116edo
| ← 115edo | 116edo | 117edo → |
116 equal divisions of the octave (abbreviated 116edo or 116ed2), also called 116-tone equal temperament (116tet) or 116 equal temperament (116et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 116 equal parts of about 10.345 ¢ each. Each step represents a frequency ratio of 21/116, or the 116th root of 2.
116edo is only consistent to the 5-odd-limit, and is not quite accurate for its size. It can be viewed as splitting 58edo's step in two, and the enfactored 116cef val comes out on top accuracy in the 7-, 11-, and 13-limit. In the 5-limit, however, the patent val ⟨116 184 269] beats the enfactored 116c val ⟨116 184 270] by a thin margin, and it tempers out 20000/19683 (tetracot comma) and 2197265625/2147483648 (wizard comma).
In the 7-, 11- and 13-limit, the patent val ⟨116 184 269 326 401 429] comes in second best after the enfactored 116cef val ⟨116 184 270 326 402 430] , and it tempers out 225/224, 15625/15309, and 51200/50421 in the 7-limit; 385/384, 540/539, 4000/3993, and 6655/6561 in the 11-limit; 169/168, 275/273, 352/351, and 640/637 in the 13-limit. 116edo provides the optimal patent val for submajor temperament in the 11- and 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +1.49 | -3.56 | +3.59 | -3.04 | -2.60 | -1.51 | +2.49 | +2.76 | +4.91 | +3.24 |
| relative (%) | +0 | +14 | -34 | +35 | -29 | -25 | -15 | +24 | +27 | +47 | +31 | |
| Steps (reduced) |
116 (0) |
184 (68) |
269 (37) |
326 (94) |
401 (53) |
429 (81) |
474 (10) |
493 (29) |
525 (61) |
564 (100) |
575 (111) | |
Subsets and supersets
Since 116 factors into 22 × 29, 116edo has subset edos 2, 4, 29, and 58. 232edo, which doubles it, is a notable tuning.