117edt
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Prime factorization
32 × 13
Step size
16.256¢
Octave
74\117edt (1202.95¢)
Consistency limit
4
Distinct consistency limit
4
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117 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 117edt or 117ed3), is a nonoctave tuning system that divides the interval of 3/1 into 117 equal parts of about 16.3 ¢ each. Each step represents a frequency ratio of 31/117, or the 117th root of 3.
Intervals
Steps | Cents | Hekts | Approximate ratios |
---|---|---|---|
0 | 0 | 0 | 1/1 |
1 | 16.3 | 11.1 | |
2 | 32.5 | 22.2 | |
3 | 48.8 | 33.3 | 37/36 |
4 | 65 | 44.4 | 27/26, 28/27 |
5 | 81.3 | 55.6 | 22/21 |
6 | 97.5 | 66.7 | 18/17 |
7 | 113.8 | 77.8 | 31/29, 47/44 |
8 | 130 | 88.9 | 14/13 |
9 | 146.3 | 100 | 37/34, 49/45 |
10 | 162.6 | 111.1 | 45/41 |
11 | 178.8 | 122.2 | |
12 | 195.1 | 133.3 | 47/42 |
13 | 211.3 | 144.4 | 26/23 |
14 | 227.6 | 155.6 | |
15 | 243.8 | 166.7 | 23/20 |
16 | 260.1 | 177.8 | 36/31, 43/37 |
17 | 276.4 | 188.9 | 27/23, 34/29 |
18 | 292.6 | 200 | |
19 | 308.9 | 211.1 | 43/36, 49/41 |
20 | 325.1 | 222.2 | |
21 | 341.4 | 233.3 | 28/23 |
22 | 357.6 | 244.4 | |
23 | 373.9 | 255.6 | 36/29, 41/33 |
24 | 390.1 | 266.7 | |
25 | 406.4 | 277.8 | 24/19, 43/34 |
26 | 422.7 | 288.9 | 23/18, 37/29 |
27 | 438.9 | 300 | |
28 | 455.2 | 311.1 | 13/10 |
29 | 471.4 | 322.2 | |
30 | 487.7 | 333.3 | |
31 | 503.9 | 344.4 | |
32 | 520.2 | 355.6 | 27/20 |
33 | 536.4 | 366.7 | 15/11 |
34 | 552.7 | 377.8 | |
35 | 569 | 388.9 | |
36 | 585.2 | 400 | |
37 | 601.5 | 411.1 | 17/12 |
38 | 617.7 | 422.2 | 10/7 |
39 | 634 | 433.3 | |
40 | 650.2 | 444.4 | |
41 | 666.5 | 455.6 | |
42 | 682.8 | 466.7 | 43/29, 46/31, 49/33 |
43 | 699 | 477.8 | |
44 | 715.3 | 488.9 | |
45 | 731.5 | 500 | 29/19 |
46 | 747.8 | 511.1 | 20/13, 37/24 |
47 | 764 | 522.2 | 14/9 |
48 | 780.3 | 533.3 | |
49 | 796.5 | 544.4 | 19/12 |
50 | 812.8 | 555.6 | |
51 | 829.1 | 566.7 | 21/13 |
52 | 845.3 | 577.8 | 44/27 |
53 | 861.6 | 588.9 | |
54 | 877.8 | 600 | |
55 | 894.1 | 611.1 | |
56 | 910.3 | 622.2 | 22/13 |
57 | 926.6 | 633.3 | 29/17 |
58 | 942.8 | 644.4 | 31/18 |
59 | 959.1 | 655.6 | 40/23, 47/27 |
60 | 975.4 | 666.7 | |
61 | 991.6 | 677.8 | 39/22 |
62 | 1007.9 | 688.9 | 34/19, 43/24 |
