3edt
| ← 2edt | 3edt | 4edt → |
(convergent)
3 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 3edt or 3ed3), is a nonoctave tuning system that divides the interval of 3/1 into 3 equal parts of about 634 ¢ each. Each step represents a frequency ratio of 31/3, or the 3rd root of 3.
Theory
3edt can be thought of as 2edo with the 3/1 made just, by stretching the octave by 67.97 cents.
Despite its small size, 3edt has an excellent approximation to the 13th harmonic: 7 steps of 3edt is only 2.63 cents flat of 13/1. One step of 3edt has two good 13-limit rational approximations, 13/9 and 75/52, both which are convergents. 3edt thus tempers out (13/9)3 / (3/1) = 2197/2187, the threedie, and (75/52)3 / (3/1) = 140625/140608, the catasma. The good approximation for 13/9 and 75/52 also implies a good approximation for 25/4, or (5/2)2.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +68 | +0 | +136 | -250 | +68 | -199 | +204 | +0 | -182 | +287 | +136 | -3 | -131 | -250 | +272 |
| Relative (%) | +10.7 | +0.0 | +21.4 | -39.5 | +10.7 | -31.4 | +32.2 | +0.0 | -28.8 | +45.2 | +21.4 | -0.4 | -20.7 | -39.5 | +42.9 | |
| Steps (reduced) |
2 (2) |
3 (0) |
4 (1) |
4 (1) |
5 (2) |
5 (2) |
6 (0) |
6 (0) |
6 (0) |
7 (1) |
7 (1) |
7 (1) |
7 (1) |
7 (1) |
8 (2) | |
Relationship to octave temperaments
One step of 3edt can represent the generator to any rank-2 octavated temperament which takes 3 generators to reach the 3rd harmonic. These are: