15ed5
15 equal divisions of the 5th harmonic (abbreviated 15ed5) is a nonoctave tuning system that divides the interval of 5/1 into 15 equal parts of about 186 ¢ each. Each step represents a frequency ratio of 51/15, or the 15th root of 5. A multiple of 5ed5 It approximates the 7th, 12th, 13th, 18th, 31st and 43rd harmonics with some accuracy (but especially the 31st!). Hyperpyth analogues of blackwood of course are warranted.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 185.8 | 10/9, 11/10, 19/17, 21/19 |
| 2 | 371.5 | 21/17 |
| 3 | 557.3 | 15/11 |
| 4 | 743 | 14/9, 17/11, 23/15 |
| 5 | 928.8 | 17/10, 19/11 |
| 6 | 1114.5 | 17/9, 19/10, 21/11 |
| 7 | 1300.3 | 15/7, 19/9, 21/10 |
| 8 | 1486 | 7/3 |
| 9 | 1671.8 | 13/5 |
| 10 | 1857.5 | |
| 11 | 2043.3 | 23/7 |
| 12 | 2229.1 | 11/3 |
| 13 | 2414.8 | |
| 14 | 2600.6 | 9/2 |
| 15 | 2786.3 | 5/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -85.5 | -44.4 | +14.8 | +0.0 | +55.9 | -25.2 | -70.7 | -88.8 | -85.5 | -64.7 | -29.6 |
| Relative (%) | -46.0 | -23.9 | +8.0 | +0.0 | +30.1 | -13.6 | -38.0 | -47.8 | -46.0 | -34.8 | -15.9 | |
| Steps (reduced) |
6 (6) |
10 (10) |
13 (13) |
15 (0) |
17 (2) |
18 (3) |
19 (4) |
20 (5) |
21 (6) |
22 (7) |
23 (8) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +17.6 | +75.0 | -44.4 | +29.6 | -75.3 | +11.5 | -82.1 | +14.8 | -69.7 | +35.6 | -41.4 |
| Relative (%) | +9.5 | +40.4 | -23.9 | +15.9 | -40.6 | +6.2 | -44.2 | +8.0 | -37.5 | +19.1 | -22.3 | |
| Steps (reduced) |
24 (9) |
25 (10) |
25 (10) |
26 (11) |
26 (11) |
27 (12) |
27 (12) |
28 (13) |
28 (13) |
29 (14) |
29 (14) | |