58edt
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58 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 58edt or 58ed3), is a nonoctave tuning system that divides the interval of 3/1 into 58 equal parts of about 32.8 ¢ each. Each step represents a frequency ratio of 31/58, or the 58th root of 3.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 32.8 | 22.4 | |
| 2 | 65.6 | 44.8 | 26/25, 27/26 |
| 3 | 98.4 | 67.2 | 18/17 |
| 4 | 131.2 | 89.7 | 27/25 |
| 5 | 164 | 112.1 | 11/10 |
| 6 | 196.8 | 134.5 | |
| 7 | 229.5 | 156.9 | |
| 8 | 262.3 | 179.3 | 7/6 |
| 9 | 295.1 | 201.7 | |
| 10 | 327.9 | 224.1 | |
| 11 | 360.7 | 246.6 | |
| 12 | 393.5 | 269 | |
| 13 | 426.3 | 291.4 | 23/18 |
| 14 | 459.1 | 313.8 | 30/23 |
| 15 | 491.9 | 336.2 | |
| 16 | 524.7 | 358.6 | 23/17 |
| 17 | 557.5 | 381 | 29/21 |
| 18 | 590.3 | 403.4 | |
| 19 | 623.1 | 425.9 | 33/23 |
| 20 | 655.8 | 448.3 | 19/13 |
| 21 | 688.6 | 470.7 | |
| 22 | 721.4 | 493.1 | |
| 23 | 754.2 | 515.5 | 17/11 |
| 24 | 787 | 537.9 | 11/7 |
| 25 | 819.8 | 560.3 | |
| 26 | 852.6 | 582.8 | 18/11 |
| 27 | 885.4 | 605.2 | 5/3 |
| 28 | 918.2 | 627.6 | 17/10 |
| 29 | 951 | 650 | 26/15 |
| 30 | 983.8 | 672.4 | 30/17 |
| 31 | 1016.6 | 694.8 | 9/5 |
| 32 | 1049.4 | 717.2 | 11/6 |
| 33 | 1082.1 | 739.7 | |
| 34 | 1114.9 | 762.1 | 21/11 |
| 35 | 1147.7 | 784.5 | 33/17 |
| 36 | 1180.5 | 806.9 | |
| 37 | 1213.3 | 829.3 | |
| 38 | 1246.1 | 851.7 | |
| 39 | 1278.9 | 874.1 | 23/11 |
| 40 | 1311.7 | 896.6 | |
| 41 | 1344.5 | 919 | |
| 42 | 1377.3 | 941.4 | |
| 43 | 1410.1 | 963.8 | |
| 44 | 1442.9 | 986.2 | 23/10 |
| 45 | 1475.7 | 1008.6 | |
| 46 | 1508.4 | 1031 | 31/13 |
| 47 | 1541.2 | 1053.4 | |
| 48 | 1574 | 1075.9 | |
| 49 | 1606.8 | 1098.3 | |
| 50 | 1639.6 | 1120.7 | 18/7 |
| 51 | 1672.4 | 1143.1 | |
| 52 | 1705.2 | 1165.5 | |
| 53 | 1738 | 1187.9 | 30/11 |
| 54 | 1770.8 | 1210.3 | 25/9 |
| 55 | 1803.6 | 1232.8 | 17/6 |
| 56 | 1836.4 | 1255.2 | 26/9 |
| 57 | 1869.2 | 1277.6 | |
| 58 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +13.3 | +0.0 | -6.2 | +1.0 | +13.3 | +8.8 | +7.2 | +0.0 | +14.4 | +13.3 | -6.2 |
| Relative (%) | +40.6 | +0.0 | -18.8 | +3.2 | +40.6 | +26.8 | +21.8 | +0.0 | +43.8 | +40.6 | -18.8 | |
| Steps (reduced) |
37 (37) |
58 (0) |
73 (15) |
85 (27) |
95 (37) |
103 (45) |
110 (52) |
116 (0) |
122 (6) |
127 (11) |
131 (15) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -13.6 | -10.7 | +1.0 | -12.3 | +13.9 | +13.3 | -14.7 | -5.1 | +8.8 | -6.2 | +15.3 |
| Relative (%) | -41.4 | -32.6 | +3.2 | -37.6 | +42.4 | +40.6 | -44.8 | -15.6 | +26.8 | -18.8 | +46.5 | |
| Steps (reduced) |
135 (19) |
139 (23) |
143 (27) |
146 (30) |
150 (34) |
153 (37) |
155 (39) |
158 (42) |
161 (45) |
163 (47) |
166 (50) | |