19ed4: Difference between revisions
Created page with "'''19ED4''' is the equal division of the double octave into 19 parts of 126.3158 cents each (every second step of 19edo). It is consistent to the no-twos..." Tags: Mobile edit Mobile web edit |
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{{Infobox ET}} | |||
{{ED intro}} 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]] | Lookalikes: [[15edt]] | ||
== Intervals == | |||
{{Interval table}} | |||
== Harmonics == | |||
{{Harmonics in equal | |||
| steps = 19 | |||
| num = 4 | |||
| denom = 1 | |||
}} | |||
{{Harmonics in equal | |||
| steps = 19 | |||
| num = 4 | |||
| denom = 1 | |||
| start = 12 | |||
| collapsed = 1 | |||
}} | |||
{{todo|expand}} | |||
Latest revision as of 04:15, 22 December 2024
| ← 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 | 126.3 | |
| 2 | 252.6 | 7/6, 15/13 |
| 3 | 378.9 | 21/17 |
| 4 | 505.3 | |
| 5 | 631.6 | 10/7, 13/9 |
| 6 | 757.9 | 17/11 |
| 7 | 884.2 | 5/3 |
| 8 | 1010.5 | 9/5 |
| 9 | 1136.8 | |
| 10 | 1263.2 | 23/11 |
| 11 | 1389.5 | |
| 12 | 1515.8 | |
| 13 | 1642.1 | 13/5, 18/7 |
| 14 | 1768.4 | |
| 15 | 1894.7 | 3/1 |
| 16 | 2021.1 | |
| 17 | 2147.4 | |
| 18 | 2273.7 | |
| 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) | |