User:MisterShafXen/11edo
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Prime factorization
11 (prime)
Step size
109.091 ¢
Fifth
6\11 (654.545 ¢)
Semitones (A1:m2)
-2:3 (-218.2 ¢ : 327.3 ¢)
Dual sharp fifth
7\11 (763.636 ¢)
Dual flat fifth
6\11 (654.545 ¢)
Dual major 2nd
2\11 (218.182 ¢)
(semiconvergent)
Consistency limit
3
Distinct consistency limit
3
| ← 10edo | 11edo | 12edo → |
(semiconvergent)
11 equal divisions of the octave (abbreviated 11edo or 11ed2), also called 11-tone equal temperament (11tet) or 11 equal temperament (11et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 11 equal parts of about 109 ¢ each. Each step represents a frequency ratio of 21/11, or the 11th root of 2.
Theory
11edo is the first edo to support a non-equalized balzano (2L 7s) MOS. This is because its best fifth is flatter than mavila.
11edo also contains an excellent approximation of 9/7. Closing a circle of 11 9/7s at the octave is associated with the comma [4 -22 0 11⟩, about 14 ¢.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 6\11) |
Ups and downs notation (Dual sharp fifth 7\11) |
This interval… |
|---|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D | is the only interval shared by all tuning systems. |
| 1 | 109.1 | 14/13, 15/14, 16/15, 17/16, 20/19, 22/21 | E | F | approximates the classical diatonic semitone of 16/15. |
| 2 | 218.2 | 8/7, 15/13, 17/15, 19/17 | ^E, F♭ | ^F, vE | is close to the mediant of 9/8 and 8/7. |
| 3 | 327.3 | 17/14, 19/16 | E♯, vF | E | is generally the sharpest possible minor third, serving as a generator for smitonic. |
| 4 | 436.4 | 13/10, 14/11, 17/13, 19/15, 22/17 | F | G | is extremely close to 9/7. |
| 5 | 545.5 | 11/8, 15/11, 19/14 | G | ^G, vvA | is close to 11/8. |
| 6 | 654.5 | 16/11, 19/13, 22/15 | A | ^^G, vA | is close to 16/11, but fills in for 3/2. |
| 7 | 763.6 | 11/7, 17/11, 20/13 | B | A | is extremely close to 14/9. |
| 8 | 872.7 | ^B, C♭ | C | is a decent submajor sixth. | |
| 9 | 981.8 | 7/4 | B♯, vC | ^C, vB | is fairly close to 7/4. |
| 10 | 1090.9 | 13/7, 15/8, 19/10, 21/11 | C | B | is a great approximation of 15/8. |
| 11 | 1200 | 2/1 | D | D | is the only interval distinct from the unison shared by all edos. |
Harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -47.4 | +50.0 | +13.0 | +14.3 | -5.9 | +32.2 | +2.6 | +4.1 | +29.8 | -34.4 | +26.3 |
| Relative (%) | -43.5 | +45.9 | +11.9 | +13.1 | -5.4 | +29.5 | +2.4 | +3.8 | +27.3 | -31.5 | +24.1 | |
| Steps (reduced) |
17 (6) |
26 (4) |
31 (9) |
35 (2) |
38 (5) |
41 (8) |
43 (10) |
45 (1) |
47 (3) |
48 (4) |
50 (6) | |