22ed5/2
22 equal divisions of 5/2 (abbreviated 22ed5/2) is a nonoctave tuning system that divides the interval of 5/2 into 22 equal parts of about 72.1 ¢ each. Each step represents a frequency ratio of (5/2)1/22, or the 22nd root of 5/2.
| ← 21ed5/2 | 22ed5/2 | 23ed5/2 → |
Theory
There is a vague similarity to 17edo. It does not approximate harmonics well, but it approximates several half-integer (or /2 in primodality terms) ratios well, in particular it has decent approximations of 7/2, 11/2 and 13/2, and could be analyzed as a 5/2.7/2.11/2.13/2 subgroup temperament. It supports frosthog temperament, equating 11/7 to the square root of 5/2.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 72.1 | 25/24 |
| 2 | 144.2 | 12/11, 13/12, 23/21 |
| 3 | 216.3 | 17/15, 25/22 |
| 4 | 288.4 | 13/11 |
| 5 | 360.5 | 21/17 |
| 6 | 432.6 | 23/18 |
| 7 | 504.7 | |
| 8 | 576.8 | 7/5 |
| 9 | 648.9 | 19/13 |
| 10 | 721.1 | |
| 11 | 793.2 | 11/7, 19/12 |
| 12 | 865.3 | |
| 13 | 937.4 | 12/7, 19/11 |
| 14 | 1009.5 | 25/14 |
| 15 | 1081.6 | 13/7 |
| 16 | 1153.7 | |
| 17 | 1225.8 | |
| 18 | 1297.9 | |
| 19 | 1370 | 11/5 |
| 20 | 1442.1 | |
| 21 | 1514.2 | 12/5 |
| 22 | 1586.3 | 5/2 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +25.8 | -27.2 | -20.5 | +25.8 | -1.4 | +20.1 | +5.3 | +17.7 | -20.5 | +30.8 | +24.4 |
| Relative (%) | +35.8 | -37.8 | -28.5 | +35.8 | -2.0 | +27.9 | +7.3 | +24.5 | -28.5 | +42.7 | +33.8 | |
| Steps (reduced) |
17 (17) |
26 (4) |
33 (11) |
39 (17) |
43 (21) |
47 (3) |
50 (6) |
53 (9) |
55 (11) |
58 (14) |
60 (16) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +30.0 | -26.2 | -1.4 | +31.0 | -1.8 | -28.7 | +22.0 | +5.3 | -7.1 | -15.5 | -20.4 |
| Relative (%) | +41.6 | -36.3 | -2.0 | +43.1 | -2.5 | -39.7 | +30.4 | +7.3 | -9.9 | -21.5 | -28.3 | |
| Steps (reduced) |
62 (18) |
63 (19) |
65 (21) |
67 (1) |
68 (2) |
69 (3) |
71 (5) |
72 (6) |
73 (7) |
74 (8) |
75 (9) | |