86ed12
86 equal divisions of the 12th harmonic (abbreviated 86ed12) is a nonoctave tuning system that divides the interval of 12/1 into 86 equal parts of about 50 ¢ each. Each step represents a frequency ratio of 121/86, or the 86th root of 12.
Theory
86ed12 is very nearly identical to 24edo (quarter-tone tuning), but with the 12th harmonic rather than the octave being just. The octave is about 0.546 cents stretched.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.5 | -1.1 | +1.1 | +15.0 | -0.5 | -17.3 | +1.6 | -2.2 | +15.5 | +0.6 | +0.0 |
| Relative (%) | +1.1 | -2.2 | +2.2 | +29.9 | -1.1 | -34.6 | +3.3 | -4.4 | +31.0 | +1.1 | +0.0 | |
| Steps (reduced) |
24 (24) |
38 (38) |
48 (48) |
56 (56) |
62 (62) |
67 (67) |
72 (72) |
76 (76) |
80 (80) |
83 (83) |
86 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +11.5 | -16.8 | +13.9 | +2.2 | -2.7 | -1.6 | +4.8 | +16.1 | -18.4 | +1.1 | +24.2 | +0.5 |
| Relative (%) | +23.0 | -33.5 | +27.7 | +4.4 | -5.5 | -3.3 | +9.6 | +32.1 | -36.8 | +2.2 | +48.4 | +1.1 | |
| Steps (reduced) |
89 (3) |
91 (5) |
94 (8) |
96 (10) |
98 (12) |
100 (14) |
102 (16) |
104 (18) |
105 (19) |
107 (21) |
109 (23) |
110 (24) | |
Subsets and supersets
Since 86 factors into primes as 2 × 43, 86ed12 contains subset ed12's 2ed12 and 43ed12.