1776edo: Difference between revisions
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1776edo is [[consistent]] in the 15-odd-limit, and has good representation of it, with errors less than 26%. In the 2.5.7.11.13 subgroup, it is [[Saturation, torsion, and contorsion#Torsion and contorsion|enfactored]] with the same mapping as [[888edo]], and corrects the latter's mapping for harmonic 3. A comma basis for the 13-limit is {4096/4095, 9801/9800, 67392/67375, 250047/250000, 531674/531441}. | 1776edo is [[consistent]] in the 15-odd-limit, and has good representation of it, with errors less than 26%. In the 2.5.7.11.13 subgroup, it is [[Saturation, torsion, and contorsion#Torsion and contorsion|enfactored]] with the same mapping as [[888edo]], and corrects the latter's mapping for harmonic 3. A comma basis for the 13-limit is {4096/4095, 9801/9800, 67392/67375, 250047/250000, 531674/531441}. | ||
Revision as of 05:52, 21 February 2025
| ← 1775edo | 1776edo | 1777edo → |
1776 equal divisions of the octave (abbreviated 1776edo or 1776ed2), also called 1776-tone equal temperament (1776tet) or 1776 equal temperament (1776et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1776 equal parts of about 0.676 ¢ each. Each step represents a frequency ratio of 21/1776, or the 1776th root of 2.
1776edo is consistent in the 15-odd-limit, and has good representation of it, with errors less than 26%. In the 2.5.7.11.13 subgroup, it is enfactored with the same mapping as 888edo, and corrects the latter's mapping for harmonic 3. A comma basis for the 13-limit is {4096/4095, 9801/9800, 67392/67375, 250047/250000, 531674/531441}.
In the 5-limit, it supports the squarschmidt temperament, tempering out the [61 4 -29⟩ comma, and it also tempers out [55 -64 -20⟩, [6, 68, -49⟩, [116, -60, -9⟩.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.072 | +0.173 | +0.093 | +0.033 | +0.013 | -0.226 | -0.216 | +0.104 | +0.153 | +0.235 |
| Relative (%) | +0.0 | +10.7 | +25.6 | +13.8 | +4.9 | +1.9 | -33.4 | -31.9 | +15.4 | +22.6 | +34.7 | |
| Steps (reduced) |
1776 (0) |
2815 (1039) |
4124 (572) |
4986 (1434) |
6144 (816) |
6572 (1244) |
7259 (155) |
7544 (440) |
8034 (930) |
8628 (1524) |
8799 (1695) | |
Subsets and supersets
Since 1776 factors as 24 × 3 × 37, it has subset edos 1, 2, 3, 4, 6, 8, 12, 16, 24, 37, 48, 74, 111, 148, 222, 296, 444, 592, 888.