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
3edf, if the attempt is made to use it as an actual scale, 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