11edt
| ← 10edt | 11edt | 12edt → |
(semiconvergent)
11 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 11edt or 11ed3), is a nonoctave tuning system that divides the interval of 3/1 into 11 equal parts of about 173 ¢ each. Each step represents a frequency ratio of 31/11, or the 11th root of 3.
Theory
From a no-two point of view, 11edt has a 5/3 major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic 7/1, which is off by almost half a step (or about a semitone), which causes it to temper out 49/45 in the 7-limit. 11edt is at the extreme end of arcturus temperament, defined by tempering out 15625/15309 in the 3.5.7 subgroup. It gives an equalized interpretation for the sub-arcturus MOS scale.
The 11th harmonic, 11/1, only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out 35/33 and 77/75 in the 11-limit. In the 3.5.11 subgroup, it tempers out 125/121.
Relation to edos
11edt can be seen as a very stretched version of 7edo, with octaves sharpened by 10.3 cents. The octave stretching makes the 3/2 perfect fifth in better tune, while preserving a just 3/1 tritave.
Prime harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.3 | +0.0 | +20.7 | -19.8 | +10.3 | -83.6 | +31.0 | +0.0 | -9.5 | -1.6 | +20.7 |
| Relative (%) | +6.0 | +0.0 | +12.0 | -11.5 | +6.0 | -48.4 | +17.9 | +0.0 | -5.5 | -0.9 | +12.0 | |
| Steps (reduced) |
7 (7) |
11 (0) |
14 (3) |
16 (5) |
18 (7) |
19 (8) |
21 (10) |
22 (0) |
23 (1) |
24 (2) |
25 (3) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.3 | +0.0 | -19.8 | -83.6 | -1.6 | +55.0 | -63.6 | -83.3 | -68.2 | +49.2 | -66.3 |
| Relative (%) | +6.0 | +0.0 | -11.5 | -48.4 | -0.9 | +31.8 | -36.8 | -48.2 | -39.5 | +28.5 | -38.3 | |
| Steps (reduced) |
7 (7) |
11 (0) |
16 (5) |
19 (8) |
24 (2) |
26 (4) |
28 (6) |
29 (7) |
31 (9) |
34 (1) |
34 (1) | |
Interval table
| # | Cents | Hekts | Approximate ratios | Arcturus nonatonic notation (J = 1/1) |
|---|---|---|---|---|
| 0 | 1/1 | J | ||
| 1 | 172.9 | 118.1 | 11/10, 10/9 | J#, Kb |
| 2 | 345.8 | 236.2 | 11/9 | K |
| 3 | 518.7 | 354.3 | 4/3, 27/20 | L |
| 4 | 691.6 | 472.4 | 3/2, 40/27 | M |
| 5 | 864.5 | 590.5 | 5/3, 28/17, 105/64 | N |
| 6 | 1037.4 | 708.6 | 29/16, 20/11, 64/35 | N#, Ob |
| 7 | 1210.3 | 826.7 | 2/1 | O |
| 8 | 1383.2 | 944.8 | P | |
| 9 | 1556.1 | 1062.9 | Q | |
| 10 | 1729 | 1181 | R | |
| 11 | 1902 | 1300 | J | |
Pieces
Mozart's sonata #11 in A Major K331 in 11 EDT (using a 11 => 12 key mapping so octaves become tritaves)
Frozen Time Occupies Wall Street by Chris Vaisvil =>information about the piece
Molly's Playground by Chris Vaisvil => information about the piece