186ed6
| ← 185ed6 | 186ed6 | 187ed6 → |
186 equal divisions of the 6th harmonic (abbreviated 186ed6) is a nonoctave tuning system that divides the interval of 6/1 into 186 equal parts of about 16.7 ¢ each. Each step represents a frequency ratio of 61/186, or the 186th root of 6.
Theory
186ed6 is closely related to 72edo, but with the 6th harmonic rather than the octave being just, which results in the octaves being stretched by about 0.757 ¢. Like 72edo, 186ed6 is consistent to the 18-integer-limit. While it tunes 2 and 11 sharp, the 3 and 5 remain flat as in 72edo but less so, and the 7 is practically pure. Moreover, the 13 and 17 are significantly improved compared to 72edo, although the 19 becomes worse.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.76 | -0.76 | +1.51 | -1.23 | +0.00 | -0.04 | +2.27 | -1.51 | -0.47 | +1.30 | +0.76 |
| Relative (%) | +4.5 | -4.5 | +9.1 | -7.3 | +0.0 | -0.2 | +13.6 | -9.1 | -2.8 | +7.8 | +4.5 | |
| Steps (reduced) |
72 (72) |
114 (114) |
144 (144) |
167 (167) |
186 (0) |
202 (16) |
216 (30) |
228 (42) |
239 (53) |
249 (63) |
258 (72) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.40 | +0.72 | -1.98 | +3.03 | -1.87 | -0.76 | +5.70 | +0.29 | -0.79 | +2.06 | -8.19 | +1.51 |
| Relative (%) | -26.4 | +4.3 | -11.9 | +18.2 | -11.2 | -4.5 | +34.2 | +1.7 | -4.8 | +12.3 | -49.1 | +9.1 | |
| Steps (reduced) |
266 (80) |
274 (88) |
281 (95) |
288 (102) |
294 (108) |
300 (114) |
306 (120) |
311 (125) |
316 (130) |
321 (135) |
325 (139) |
330 (144) | |
Subsets and supersets
Since 186 factors into primes as 2 × 3 × 31, 186ed6 contains subset ed6's 2, 3, 6, 31, 62, and 93.