2777edo
| ← 2776edo | 2777edo | 2778edo → |
2777 equal divisions of the octave (abbreviated 2777edo or 2777ed2), also called 2777-tone equal temperament (2777tet) or 2777 equal temperament (2777et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2777 equal parts of about 0.432 ¢ each. Each step represents a frequency ratio of 21/2777, or the 2777th root of 2.
Theory
2777edo is consistent to the 7-odd-limit and its harmonic 3 is about halfway its steps. Using the 2.9.5.7.11.13.17.23.29.31 subgroup, it tempers out 12376/12375, 14400/14399, 25025/25024, 123201/123200, 20736/20735, 194481/194480, 16445/16443, 27625/27621 and 23716/23715. Using the 2.5.7.11.13.17.23 subgroup, it tempers out 25025/25024.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.191 | +0.002 | -0.011 | +0.051 | +0.068 | -0.052 | -0.188 | +0.050 | -0.214 | -0.201 | +0.030 |
| Relative (%) | -44.1 | +0.6 | -2.5 | +11.8 | +15.8 | -12.1 | -43.5 | +11.6 | -49.5 | -46.5 | +6.8 | |
| Steps (reduced) |
4401 (1624) |
6448 (894) |
7796 (2242) |
8803 (472) |
9607 (1276) |
10276 (1945) |
10849 (2518) |
11351 (243) |
11796 (688) |
12197 (1089) |
12562 (1454) | |
Subsets and supersets
2777edo is the 404th prime EDO. 5554edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [8803 -2777⟩ | [⟨2777 8803]] | -0.0081 | 0.0081 | 1.87 |
| 2.9.5 | [146 -38 -11⟩, [-89 -21 67⟩ | [⟨2777 8803 6448]] | -0.0057 | 0.0074 | 1.71 |
| 2.9.5.7 | [2 -10 14 -1⟩, [-1 -9 -3 13⟩, [-48 0 11 8⟩ | [⟨2777 8803 6448 7796]] | -0.0033 | 0.0076 | 1.76 |
| 2.9.5.7.11 | 151263/151250, 184549376/184528125, 35156250/35153041, 3487704605/3486784401 | [⟨2777 8803 6448 7796 9607]] | -0.0066 | 0.0095 | 2.20 |
| 2.9.5.7.11.13 | 123201/123200, 6656/6655, 151263/151250, 8859375/8859136, 43061200/43046721 | [⟨2777 8803 6448 7796 9607 10276]] | -0.0032 | 0.0116 | 2.68 |
| 2.9.5.7.11.13.17 | 12376/12375, 14400/14399, 123201/123200, 194481/194480, 4685824/4685625, 81331250/81310473 | [⟨2777 8803 6448 7796 9607 10276 11351]] | -0.0045 | 0.0112 | 2.59 |