11edf
| ← 10edf | 11edf | 12edf → |
11 equal divisions of the perfect fifth (abbreviated 11edf or 11ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 11 equal parts of about 63.8 ¢ each. Each step represents a frequency ratio of (3/2)1/11, or the 11th root of 3/2.
Theory
11edf corresponds to 18.8046…edo. It is similar to 19edo, and nearly identical to Carlos Beta. Unlike 19edo, which is consistent to the 10-integer-limit, 11edf is only consistent to the 7-integer-limit.
While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51 ¢, it is 12.47 ¢ sharper than just and 3.7 ¢ flat of that of 7edo.
11edf represents the upper bound of the phoenix tuning range. It benefits from all the desirable properties of phoenix tuning systems.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +12.5 | +12.5 | +24.9 | +21.5 | +24.9 | +13.3 | -26.4 | +24.9 | -29.8 | -3.4 | -26.4 |
| Relative (%) | +19.5 | +19.5 | +39.1 | +33.7 | +39.1 | +20.9 | -41.4 | +39.1 | -46.8 | -5.3 | -41.4 | |
| Steps (reduced) |
19 (8) |
30 (8) |
38 (5) |
44 (0) |
49 (5) |
53 (9) |
56 (1) |
60 (5) |
62 (7) |
65 (10) |
67 (1) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +26.5 | +25.8 | -29.8 | -13.9 | +8.7 | -26.4 | +7.6 | -17.4 | +25.8 | +9.1 | -4.1 | -13.9 |
| Relative (%) | +41.5 | +40.4 | -46.8 | -21.8 | +13.7 | -41.4 | +11.9 | -27.2 | +40.4 | +14.2 | -6.4 | -21.8 | |
| Steps (reduced) |
70 (4) |
72 (6) |
73 (7) |
75 (9) |
77 (0) |
78 (1) |
80 (3) |
81 (4) |
83 (6) |
84 (7) |
85 (8) |
86 (9) | |
Subsets and supersets
11edf is the fifth prime edf, past 7edf and before 13edf. It does not contain any nontrivial subset edfs.
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 63.8 | 21/20, 25/24, 27/26, 28/27 |
| 2 | 127.6 | 13/12, 14/13, 15/14, 16/15 |
| 3 | 191.4 | 9/8, 10/9 |
| 4 | 255.3 | 7/6, 8/7 |
| 5 | 319.1 | 6/5 |
| 6 | 382.9 | 5/4 |
| 7 | 446.7 | 9/7 |
| 8 | 510.5 | 4/3 |
| 9 | 574.3 | 7/5 |
| 10 | 638.1 | 13/9 |
| 11 | 702.0 | 3/2 |
| 12 | 765.8 | 14/9 |
| 13 | 828.6 | 8/5, 13/8, 21/13 |
| 14 | 893.4 | 5/3 |
| 15 | 956.2 | 7/4 |
| 16 | 1020.0 | 9/5 |
| 17 | 1084.8 | 15/8 |
| 18 | 1148.7 | 27/14, 35/18 |
| 19 | 1211.5 | 2/1 |
| 20 | 1276.3 | 21/10, 25/12, 27/13 |
| 21 | 1340.1 | 13/6 |
| 22 | 1403.9 | 9/4 |