13ed29/16
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(semiconvergent)
(convergent)
13 equal divisions of 29/16 (abbreviated 13ed29/16) is a nonoctave tuning system that divides the interval of 29/16 into 13 equal parts of about 79.2 ¢ each. Each step represents a frequency ratio of (29/16)1/13, or the 13th root of 29/16.
Theory
13ed29/16 is like 15edo, but with its intervals compressed, resulting in a flat tendency except for the 7th harmonic, which is highly sharp (perhaps warranting a wart), until reaching the 16th harmonic, after which harmonics are mixed sharp and flat. With the 2nd harmonic being flat and the 29th harmonic being sharp, 13ed29/16 maps its own equave inconsistently unless one (but not both) of the following is done: the 29th harmonic (which is sharp) is altered with a wart (to make it flat), or the 16th harmonic is directly approximated (making it sharp instead of flat).
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -12.0 | -1.2 | -24.1 | -14.4 | -13.2 | +36.7 | -36.1 | -2.4 | -26.4 | -33.0 | -25.2 |
| Relative (%) | -15.2 | -1.5 | -30.4 | -18.2 | -16.7 | +46.3 | -45.6 | -3.0 | -33.3 | -41.7 | -31.9 | |
| Steps (reduced) |
15 (2) |
24 (11) |
30 (4) |
35 (9) |
39 (0) |
43 (4) |
45 (6) |
48 (9) |
50 (11) |
52 (0) |
54 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.4 | +24.7 | -15.6 | +31.1 | +5.3 | -14.4 | -28.8 | -38.4 | +35.5 | +34.2 | +36.4 |
| Relative (%) | -6.9 | +31.2 | -19.7 | +39.3 | +6.7 | -18.2 | -36.4 | -48.5 | +44.8 | +43.1 | +46.0 | |
| Steps (reduced) |
56 (4) |
58 (6) |
59 (7) |
61 (9) |
62 (10) |
63 (11) |
64 (12) |
65 (0) |
67 (2) |
68 (3) |
69 (4) | |
| Harmonic | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -37.3 | -28.8 | -17.5 | -3.6 | +12.6 | +31.1 | -27.6 | -5.2 | +19.1 | -34.2 | -6.7 |
| Relative (%) | -47.1 | -36.3 | -22.0 | -4.5 | +16.0 | +39.3 | -34.8 | -6.5 | +24.1 | -43.2 | -8.4 | |
| Steps (reduced) |
69 (4) |
70 (5) |
71 (6) |
72 (7) |
73 (8) |
74 (9) |
74 (9) |
75 (10) |
76 (11) |
76 (11) |
77 (12) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 79.2 | 18/17, 19/18, 20/19 |
| 2 | 158.4 | 10/9, 11/10, 12/11, 13/12, 23/21 |
| 3 | 237.6 | 15/13, 17/15, 22/19 |
| 4 | 316.8 | 6/5, 17/14, 19/16 |
| 5 | 396 | 5/4, 19/15 |
| 6 | 475.2 | 4/3, 13/10, 17/13 |
| 7 | 554.4 | 11/8, 15/11, 18/13 |
| 8 | 633.6 | 13/9, 16/11, 19/13 |
| 9 | 712.8 | 3/2 |
| 10 | 792 | 8/5, 19/12 |
| 11 | 871.2 | 5/3, 18/11, 23/14 |
| 12 | 950.4 | 19/11 |
| 13 | 1029.6 | 9/5, 11/6, 20/11 |
Instruments
A Lumatone mapping for 13ed29/16 is available, although it uses a mapping for 15edo (the relative edo for 13ed29/16).
Music
- 13ed(29/16) improv (2026)
See also
- 15edo – relative edo