153edt
| ← 152edt | 153edt | 154edt → |
153 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 153edt or 153ed3), is a nonoctave tuning system that divides the interval of 3/1 into 153 equal parts of about 12.4 ¢ each. Each step represents a frequency ratio of 31/153, or the 153rd root of 3.
153edt is notable for being the denominator of a convergent to log3(7/3), after 9edt, 13edt and 35edt, and the last before 3401edt, and therefore has an extremely accurate approximation to 7/3, a mere 0.0036 cents flat. In fact, 153edt demonstrates 11-strong 7-3 telicity, due to the next term in the continued fraction expansion being large (note how much larger 3401 is than 153), although 3401edt in fact surpasses it, demonstrating 16-strong 7-3 telicity.
In the no-twos 7-limit, 153edt supports canopus temperament, which gives it a rather accurate approximation of the 5th harmonic; and it additionally is accurate in the 11-limit, tempering out the comma 387420489/386683451 in the 3.7.11 subgroup. Harmonics 19 and 29 are also notably good.
However, 153edt's approximation of 2/1 is close to maximally bad, meaning that it is as far from an octave-equivalent tuning that an EDT of this size can be (though by this point, it is only 6 or so cents off).
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 12.4 | 8.5 | |
| 2 | 24.9 | 17 | |
| 3 | 37.3 | 25.5 | |
| 4 | 49.7 | 34 | |
| 5 | 62.2 | 42.5 | 57/55 |
| 6 | 74.6 | 51 | 47/45 |
| 7 | 87 | 59.5 | 41/39 |
| 8 | 99.4 | 68 | 18/17 |
| 9 | 111.9 | 76.5 | |
| 10 | 124.3 | 85 | 29/27 |
| 11 | 136.7 | 93.5 | |
| 12 | 149.2 | 102 | |
| 13 | 161.6 | 110.5 | 45/41 |
| 14 | 174 | 119 | 21/19 |
| 15 | 186.5 | 127.5 | 39/35 |
| 16 | 198.9 | 135.9 | 37/33, 55/49 |
| 17 | 211.3 | 144.4 | 26/23, 35/31 |
| 18 | 223.8 | 152.9 | 33/29, 58/51 |
| 19 | 236.2 | 161.4 | 47/41 |
| 20 | 248.6 | 169.9 | 15/13 |
| 21 | 261.1 | 178.4 | 43/37, 50/43, 57/49 |
| 22 | 273.5 | 186.9 | 41/35, 55/47 |
| 23 | 285.9 | 195.4 | |
| 24 | 298.3 | 203.9 | |
| 25 | 310.8 | 212.4 | |
| 26 | 323.2 | 220.9 | 47/39 |
| 27 | 335.6 | 229.4 | 17/14 |
| 28 | 348.1 | 237.9 | 11/9 |
| 29 | 360.5 | 246.4 | |
| 30 | 372.9 | 254.9 | 31/25 |
| 31 | 385.4 | 263.4 | |
| 32 | 397.8 | 271.9 | 39/31 |
| 33 | 410.2 | 280.4 | 19/15 |
| 34 | 422.7 | 288.9 | 37/29 |
| 35 | 435.1 | 297.4 | 9/7 |
| 36 | 447.5 | 305.9 | 22/17 |
| 37 | 459.9 | 314.4 | 30/23 |
| 38 | 472.4 | 322.9 | |
| 39 | 484.8 | 331.4 | 41/31, 49/37 |
| 40 | 497.2 | 339.9 | |
| 41 | 509.7 | 348.4 | 47/35, 51/38, 55/41 |
| 42 | 522.1 | 356.9 | 23/17, 50/37 |
| 43 | 534.5 | 365.4 | |
| 44 | 547 | 373.9 | 37/27 |
| 45 | 559.4 | 382.4 | 29/21 |
| 46 | 571.8 | 390.8 | 57/41 |
| 47 | 584.3 | 399.3 | |
| 48 | 596.7 | 407.8 | |
| 49 | 609.1 | 416.3 | 27/19 |
| 50 | 621.6 | 424.8 | |
| 51 | 634 | 433.3 | |
| 52 | 646.4 | 441.8 | 45/31 |
| 53 | 658.8 | 450.3 | |
| 54 | 671.3 | 458.8 | |
| 55 | 683.7 | 467.3 | 49/33 |
| 56 | 696.1 | 475.8 | |
| 57 | 708.6 | 484.3 | |
| 58 | 721 | 492.8 | 47/31 |
| 59 | 733.4 | 501.3 | 29/19 |
| 60 | 745.9 | 509.8 | |
| 61 | 758.3 | 518.3 | |
| 62 | 770.7 | 526.8 | 39/25 |
| 63 | 783.2 | 535.3 | 11/7 |
| 64 | 795.6 | 543.8 | |
| 65 | 808 | 552.3 | |
| 66 | 820.5 | 560.8 | |
| 67 | 832.9 | 569.3 | |
| 68 | 845.3 | 577.8 | 57/35 |
