3edf
3 equal divisions of the perfect fifth (abbreviated 3edf or 3ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 3 equal parts of about 234 ¢ each. Each step represents a frequency ratio of (3/2)1/3, or the cube root of 3/2.
Theory
3ed, would divide the just perfect fifth into three equal parts, each of size 233.985 cents, which is to say (3/2)1/3 as a frequency ratio. It corresponds to 5.1285 edo.
It can also be treated as a heavily compressed version of 5edo, with the octave compressed by about 30 cents. Its patent val matches that of 5edo up to the 7-limit, and thus tempers out the same commas.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -30 | -30 | -60 | +22 | -60 | -93 | -90 | -60 | -9 | +60 | -90 |
| Relative (%) | -12.9 | -12.9 | -25.7 | +9.2 | -25.7 | -39.8 | -38.6 | -25.7 | -3.7 | +25.8 | -38.6 | |
| Steps (reduced) |
5 (2) |
8 (2) |
10 (1) |
12 (0) |
13 (1) |
14 (2) |
15 (0) |
16 (1) |
17 (2) |
18 (0) |
18 (0) | |
Factoids about 3edf
3edf is essentially equivalent to the slendric temperament, which tempers out 1029/1024 in the 2.3.7 subgroup, without octave repetition, and its step size is the 1/3-comma tuning of the slendric generator (approximated by, for instance, 8\41 and 39\200). It also works well as a tuning for arto and tendo chords.
Intervals
| # | Cents | Approximate JI Ratios |
|---|---|---|
| 1 | 233.99 | 8/7 |
| 2 | 467.97 | 21/16, 17/13 |
| 3 | 701.96 | exact 3/2 |
Music
- Sequences & Chaos by Bazil Müzik