21ed4
21 equal divisions of the 4th harmonic (abbreviated 21ed4) is a nonoctave tuning system that divides the interval of 4/1 into 21 equal parts of about 114 ¢ each. Each step represents a frequency ratio of 41/21, or the 21st root of 4. It represents every second step of 21edo.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 114.3 | 14/13, 15/14 |
| 2 | 228.6 | 17/15 |
| 3 | 342.9 | 17/14 |
| 4 | 457.1 | 13/10, 17/13, 22/17 |
| 5 | 571.4 | 7/5 |
| 6 | 685.7 | |
| 7 | 800 | |
| 8 | 914.3 | 17/10, 22/13 |
| 9 | 1028.6 | |
| 10 | 1142.9 | |
| 11 | 1257.1 | |
| 12 | 1371.4 | 11/5, 20/9 |
| 13 | 1485.7 | |
| 14 | 1600 | |
| 15 | 1714.3 | |
| 16 | 1828.6 | |
| 17 | 1942.9 | |
| 18 | 2057.1 | 23/7 |
| 19 | 2171.4 | |
| 20 | 2285.7 | |
| 21 | 2400 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +57.1 | +40.9 | +0.0 | -43.5 | -16.2 | -54.5 | +57.1 | -32.5 | +13.7 | -37.0 | +40.9 |
| Relative (%) | +50.0 | +35.8 | +0.0 | -38.0 | -14.2 | -47.7 | +50.0 | -28.4 | +12.0 | -32.4 | +35.8 | |
| Steps (reduced) |
11 (11) |
17 (17) |
21 (0) |
24 (3) |
27 (6) |
29 (8) |
32 (11) |
33 (12) |
35 (14) |
36 (15) |
38 (17) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +16.6 | +2.6 | -2.6 | +0.0 | +9.3 | +24.7 | +45.3 | -43.5 | -13.6 | +20.1 | -56.8 |
| Relative (%) | +14.5 | +2.3 | -2.2 | +0.0 | +8.2 | +21.6 | +39.7 | -38.0 | -11.9 | +17.6 | -49.7 | |
| Steps (reduced) |
39 (18) |
40 (19) |
41 (20) |
42 (0) |
43 (1) |
44 (2) |
45 (3) |
45 (3) |
46 (4) |
47 (5) |
47 (5) | |
Music
- Sister equal by Chris Vaisvil — 3rd Song on that page is in 21ED4