| 69 | 857.7 | 586.3 | 41/25 |
| 70 | 870.2 | 594.8 | 38/23, 43/26 |
| 71 | 882.6 | 603.3 | |
| 72 | 895 | 611.8 | |
| 73 | 907.5 | 620.3 | 49/29 |
| 74 | 919.9 | 628.8 | 17/10 |
| 75 | 932.3 | 637.3 | |
| 76 | 944.8 | 645.8 | 19/11 |
| 77 | 957.2 | 654.2 | 33/19 |
| 78 | 969.6 | 662.7 | |
| 79 | 982.1 | 671.2 | 30/17, 37/21 |
| 80 | 994.5 | 679.7 | |
| 81 | 1006.9 | 688.2 | |
| 82 | 1019.3 | 696.7 | |
| 83 | 1031.8 | 705.2 | 49/27 |
| 84 | 1044.2 | 713.7 | |
| 85 | 1056.6 | 722.2 | 35/19 |
| 86 | 1069.1 | 730.7 | |
| 87 | 1081.5 | 739.2 | |
| 88 | 1093.9 | 747.7 | 47/25 |
| 89 | 1106.4 | 756.2 | |
| 90 | 1118.8 | 764.7 | 21/11 |
| 91 | 1131.2 | 773.2 | 25/13 |
| 92 | 1143.7 | 781.7 | |
| 93 | 1156.1 | 790.2 | |
| 94 | 1168.5 | 798.7 | 57/29 |
| 95 | 1181 | 807.2 | |
| 96 | 1193.4 | 815.7 | |
| 97 | 1205.8 | 824.2 | |
| 98 | 1218.2 | 832.7 | |
| 99 | 1230.7 | 841.2 | 55/27 |
| 100 | 1243.1 | 849.7 | |
| 101 | 1255.5 | 858.2 | 31/15 |
| 102 | 1268 | 866.7 | |
| 103 | 1280.4 | 875.2 | |
| 104 | 1292.8 | 883.7 | 19/9 |
| 105 | 1305.3 | 892.2 | |
| 106 | 1317.7 | 900.7 | |
| 107 | 1330.1 | 909.2 | 41/19 |
| 108 | 1342.6 | 917.6 | |
| 109 | 1355 | 926.1 | |
| 110 | 1367.4 | 934.6 | |
| 111 | 1379.8 | 943.1 | 51/23 |
| 112 | 1392.3 | 951.6 | 38/17 |
| 113 | 1404.7 | 960.1 | |
| 114 | 1417.1 | 968.6 | |
| 115 | 1429.6 | 977.1 | |
| 116 | 1442 | 985.6 | 23/10 |
| 117 | 1454.4 | 994.1 | 51/22 |
| 118 | 1466.9 | 1002.6 | 7/3 |
| 119 | 1479.3 | 1011.1 | |
| 120 | 1491.7 | 1019.6 | 45/19 |
| 121 | 1504.2 | 1028.1 | 31/13 |
| 122 | 1516.6 | 1036.6 | |
| 123 | 1529 | 1045.1 | |
| 124 | 1541.5 | 1053.6 | |
| 125 | 1553.9 | 1062.1 | 27/11 |
| 126 | 1566.3 | 1070.6 | 42/17 |
| 127 | 1578.7 | 1079.1 | |
| 128 | 1591.2 | 1087.6 | |
| 129 | 1603.6 | 1096.1 | |
| 130 | 1616 | 1104.6 | |
| 131 | 1628.5 | 1113.1 | |
| 132 | 1640.9 | 1121.6 | 49/19 |
| 133 | 1653.3 | 1130.1 | 13/5 |
| 134 | 1665.8 | 1138.6 | 55/21 |
| 135 | 1678.2 | 1147.1 | 29/11 |
| 136 | 1690.6 | 1155.6 | |
| 137 | 1703.1 | 1164.1 | |
| 138 | 1715.5 | 1172.5 | 35/13 |
| 139 | 1727.9 | 1181 | 19/7 |
| 140 | 1740.4 | 1189.5 | 41/15 |
| 141 | 1752.8 | 1198 | |
| 142 | 1765.2 | 1206.5 | |
| 143 | 1777.6 | 1215 | |
| 144 | 1790.1 | 1223.5 | |
| 145 | 1802.5 | 1232 | 17/6 |
| 146 | 1814.9 | 1240.5 | |
| 147 | 1827.4 | 1249 | |
| 148 | 1839.8 | 1257.5 | 55/19 |
| 149 | 1852.2 | 1266 | |
| 150 | 1864.7 | 1274.5 | |
| 151 | 1877.1 | 1283 | |
| 152 | 1889.5 | 1291.5 | |
| 153 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.81 | +0.00 | -0.80 | -1.75 | +5.81 | -0.00 | +5.01 | +0.00 | +4.06 | +0.66 | -0.80 |
| Relative (%) | +46.8 | +0.0 | -6.5 | -14.1 | +46.8 | -0.0 | +40.3 | +0.0 | +32.7 | +5.3 | -6.5 | |
| Steps (reduced) |
97 (97) |
153 (0) |
193 (40) |
224 (71) |
250 (97) |
271 (118) |
290 (137) |
306 (0) |
321 (15) |
334 (28) |
346 (40) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.63 | +5.81 | -1.75 | -1.60 | +5.32 | +5.81 | -0.77 | -2.55 | -0.00 | -5.95 | +4.11 |
| Relative (%) | -21.2 | +46.7 | -14.1 | -12.9 | +42.8 | +46.8 | -6.2 | -20.5 | -0.0 | -47.9 | +33.0 | |
| Steps (reduced) |
357 (51) |
368 (62) |
377 (71) |
386 (80) |
395 (89) |
403 (97) |
410 (104) |
417 (111) |
424 (118) |
430 (124) |
437 (131) | |