1778edo
| ← 1777edo | 1778edo | 1779edo → |
1778 equal divisions of the octave (abbreviated 1778edo or 1778ed2), also called 1778-tone equal temperament (1778tet) or 1778 equal temperament (1778et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1778 equal parts of about 0.675 ¢ each. Each step represents a frequency ratio of 21/1778, or the 1778th root of 2.
1778edo is consistent to the 9-odd-limit, but the errors of both harmonics 5 and 7 are quite large. It is enfactored in the 5-limit, with the same tuning as 889edo, tempering out [-29 -11 20⟩ (gammic comma) and [-69 45 -1⟩ (counterschisma). In the 7-limit, the equal temperament tempers out 2401/2400 (breedsma) and 48828125/48771072 (neptunisma). It provides the optimal patent val for the 7-limit neptune temperament.
For higher harmonics, it is suitable for a 2.3.11.19.23.43.47.61 subgroup interpretation.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -0.043 | -0.262 | -0.322 | -0.085 | +0.088 | -0.258 | -0.305 | +0.331 | +0.125 | +0.310 | +0.072 |
| relative (%) | -6 | -39 | -48 | -13 | +13 | -38 | -45 | +49 | +18 | +46 | +11 | |
| Steps (reduced) |
2818 (1040) |
4128 (572) |
4991 (1435) |
5636 (302) |
6151 (817) |
6579 (1245) |
6946 (1612) |
7268 (156) |
7553 (441) |
7810 (698) |
8043 (931) | |
Subsets and supersets
Since 1778 factors into 2 × 7 × 127, 1778edo has subset edos 2, 7, 14, 127, 254, and 889.