51edt
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51 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 51edt or 51ed3), is a nonoctave tuning system that divides the interval of 3/1 into 51 equal parts of about 37.3 ¢ each. Each step represents a frequency ratio of 31/51, or the 51st root of 3.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 37.3 | 25.5 | |
| 2 | 74.6 | 51 | |
| 3 | 111.9 | 76.5 | 31/29 |
| 4 | 149.2 | 102 | 12/11, 25/23 |
| 5 | 186.5 | 127.5 | 10/9, 29/26 |
| 6 | 223.8 | 152.9 | |
| 7 | 261.1 | 178.4 | |
| 8 | 298.3 | 203.9 | |
| 9 | 335.6 | 229.4 | 23/19 |
| 10 | 372.9 | 254.9 | 26/21 |
| 11 | 410.2 | 280.4 | 19/15 |
| 12 | 447.5 | 305.9 | 31/24 |
| 13 | 484.8 | 331.4 | |
| 14 | 522.1 | 356.9 | 23/17, 27/20 |
| 15 | 559.4 | 382.4 | 18/13, 29/21 |
| 16 | 596.7 | 407.8 | 31/22 |
| 17 | 634 | 433.3 | 13/9 |
| 18 | 671.3 | 458.8 | 25/17, 31/21 |
| 19 | 708.6 | 484.3 | |
| 20 | 745.9 | 509.8 | 20/13 |
| 21 | 783.2 | 535.3 | 11/7 |
| 22 | 820.5 | 560.8 | 29/18 |
| 23 | 857.7 | 586.3 | 18/11 |
| 24 | 895 | 611.8 | |
| 25 | 932.3 | 637.3 | 12/7 |
| 26 | 969.6 | 662.7 | 7/4 |
| 27 | 1006.9 | 688.2 | |
| 28 | 1044.2 | 713.7 | 11/6 |
| 29 | 1081.5 | 739.2 | |
| 30 | 1118.8 | 764.7 | 21/11 |
| 31 | 1156.1 | 790.2 | |
| 32 | 1193.4 | 815.7 | |
| 33 | 1230.7 | 841.2 | |
| 34 | 1268 | 866.7 | 27/13 |
| 35 | 1305.3 | 892.2 | |
| 36 | 1342.6 | 917.6 | 13/6 |
| 37 | 1379.8 | 943.1 | 20/9, 31/14 |
| 38 | 1417.1 | 968.6 | |
| 39 | 1454.4 | 994.1 | |
| 40 | 1491.7 | 1019.6 | 26/11 |
| 41 | 1529 | 1045.1 | 29/12 |
| 42 | 1566.3 | 1070.6 | |
| 43 | 1603.6 | 1096.1 | |
| 44 | 1640.9 | 1121.6 | 31/12 |
| 45 | 1678.2 | 1147.1 | 29/11 |
| 46 | 1715.5 | 1172.5 | 27/10 |
| 47 | 1752.8 | 1198 | 11/4 |
| 48 | 1790.1 | 1223.5 | 31/11 |
| 49 | 1827.4 | 1249 | |
| 50 | 1864.7 | 1274.5 | |
| 51 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -6.6 | +0.0 | -13.2 | +10.7 | -6.6 | -12.4 | +17.4 | +0.0 | +4.1 | -11.8 | -13.2 |
| Relative (%) | -17.7 | +0.0 | -35.5 | +28.6 | -17.7 | -33.3 | +46.8 | +0.0 | +10.9 | -31.6 | -35.5 | |
| Steps (reduced) |
32 (32) |
51 (0) |
64 (13) |
75 (24) |
83 (32) |
90 (39) |
97 (46) |
102 (0) |
107 (5) |
111 (9) |
115 (13) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.6 | +18.2 | +10.7 | +10.8 | +17.8 | -6.6 | +11.7 | -2.6 | -12.4 | -18.4 | +16.5 |
| Relative (%) | -7.1 | +48.9 | +28.6 | +29.0 | +47.6 | -17.7 | +31.3 | -6.8 | -33.3 | -49.3 | +44.3 | |
| Steps (reduced) |
119 (17) |
123 (21) |
126 (24) |
129 (27) |
132 (30) |
134 (32) |
137 (35) |
139 (37) |
141 (39) |
143 (41) |
146 (44) | |