93ed6
| ← 92ed6 | 93ed6 | 94ed6 → |
93 equal divisions of the 6th harmonic (abbreviated 93ed6) is a nonoctave tuning system that divides the interval of 6/1 into 93 equal parts of about 33.4 ¢ each. Each step represents a frequency ratio of 61/93, or the 93rd root of 6.
Theory
93ed6 is nearly identical to 36edo, but with the 6th harmonic rather than the octave being just. The octave is stretched by about 0.757 cents (almost identical to 101ed7, where the octave is stretched by about 0.770 cents). Like 36edo, 93ed6 is consistent to the 8-integer-limit.
Compared to 36edo, 93ed6 is pretty well optimized for the 2.3.7.13.17 subgroup, with slightly better 3, 7, 13 and 17, and a slightly worse 2 versus 36edo. Using the patent val, the 5 is also less accurate. Overall this means 36edo is still better in the 5-limit, but 93ed6 is better in the 13- and 17-limit, especially when treating it as a dual-5 dual-11 tuning.
The local zeta peak around 36 is located at 35.982388, which has a step size of 33.3496 ¢ and has octaves stretched by 0.587 ¢, making 93ed6 very close to optimal for 36edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.8 | -0.8 | +1.5 | +15.5 | +0.0 | -0.0 | +2.3 | -1.5 | +16.2 | -15.4 | +0.8 |
| Relative (%) | +2.3 | -2.3 | +4.5 | +46.3 | +0.0 | -0.1 | +6.8 | -4.5 | +48.6 | -46.1 | +2.3 | |
| Steps (reduced) |
36 (36) |
57 (57) |
72 (72) |
84 (84) |
93 (0) |
101 (8) |
108 (15) |
114 (21) |
120 (27) |
124 (31) |
129 (36) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.4 | +0.7 | +14.7 | +3.0 | -1.9 | -0.8 | +5.7 | -16.4 | -0.8 | -14.6 | +8.5 | +1.5 |
| Relative (%) | -13.2 | +2.2 | +44.1 | +9.1 | -5.6 | -2.3 | +17.1 | -49.1 | -2.4 | -43.8 | +25.4 | +4.5 | |
| Steps (reduced) |
133 (40) |
137 (44) |
141 (48) |
144 (51) |
147 (54) |
150 (57) |
153 (60) |
155 (62) |
158 (65) |
160 (67) |
163 (70) |
165 (72) | |
Subsets and supersets
Since 93 factors into primes as 3 × 31, 93ed6 contains subset ed6's 3ed6 and 31ed6.