162edt
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162 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 162edt or 162ed3), is a nonoctave tuning system that divides the interval of 3/1 into 162 equal parts of about 11.7 ¢ each. Each step represents a frequency ratio of 31/162, or the 162nd root of 3.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 11.7 | 8 | |
| 2 | 23.5 | 16 | |
| 3 | 35.2 | 24.1 | |
| 4 | 47 | 32.1 | 37/36, 38/37 |
| 5 | 58.7 | 40.1 | |
| 6 | 70.4 | 48.1 | 25/24, 51/49 |
| 7 | 82.2 | 56.2 | 43/41 |
| 8 | 93.9 | 64.2 | 19/18 |
| 9 | 105.7 | 72.2 | |
| 10 | 117.4 | 80.2 | |
| 11 | 129.1 | 88.3 | 14/13 |
| 12 | 140.9 | 96.3 | 51/47 |
| 13 | 152.6 | 104.3 | 47/43 |
| 14 | 164.4 | 112.3 | |
| 15 | 176.1 | 120.4 | 31/28 |
| 16 | 187.8 | 128.4 | 39/35 |
| 17 | 199.6 | 136.4 | 55/49 |
| 18 | 211.3 | 144.4 | 26/23, 35/31 |
| 19 | 223.1 | 152.5 | 33/29, 58/51 |
| 20 | 234.8 | 160.5 | |
| 21 | 246.5 | 168.5 | 15/13 |
| 22 | 258.3 | 176.5 | 36/31 |
| 23 | 270 | 184.6 | |
| 24 | 281.8 | 192.6 | |
| 25 | 293.5 | 200.6 | 45/38 |
| 26 | 305.3 | 208.6 | 31/26, 37/31 |
| 27 | 317 | 216.7 | 6/5 |
| 28 | 328.7 | 224.7 | |
| 29 | 340.5 | 232.7 | 28/23 |
| 30 | 352.2 | 240.7 | 38/31 |
| 31 | 364 | 248.8 | 37/30, 58/47 |
| 32 | 375.7 | 256.8 | 41/33, 46/37 |
| 33 | 387.4 | 264.8 | 5/4 |
| 34 | 399.2 | 272.8 | 34/27 |
| 35 | 410.9 | 280.9 | |
| 36 | 422.7 | 288.9 | |
| 37 | 434.4 | 296.9 | 9/7 |
| 38 | 446.1 | 304.9 | 22/17 |
| 39 | 457.9 | 313 | 43/33 |
| 40 | 469.6 | 321 | |
| 41 | 481.4 | 329 | 37/28 |
| 42 | 493.1 | 337 | |
| 43 | 504.8 | 345.1 | |
| 44 | 516.6 | 353.1 | 31/23 |
| 45 | 528.3 | 361.1 | 19/14 |
| 46 | 540.1 | 369.1 | |
| 47 | 551.8 | 377.2 | |
| 48 | 563.5 | 385.2 | 18/13 |
| 49 | 575.3 | 393.2 | |
| 50 | 587 | 401.2 | |
| 51 | 598.8 | 409.3 | 41/29 |
| 52 | 610.5 | 417.3 | 37/26 |
| 53 | 622.2 | 425.3 | |
| 54 | 634 | 433.3 | 49/34 |
| 55 | 645.7 | 441.4 | 45/31 |
| 56 | 657.5 | 449.4 | 19/13 |
| 57 | 669.2 | 457.4 | |
| 58 | 680.9 | 465.4 | 43/29 |
| 59 | 692.7 | 473.5 | |
| 60 | 704.4 | 481.5 | |
| 61 | 716.2 | 489.5 | 56/37 |
| 62 | 727.9 | 497.5 | 35/23 |
| 63 | 739.6 | 505.6 | 23/15 |
| 64 | 751.4 | 513.6 | 54/35 |
| 65 | 763.1 | 521.6 | |
| 66 | 774.9 | 529.6 | 36/23 |
| 67 | 786.6 | 537.7 | |
| 68 | 798.4 | 545.7 | |
| 69 | 810.1 | 553.7 | |
| 70 | 821.8 | 561.7 | 37/23, 45/28 |
| 71 | 833.6 | 569.8 | 34/21, 55/34 |
| 72 | 845.3 | 577.8 | 44/27, 57/35 |
| 73 | 857.1 | 585.8 | |
| 74 | 868.8 | 593.8 | 38/23 |
| 75 | 880.5 | 601.9 | |
| 76 | 892.3 | 609.9 | |
| 77 | 904 | 617.9 | |
| 78 | 915.8 | 625.9 | |
| 79 | 927.5 | 634 | |
| 80 | 939.2 | 642 | |
| 81 | 951 | 650 | 26/15, 45/26 |
| 82 | 962.7 | 658 | |
| 83 | 974.5 | 666 | |
| 84 | 986.2 | 674.1 | |
| 85 | 997.9 | 682.1 | |
| 86 | 1009.7 | 690.1 | |
| 87 | 1021.4 | 698.1 | |
| 88 | 1033.2 | 706.2 | 49/27 |
| 89 | 1044.9 | 714.2 | |
