1193edo
| ← 1192edo | 1193edo | 1194edo → |
1193 equal divisions of the octave (abbreviated 1193edo or 1193ed2), also called 1193-tone equal temperament (1193tet) or 1193 equal temperament (1193et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1193 equal parts of about 1.01 ¢ each. Each step represents a frequency ratio of 21/1193, or the 1193rd root of 2.
Theory
1193edo is consistent to the 11-odd-limit. As an equal temperament, it tempers out [144 -22 -47⟩ (gross comma) in the 5-limit; 2401/2400 in the 7-limit; 3025/3024, 41503/41472, 151263/151250, 17561600/17537553, 26214400/26198073, 95703125/95664294, and 117440512/117406179 in the 11-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.141 | -0.061 | -0.175 | -0.103 | +0.378 | -0.345 | +0.224 | +0.393 | +0.431 | -0.358 |
| Relative (%) | +0.0 | +14.0 | -6.0 | -17.4 | -10.2 | +37.5 | -34.3 | +22.2 | +39.1 | +42.9 | -35.6 | |
| Steps (reduced) |
1193 (0) |
1891 (698) |
2770 (384) |
3349 (963) |
4127 (548) |
4415 (836) |
4876 (104) |
5068 (296) |
5397 (625) |
5796 (1024) |
5910 (1138) | |
Subsets and supersets
1193edo is the 196th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1891 -1193⟩ | [⟨1193 1891]] | −0.0443 | 0.0443 | 4.41 |
| 2.3.5 | [10 -40 -23⟩, [77 -31 -12⟩ | [⟨1193 1891 2770]] | −0.0209 | 0.0491 | 4.88 |
| 2.3.5.7 | 2401/2400, [-13 -12 15 -1⟩, [41 -15 -5 -2⟩ | [⟨1193 1891 2770 3349]] | −0.000028 | 0.0558 | 5.55 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 17561600/17537553, 95703125/95664294 | [⟨1193 1891 2770 3349 4127]] | +0.0059 | 0.0513 | 5.10 |
| 2.3.5.7.11.13 | 2401/2400, 3025/3024, 4096/4095, 78125/78078, 114345/114244 | [⟨1193 1891 2770 3349 4127 4415]] | −0.0121 | 0.0617 | 6.14 |
| 2.3.5.7.11.13.17 | 2401/2400, 2431/2430, 2601/2600, 3025/3024, 4096/4095, 42500/42471 | [⟨1193 1891 2770 3349 4127 4415 4876]] | +0.0017 | 0.0664 | 6.60 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 91\1193 | 91.534 | [144 -22 -47⟩ | Gross |
| 1 | 257\1193 | 258.508 | 125/108 | Lafa |
| 1 | 355\1193 | 357.083 | 768/625 | Dodifo |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct