21, 31/26
15 324.3 35/29, 47/39
16 345.9 11/9
17 367.6 21/17, 26/21
18 389.2
19 410.8 19/15, 33/26
20 432.4
21 454.1
22 475.7 25/19
23 497.3
24 518.9 31/23
25 540.5
26 562.2
27 583.8 7/5
28 605.4
29 627 33/23
30 648.6
31 670.3
32 691.9
33 713.5
34 735.1 26/17
35 756.8
36 778.4
37 800 27/17, 46/29
38 821.6 37/23
39 843.2
40 864.9
41 886.5 5/3
42 908.1
43 929.7
44 951.4 26/15, 45/26
45 973
46 994.6
47 1016.2 9/5
48 1037.8
49 1059.5 35/19
50 1081.1 43/23
51 1102.7 17/9
52 1124.3
53 1145.9
54 1167.6
55 1189.2
56 1210.8
57 1232.4
58 1254.1
59 1275.7 23/11
60 1297.3
61 1318.9 15/7
62 1340.5
63 1362.2
64 1383.8
65 1405.4
66 1427
67 1448.6
68 1470.3
69 1491.9 45/19
70 1513.5
71 1535.1 17/7
72 1556.8
73 1578.4
74 1600
75 1621.6
76 1643.2
77 1664.9
78 1686.5 45/17
79 1708.1
80 1729.7 19/7
81 1751.4
82 1773
83 1794.6 31/11
84 1816.2
85 1837.8 26/9
86 1859.5
87 1881.1
88 1902.7 3/1
89 1924.3
90 1945.9
91 1967.6
92 1989.2 41/13
93 2010.8
94 2032.4
95 2054.1
96 2075.7
97 2097.3
98 2118.9 17/5
99 2140.5 31/9
100 2162.2
101 2183.8
102 2205.4 25/7
103 2227 47/13
104 2248.6 11/3
105 2270.3 26/7
106 2291.9
107 2313.5
108 2335.1 27/7
109 2356.8
110 2378.4
111 2400

Harmonics

Approximation of harmonics in 111ed4
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +10.81 +0.75 +0.00 +2.88 -10.06 +4.15 +10.81 +1.50 -7.94 +0.03 +0.75
Relative (%) +50.0 +3.5 +0.0 +13.3 -46.5 +19.2 +50.0 +6.9 -36.7 +0.2 +3.5
Steps
(reduced)
56
(56)
88
(88)
111
(0)
129
(18)
143
(32)
156
(45)
167
(56)
176
(65)
184
(73)
192
(81)
199
(88)
Approximation of harmonics in 111ed4
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -8.10 -6.66 +3.62 +0.00 +3.15 -9.32 +5.19 +2.88 +4.89 -10.78 -1.25
Relative (%) -37.4 -30.8 +16.8 +0.0 +14.6 -43.1 +24.0 +13.3 +22.6 -49.8 -5.8
Steps
(reduced)
205
(94)
211
(100)
217
(106)
222
(0)
227
(5)
231
(9)
236
(14)
240
(18)
244
(22)
247
(25)
251
(29)