13ed5/3
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(convergent)
13 equal divisions of 5/3 (abbreviated 13ed5/3) is a nonoctave tuning system that divides the interval of 5/3 into 13 equal parts of about 68 ¢ each. Each step represents a frequency ratio of (5/3)1/13, or the 13th root of 5/3. This tuning can be seen as extremely streched 18edo, with the flat 3/2 coming a lot closer to being a diatonic fifth (although it misses by about 5.5 cents. In fact, this fifth is almost exactly 40/27.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 68 | 19/18, 21/20, 22/21, 23/22 |
| 2 | 136.1 | 23/21 |
| 3 | 204.1 | 10/9, 17/15, 19/17 |
| 4 | 272.1 | 7/6, 13/11, 15/13, 22/19 |
| 5 | 340.1 | 6/5, 11/9, 23/19 |
| 6 | 408.2 | 19/15, 23/18 |
| 7 | 476.2 | 17/13 |
| 8 | 544.2 | 15/11, 23/17 |
| 9 | 612.2 | 10/7, 13/9 |
| 10 | 680.3 | 3/2, 19/13, 22/15 |
| 11 | 748.3 | 17/11, 23/15 |
| 12 | 816.3 | |
| 13 | 884.4 | 5/3 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +24.5 | +2.8 | -19.0 | +2.8 | +27.3 | +32.6 | +5.5 | +5.6 | +27.3 | -1.6 | -16.2 |
| Relative (%) | +36.0 | +4.1 | -28.0 | +4.1 | +40.2 | +47.9 | +8.0 | +8.3 | +40.2 | -2.4 | -23.8 | |
| Steps (reduced) |
18 (5) |
28 (2) |
35 (9) |
41 (2) |
46 (7) |
50 (11) |
53 (1) |
56 (4) |
59 (7) |
61 (9) |
63 (11) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -18.7 | -11.0 | +5.6 | +30.0 | -7.0 | +30.1 | +4.6 | -16.2 | -32.7 | +22.9 | +13.9 |
| Relative (%) | -27.5 | -16.1 | +8.3 | +44.0 | -10.2 | +44.3 | +6.7 | -23.8 | -48.0 | +33.6 | +20.5 | |
| Steps (reduced) |
65 (0) |
67 (2) |
69 (4) |
71 (6) |
72 (7) |
74 (9) |
75 (10) |
76 (11) |
77 (12) |
79 (1) |
80 (2) | |