159edt
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159 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 159edt or 159ed3), is a nonoctave tuning system that divides the interval of 3/1 into 159 equal parts of about 12 ¢ each. Each step represents a frequency ratio of 31/159, or the 159th root of 3.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 12 | 8.2 | |
| 2 | 23.9 | 16.4 | |
| 3 | 35.9 | 24.5 | 47/46 |
| 4 | 47.8 | 32.7 | |
| 5 | 59.8 | 40.9 | 29/28, 30/29 |
| 6 | 71.8 | 49.1 | |
| 7 | 83.7 | 57.2 | 43/41 |
| 8 | 95.7 | 65.4 | 37/35 |
| 9 | 107.7 | 73.6 | 33/31, 50/47 |
| 10 | 119.6 | 81.8 | 15/14 |
| 11 | 131.6 | 89.9 | 41/38, 55/51 |
| 12 | 143.5 | 98.1 | 25/23 |
| 13 | 155.5 | 106.3 | |
| 14 | 167.5 | 114.5 | |
| 15 | 179.4 | 122.6 | 51/46 |
| 16 | 191.4 | 130.8 | 19/17 |
| 17 | 203.4 | 139 | |
| 18 | 215.3 | 147.2 | 17/15, 43/38 |
| 19 | 227.3 | 155.3 | 57/50 |
| 20 | 239.2 | 163.5 | 31/27, 54/47 |
| 21 | 251.2 | 171.7 | |
| 22 | 263.2 | 179.9 | |
| 23 | 275.1 | 188.1 | 34/29 |
| 24 | 287.1 | 196.2 | 46/39 |
| 25 | 299 | 204.4 | |
| 26 | 311 | 212.6 | |
| 27 | 323 | 220.8 | 41/34, 47/39 |
| 28 | 334.9 | 228.9 | 17/14, 57/47 |
| 29 | 346.9 | 237.1 | 11/9 |
| 30 | 358.9 | 245.3 | |
| 31 | 370.8 | 253.5 | 57/46 |
| 32 | 382.8 | 261.6 | |
| 33 | 394.7 | 269.8 | 54/43 |
| 34 | 406.7 | 278 | 43/34 |
| 35 | 418.7 | 286.2 | 14/11 |
| 36 | 430.6 | 294.3 | 50/39 |
| 37 | 442.6 | 302.5 | |
| 38 | 454.6 | 310.7 | 13/10 |
| 39 | 466.5 | 318.9 | 38/29, 55/42 |
| 40 | 478.5 | 327 | 29/22 |
| 41 | 490.4 | 335.2 | |
| 42 | 502.4 | 343.4 | |
| 43 | 514.4 | 351.6 | |
| 44 | 526.3 | 359.7 | 42/31 |
| 45 | 538.3 | 367.9 | 15/11 |
| 46 | 550.3 | 376.1 | |
| 47 | 562.2 | 384.3 | 18/13 |
| 48 | 574.2 | 392.5 | 39/28, 46/33 |
| 49 | 586.1 | 400.6 | |
| 50 | 598.1 | 408.8 | 41/29 |
| 51 | 610.1 | 417 | |
| 52 | 622 | 425.2 | 43/30 |
| 53 | 634 | 433.3 | |
| 54 | 645.9 | 441.5 | 45/31 |
| 55 | 657.9 | 449.7 | 19/13 |
| 56 | 669.9 | 457.9 | 28/19 |
| 57 | 681.8 | 466 | 43/29, 46/31 |
| 58 | 693.8 | 474.2 | |
| 59 | 705.8 | 482.4 | |
| 60 | 717.7 | 490.6 | |
| 61 | 729.7 | 498.7 | |
| 62 | 741.6 | 506.9 | 43/28 |
| 63 | 753.6 | 515.1 | 17/11 |
| 64 | 765.6 | 523.3 | 14/9 |
| 65 | 777.5 | 531.4 | 47/30 |
| 66 | 789.5 | 539.6 | 30/19, 41/26 |
| 67 | 801.5 | 547.8 | 27/17 |
| 68 | 813.4 | 556 | |
| 69 | 825.4 | 564.2 | 29/18 |
| 70 | 837.3 | 572.3 | 47/29 |
| 71 | 849.3 | 580.5 | |
| 72 | 861.3 | 588.7 | 51/31 |
| 73 | 873.2 | 596.9 | |
| 74 | 885.2 | 605 | 5/3 |
| 75 | 897.1 | 613.2 | 42/25, 47/28 |
| 76 | 909.1 | 621.4 | |
| 77 | 921.1 | 629.6 | 46/27 |
| 78 | 933 | 637.7 | |
| 79 | 945 | 645.9 | 19/11 |
| 80 | 957 | 654.1 | 33/19 |
| 81 | 968.9 | 662.3 | |
| 82 | 980.9 | 670.4 | 37/21 |
| 83 | 992.8 | 678.6 | 55/31 |
| 84 | 1004.8 | 686.8 | 25/14 |
| 85 | 1016.8 | 695 | 9/5 |
| 86 | 1028.7 | 703.1 | |
| 87 | 1040.7 | 711.3 | 31/17 |
| 88 | 1052.7 | 719.5 | |
| 89 | 1064.6 | 727.7 | |
| 90 | 1076.6 | 735.8 | 41/22, 54/29 |
| 91 | 1088.5 | 744 | |
| 92 | 1100.5 | 752.2 | 17/9 |
| 93 | 1112.5 | 760.4 | 19/10 |
| 94 | 1124.4 | 768.6 | |
| 95 | 1136.4 | 776.7 | 27/14 |
| 96 | 1148.4 | 784.9 | 33/17 |
| 97 | 1160.3 | 793.1 | 43/22 |
| 98 | 1172.3 | 801.3 | |
| 99 | 1184.2 | 809.4 | |
| 100 | 1196.2 | 817.6 | |
| 101 | 1208.2 | 825.8 | |
| 102 | 1220.1 | 834 | |
| 103 | 1232.1 | 842.1 | 55/27, 57/28 |
| 104 | 1244 | 850.3 | 39/19, 41/20 |
| 105 | 1256 | 858.5 | 31/15 |
| 106 | 1268 | 866.7 | |
| 107 | 1279.9 | 874.8 | |
| 108 | 1291.9 | 883 | |
| 109 | 1303.9 | 891.2 | |
| 110 | 1315.8 | 899.4 | |
| 111 | 1327.8 | 907.5 | 28/13 |
| 112 | 1339.7 | 915.7 | 13/6 |
| 113 | 1351.7 | 923.9 | |
| 114 | 1363.7 | 932.1 | 11/5 |
| 115 | 1375.6 | 940.3 | 31/14 |
| 116 | 1387.6 | 948.4 | |
| 117 | 1399.6 | 956.6 | |
| 118 | 1411.5 | 964.8 | |
| 119 | 1423.5 | 973 | |
| 120 | 1435.4 | 981.1 | |
| 121 | 1447.4 | 989.3 | 30/13 |
| 122 | 1459.4 | 997.5 | |
| 123 | 1471.3 | 1005.7 | |
| 124 | 1483.3 | 1013.8 | 33/14 |
| 125 | 1495.2 | 1022 | |
| 126 | 1507.2 | 1030.2 | 43/18 |
| 127 | 1519.2 | 1038.4 | |
| 128 | 1531.1 | 1046.5 | 46/19 |
| 129 | 1543.1 | 1054.7 | |
| 130 | 1555.1 | 1062.9 | 27/11 |
| 131 | 1567 | 1071.1 | 42/17, 47/19 |
| 132 | 1579 | 1079.2 | |
| 133 | 1590.9 | 1087.4 | |
| 134 | 1602.9 | 1095.6 | |
| 135 | 1614.9 | 1103.8 | |
| 136 | 1626.8 | 1111.9 | |
| 137 | 1638.8 | 1120.1 | |
| 138 | 1650.8 | 1128.3 | |
| 139 | 1662.7 | 1136.5 | 47/18 |
| 140 | 1674.7 | 1144.7 | 50/19 |
| 141 | 1686.6 | 1152.8 | 45/17 |
| 142 | 1698.6 | 1161 | |
| 143 | 1710.6 | 1169.2 | 51/19 |
| 144 | 1722.5 | 1177.4 | 46/17 |
| 145 | 1734.5 | 1185.5 | |
| 146 | 1746.4 | 1193.7 | |
| 147 | 1758.4 | 1201.9 | |
| 148 | 1770.4 | 1210.1 | |
| 149 | 1782.3 | 1218.2 | 14/5 |
| 150 | 1794.3 | 1226.4 | 31/11 |
| 151 | 1806.3 | 1234.6 | |
| 152 | 1818.2 | 1242.8 | |
| 153 | 1830.2 | 1250.9 | |
| 154 | 1842.1 | 1259.1 | 29/10 |
| 155 | 1854.1 | 1267.3 | |
| 156 | 1866.1 | 1275.5 | |
| 157 | 1878 | 1283.6 | |
| 158 | 1890 | 1291.8 | |
| 159 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.80 | +0.00 | +4.36 | +0.83 | -3.80 | +4.45 | +0.56 | +0.00 | -2.97 | -0.51 | +4.36 |
| Relative (%) | -31.8 | +0.0 | +36.4 | +6.9 | -31.8 | +37.2 | +4.7 | +0.0 | -24.9 | -4.3 | +36.4 | |
| Steps (reduced) |
100 (100) |
159 (0) |
201 (42) |
233 (74) |
259 (100) |
282 (123) |
301 (142) |
318 (0) |
333 (15) |
347 (29) |
360 (42) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.63 | +0.65 | +0.83 | -3.25 | -0.54 | -3.80 | -1.71 | +5.19 | +4.45 | -4.31 | +2.47 |
| Relative (%) | -22.0 | +5.4 | +6.9 | -27.1 | -4.5 | -31.8 | -14.3 | +43.4 | +37.2 | -36.1 | +20.6 | |
| Steps (reduced) |
371 (53) |
382 (64) |
392 (74) |
401 (83) |
410 (92) |
418 (100) |
426 (108) |
434 (116) |
441 (123) |
447 (129) |
454 (136) | |