155edt
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155 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 155edt or 155ed3), is a nonoctave tuning system that divides the interval of 3/1 into 155 equal parts of about 12.3 ¢ each. Each step represents a frequency ratio of 31/155, or the 155th root of 3.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 12.3 | 8.4 | |
| 2 | 24.5 | 16.8 | |
| 3 | 36.8 | 25.2 | 46/45, 47/46, 49/48 |
| 4 | 49.1 | 33.5 | 35/34, 36/35 |
| 5 | 61.4 | 41.9 | 57/55 |
| 6 | 73.6 | 50.3 | |
| 7 | 85.9 | 58.7 | 21/20, 41/39 |
| 8 | 98.2 | 67.1 | 18/17 |
| 9 | 110.4 | 75.5 | |
| 10 | 122.7 | 83.9 | 29/27, 44/41 |
| 11 | 135 | 92.3 | |
| 12 | 147.2 | 100.6 | |
| 13 | 159.5 | 109 | |
| 14 | 171.8 | 117.4 | |
| 15 | 184.1 | 125.8 | |
| 16 | 196.3 | 134.2 | |
| 17 | 208.6 | 142.6 | 44/39 |
| 18 | 220.9 | 151 | 25/22 |
| 19 | 233.1 | 159.4 | |
| 20 | 245.4 | 167.7 | 38/33 |
| 21 | 257.7 | 176.1 | 29/25 |
| 22 | 270 | 184.5 | |
| 23 | 282.2 | 192.9 | 20/17 |
| 24 | 294.5 | 201.3 | 51/43 |
| 25 | 306.8 | 209.7 | 37/31, 43/36 |
| 26 | 319 | 218.1 | |
| 27 | 331.3 | 226.5 | 23/19 |
| 28 | 343.6 | 234.8 | 50/41 |
| 29 | 355.8 | 243.2 | 27/22, 43/35 |
| 30 | 368.1 | 251.6 | 47/38 |
| 31 | 380.4 | 260 | |
| 32 | 392.7 | 268.4 | |
| 33 | 404.9 | 276.8 | |
| 34 | 417.2 | 285.2 | |
| 35 | 429.5 | 293.5 | 50/39 |
| 36 | 441.7 | 301.9 | |
| 37 | 454 | 310.3 | 13/10 |
| 38 | 466.3 | 318.7 | |
| 39 | 478.6 | 327.1 | 29/22 |
| 40 | 490.8 | 335.5 | |
| 41 | 503.1 | 343.9 | |
| 42 | 515.4 | 352.3 | 31/23, 35/26 |
| 43 | 527.6 | 360.6 | |
| 44 | 539.9 | 369 | 41/30 |
| 45 | 552.2 | 377.4 | |
| 46 | 564.5 | 385.8 | 18/13 |
| 47 | 576.7 | 394.2 | |
| 48 | 589 | 402.6 | |
| 49 | 601.3 | 411 | 58/41 |
| 50 | 613.5 | 419.4 | 47/33 |
| 51 | 625.8 | 427.7 | 33/23 |
| 52 | 638.1 | 436.1 | 13/9 |
| 53 | 650.3 | 444.5 | 51/35 |
| 54 | 662.6 | 452.9 | 22/15 |
| 55 | 674.9 | 461.3 | |
| 56 | 687.2 | 469.7 | 55/37, 58/39 |
| 57 | 699.4 | 478.1 | |
| 58 | 711.7 | 486.5 | |
| 59 | 724 | 494.8 | 38/25, 41/27 |
| 60 | 736.2 | 503.2 | 26/17, 49/32 |
| 61 | 748.5 | 511.6 | 57/37 |
| 62 | 760.8 | 520 | 45/29 |
| 63 | 773.1 | 528.4 | |
| 64 | 785.3 | 536.8 | |
| 65 | 797.6 | 545.2 | 46/29 |
| 66 | 809.9 | 553.5 | |
| 67 | 822.1 | 561.9 | 37/23 |
| 68 | 834.4 | 570.3 | 34/21 |
| 69 | 846.7 | 578.7 | 31/19, 44/27 |
| 70 | 858.9 | 587.1 | |
| 71 | 871.2 | 595.5 | 43/26 |
| 72 | 883.5 | 603.9 | 5/3 |
| 73 | 895.8 | 612.3 | |
| 74 | 908 | 620.6 | |
| 75 | 920.3 | 629 | |
| 76 | 932.6 | 637.4 | 12/7 |
| 77 | 944.8 | 645.8 | 19/11 |
| 78 | 957.1 | 654.2 | 33/19 |
| 79 | 969.4 | 662.6 | 7/4 |
| 80 | 981.7 | 671 | |
| 81 | 993.9 | 679.4 | 55/31 |
| 82 | 1006.2 | 687.7 | |
| 83 | 1018.5 | 696.1 | 9/5 |
| 84 | 1030.7 | 704.5 | |
| 85 | 1043 | 712.9 | |
