40ed5
40 equal divisions of the 5th harmonic (abbreviated 40ed5) is a nonoctave tuning system that divides the interval of 5/1 into 40 equal parts of about 69.7 ¢ each. Each step represents a frequency ratio of 51/40, or the 40th root of 5.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 69.7 | 26/25, 28/27 |
| 2 | 139.3 | 25/23 |
| 3 | 209 | 9/8, 26/23 |
| 4 | 278.6 | 20/17 |
| 5 | 348.3 | |
| 6 | 417.9 | |
| 7 | 487.6 | |
| 8 | 557.3 | |
| 9 | 626.9 | |
| 10 | 696.6 | 3/2 |
| 11 | 766.2 | 14/9 |
| 12 | 835.9 | |
| 13 | 905.6 | 22/13, 27/16 |
| 14 | 975.2 | 7/4 |
| 15 | 1044.9 | |
| 16 | 1114.5 | 19/10 |
| 17 | 1184.2 | |
| 18 | 1253.8 | |
| 19 | 1323.5 | 15/7 |
| 20 | 1393.2 | 29/13 |
| 21 | 1462.8 | 7/3 |
| 22 | 1532.5 | 17/7 |
| 23 | 1602.1 | |
| 24 | 1671.8 | 21/8, 29/11 |
| 25 | 1741.4 | |
| 26 | 1811.1 | 20/7 |
| 27 | 1880.8 | |
| 28 | 1950.4 | |
| 29 | 2020.1 | |
| 30 | 2089.7 | 10/3 |
| 31 | 2159.4 | |
| 32 | 2229.1 | |
| 33 | 2298.7 | |
| 34 | 2368.4 | |
| 35 | 2438 | |
| 36 | 2507.7 | 17/4 |
| 37 | 2577.3 | |
| 38 | 2647 | 23/5 |
| 39 | 2716.7 | |
| 40 | 2786.3 | 5/1 |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -15.8 | -21.2 | +0.0 | -25.2 | +28.2 | +17.6 | -28.9 | -12.5 | +5.0 | +21.7 | -24.1 |
| Relative (%) | -22.7 | -30.4 | +0.0 | -36.2 | +40.4 | +25.2 | -41.5 | -17.9 | +7.2 | +31.1 | -34.6 | |
| Steps (reduced) |
17 (17) |
27 (27) |
40 (0) |
48 (8) |
60 (20) |
64 (24) |
70 (30) |
73 (33) |
78 (38) |
84 (4) |
85 (5) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | +3.9 | -33.4 | +19.4 | +13.4 | +6.5 | -34.4 | -15.2 | +7.0 | +29.2 | -15.6 |
| Relative (%) | +0.0 | +5.6 | -47.3 | +27.5 | +19.0 | +9.3 | -48.7 | -21.5 | +9.9 | +41.4 | -22.1 | |
| Steps (reduced) |
17 (0) |
27 (10) |
39 (5) |
48 (14) |
59 (8) |
63 (12) |
69 (1) |
72 (4) |
77 (9) |
83 (15) |
84 (16) | |
Comparison with 17edo
40ed5 has the same scale shape as 17edo but with a very compressed octave about 16 ¢ flat. This dramatically changes the nature of the tuning:
- 17edo's strongest simple primes are 2, 3, 13 and 23
- 17edo's weakest are 5, 17 and 29
- 40ed5's strongest simple primes are 5, 19 and 23
- 40ed5's weakest are 7, 11 and 17
- The mappings of prime 5, 11, 13, 17, 19, 23, 29 and 31 are all changed from 17edo
17edo can be thought of as a tuning for either:
- the no-5s 13-limit (high accuracy for its size)
- the no-5s no-17s 23-limit (high accuracy for its size)
40ed5 can be thought of as a tuning for either:
- the full 13-limit (low accuracy for its size)
- the no-7s no-11s 41-limit (moderate accuracy for its size)
Music
- escape yourself - Merct (2025)
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