37ed5
Jump to navigation
Jump to search
| ← 36ed5 | 37ed5 | 38ed5 → |
(semiconvergent)
37 equal divisions of the 5th harmonic (abbreviated 37ed5) is a nonoctave tuning system that divides the interval of 5/1 into 37 equal parts of about 75.3 ¢ each. Each step represents a frequency ratio of 51/37, or the 37th root of 5.
Theory
37ed5 is related to 16edo, but with the 5/1 rather than 2/1 being just. The octave is stretched by about 4.893 cents. 37ed5 is consistent to the 6-integer-limit. 37ed5 and 16edo share their patent vals until the 23-limit, although 37ed5 has a better 3/1, 5/1, 11/1 and 17/1, at expense of a worse, but still acceptable 2/1.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +4.9 | -19.3 | +0.0 | +19.9 | -9.5 | +2.5 | -10.1 | +23.3 | -6.3 | -31.0 | +4.1 |
| Relative (%) | +6.5 | -25.6 | +0.0 | +26.5 | -12.6 | +3.3 | -13.4 | +30.9 | -8.3 | -41.2 | +5.5 | |
| Steps (reduced) |
16 (16) |
25 (25) |
37 (0) |
45 (8) |
55 (18) |
59 (22) |
65 (28) |
68 (31) |
72 (35) |
77 (3) |
79 (5) | |
16edo for comparison:
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -27.0 | -11.3 | +6.2 | -26.3 | -15.5 | -30.0 | +2.5 | -28.3 | +20.4 | -20.0 |
| Relative (%) | +0.0 | -35.9 | -15.1 | +8.2 | -35.1 | -20.7 | -39.9 | +3.3 | -37.7 | +27.2 | -26.7 | |
| Steps (reduced) |
16 (0) |
25 (9) |
37 (5) |
45 (13) |
55 (7) |
59 (11) |
65 (1) |
68 (4) |
72 (8) |
78 (14) |
79 (15) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 75.3 | 22/21, 23/22, 24/23, 25/24, 26/25 |
| 2 | 150.6 | 12/11, 23/21, 25/23 |
| 3 | 225.9 | 8/7, 25/22 |
| 4 | 301.2 | 19/16, 25/21 |
| 5 | 376.5 | 26/21 |
| 6 | 451.8 | 13/10, 22/17 |
| 7 | 527.1 | 19/14, 23/17 |
| 8 | 602.4 | 17/12, 24/17 |
| 9 | 677.8 | 28/19 |
| 10 | 753.1 | 17/11, 20/13 |
| 11 | 828.4 | 21/13, 29/18 |
| 12 | 903.7 | 22/13 |
| 13 | 979 | |
| 14 | 1054.3 | 11/6, 24/13 |
| 15 | 1129.6 | 23/12, 25/13 |
| 16 | 1204.9 | 2/1 |
| 17 | 1280.2 | 21/10, 23/11 |
| 18 | 1355.5 | 24/11 |
| 19 | 1430.8 | 16/7 |
| 20 | 1506.1 | |
| 21 | 1581.4 | 5/2 |
| 22 | 1656.7 | 13/5 |
| 23 | 1732 | 19/7 |
| 24 | 1807.3 | 17/6 |
| 25 | 1882.6 | |
| 26 | 1958 | |
| 27 | 2033.3 | 13/4, 29/9 |
| 28 | 2108.6 | |
| 29 | 2183.9 | |
| 30 | 2259.2 | |
| 31 | 2334.5 | |
| 32 | 2409.8 | |
| 33 | 2485.1 | 21/5 |
| 34 | 2560.4 | 22/5 |
| 35 | 2635.7 | 23/5 |
| 36 | 2711 | 24/5 |
| 37 | 2786.3 | 5/1 |