13ed11/5
| ← 12ed11/5 | 13ed11/5 | 14ed11/5 → |
(semiconvergent)
13 equal divisions of 11/5 (abbreviated 13ed11/5) is a nonoctave tuning system that divides the interval of 11/5 into 13 equal parts of about 105 ¢ each. Each step represents a frequency ratio of (11/5)1/13, or the 13th root of 11/5.
For its size, 13ed11/5 has decent approximations of the 3rd and 4th harmonics, and a great approximation of the 7th harmonic.
It completely misses the 5th harmonic, however it nails the 10th harmonic almost exactly.
Given this, it could be approached as a 3.4.7 subgroup tuning, or even as a 3.4.7.10 tuning, which is quite the quirky combination.
When playing by ear, the 4/1 approximation, roughly 2415 cents, tends to sound like an equivalence or a tonal centre. This makes 13ed11/5 sound similar to 23ed4, or every second note of 46edo.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 105 | 14/13, 21/20, 22/21 |
| 2 | 210 | 17/15, 19/17 |
| 3 | 315 | 23/19 |
| 4 | 420 | 9/7, 19/15 |
| 5 | 525 | 15/11, 23/17 |
| 6 | 630 | 10/7, 13/9 |
| 7 | 735 | 17/11, 20/13, 23/15 |
| 8 | 840 | 21/13 |
| 9 | 945 | 19/11 |
| 10 | 1050 | 13/7 |
| 11 | 1155 | |
| 12 | 1260 | 21/10, 23/11 |
| 13 | 1365 | 11/5, 20/9 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -45.0 | -11.9 | +15.0 | +48.7 | +48.1 | -8.8 | -30.0 | -23.9 | +3.7 | +48.7 | +3.1 |
| Relative (%) | -42.9 | -11.4 | +14.3 | +46.4 | +45.8 | -8.4 | -28.6 | -22.8 | +3.5 | +46.4 | +2.9 | |
| Steps (reduced) |
11 (11) |
18 (5) |
23 (10) |
27 (1) |
30 (4) |
32 (6) |
34 (8) |
36 (10) |
38 (12) |
40 (1) |
41 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -30.5 | +51.2 | +36.7 | +30.0 | +30.1 | +36.1 | +47.5 | -41.3 | -20.8 | +3.7 | +31.7 |
| Relative (%) | -29.1 | +48.8 | +35.0 | +28.6 | +28.6 | +34.4 | +45.2 | -39.3 | -19.8 | +3.5 | +30.2 | |
| Steps (reduced) |
42 (3) |
44 (5) |
45 (6) |
46 (7) |
47 (8) |
48 (9) |
49 (10) |
49 (10) |
50 (11) |
51 (12) |
52 (0) | |