4ed5/3
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Prime factorization
22
Step size
221.09¢
Octave
5\4ed5/3 (1105.45¢)
Twelfth
9\4ed5/3 (1989.81¢)
Consistency limit
2
Distinct consistency limit
2
Special properties
| ← 3ed5/3 | 4ed5/3 | 5ed5/3 → |
4 equal divisions of 5/3 (abbreviated 4ed5/3) is a nonoctave tuning system that divides the interval of 5/3 into 4 equal parts of about 221 ¢ each. Each step represents a frequency ratio of (5/3)1/4, or the 4th root of 5/3.
Theory
This tuning tempers out 10/9 in the 5-limit; 8/7 in the 7-limit; 12/11 and 22/21 in the 11-limit; 14/13 and 16/13 in the 13-limit; 17/15 in the 17-limit; 19/18 and 20/19 in the 19-limit; 26/23 and 28/23 in the 23-limit; 29/25 in the 29-limit; 31/30 in the 31-limit; and 43/41 in the 41-limit.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1, 19/18, 20/19 |
| 1 | 221.1 | 6/5, 7/6, 11/10, 13/11, 13/12, 19/17, 20/17, 21/19 |
| 2 | 442.2 | 13/10, 17/13, 21/17 |
| 3 | 663.3 | 10/7, 17/11, 17/12, 19/13, 20/13 |
| 4 | 884.4 | 5/3, 12/7, 17/10, 18/11, 19/11, 19/12, 21/13 |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -94.6 | +87.9 | +87.9 | -52.5 | +49.4 | -18.7 | -41.0 | -12.5 | +99.0 | -81.2 | +24.4 |
| Relative (%) | -42.8 | +39.7 | +39.7 | -23.7 | +22.3 | -8.5 | -18.5 | -5.6 | +44.8 | -36.7 | +11.0 | |
| Steps (reduced) |
5 (1) |
9 (1) |
13 (1) |
15 (3) |
19 (3) |
20 (0) |
22 (2) |
23 (3) |
25 (1) |
26 (2) |
27 (3) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -60.8 | -17.5 | -99.9 | -32.8 | -19.7 | +15.7 | -42.0 | +16.7 | -83.7 | +89.3 | -47.5 |
| Relative (%) | -27.5 | -7.9 | -45.2 | -14.8 | -8.9 | +7.1 | -19.0 | +7.5 | -37.9 | +40.4 | -21.5 | |
| Steps (reduced) |
28 (0) |
29 (1) |
29 (1) |
30 (2) |
31 (3) |
32 (0) |
32 (0) |
33 (1) |
33 (1) |
34 (2) |
34 (2) | |