User:Francium/6689edo
| ← 6688edo | 6689edo | 6690edo → |
6689 equal divisions of the octave (abbreviated 6689edo or 6689ed2), also called 6689-tone equal temperament (6689tet) or 6689 equal temperament (6689et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 6689 equal parts of about 0.179 ¢ each. Each step represents a frequency ratio of 21/6689, or the 6689th root of 2.
Theory
6689edo is inconsistent to the 5-limit. It is strong in the 2.3.11.17.23.29 subgroup, tempering out 387571712/387420489, 33554432/33550767, 39137536/39135393, 15555136/15554187 and 3382124478464/3381592485051. As an equal temperament, it supports hectosaros leap week.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0333 | -0.0676 | -0.0712 | -0.0248 | -0.0433 | -0.0070 | -0.0695 | -0.0190 | -0.0063 | +0.0683 |
| Relative (%) | +0.0 | +18.6 | -37.7 | -39.7 | -13.8 | -24.1 | -3.9 | -38.7 | -10.6 | -3.5 | +38.1 | |
| Steps (reduced) |
6689 (0) |
10602 (3913) |
15531 (2153) |
18778 (5400) |
23140 (3073) |
24752 (4685) |
27341 (585) |
28414 (1658) |
30258 (3502) |
32495 (5739) |
33139 (6383) | |
Subsets and supersets
6689edo is the 862nd prime edo. 13378edo, which doubles it, gives a good correction to its harmonics 5 and 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [10602 -6689⟩ | [⟨6689 10602]] | −0.0105 | 0.0105 | 5.85 |