62edt
62 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 62edt or 62ed3), is a nonoctave tuning system that divides the interval of 3/1 into 62 equal parts of about 30.7 ¢ each. Each step represents a frequency ratio of 31/62, or the 62nd root of 3.
Theory
62edt is related to 39edo, but with the 3/1 rather than the 2/1 being just. The octave is about 3.61 cents compressed and the step size is about 30.6767 cents. 62edt is consistent to the 7-integer-limit, but not to the 8-integer-limit. In comparison, 39edo is only consistent up to the 6-integer-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.6 | +0.0 | -7.2 | +5.3 | -3.6 | +5.6 | -10.8 | +0.0 | +1.7 | -10.0 | -7.2 |
| Relative (%) | -11.8 | +0.0 | -23.5 | +17.2 | -11.8 | +18.3 | -35.3 | +0.0 | +5.4 | -32.5 | -23.5 | |
| Steps (reduced) |
39 (39) |
62 (0) |
78 (16) |
91 (29) |
101 (39) |
110 (48) |
117 (55) |
124 (0) |
130 (6) |
135 (11) |
140 (16) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.6 | +2.0 | +5.3 | -14.4 | +3.3 | -3.6 | -5.2 | -2.0 | +5.6 | -13.6 | +1.5 | -10.8 |
| Relative (%) | +24.8 | +6.5 | +17.2 | -47.1 | +10.8 | -11.8 | -16.9 | -6.4 | +18.3 | -44.2 | +4.9 | -35.3 | |
| Steps (reduced) |
145 (21) |
149 (25) |
153 (29) |
156 (32) |
160 (36) |
163 (39) |
166 (42) |
169 (45) |
172 (48) |
174 (50) |
177 (53) |
179 (55) | |
Subsets and supersets
Since 62 factors into primes as 2 × 31, 62edt contains 2edt and 31edt as subset edts.
Intervals
| # | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0.0 | 0.0 | 1/1 |
| 1 | 30.7 | 21.0 | 36/35, 50/49, 55/54, 56/55, 81/80 |
| 2 | 61.4 | 41.9 | 22/21, 28/27, 33/32, 45/44, 49/48 |
| 3 | 92.0 | 62.9 | 16/15, 19/18, 20/19, 21/20, 25/24 |
| 4 | 122.7 | 83.9 | 15/14 |
| 5 | 153.4 | 104.8 | 11/10, 12/11 |
| 6 | 184.1 | 125.8 | 10/9 |
| 7 | 214.7 | 146.8 | 8/7, 9/8 |
| 8 | 245.4 | 167.7 | 22/19 |
| 9 | 276.1 | 188.7 | 7/6 |
| 10 | 306.8 | 209.7 | 6/5 |
| 11 | 337.4 | 230.6 | 11/9 |
| 12 | 368.1 | 251.6 | 27/22 |
| 13 | 398.8 | 272.6 | 5/4 |
| 14 | 429.5 | 293.5 | 9/7, 14/11 |
| 15 | 460.2 | 314.5 | 35/27, 57/44 |
| 16 | 490.8 | 335.5 | 4/3 |
| 17 | 521.5 | 356.5 | 19/14, 27/20 |
| 18 | 552.2 | 377.5 | 11/8, 15/11 |
| 19 | 582.9 | 398.4 | 7/5 |
| 20 | 613.5 | 419.4 | 10/7 |
| 21 | 644.2 | 440.3 | 16/11, 22/15 |
| 22 | 674.9 | 461.3 | 28/19, 35/24 |
| 23 | 705.6 | 482.3 | 3/2 |
| 24 | 736.2 | 503.2 | 54/35, 88/57 |
| 25 | 766.9 | 524.2 | 11/7, 14/9 |
| 26 | 797.6 | 545.2 | 8/5, 19/12 |
| 27 | 828.3 | 566.1 | 44/27 |
| 28 | 858.9 | 587.1 | 18/11 |
| 29 | 889.6 | 608.1 | 5/3 |
| 30 | 920.3 | 629.0 | 12/7 |
| 31 | 951.0 | 650.0 | 19/11 |
| 32 | 981.7 | 671.0 | 7/4, 16/9 |
| 33 | 1012.3 | 691.9 | 9/5 |
| 34 | 1043.0 | 712.9 | 11/6, 20/11 |
| 35 | 1073.7 | 733.9 | 28/15 |
| 36 | 1104.4 | 754.8 | 15/8, 19/10 |
| 37 | 1135.0 | 775.8 | 21/11, 27/14 |
| 38 | 1165.7 | 796.8 | 35/18, 49/25, 55/28 |
| 39 | 1196.4 | 817.7 | 2/1 |
| 40 | 1227.1 | 838.7 | 55/27, 81/40 |
| 41 | 1257.7 | 859.7 | 33/16, 45/22, 49/24 |
| 42 | 1288.4 | 880.6 | 19/9, 21/10, 25/12 |
| 43 | 1319.1 | 901.6 | 15/7 |
| 44 | 1349.8 | 922.6 | 11/5 |
| 45 | 1380.5 | 943.5 | 20/9 |
| 46 | 1411.1 | 964.5 | 9/4 |
| 47 | 1441.8 | 985.5 | 44/19 |
| 48 | 1472.5 | 1006.5 | 7/3 |
| 49 | 1503.2 | 1027.4 | 12/5, 19/8 |
| 50 | 1533.8 | 1048.4 | 22/9 |
| 51 | 1564.5 | 1069.4 | 27/11 |
| 52 | 1595.2 | 1090.3 | 5/2 |
| 53 | 1625.9 | 1111.3 | 18/7, 28/11 |
| 54 | 1656.5 | 1132.3 | 57/22, 70/27 |
| 55 | 1687.2 | 1153.2 | 8/3 |
| 56 | 1717.9 | 1174.2 | 19/7, 27/10 |
| 57 | 1748.6 | 1195.2 | 11/4 |
| 58 | 1779.2 | 1216.1 | 14/5 |
| 59 | 1809.9 | 1237.1 | 20/7 |
| 60 | 1840.6 | 1258.1 | 32/11 |
| 61 | 1871.3 | 1279.0 | 35/12 |
| 62 | 1902.0 | 1300.0 | 3/1 |