76ed7/3
| ← 75ed7/3 | 76ed7/3 | 77ed7/3 → |
76 equal divisions of 7/3 (abbreviated 76ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 76 equal parts of about 19.3 ¢ each. Each step represents a frequency ratio of (7/3)1/76, or the 76th root of 7/3.
While it fails to accurately represent the 3rd, 5th, or 7th harmonics, it inherits great approximations of the 11th, 13th, 17th, and 19th harmonics from its cousin 197edt, notable for its strong representation of the no-twos, no-fives JI subgroup. 76ed7/3 additionally provides an equave stretch appropriate for producing, at the cost of a flat tendency for most well-represented prime harmonics as well as the 9th harmonic, a passable approximation to 5/3 and interesting approximations to many higher primes; however, 76ed7/3 should also be noted for the exceptional quality of its approximation to 11/9, inherited from 38ed7/3, which is a mere 0.0088 cents off from just. Its natural subgroup in the 19-limit is 7/3.9.11.13.15.17.19, but this can extend to include higher primes, especially 29 and 31.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.34 | +8.84 | -6.98 | +8.84 | -1.62 | -1.31 | -2.52 | -2.07 | -4.71 | -0.70 | -0.35 | +2.16 | -1.85 | -7.10 | -6.68 |
| Relative (%) | -17.3 | +45.8 | -36.2 | +45.8 | -8.4 | -6.8 | -13.0 | -10.7 | -24.4 | -3.6 | -1.8 | +11.2 | -9.6 | -36.8 | -34.6 | |
| Steps (reduced) |
62 (62) |
99 (23) |
144 (68) |
175 (23) |
215 (63) |
230 (2) |
254 (26) |
264 (36) |
281 (53) |
302 (74) |
308 (4) |
324 (20) |
333 (29) |
337 (33) |
345 (41) | |
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.84 | -6.98 | +8.84 | -1.63 | -1.62 | -1.31 | +1.86 | -2.52 | -2.07 | -1.63 | -4.71 | +5.34 | +7.21 | -0.70 | -0.35 |
| Relative (%) | +45.8 | -36.2 | +45.8 | -8.4 | -8.4 | -6.8 | +9.6 | -13.0 | -10.7 | -8.4 | -24.4 | +27.7 | +37.4 | -3.6 | -1.8 | |
| Steps (reduced) |
99 (23) |
144 (68) |
175 (23) |
197 (45) |
215 (63) |
230 (2) |
243 (15) |
254 (26) |
264 (36) |
273 (45) |
281 (53) |
289 (61) |
296 (68) |
302 (74) |
308 (4) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 19.3 | |
| 2 | 38.6 | |
| 3 | 57.9 | 29/28, 30/29, 31/30 |
| 4 | 77.2 | 23/22 |
| 5 | 96.5 | 37/35 |
| 6 | 115.8 | 31/29 |
| 7 | 135.1 | |
| 8 | 154.4 | |
| 9 | 173.7 | 31/28 |
| 10 | 193 | 19/17 |
| 11 | 212.3 | 26/23, 35/31 |
| 12 | 231.6 | |
| 13 | 250.9 | 15/13, 22/19 |
| 14 | 270.2 | 7/6 |
| 15 | 289.5 | 13/11 |
| 16 | 308.8 | 37/31 |
| 17 | 328.1 | 23/19, 29/24, 35/29 |
| 18 | 347.4 | |
| 19 | 366.7 | |
| 20 | 386 | 5/4 |
| 21 | 405.3 | 24/19 |
| 22 | 424.6 | 32/25, 37/29 |
| 23 | 443.9 | 22/17, 31/24 |
| 24 | 463.2 | 17/13, 30/23 |
| 25 | 482.5 | 37/28 |
| 26 | 501.8 | |
| 27 | 521.1 | 23/17 |
| 28 | 540.4 | 26/19 |
| 29 | 559.7 | |
| 30 | 579 | |
| 31 | 598.3 | 24/17 |
| 32 | 617.6 | |
| 33 | 636.9 | |
| 34 | 656.2 | 19/13, 35/24 |
| 35 | 675.5 | 34/23 |
| 36 | 694.8 | |
| 37 | 714.1 | |
| 38 | 733.4 | 26/17, 29/19 |
| 39 | 752.7 | 17/11, 37/24 |
| 40 | 772 | 25/16 |
| 41 | 791.3 | 30/19 |
| 42 | 810.6 | 8/5 |
| 43 | 829.9 | |
| 44 | 849.2 | 31/19 |
| 45 | 868.5 | 38/23 |
| 46 | 887.8 | |
| 47 | 907.1 | |
| 48 | 926.4 | 29/17 |
| 49 | 945.7 | 19/11 |
| 50 | 965 | |
| 51 | 984.3 | 23/13, 30/17 |
| 52 | 1003.6 | |
| 53 | 1022.9 | |
| 54 | 1042.3 | 31/17 |
| 55 | 1061.6 | 24/13 |
| 56 | 1080.9 | 28/15 |
| 57 | 1100.2 | |
| 58 | 1119.5 | |
| 59 | 1138.8 | 29/15 |
| 60 | 1158.1 | |
| 61 | 1177.4 | |
| 62 | 1196.7 | 2/1 |
| 63 | 1216 | |
| 64 | 1235.3 | |
| 65 | 1254.6 | 31/15 |
| 66 | 1273.9 | 23/11 |
| 67 | 1293.2 | |
| 68 | 1312.5 | |
| 69 | 1331.8 | |
| 70 | 1351.1 | 24/11 |
| 71 | 1370.4 | |
| 72 | 1389.7 | 29/13, 38/17 |
| 73 | 1409 | |
| 74 | 1428.3 | |
| 75 | 1447.6 | 30/13 |
| 76 | 1466.9 | 7/3 |