10ed8/3
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| ← 9ed8/3 | 10ed8/3 | 11ed8/3 → |
(semiconvergent)
10 equal divisions of 8/3 (abbreviated 10ed8/3) is a nonoctave tuning system that divides the interval of 8/3 into 10 equal parts of about 170 ¢ each. Each step represents a frequency ratio of (8/3)1/10, or the 10th root of 8/3.
Theory
10ed8/3 can be seen as a very compressed version of 7edo. The octave compression results in a more accurate perfect fourth, at the expense of the fifth, which becomes a sharp Mavila fifth.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 169.8 | 9/8, 10/9, 11/10, 12/11, 13/12, 19/17, 21/19 |
| 2 | 339.6 | 6/5, 11/9, 16/13, 17/14, 21/17 |
| 3 | 509.4 | 4/3, 15/11, 19/14 |
| 4 | 679.2 | 3/2, 16/11, 19/13, 22/15 |
| 5 | 849 | 13/8, 18/11, 21/13 |
| 6 | 1018.8 | 9/5, 11/6, 16/9, 20/11 |
| 7 | 1188.6 | 2/1 |
| 8 | 1358.4 | 11/5, 13/6, 20/9 |
| 9 | 1528.2 | 12/5, 17/7, 19/8, 22/9 |
| 10 | 1698 | 8/3, 19/7, 21/8 |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -11.4 | -34.1 | -69.4 | +27.3 | -76.0 | -25.6 | +19.4 | -3.4 | +5.5 | -56.2 | -1.9 |
| Relative (%) | -6.7 | -20.1 | -40.9 | +16.1 | -44.8 | -15.1 | +11.4 | -2.0 | +3.2 | -33.1 | -1.1 | |
| Steps (reduced) |
7 (7) |
11 (1) |
16 (6) |
20 (0) |
24 (4) |
26 (6) |
29 (9) |
30 (0) |
32 (2) |
34 (4) |
35 (5) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +31.4 | +23.5 | -58.9 | -43.1 | -81.3 | +72.6 | +14.9 | +22.3 | -78.1 | +43.6 | +76.7 |
| Relative (%) | +18.5 | +13.8 | -34.7 | -25.4 | -47.9 | +42.8 | +8.8 | +13.1 | -46.0 | +25.7 | +45.1 | |
| Steps (reduced) |
37 (7) |
38 (8) |
38 (8) |
39 (9) |
40 (0) |
42 (2) |
42 (2) |
43 (3) |
43 (3) |
44 (4) |
45 (5) | |
Music
Cole