7ed5
| ← 6ed5 | 7ed5 | 8ed5 → |
(convergent)
(semiconvergent)
7 equal divisions of the 5th harmonic (abbreviated 7ed5) is a nonoctave tuning system that divides the interval of 5/1 into 7 equal parts of about 398 ¢ each. Each step represents a frequency ratio of 51/7, or the 7th root of 5.
Theory
7ed5 is related to 3edo, but with the 5/1 rather than the 2/1 being just. The octave is about 6 cents compressed and the step size is about 398 cents. It is present (though possibly tempered) in any regular temperament which tempers out 441/440 and 244515348/244140625 in the 11-limit, such as equal temperaments 3, 12, 15, 175, 190, 202, and 217edo.
Due to Kirnberger's atom, its step is 100.0002¢ flat of 4/3[clarification needed].
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -6 | +88 | -12 | +0 | +82 | -184 | -18 | +177 | -6 | -171 | +77 |
| Relative (%) | -1.5 | +22.2 | -2.9 | +0.0 | +20.7 | -46.3 | -4.4 | +44.4 | -1.5 | -42.9 | +19.2 | |
| Steps (reduced) |
3 (3) |
5 (5) |
6 (6) |
7 (0) |
8 (1) |
8 (1) |
9 (2) |
10 (3) |
10 (3) |
10 (3) |
11 (4) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -62 | -190 | +88 | -23 | -128 | +171 | +77 | -12 | -96 | -177 | +144 | +71 |
| Relative (%) | -15.6 | -47.8 | +22.2 | -5.9 | -32.3 | +42.9 | +19.4 | -2.9 | -24.2 | -44.4 | +36.3 | +17.8 | |
| Steps (reduced) |
11 (4) |
11 (4) |
12 (5) |
12 (5) |
12 (5) |
13 (6) |
13 (6) |
13 (6) |
13 (6) |
13 (6) |
14 (0) |
14 (0) | |
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 398 | 5/4, 34/27 |
| 2 | 796 | 19/12 |
| 3 | 1194 | 2/1, 255/128 |
| 4 | 1592 | 5/2, 128/51 |
| 5 | 1990 | 60/19 |
| 6 | 2388 | 4/1, 135/34 |
| 7 | 2786 | 5/1 |