5edt
| ← 4edt | 5edt | 6edt → |
(semiconvergent)
5 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 5edt or 5ed3), is a nonoctave tuning system that divides the interval of 3/1 into 5 equal parts of about 380 ¢ each. Each step represents a frequency ratio of 31/5, or the 5th root of 3.
Theory
5edt has steps too large to be used melodically, though it has some notable harmonic properties shared by other 5n-edts (such as 10edt, 15edt, etc.). It has a surprisingly accurate 5/4 major third (only 5.92 cents flat), five of them making a tritave. (This phenomenon cannot be seen in single-digit edos, and it is not until 19edo that an approximation of 5/4 to within 10 cents can be seen.) 5edt therefore tempers out 3125/3072, the magic comma. Two of these major thirds give a septimal minor sixth, meaning that it also tempers out the marvel comma 225/224.
5edt is the lowest equal division of the tritave to encompass elements of 5-limit harmony. The available chords are 4:5:12 and 5:12:15; being inversions of each other, they essentially act as "major" and "minor" chords. This is similar to 6edt, which contains the same chords but with the major thirds transposed an octave higher, and vice versa. Also available is the chord 9:14:27, which can express the single 7-limit consonance found in 5edt.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -59 | +0 | -118 | -124 | -59 | +55 | -176 | +0 | -182 | +33 | -118 | +124 | -4 | -124 | +145 |
| Relative (%) | -15.5 | +0.0 | -30.9 | -32.5 | -15.5 | +14.4 | -46.4 | +0.0 | -48.0 | +8.7 | -30.9 | +32.6 | -1.1 | -32.5 | +38.1 | |
| Steps (reduced) |
3 (3) |
5 (0) |
6 (1) |
7 (2) |
8 (3) |
9 (4) |
9 (4) |
10 (0) |
10 (0) |
11 (1) |
11 (1) |
12 (2) |
12 (2) |
12 (2) |
13 (3) | |
Steps
| Degrees | Cents | Hekts | Approximate Ratios |
|---|---|---|---|
| 0 | 0.000 | 0 | 1/1 |
| 1 | 380.391 | 260 | 5/4 |
| 2 | 760.782 | 520 | 14/9 |
| 3 | 1141.173 | 780 | 27/14 |
| 4 | 1521.564 | 1040 | 12/5 (6/5 plus an octave) |
| 5 | 1901.955 | 1300 | 3/1 |