61edt
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Prime factorization
61 (prime)
Step size
31.1796¢
Octave
38\61edt (1184.82¢)
Consistency limit
2
Distinct consistency limit
2
| ← 60edt | 61edt | 62edt → |
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.
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 lacking an accurate approximation to the 5th harmonic. 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 fourth 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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