111edt

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← 110edt 111edt 112edt →
Prime factorization 3 × 37
Step size 17.1347¢ 
Octave 70\111edt (1199.43¢)
Consistency limit 10
Distinct consistency limit 10

111 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 111edt or 111ed3), is a nonoctave tuning system that divides the interval of 3/1 into 111 equal parts of about 17.1⁠ ⁠¢ each. Each step represents a frequency ratio of 31/111, or the 111th root of 3.

Intervals

Steps Cents Hekts Approximate ratios
0 0 0 1/1
1 17.1 11.7
2 34.3 23.4
3 51.4 35.1 33/32, 34/33
4 68.5 46.8
5 85.7 58.6 21/20, 41/39
6 102.8 70.3 17/16
7 119.9 82 15/14
8 137.1 93.7 13/12, 40/37
9 154.2 105.4 47/43
10 171.3 117.1 32/29
11 188.5 128.8 29/26
12 205.6 140.5 9/8
13 222.8 152.3 33/29
14 239.9 164 23/20, 31/27
15 257 175.7 36/31
16 274.2 187.4 34/29
17 291.3 199.1 13/11
18 308.4 210.8 37/31, 43/36
19 325.6 222.5 29/24, 41/34
20 342.7 234.2 28/23, 39/32
21 359.8 245.9 16/13
22 377 257.7 41/33, 46/37
23 394.1 269.4
24 411.2 281.1 33/26
25 428.4 292.8 41/32
26 445.5 304.5 22/17
27 462.6 316.2 17/13, 47/36
28 479.8 327.9 29/22
29 496.9 339.6 4/3
30 514 351.4 39/29
31 531.2 363.1
32 548.3 374.8
33 565.4 386.5 18/13, 43/31
34 582.6 398.2 7/5
35 599.7 409.9 41/29
36 616.9 421.6 10/7
37 634 433.3
38 651.1 445
39 668.3 456.8
40 685.4 468.5 46/31
41 702.5 480.2 3/2
42 719.7 491.9 44/29, 47/31
43 736.8 503.6 26/17
44 753.9 515.3 17/11
45 771.1 527
46 788.2 538.7 41/26
47 805.3 550.5 43/27
48 822.5 562.2 37/23, 45/28
49 839.6 573.9 13/8
50 856.7 585.6
51 873.9 597.3
52 891 609
53 908.1 620.7
54 925.3 632.4 29/17, 41/24
55 942.4 644.1 31/18
56 959.5 655.9 40/23, 47/27
57 976.7 667.6
58 993.8 679.3
59 1010.9 691 43/24
60 1028.1 702.7 29/16
61 1045.2 714.4
62 1062.4 726.1 24/13
63 1079.5 737.8 28/15, 41/22
64 1096.6 749.5 32/17
65 1113.8 761.3 40/21
66 1130.9 773
67 1148 784.7 33/17
68 1165.2 796.4 47/24
69 1182.3 808.1
70 1199.4 819.8 2/1
71 1216.6 831.5
72 1233.7 843.2
73 1250.8 855
74 1268 866.7
75 1285.1 878.4 21/10
76 1302.2 890.1
77 1319.4 901.8 15/7
78 1336.5 913.5 13/6
79 1353.6 925.2
80 1370.8 936.9
81 1387.9 948.6 29/13
82 1405 960.4 9/4
83 1422.2 972.1
84 1439.3 983.8 39/17
85 1456.5 995.5
86 1473.6 1007.2
87 1490.7 1018.9 26/11
88 1507.9 1030.6 43/18
89 1525 1042.3 41/17
90 1542.1 1054.1 39/16
91 1559.3 1065.8 32/13
92 1576.4 1077.5
93 1593.5 1089.2
94 1610.7 1100.9 33/13
95 1627.8 1112.6 41/16
96 1644.9 1124.3 31/12, 44/17
97 1662.1 1136 47/18
98 1679.2 1147.7 29/11
99 1696.3 1159.5 8/3
100 1713.5 1171.2 43/16
101 1730.6 1182.9
102 1747.7 1194.6
103 1764.9 1206.3 36/13
104 1782 1218 14/5
105 1799.1 1229.7
106 1816.3 1241.4 20/7
107 1833.4 1253.2
108 1850.6 1264.9 32/11
109 1867.7 1276.6 47/16
110 1884.8 1288.3
111 1902 1300 3/1

Harmonics

Approximation of harmonics in 111edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.57 +0.00 -1.14 +6.65 -0.57 +6.72 -1.71 +0.00 +6.08 -4.71 -1.14
Relative (%) -3.3 +0.0 -6.6 +38.8 -3.3 +39.2 -10.0 +0.0 +35.5 -27.5 -6.6
Steps
(reduced)
70
(70)
111
(0)
140
(29)
163
(52)
181
(70)
197
(86)
210
(99)
222
(0)
233
(11)
242
(20)
251
(29)
Approximation of harmonics in 111edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -2.63 +6.15 +6.65 -2.28 -4.42 -0.57 -8.50 +5.51 +6.72 -5.28 +3.43
Relative (%) -15.4 +35.9 +38.8 -13.3 -25.8 -3.3 -49.6 +32.2 +39.2 -30.8 +20.0
Steps
(reduced)
259
(37)
267
(45)
274
(52)
280
(58)
286
(64)
292
(70)
297
(75)
303
(81)
308
(86)
312
(90)
317
(95)