4edt
| ← 3edt | 4edt | 5edt → |
4 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 4edt or 4ed3), is a nonoctave tuning system that divides the interval of 3/1 into 4 equal parts of about 475 ¢ each. Each step represents a frequency ratio of 31/4, or the 4th root of 3.
Theory
4edt fails to approximate a lot of low prime harmonics well -- the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to 3edt, which has the 13th harmonic, and 5edt, which has the 5th harmonic.) Nevertheless, in terms of convergents, 4edt manages to accurately approximate 25/19 with one step, and a less accurate 16/7 with three steps (hence 21/16 with one step). However, it is not until edts with further multiples of 4 (i.e. 8edt, 12edt, etc.) that these intervals see practical use.
4edt can be viewed as a "collapsed" version of the Bohlen-Pierce lambda scale, analogous to how 5edo is a collapsed version of the diatonic scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale[clarification needed]. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like 8edt.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +226 | +0 | -23 | +67 | +226 | -40 | +204 | +0 | -182 | +128 | -23 |
| relative (%) | +48 | +0 | -5 | +14 | +48 | -8 | +43 | +0 | -38 | +27 | -5 | |
| Steps (reduced) |
3 (3) |
4 (0) |
5 (1) |
6 (2) |
7 (3) |
7 (3) |
8 (0) |
8 (0) |
8 (0) |
9 (1) |
9 (1) | |
Approximation of intervals
| Steps | Cents | Hekts | Corresponding JI intervals |
Comments |
|---|---|---|---|---|
| 0 | 0.0000 | 0 | 1/1 | perfect unison |
| 1 | 475.4888 | 325 | 17/13, 21/16, 25/19, 33/25 | |
| 2 | 950.9775 | 650 | 19/11, 45/26, 26/15, (85/49), 33/19 | one step of 2edt |
| 3 | 1426.4663 | 975 | 25/11, 57/25, 16/7, 39/17 | |
| 4 | 1901.9550 | 1300 | 3/1 | just perfect fifth plus an octave |
Related regular temperaments
4EDT is a generator of the vulture temperament, which tempers out 10485760000/10460353203 in the 5-limit.