| 86 | 1055.3 | 721.3 | 46/25, 57/31 |
| 87 | 1067.5 | 729.7 | 50/27 |
| 88 | 1079.8 | 738.1 | |
| 89 | 1092.1 | 746.5 | 47/25 |
| 90 | 1104.4 | 754.8 | |
| 91 | 1116.6 | 763.2 | 40/21 |
| 92 | 1128.9 | 771.6 | |
| 93 | 1141.2 | 780 | 29/15 |
| 94 | 1153.4 | 788.4 | 37/19 |
| 95 | 1165.7 | 796.8 | 51/26 |
| 96 | 1178 | 805.2 | |
| 97 | 1190.3 | 813.5 | |
| 98 | 1202.5 | 821.9 | |
| 99 | 1214.8 | 830.3 | |
| 100 | 1227.1 | 838.7 | |
| 101 | 1239.3 | 847.1 | 43/21, 45/22 |
| 102 | 1251.6 | 855.5 | 35/17 |
| 103 | 1263.9 | 863.9 | 27/13 |
| 104 | 1276.2 | 872.3 | 23/11 |
| 105 | 1288.4 | 880.6 | |
| 106 | 1300.7 | 889 | |
| 107 | 1313 | 897.4 | 47/22 |
| 108 | 1325.2 | 905.8 | 43/20 |
| 109 | 1337.5 | 914.2 | 13/6 |
| 110 | 1349.8 | 922.6 | |
| 111 | 1362 | 931 | |
| 112 | 1374.3 | 939.4 | |
| 113 | 1386.6 | 947.7 | |
| 114 | 1398.9 | 956.1 | |
| 115 | 1411.1 | 964.5 | |
| 116 | 1423.4 | 972.9 | |
| 117 | 1435.7 | 981.3 | |
| 118 | 1447.9 | 989.7 | 30/13 |
| 119 | 1460.2 | 998.1 | |
| 120 | 1472.5 | 1006.5 | |
| 121 | 1484.8 | 1014.8 | |
| 122 | 1497 | 1023.2 | |
| 123 | 1509.3 | 1031.6 | 55/23 |
| 124 | 1521.6 | 1040 | |
| 125 | 1533.8 | 1048.4 | |
| 126 | 1546.1 | 1056.8 | 22/9 |
| 127 | 1558.4 | 1065.2 | |
| 128 | 1570.6 | 1073.5 | 52/21, 57/23 |
| 129 | 1582.9 | 1081.9 | |
| 130 | 1595.2 | 1090.3 | |
| 131 | 1607.5 | 1098.7 | 43/17 |
| 132 | 1619.7 | 1107.1 | 51/20 |
| 133 | 1632 | 1115.5 | |
| 134 | 1644.3 | 1123.9 | |
| 135 | 1656.5 | 1132.3 | |
| 136 | 1668.8 | 1140.6 | |
| 137 | 1681.1 | 1149 | |
| 138 | 1693.4 | 1157.4 | |
| 139 | 1705.6 | 1165.8 | |
| 140 | 1717.9 | 1174.2 | |
| 141 | 1730.2 | 1182.6 | |
| 142 | 1742.4 | 1191 | |
| 143 | 1754.7 | 1199.4 | |
| 144 | 1767 | 1207.7 | |
| 145 | 1779.2 | 1216.1 | |
| 146 | 1791.5 | 1224.5 | |
| 147 | 1803.8 | 1232.9 | 17/6 |
| 148 | 1816.1 | 1241.3 | 20/7 |
| 149 | 1828.3 | 1249.7 | |
| 150 | 1840.6 | 1258.1 | 55/19 |
| 151 | 1852.9 | 1266.5 | 35/12 |
| 152 | 1865.1 | 1274.8 | |
| 153 | 1877.4 | 1283.2 | |
| 154 | 1889.7 | 1291.6 | |
| 155 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.53 | +0.00 | +5.05 | -0.87 | +2.53 | +5.61 | -4.69 | +0.00 | +1.66 | -3.83 | +5.05 |
| Relative (%) | +20.6 | +0.0 | +41.2 | -7.1 | +20.6 | +45.7 | -38.2 | +0.0 | +13.5 | -31.2 | +41.2 | |
| Steps (reduced) |
98 (98) |
155 (0) |
196 (41) |
227 (72) |
253 (98) |
275 (120) |
293 (138) |
310 (0) |
325 (15) |
338 (28) |
351 (41) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.46 | -4.13 | -0.87 | -2.17 | +3.32 | +2.53 | -5.18 | +4.18 | +5.61 | -1.30 | -4.63 |
| Relative (%) | +11.9 | -33.7 | -7.1 | -17.6 | +27.0 | +20.6 | -42.2 | +34.1 | +45.7 | -10.6 | -37.8 | |
| Steps (reduced) |
362 (52) |
372 (62) |
382 (72) |
391 (81) |
400 (90) |
408 (98) |
415 (105) |
423 (113) |
430 (120) |
436 (126) |
442 (132) | |