186edo
| ← 185edo | 186edo | 187edo → |
186 equal divisions of the octave (abbreviated 186edo or 186ed2), also called 186-tone equal temperament (186tet) or 186 equal temperament (186et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 186 equal parts of about 6.45 ¢ each. Each step represents a frequency ratio of 21/186, or the 186th root of 2.
186edo is closely related to 31edo, but the patent vals differ on the mapping for 3. The equal temperament tempers out 67108864/66430125 (misty comma) and 390625000/387420489 (quartonic comma) in the 5-limit, as well as 6115295232/6103515625 (vishnuzma); 3136/3125, 5120/5103 and 117649/116640 in the 7-limit. Using the patent val, it tempers out 385/384, 1331/1323, 2200/2187, and 3773/3750 in the 11-limit; 325/324, 352/351, 847/845, and 1573/1568 in the 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +1.27 | +0.78 | -1.08 | -2.93 | -1.82 | -1.73 | -0.74 | -2.47 | +2.68 | -3.10 |
| relative (%) | +0 | +20 | +12 | -17 | -45 | -28 | -27 | -11 | -38 | +42 | -48 | |
| Steps (reduced) |
186 (0) |
295 (109) |
432 (60) |
522 (150) |
643 (85) |
688 (130) |
760 (16) |
790 (46) |
841 (97) |
904 (160) |
921 (177) | |
Subsets and supersets
Since 186 factors into 2 × 3 × 31, 186edo has subset edos 2, 3, 6, 31, 62, and 93.