63 | 1024.1 | 700 | 47/26 |
64 | 1040.4 | 711.1 | 31/17 |
65 | 1056.6 | 722.2 | |
66 | 1072.9 | 733.3 | 13/7 |
67 | 1089.2 | 744.4 | |
68 | 1105.4 | 755.6 | 36/19 |
69 | 1121.7 | 766.7 | 44/23 |
70 | 1137.9 | 777.8 | 27/14 |
71 | 1154.2 | 788.9 | 37/19, 39/20 |
72 | 1170.4 | 800 | |
73 | 1186.7 | 811.1 | |
74 | 1202.9 | 822.2 | |
75 | 1219.2 | 833.3 | |
76 | 1235.5 | 844.4 | 47/23 |
77 | 1251.7 | 855.6 | |
78 | 1268 | 866.7 | |
79 | 1284.2 | 877.8 | 21/10 |
80 | 1300.5 | 888.9 | 36/17 |
81 | 1316.7 | 900 | |
82 | 1333 | 911.1 | |
83 | 1349.3 | 922.2 | |
84 | 1365.5 | 933.3 | 11/5 |
85 | 1381.8 | 944.4 | 20/9 |
86 | 1398 | 955.6 | |
87 | 1414.3 | 966.7 | 43/19 |
88 | 1430.5 | 977.8 | |
89 | 1446.8 | 988.9 | 30/13 |
90 | 1463 | 1000 | |
91 | 1479.3 | 1011.1 | 40/17, 47/20 |
92 | 1495.6 | 1022.2 | 19/8 |
93 | 1511.8 | 1033.3 | |
94 | 1528.1 | 1044.4 | 29/12 |
95 | 1544.3 | 1055.6 | |
96 | 1560.6 | 1066.7 | |
97 | 1576.8 | 1077.8 | |
98 | 1593.1 | 1088.9 | |
99 | 1609.3 | 1100 | |
100 | 1625.6 | 1111.1 | 23/9 |
101 | 1641.9 | 1122.2 | 31/12 |
102 | 1658.1 | 1133.3 | |
103 | 1674.4 | 1144.4 | |
104 | 1690.6 | 1155.6 | |
105 | 1706.9 | 1166.7 | |
106 | 1723.1 | 1177.8 | 46/17 |
107 | 1739.4 | 1188.9 | 41/15 |
108 | 1755.7 | 1200 | |
109 | 1771.9 | 1211.1 | 39/14 |
110 | 1788.2 | 1222.2 | |
111 | 1804.4 | 1233.3 | 17/6 |
112 | 1820.7 | 1244.4 | |
113 | 1836.9 | 1255.6 | 26/9 |
114 | 1853.2 | 1266.7 | |
115 | 1869.4 | 1277.8 | |
116 | 1885.7 | 1288.9 | |
117 | 1902 | 1300 | 3/1 |
Harmonics
Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +2.95 | +0.00 | +5.89 | -6.53 | +2.95 | -3.83 | -7.42 | +0.00 | -3.59 | -6.03 | +5.89 |
Relative (%) | +18.1 | +0.0 | +36.2 | -40.2 | +18.1 | -23.6 | -45.6 | +0.0 | -22.1 | -37.1 | +36.2 | |
Steps (reduced) |
74 (74) |
117 (0) |
148 (31) |
171 (54) |
191 (74) |
207 (90) |
221 (104) |
234 (0) |
245 (11) |
255 (21) |
265 (31) |
Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | -2.63 | -0.88 | -6.53 | -4.47 | +4.36 | +2.95 | +6.88 | -0.64 | -3.83 | -3.09 | +1.24 |
Relative (%) | -16.2 | -5.4 | -40.2 | -27.5 | +26.8 | +18.1 | +42.3 | -3.9 | -23.6 | -19.0 | +7.6 | |
Steps (reduced) |
273 (39) |
281 (47) |
288 (54) |
295 (61) |
302 (68) |
308 (74) |
314 (80) |
319 (85) |
324 (90) |
329 (95) |
334 (100) |