18ed5
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Prime factorization
2 × 32
Step size
154.795¢
Octave
8\18ed5 (1238.36¢) (→4\9ed5)
Twelfth
12\18ed5 (1857.54¢) (→2\3ed5)
Consistency limit
2
Distinct consistency limit
2
 |
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| ← 17ed5 | 18ed5 | 19ed5 → |
18 equal divisions of the 5th harmonic (abbreviated 18ed5) is a nonoctave tuning system that divides the interval of 5/1 into 18 equal parts of about 155 ¢ each. Each step represents a frequency ratio of 51/18, or the 18th root of 5.
18ed5 is an impressive tuning for the quite kooky 5.6.11.16.19.21.23 subgroup.
Or it can approached more simply as a no-2s, no-3s 11-limit temperament. Or as a no-2s, no-3s, no-13s, no-17s 23-limit temperament. It’s a bit more mediocre for these purposes, because they don’t take advantage of its strong 6/1 and they rely on its pretty average 7/1. But still, for its size, it’s reasonably good.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 154.795 | 11/10, 12/11, 13/12, 23/21 |
| 2 | 309.59 | 6/5 |
| 3 | 464.386 | 13/10, 17/13, 22/17 |
| 4 | 619.181 | 10/7, 17/12 |
| 5 | 773.976 | 11/7 |
| 6 | 928.771 | 12/7, 17/10, 19/11, 22/13 |
| 7 | 1083.566 | 13/7 |
| 8 | 1238.362 | |
| 9 | 1393.157 | |
| 10 | 1547.952 | 17/7 |
| 11 | 1702.747 | |
| 12 | 1857.542 | |
| 13 | 2012.338 | 19/6 |
| 14 | 2167.133 | 7/2 |
| 15 | 2321.928 | 19/5, 23/6 |
| 16 | 2476.723 | 21/5 |
| 17 | 2631.519 | 23/5 |
| 18 | 2786.314 | 5/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +38.4 | -44.4 | +76.7 | +0.0 | -6.1 | +36.7 | -39.7 | +66.0 | +38.4 | +28.2 | +32.3 |
| Relative (%) | +24.8 | -28.7 | +49.6 | +0.0 | -3.9 | +23.7 | -25.7 | +42.6 | +24.8 | +18.2 | +20.9 | |
| Steps (reduced) |
8 (8) |
12 (12) |
16 (16) |
18 (0) |
20 (2) |
22 (4) |
23 (5) |
25 (7) |
26 (8) |
27 (9) |
28 (10) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +48.5 | +75.0 | -44.4 | -1.3 | +48.5 | -50.5 | +10.7 | +76.7 | -7.7 | +66.5 | -10.4 |
| Relative (%) | +31.4 | +48.5 | -28.7 | -0.9 | +31.3 | -32.6 | +6.9 | +49.6 | -5.0 | +43.0 | -6.7 | |
| Steps (reduced) |
29 (11) |
30 (12) |
30 (12) |
31 (13) |
32 (14) |
32 (14) |
33 (15) |
34 (16) |
34 (16) |
35 (17) |
35 (17) | |
| Harmonic | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +70.7 | +0.0 | -67.9 | +21.6 | -41.4 | +52.6 | -6.1 | -62.8 | +37.0 | -16.3 | -67.9 |
| Relative (%) | +45.7 | +0.0 | -43.9 | +13.9 | -26.7 | +34.0 | -3.9 | -40.6 | +23.9 | -10.5 | -43.9 | |
| Steps (reduced) |
36 (0) |
36 (0) |
36 (0) |
37 (1) |
37 (1) |
38 (2) |
38 (2) |
38 (2) |
39 (3) |
39 (3) |
39 (3) | |