61edt
61 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 61edt or 61ed3), is a nonoctave tuning system that divides the interval of 3/1 into 61 equal parts of about 31.2 ¢ each. Each step represents a frequency ratio of 31/61, or the 61st root of 3.
| ← 60edt | 61edt | 62edt → |
61edt provides a good tuning of mintaka temperament in the 3.7.11 subgroup, and contains an intersection of it with Bohlen-Pierce-Stearns, despite the 5th harmonic being rather far from accurate. Notably, the octave is almost halfway in between steps, and therefore this system minimizes the prospect of a shimmering octave appearing, although it has a good 4th harmonic.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 31.2 | 21.3 | |
| 2 | 62.4 | 42.6 | |
| 3 | 93.5 | 63.9 | 19/18 |
| 4 | 124.7 | 85.2 | 14/13, 29/27 |
| 5 | 155.9 | 106.6 | 23/21 |
| 6 | 187.1 | 127.9 | |
| 7 | 218.3 | 149.2 | 17/15, 25/22 |
| 8 | 249.4 | 170.5 | 15/13, 22/19 |
| 9 | 280.6 | 191.8 | 27/23 |
| 10 | 311.8 | 213.1 | 6/5 |
| 11 | 343 | 234.4 | |
| 12 | 374.2 | 255.7 | |
| 13 | 405.3 | 277 | 19/15, 29/23 |
| 14 | 436.5 | 298.4 | 9/7 |
| 15 | 467.7 | 319.7 | 17/13 |
| 16 | 498.9 | 341 | |
| 17 | 530.1 | 362.3 | 19/14, 34/25 |
| 18 | 561.2 | 383.6 | 18/13, 29/21 |
| 19 | 592.4 | 404.9 | |
| 20 | 623.6 | 426.2 | 33/23 |
| 21 | 654.8 | 447.5 | 19/13 |
| 22 | 686 | 468.9 | |
| 23 | 717.1 | 490.2 | |
| 24 | 748.3 | 511.5 | |
| 25 | 779.5 | 532.8 | 11/7 |
| 26 | 810.7 | 554.1 | |
| 27 | 841.8 | 575.4 | |
| 28 | 873 | 596.7 | |
| 29 | 904.2 | 618 | |
| 30 | 935.4 | 639.3 | |
| 31 | 966.6 | 660.7 | |
| 32 | 997.7 | 682 | |
| 33 | 1028.9 | 703.3 | |
| 34 | 1060.1 | 724.6 | 35/19 |
| 35 | 1091.3 | 745.9 | |
| 36 | 1122.5 | 767.2 | 21/11 |
| 37 | 1153.6 | 788.5 | 35/18 |
| 38 | 1184.8 | 809.8 | |
| 39 | 1216 | 831.1 | |
| 40 | 1247.2 | 852.5 | 35/17 |
| 41 | 1278.4 | 873.8 | 23/11 |
| 42 | 1309.5 | 895.1 | |
| 43 | 1340.7 | 916.4 | 13/6 |
| 44 | 1371.9 | 937.7 | |
| 45 | 1403.1 | 959 | |
| 46 | 1434.3 | 980.3 | |
| 47 | 1465.4 | 1001.6 | 7/3 |
| 48 | 1496.6 | 1023 | |
| 49 | 1527.8 | 1044.3 | |
| 50 | 1559 | 1065.6 | |
| 51 | 1590.2 | 1086.9 | 5/2 |
| 52 | 1621.3 | 1108.2 | 23/9 |
| 53 | 1652.5 | 1129.5 | 13/5 |
| 54 | 1683.7 | 1150.8 | |
| 55 | 1714.9 | 1172.1 | 35/13 |
| 56 | 1746.1 | 1193.4 | |
| 57 | 1777.2 | 1214.8 | |
| 58 | 1808.4 | 1236.1 | |
| 59 | 1839.6 | 1257.4 | |
| 60 | 1870.8 | 1278.7 | |
| 61 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -15.2 | +0.0 | -11.3 | -1.4 | -4.4 | -13.0 | -9.8 | -15.2 | -3.0 |
| Relative (%) | -48.7 | +0.0 | -36.3 | -4.6 | -14.2 | -41.8 | -31.3 | -48.9 | -9.7 | |
| Steps (reduced) |
38 (38) |
61 (0) |
89 (28) |
108 (47) |
133 (11) |
142 (20) |
157 (35) |
163 (41) |
174 (52) | |
| Harmonic | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.5 | +0.0 | +1.0 | +10.3 | -4.4 | -12.8 | -15.4 | -13.0 | -6.1 | +5.0 | -11.3 |
| Relative (%) | +27.3 | +0.0 | +3.2 | +32.9 | -14.2 | -40.9 | -49.5 | -41.8 | -19.5 | +16.1 | -36.3 | |
| Steps (reduced) |
179 (57) |
183 (0) |
187 (4) |
191 (8) |
194 (11) |
197 (14) |
200 (17) |
203 (20) |
206 (23) |
209 (26) |
211 (28) | |
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