13edf
| ← 12edf | 13edf | 14edf → |
13 equal divisions of the perfect fifth (abbreviated 13edf or 13ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 13 equal parts of about 54 ¢ each. Each step represents a frequency ratio of (3/2)1/13, or the 13th root of 3/2.
Theory
13edf corresponds to 22.2236edo. It is nearly identical to every ninth step of 200edo, but not quite similar to 22edo; the octave is compressed by 12.076 ¢, a deviation that is small but significant enough to create a discrepancy for the 7th and 11th harmonics.
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | |
|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -12.1 | -12.1 | +21.5 | -21.0 | +6.4 | -12.8 | +8.7 | -21.8 |
| Relative (%) | -22.4 | -22.4 | +39.8 | -39.0 | +11.9 | -23.7 | +16.2 | -40.4 | |
| Steps (reduced) |
22 (9) |
35 (9) |
52 (0) |
62 (10) |
77 (12) |
82 (4) |
91 (0) |
94 (3) | |
| Harmonic | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | |
|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +25.4 | +2.0 | -5.4 | +12.3 | -3.5 | +22.1 | -23.9 | -15.9 |
| Relative (%) | +47.0 | +3.8 | -10.0 | +22.7 | -6.4 | +40.9 | -44.3 | -29.5 | |
| Steps (reduced) |
101 (10) |
108 (4) |
110 (6) |
116 (12) |
119 (2) |
121 (4) |
123 (6) |
127 (10) | |
Intervals
| Degree | Cents | Corresponding JI intervals
(even if inconsistent) |
Comments |
|---|---|---|---|
| 0 | exact 1/1 | ||
| 1 | 53.9965 | 33/32 | pseudo-25/24 |
| 2 | 107.9931 | 17/16, 117/110, 16/15 | |
| 3 | 161.9896 | 11/10 | |
| 4 | 215.9862 | 17/15 | |
| 5 | 269.9827 | 7/6 | |
| 6 | 323.9792 | 77/64 | pseudo-6/5 |
| 7 | 377.9758 | 56/45 | pseudo-5/4 |
| 8 | 431.9723 | 9/7 | |
| 9 | 485.9688 | 45/34 | pseudo-4/3 |
| 10 | 539.9654 | 15/11 | |
| 11 | 593.9619 | 55/39, 24/17 | |
| 12 | 647.9585 | 16/11 | |
| 13 | 701.9550 | exact 3/2 | just perfect fifth |
| 14 | 755.9515 | 99/64 | |
| 15 | 809.9481 | 51/32, 8/5 | |
| 16 | 863.9446 | 33/20 | |
| 17 | 917.9412 | 17/10 | |
| 18 | 971.9377 | 7/4 | |
| 19 | 1025.9342 | 29/16 | pseudo-9/5 |
| 20 | 1079.9308 | 28/15 | pseudo-15/8 |
| 21 | 1133.9273 | 52/27, 27/14 | |
| 22 | 1187.9238 | 135/68 | pseudo-octave |
| 23 | 1241.9204 | 45/22 | |
| 24 | 1295.9169 | 19/9, 36/17 | |
| 25 | 1349.9135 | 24/11 | |
| 26 | 1403.9100 | exact 9/4 | pythagorean major ninth |
Instruments
A Lumatone mapping for 13edf is available.
Music
- 13ed(3/2) improv (2026)
See also
- 22edo — 13edf is fairly close to 22.2edo.
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