19ed18/5
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| ← 18ed18/5 | 19ed18/5 | 20ed18/5 → |
19 equal divisions of the 18/5 (abbreviated 19ed18/5), when viewed under a regular temperament perspective, is the tuning system that divides the 18/5 interval into 19 equal parts of about 116.7 ¢ each. Each step of 19ed18/5 represents a frequency ratio of (18/5)1/19, or the 19th root of 18/5.
19ed18/5 is most notable for the fact that its one step is defined as secor.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 116.7 | 14/13, 15/14 |
| 2 | 233.4 | 15/13, 17/15 |
| 3 | 350.1 | 17/14, 21/17 |
| 4 | 466.9 | 13/10, 17/13 |
| 5 | 583.6 | 7/5 |
| 6 | 700.3 | 3/2 |
| 7 | 817 | 21/13 |
| 8 | 933.7 | 17/10, 19/11 |
| 9 | 1050.4 | |
| 10 | 1167.2 | |
| 11 | 1283.9 | 21/10, 23/11 |
| 12 | 1400.6 | 9/4 |
| 13 | 1517.3 | |
| 14 | 1634 | |
| 15 | 1750.7 | |
| 16 | 1867.4 | |
| 17 | 1984.2 | 22/7 |
| 18 | 2100.9 | 10/3 |
| 19 | 2217.6 |
Theory
If considered in its own right, the regular temperament has good approximations for harmonics 5, 7, 8, and 12, all being sharp by roughly the same amount, therefore making the 18/5.5.7.8.12 subgroup the best for this tuning. There, it tempers out 81/80, 126/125, 225/224, 1728/1715, 5103/5000.
Integer harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -32.8 | -34.5 | +51.0 | +14.9 | +49.4 | +15.9 | +18.2 | +47.7 | -18.0 | +50.4 | +16.5 |
| Relative (%) | -28.1 | -29.6 | +43.7 | +12.7 | +42.3 | +13.6 | +15.6 | +40.9 | -15.4 | +43.2 | +14.2 | |
| Steps (reduced) |
10 (10) |
16 (16) |
21 (2) |
24 (5) |
27 (8) |
29 (10) |
31 (12) |
33 (14) |
34 (15) |
36 (17) |
37 (18) | |