23ed4
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Prime factorization
23 (prime)
Step size
104.348¢
Octave
12\23ed4 (1252.17¢)
Twelfth
18\23ed4 (1878.26¢)
Consistency limit
1
Distinct consistency limit
1
 |
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| ← 21ed4 | 23ed4 | 25ed4 → |
23 equal divisions of the 4th harmonic (abbreviated 23ed4) is a nonoctave tuning system that divides the interval of 4/1 into 23 equal parts of about 104 ¢ each. Each step represents a frequency ratio of 41/23, or the 23rd root of 4.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 104.3 | 18/17, 19/18 |
| 2 | 208.7 | 17/15, 25/22 |
| 3 | 313 | 6/5 |
| 4 | 417.4 | 14/11, 19/15, 23/18 |
| 5 | 521.7 | 19/14, 23/17 |
| 6 | 626.1 | |
| 7 | 730.4 | 23/15 |
| 8 | 834.8 | |
| 9 | 939.1 | 19/11 |
| 10 | 1043.5 | 11/6 |
| 11 | 1147.8 | |
| 12 | 1252.2 | |
| 13 | 1356.5 | 11/5 |
| 14 | 1460.9 | 7/3 |
| 15 | 1565.2 | |
| 16 | 1669.6 | |
| 17 | 1773.9 | 14/5 |
| 18 | 1878.3 | |
| 19 | 1982.6 | 19/6 |
| 20 | 2087 | |
| 21 | 2191.3 | |
| 22 | 2295.7 | |
| 23 | 2400 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +52.2 | -23.7 | +0.0 | +31.1 | +28.5 | -29.7 | +52.2 | -47.4 | -21.1 | +22.6 | -23.7 |
| Relative (%) | +50.0 | -22.7 | +0.0 | +29.8 | +27.3 | -28.5 | +50.0 | -45.4 | -20.2 | +21.7 | -22.7 | |
| Steps (reduced) |
12 (12) |
18 (18) |
23 (0) |
27 (4) |
30 (7) |
32 (9) |
35 (12) |
36 (13) |
38 (15) |
40 (17) |
41 (18) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +46.4 | +22.5 | +7.4 | +0.0 | -0.6 | +4.8 | +15.5 | +31.1 | +51.0 | -29.6 | -2.2 |
| Relative (%) | +44.5 | +21.5 | +7.1 | +0.0 | -0.6 | +4.6 | +14.9 | +29.8 | +48.8 | -28.3 | -2.1 | |
| Steps (reduced) |
43 (20) |
44 (21) |
45 (22) |
46 (0) |
47 (1) |
48 (2) |
49 (3) |
50 (4) |
51 (5) |
51 (5) |
52 (6) | |