19ed4
| ← 17ed4 | 19ed4 | 21ed4 → |
(semiconvergent)
19 equal divisions of the 4th harmonic (abbreviated 19ed4) is a nonoctave tuning system that divides the interval of 4/1 into 19 equal parts of about 126 ¢ each. Each step represents a frequency ratio of 41/19, or the 19th root of 4. It represents every second step of 19edo.
Theory
It is consistent to the no-twos 17-integer-limit. Using the patent val, it tempers out 375/343 and 6561/6125 in the 7-limit; 81/77, 125/121, and 363/343 in the 11-limit; 65/63, 169/165, 585/539, and 1287/1225 in the 13-limit; 51/49, 121/119, 125/119, 189/187, and 195/187 in the 17-limit (no-twos subgroup). It tempers out 36/35, 64/63, and 375/343 in the 3.4.5.7 subgroup; 45/44, 80/77, 81/77, and 363/343 in the 3.4.5.7.11 subgroup; 52/49, 65/63, 65/64, 143/140, and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, 52/51, 85/84, and 121/119 in the 3.4.5.7.11.13.17 subgroup.
Lookalikes: 15edt
Intervals
| Steps | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 126.316 | |
| 2 | 252.632 | 7/6, 15/13 |
| 3 | 378.947 | 21/17 |
| 4 | 505.263 | |
| 5 | 631.579 | 10/7, 13/9 |
| 6 | 757.895 | 17/11 |
| 7 | 884.211 | 5/3 |
| 8 | 1010.526 | 9/5 |
| 9 | 1136.842 | |
| 10 | 1263.158 | 23/11 |
| 11 | 1389.474 | |
| 12 | 1515.789 | |
| 13 | 1642.105 | 13/5, 18/7 |
| 14 | 1768.421 | |
| 15 | 1894.737 | 3/1 |
| 16 | 2021.053 | |
| 17 | 2147.368 | |
| 18 | 2273.684 | |
| 19 | 2400 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +63.2 | -7.2 | +0.0 | -7.4 | +55.9 | +41.7 | +63.2 | -14.4 | +55.8 | +17.1 | -7.2 |
| Relative (%) | +50.0 | -5.7 | +0.0 | -5.8 | +44.3 | +33.0 | +50.0 | -11.4 | +44.2 | +13.5 | -5.7 | |
| Steps (reduced) |
10 (10) |
15 (15) |
19 (0) |
22 (3) |
25 (6) |
27 (8) |
29 (10) |
30 (11) |
32 (13) |
33 (14) |
34 (15) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -19.5 | -21.5 | -14.6 | +0.0 | +21.4 | +48.7 | -44.9 | -7.4 | +34.5 | -46.1 | +3.3 |
| Relative (%) | -15.4 | -17.0 | -11.5 | +0.0 | +16.9 | +38.6 | -35.5 | -5.8 | +27.3 | -36.5 | +2.6 | |
| Steps (reduced) |
35 (16) |
36 (17) |
37 (18) |
38 (0) |
39 (1) |
40 (2) |
40 (2) |
41 (3) |
42 (4) |
42 (4) |
43 (5) | |