11ed6: Difference between revisions
→Theory: +subsets and supersets |
m Mathematical interest |
||
| Line 1: | Line 1: | ||
{{ | {{Mathematical interest}} | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | {{ED intro}} | ||
Latest revision as of 22:15, 10 August 2025
|
This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
| ← 10ed6 | 11ed6 | 12ed6 → |
11 equal divisions of the 6th harmonic (abbreviated 11ed6) is a nonoctave tuning system that divides the interval of 6/1 into 11 equal parts of about 282 ¢ each. Each step represents a frequency ratio of 61/11, or the 11th root of 6.
Theory
11ed6 corresponds to 4.2554…edo. It is related to the temperaments which temper out 28561/28512 and 85293/85184 in the 13-limit, which is supported by edos 17, 149, 166, 183, 200, 217, and 234.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -72 | +72 | +138 | +34 | +0 | +15 | +66 | -138 | -38 | +79 | -72 |
| Relative (%) | -25.5 | +25.5 | +48.9 | +11.9 | +0.0 | +5.4 | +23.4 | -48.9 | -13.6 | +27.9 | -25.5 | |
| Steps (reduced) |
4 (4) |
7 (7) |
9 (9) |
10 (10) |
11 (0) |
12 (1) |
13 (2) |
13 (2) |
14 (3) |
15 (4) |
15 (4) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +71 | -57 | +106 | -6 | -111 | +72 | -22 | -110 | +87 | +7 | -70 | +138 |
| Relative (%) | +25.3 | -20.2 | +37.5 | -2.2 | -39.4 | +25.5 | -7.7 | -39.1 | +30.9 | +2.3 | -24.9 | +48.9 | |
| Steps (reduced) |
16 (5) |
16 (5) |
17 (6) |
17 (6) |
17 (6) |
18 (7) |
18 (7) |
18 (7) |
19 (8) |
19 (8) |
19 (8) |
20 (9) | |
Subsets and supersets
11ed6 is the 5th prime ed6, following 7ed6 and before 13ed6, so it does not contain any nontrivial subset ed6's.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 282 | 7/6, 13/11, 20/17, 22/19 |
| 2 | 564 | 7/5, 15/11, 18/13 |
| 3 | 846 | 18/11, 21/13 |
| 4 | 1128 | 19/10, 21/11 |
| 5 | 1410 | |
| 6 | 1692 | |
| 7 | 1974 | 19/6, 22/7 |
| 8 | 2256 | 11/3 |
| 9 | 2538 | 13/3, 22/5 |
| 10 | 2820 | |
| 11 | 3102 | 6/1 |
Related temperament
2.3.11 subgroup 17 & 183
Comma: |-19 36 0 0 -11>
POTE generator: ~|7 -13 0 0 4> = 281.9832
Mapping: [<1 -1 -5|, <0 11 36|]
EDOs: 17, 34, 166, 183, 200, 217, 366, 383, 400, 566
2.3.11.13 subgroup 17 & 183
Commas: 28561/28512, 85293/85184
POTE generator: ~286/243 = 281.9821
Mapping: [<1 -1 -5 -1|, <0 11 36 20|]