User:Francium/1747edo
| ← 1746edo | 1747edo | 1748edo → |
1747 equal divisions of the octave (abbreviated 1747edo or 1747ed2), also called 1747-tone equal temperament (1747tet) or 1747 equal temperament (1747et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1747 equal parts of about 0.687 ¢ each. Each step represents a frequency ratio of 21/1747, or the 1747th root of 2.
Theory
1747edo is consistent to the 5-limit, tempering out [39 -29 3⟩ and [-78 -46 65⟩. Its lower harmonics have a higher error than its higher ones, while the harmonics 5 and 7 are about halfway its steps, its harmonic 31 has a relative error of only 1.9 percent. 1747edo is strong in the 2.3.19.29.31 subgroup, tempering out 191153471/191102976, 3122289/3121792, 324547248128/324270949293 and 1870256930816/1869963909147. Using the 2.3.5.7.17 subgroup, it tempers out 1225/1224.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.048 | -0.281 | -0.308 | +0.256 | +0.228 | +0.139 | -0.089 | +0.232 | +0.074 | +0.013 |
| Relative (%) | +0.0 | +7.1 | -40.8 | -44.9 | +37.3 | +33.2 | +20.2 | -12.9 | +33.7 | +10.7 | +1.9 | |
| Steps (reduced) |
1747 (0) |
2769 (1022) |
4056 (562) |
4904 (1410) |
6044 (803) |
6465 (1224) |
7141 (153) |
7421 (433) |
7903 (915) |
8487 (1499) |
8655 (1667) | |
Subsets and supersets
1747edo is the 272nd prime edo. 3494edo, which doubles it, gives a good correction to the harmonics 5 and 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2769 -1747⟩ | [⟨1747 2769]] | −0.0153 | 0.0153 | 2.23 |