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