5ed7/3
5 equal divisions of 7/3 (abbreviated 5ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 5 equal parts of about 293 ¢ each. Each step represents a frequency ratio of (7/3)1/5, or the 5th root of 7/3.
Theory
5ed7/3 is related to sirius temperament, and approximates 5/3, 7/5, and 13/11 accurately, although one step of this tuning is in fact closest to 77/65, a mere 0.07 cents sharp, the relevant comma being 1160290625/1160050353.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -27 | -142 | -53 | -146 | +125 | -142 | -80 | +10 | +121 | -44 | +99 |
| Relative (%) | -9.0 | -48.3 | -18.1 | -49.7 | +42.7 | -48.3 | -27.1 | +3.4 | +41.2 | -15.0 | +33.6 | |
| Steps (reduced) |
4 (4) |
6 (1) |
8 (3) |
9 (4) |
11 (1) |
11 (1) |
12 (2) |
13 (3) |
14 (4) |
14 (4) |
15 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -40 | +125 | +6 | -106 | +82 | -17 | -110 | +94 | +10 | -71 | +146 | +72 |
| Relative (%) | -13.6 | +42.7 | +1.9 | -36.1 | +28.1 | -5.6 | -37.5 | +32.2 | +3.4 | -24.1 | +49.7 | +24.6 | |
| Steps (reduced) |
15 (0) |
16 (1) |
16 (1) |
16 (1) |
17 (2) |
17 (2) |
17 (2) |
18 (3) |
18 (3) |
18 (3) |
19 (4) |
19 (4) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 293.4 | 6/5, 7/6, 8/7, 13/11, 16/13, 19/16 |
| 2 | 586.7 | 7/5, 10/7, 11/8, 16/11, 19/13, 19/14 |
| 3 | 880.1 | 5/3, 8/5, 12/7, 13/8, 19/11 |
| 4 | 1173.5 | 2/1, 19/10 |
| 5 | 1466.9 | 7/3, 12/5, 16/7, 19/8 |