| 90 | 1056.6 | 722.2 | 35/19, 46/25 |
| 91 | 1068.4 | 730.2 | |
| 92 | 1080.1 | 738.3 | 28/15 |
| 93 | 1091.9 | 746.3 | |
| 94 | 1103.6 | 754.3 | |
| 95 | 1115.3 | 762.3 | |
| 96 | 1127.1 | 770.4 | 23/12 |
| 97 | 1138.8 | 778.4 | |
| 98 | 1150.6 | 786.4 | 35/18 |
| 99 | 1162.3 | 794.4 | 45/23 |
| 100 | 1174 | 802.5 | |
| 101 | 1185.8 | 810.5 | |
| 102 | 1197.5 | 818.5 | |
| 103 | 1209.3 | 826.5 | |
| 104 | 1221 | 834.6 | |
| 105 | 1232.7 | 842.6 | 55/27 |
| 106 | 1244.5 | 850.6 | 39/19 |
| 107 | 1256.2 | 858.6 | 31/15 |
| 108 | 1268 | 866.7 | 52/25 |
| 109 | 1279.7 | 874.7 | 44/21 |
| 110 | 1291.5 | 882.7 | |
| 111 | 1303.2 | 890.7 | |
| 112 | 1314.9 | 898.8 | 47/22 |
| 113 | 1326.7 | 906.8 | |
| 114 | 1338.4 | 914.8 | 13/6 |
| 115 | 1350.2 | 922.8 | |
| 116 | 1361.9 | 930.9 | |
| 117 | 1373.6 | 938.9 | 42/19 |
| 118 | 1385.4 | 946.9 | 49/22 |
| 119 | 1397.1 | 954.9 | 56/25 |
| 120 | 1408.9 | 963 | |
| 121 | 1420.6 | 971 | |
| 122 | 1432.3 | 979 | |
| 123 | 1444.1 | 987 | |
| 124 | 1455.8 | 995.1 | 51/22 |
| 125 | 1467.6 | 1003.1 | 7/3 |
| 126 | 1479.3 | 1011.1 | |
| 127 | 1491 | 1019.1 | |
| 128 | 1502.8 | 1027.2 | |
| 129 | 1514.5 | 1035.2 | 12/5 |
| 130 | 1526.3 | 1043.2 | |
| 131 | 1538 | 1051.2 | |
| 132 | 1549.7 | 1059.3 | |
| 133 | 1561.5 | 1067.3 | |
| 134 | 1573.2 | 1075.3 | |
| 135 | 1585 | 1083.3 | 5/2 |
| 136 | 1596.7 | 1091.4 | |
| 137 | 1608.4 | 1099.4 | 38/15 |
| 138 | 1620.2 | 1107.4 | |
| 139 | 1631.9 | 1115.4 | |
| 140 | 1643.7 | 1123.5 | 31/12 |
| 141 | 1655.4 | 1131.5 | 13/5 |
| 142 | 1667.1 | 1139.5 | 55/21 |
| 143 | 1678.9 | 1147.5 | 29/11 |
| 144 | 1690.6 | 1155.6 | |
| 145 | 1702.4 | 1163.6 | |
| 146 | 1714.1 | 1171.6 | 35/13 |
| 147 | 1725.8 | 1179.6 | |
| 148 | 1737.6 | 1187.7 | |
| 149 | 1749.3 | 1195.7 | |
| 150 | 1761.1 | 1203.7 | 47/17 |
| 151 | 1772.8 | 1211.7 | 39/14 |
| 152 | 1784.6 | 1219.8 | |
| 153 | 1796.3 | 1227.8 | |
| 154 | 1808 | 1235.8 | 54/19 |
| 155 | 1819.8 | 1243.8 | |
| 156 | 1831.5 | 1251.9 | 49/17 |
| 157 | 1843.3 | 1259.9 | |
| 158 | 1855 | 1267.9 | |
| 159 | 1866.7 | 1275.9 | |
| 160 | 1878.5 | 1284 | |
| 161 | 1890.2 | 1292 | |
| 162 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.47 | +0.00 | -4.95 | -3.82 | -2.47 | +0.69 | +4.32 | +0.00 | +5.44 | +4.81 | -4.95 |
| Relative (%) | -21.1 | +0.0 | -42.1 | -32.6 | -21.1 | +5.9 | +36.8 | +0.0 | +46.4 | +40.9 | -42.1 | |
| Steps (reduced) |
102 (102) |
162 (0) |
204 (42) |
237 (75) |
264 (102) |
287 (125) |
307 (145) |
324 (0) |
340 (16) |
354 (30) |
366 (42) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.63 | -1.79 | -3.82 | +1.85 | +2.56 | -2.47 | -2.15 | +2.97 | +0.69 | +2.33 | -4.18 |
| Relative (%) | -22.4 | -15.2 | -32.6 | +15.8 | +21.8 | -21.1 | -18.3 | +25.3 | +5.9 | +19.9 | -35.6 | |
| Steps (reduced) |
378 (54) |
389 (65) |
399 (75) |
409 (85) |
418 (94) |
426 (102) |
434 (110) |
442 (118) |
449 (125) |
456 (132) |
462 (138) | |