1106edo
| ← 1105edo | 1106edo | 1107edo → |
1106 equal divisions of the octave (abbreviated 1106edo or 1106ed2), also called 1106-tone equal temperament (1106tet) or 1106 equal temperament (1106et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1106 equal parts of about 1.08 ¢ each. Each step represents a frequency ratio of 21/1106, or the 1106th root of 2.
Theory
1106edo is a zeta peak edo. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit relative error being 171, 270, 342, 441 and 612. It is even stronger in the 11-limit; the only ones beating it out now being 270, 342 and 612. It is less strong in the 13- and 17-limit, but even so is distinctly consistent through the 17-odd-limit.
The equal temperament tempers out [-53 10 16⟩ (kwazy comma) and [-13 -46 37⟩ (supermajor comma) in the 5-limit; 4375/4374 and 52734375/52706752 in the 7-limit; 3025/3024 and 9801/9800 in the 11-limit; 4096/4095, 78125/78078, and 105644/105625 in the 13-limit; 2500/2499, 4914/4913, and 8624/8619 in the 17-limit. It notably supports supermajor, brahmagupta, and orga in the 7-limit, and semisupermajor in the 11-limit. In the higher limits, it supports the 79th-octave temperament gold.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.034 | -0.057 | +0.071 | -0.143 | +0.340 | +0.289 | -0.225 | -0.065 | +0.079 | -0.370 | +0.374 |
| Relative (%) | +0.0 | +3.1 | -5.2 | +6.5 | -13.1 | +31.4 | +26.6 | -20.8 | -6.0 | +7.3 | -34.1 | +34.5 | |
| Steps (reduced) |
1106 (0) |
1753 (647) |
2568 (356) |
3105 (893) |
3826 (508) |
4093 (775) |
4521 (97) |
4698 (274) |
5003 (579) |
5373 (949) |
5479 (1055) |
5762 (232) | |
Subsets and supersets
Since 1106 factors into 2 × 7 × 79, it has subset edos 2, 7, 14, 79, 158, and 553.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1753 -1106⟩ | [⟨1106 1753]] | −0.010 | 0.010 | 0.99 |
| 2.3.5 | [-53 10 16⟩, [-13 -46 37⟩ | [⟨1106 1753 2568]] | +0.001 | 0.019 | 1.73 |
| 2.3.5.7 | 4375/4374, 52734375/52706752, [46 -14 -3 -6⟩ | [⟨1106 1753 2568 3105]] | −0.006 | 0.020 | 1.83 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041 | [⟨1106 1753 2568 3105 3826]] | +0.004 | 0.026 | 2.38 |
| 2.3.5.7.11.13 | 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625 | [⟨1106 1753 2568 3105 3826 4093]] | −0.012 | 0.043 | 3.94 |
| 2.3.5.7.11.13.17 | 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619 | [⟨1106 1753 2568 3105 3826 4093 4521]] | −0.021 | 0.045 | 4.11 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 213\1106 | 231.103 | 8/7 | Orga (11-limit) |
| 1 | 401\1106 | 435.081 | 9/7 | Supermajor |
| 2 | 150\1106 | 162.749 | 1125/1024 | Kwazy |
| 2 | 401\1106 (152\1106) |
435.081 (164.919) |
9/7 (11/10) |
Semisupermajor |
| 7 | 479\1106 (5\1106) |
519.711 (5.424) |
27/20 (5120/5103) |
Brahmagupta (7-limit) |
| 79 | 459\1106 (11\1106) |
498.011 (11.935) |
4/3 (?) |
Gold |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct