146edt
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146 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 146edt or 146ed3), is a nonoctave tuning system that divides the interval of 3/1 into 146 equal parts of about 13 ¢ each. Each step represents a frequency ratio of 31/146, or the 146th root of 3.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 13 | 8.9 | |
| 2 | 26.1 | 17.8 | |
| 3 | 39.1 | 26.7 | 44/43, 45/44, 46/45 |
| 4 | 52.1 | 35.6 | 34/33 |
| 5 | 65.1 | 44.5 | 27/26 |
| 6 | 78.2 | 53.4 | 23/22, 45/43 |
| 7 | 91.2 | 62.3 | 39/37 |
| 8 | 104.2 | 71.2 | |
| 9 | 117.2 | 80.1 | 46/43 |
| 10 | 130.3 | 89 | 55/51 |
| 11 | 143.3 | 97.9 | 25/23 |
| 12 | 156.3 | 106.8 | 23/21 |
| 13 | 169.4 | 115.8 | 32/29, 43/39 |
| 14 | 182.4 | 124.7 | 10/9 |
| 15 | 195.4 | 133.6 | 28/25, 47/42 |
| 16 | 208.4 | 142.5 | 44/39 |
| 17 | 221.5 | 151.4 | 25/22 |
| 18 | 234.5 | 160.3 | |
| 19 | 247.5 | 169.2 | 15/13 |
| 20 | 260.5 | 178.1 | 43/37, 50/43 |
| 21 | 273.6 | 187 | 41/35, 55/47 |
| 22 | 286.6 | 195.9 | 46/39 |
| 23 | 299.6 | 204.8 | 44/37 |
| 24 | 312.7 | 213.7 | |
| 25 | 325.7 | 222.6 | 41/34 |
| 26 | 338.7 | 231.5 | 45/37 |
| 27 | 351.7 | 240.4 | 38/31 |
| 28 | 364.8 | 249.3 | 21/17 |
| 29 | 377.8 | 258.2 | 46/37, 51/41, 56/45 |
| 30 | 390.8 | 267.1 | |
| 31 | 403.8 | 276 | 24/19 |
| 32 | 416.9 | 284.9 | 14/11 |
| 33 | 429.9 | 293.8 | 50/39 |
| 34 | 442.9 | 302.7 | 31/24 |
| 35 | 455.9 | 311.6 | 56/43 |
| 36 | 469 | 320.5 | 38/29 |
| 37 | 482 | 329.5 | 33/25, 37/28 |
| 38 | 495 | 338.4 | |
| 39 | 508.1 | 347.3 | 55/41 |
| 40 | 521.1 | 356.2 | 27/20, 50/37 |
| 41 | 534.1 | 365.1 | |
| 42 | 547.1 | 374 | |
| 43 | 560.2 | 382.9 | 47/34 |
| 44 | 573.2 | 391.8 | 39/28 |
| 45 | 586.2 | 400.7 | |
| 46 | 599.2 | 409.6 | |
| 47 | 612.3 | 418.5 | 37/26, 47/33 |
| 48 | 625.3 | 427.4 | 33/23, 56/39 |
| 49 | 638.3 | 436.3 | |
| 50 | 651.4 | 445.2 | 51/35 |
| 51 | 664.4 | 454.1 | 22/15 |
| 52 | 677.4 | 463 | 34/23, 37/25 |
| 53 | 690.4 | 471.9 | |
| 54 | 703.5 | 480.8 | 3/2 |
| 55 | 716.5 | 489.7 | 56/37 |
| 56 | 729.5 | 498.6 | |
| 57 | 742.5 | 507.5 | 43/28 |
| 58 | 755.6 | 516.4 | 48/31 |
| 59 | 768.6 | 525.3 | 39/25 |
| 60 | 781.6 | 534.2 | 11/7 |
| 61 | 794.7 | 543.2 | 19/12 |
| 62 | 807.7 | 552.1 | |
| 63 | 820.7 | 561 | 45/28 |
| 64 | 833.7 | 569.9 | 34/21, 55/34 |
| 65 | 846.8 | 578.8 | 31/19, 44/27 |
| 66 | 859.8 | 587.7 | 23/14 |
| 67 | 872.8 | 596.6 | 48/29 |
| 68 | 885.8 | 605.5 | 5/3 |
| 69 | 898.9 | 614.4 | 37/22, 42/25 |
| 70 | 911.9 | 623.3 | 22/13 |
| 71 | 924.9 | 632.2 | |
| 72 | 938 | 641.1 | 43/25 |
| 73 | 951 | 650 | 26/15, 45/26 |
| 74 | 964 | 658.9 | |
| 75 | 977 | 667.8 | |
| 76 | 990.1 | 676.7 | 39/22 |
| 77 | 1003.1 | 685.6 | 25/14 |
| 78 | 1016.1 | 694.5 | 9/5 |
| 79 | 1029.1 | 703.4 | 29/16 |
| 80 | 1042.2 | 712.3 | 42/23 |
| 81 | 1055.2 | 721.2 | 46/25 |
| 82 | 1068.2 | 730.1 | 50/27 |
| 83 | 1081.2 | 739 | 28/15 |
| 84 | 1094.3 | 747.9 | 47/25 |
| 85 | 1107.3 | 756.8 | 36/19 |
| 86 | 1120.3 | 765.8 | 21/11 |
| 87 | 1133.4 | 774.7 | 25/13, 52/27 |
| 88 | 1146.4 | 783.6 | 31/16 |
| 89 | 1159.4 | 792.5 | 41/21, 43/22 |
| 90 | 1172.4 | 801.4 | |
| 91 | 1185.5 | 810.3 | |
| 92 | 1198.5 | 819.2 | 2/1 |
| 93 | 1211.5 | 828.1 | |
| 94 | 1224.5 | 837 | |
| 95 | 1237.6 | 845.9 | 45/22, 47/23 |
| 96 | 1250.6 | 854.8 | 35/17 |
| 97 | 1263.6 | 863.7 | 56/27 |
| 98 | 1276.7 | 872.6 | 23/11 |
| 99 | 1289.7 | 881.5 | 40/19 |
| 100 | 1302.7 | 890.4 | |
| 101 | 1315.7 | 899.3 | 47/22 |
| 102 | 1328.8 | 908.2 | 28/13 |
| 103 | 1341.8 | 917.1 | |
| 104 | 1354.8 | 926 | |
| 105 | 1367.8 | 934.9 | |
| 106 | 1380.9 | 943.8 | 20/9 |
| 107 | 1393.9 | 952.7 | 47/21 |
| 108 | 1406.9 | 961.6 | |
| 109 | 1420 | 970.5 | 25/11 |
| 110 | 1433 | 979.5 | |
| 111 | 1446 | 988.4 | |
| 112 | 1459 | 997.3 | |
| 113 | 1472.1 | 1006.2 | |
| 114 | 1485.1 | 1015.1 | 33/14 |
| 115 | 1498.1 | 1024 | 19/8 |
| 116 | 1511.1 | 1032.9 | |
| 117 | 1524.2 | 1041.8 | 41/17 |
| 118 | 1537.2 | 1050.7 | 17/7 |
| 119 | 1550.2 | 1059.6 | |
| 120 | 1563.3 | 1068.5 | 37/15 |
| 121 | 1576.3 | 1077.4 | |
| 122 | 1589.3 | 1086.3 | |
| 123 | 1602.3 | 1095.2 | |
| 124 | 1615.4 | 1104.1 | |
| 125 | 1628.4 | 1113 | |
| 126 | 1641.4 | 1121.9 | |
| 127 | 1654.4 | 1130.8 | 13/5 |
| 128 | 1667.5 | 1139.7 | 55/21 |
| 129 | 1680.5 | 1148.6 | |
| 130 | 1693.5 | 1157.5 | |
| 131 | 1706.5 | 1166.4 | |
| 132 | 1719.6 | 1175.3 | 27/10 |
| 133 | 1732.6 | 1184.2 | |
| 134 | 1745.6 | 1193.2 | |
| 135 | 1758.7 | 1202.1 | |
| 136 | 1771.7 | 1211 | |
| 137 | 1784.7 | 1219.9 | |
| 138 | 1797.7 | 1228.8 | |
| 139 | 1810.8 | 1237.7 | 37/13 |
| 140 | 1823.8 | 1246.6 | 43/15 |
| 141 | 1836.8 | 1255.5 | 26/9 |
| 142 | 1849.8 | 1264.4 | |
| 143 | 1862.9 | 1273.3 | 44/15 |
| 144 | 1875.9 | 1282.2 | |
| 145 | 1888.9 | 1291.1 | |
| 146 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.51 | +0.00 | -3.02 | +1.48 | -1.51 | +5.19 | -4.52 | +0.00 | -0.02 | +4.32 | -3.02 |
| Relative (%) | -11.6 | +0.0 | -23.1 | +11.4 | -11.6 | +39.8 | -34.7 | +0.0 | -0.2 | +33.2 | -23.1 | |
| Steps (reduced) |
92 (92) |
146 (0) |
184 (38) |
214 (68) |
238 (92) |
259 (113) |
276 (130) |
292 (0) |
306 (14) |
319 (27) |
330 (38) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.71 | +3.68 | +1.48 | -6.03 | +6.26 | -1.51 | -3.92 | -1.53 | +5.19 | +2.82 | +4.02 |
| Relative (%) | +13.1 | +28.3 | +11.4 | -46.3 | +48.0 | -11.6 | -30.1 | -11.8 | +39.8 | +21.6 | +30.9 | |
| Steps (reduced) |
341 (49) |
351 (59) |
360 (68) |
368 (76) |
377 (85) |
384 (92) |
391 (99) |
398 (106) |
405 (113) |
411 (119) |
417 (125) | |