106ed6
| ← 105ed6 | 106ed6 | 107ed6 → |
(convergent)
(convergent)
106 equal divisions of the 6th harmonic (abbreviated 106ed6) is a nonoctave tuning system that divides the interval of 6/1 into 106 equal parts of about 29.3 ¢ each. Each step represents a frequency ratio of 61/106, or the 106th root of 6.
Theory
106ed6 is very nearly identical to 41edo, but with the 6th harmonic rather than the octave being just. The octave is about 0.187 cents compressed. Like 41edo, 106ed6 is consistent to the 16-integer-limit, and in comparison, it slightly improves the intonation of primes 3, 11, 13, and 17 at the expense of barely less accurate intonations of 2, 5, 7, and 19, commending itself as a suitable tuning for 13- and 17-limit-focused harmonies.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.2 | +0.2 | -0.4 | -6.3 | +0.0 | -3.5 | -0.6 | +0.4 | -6.4 | +4.1 | -0.2 |
| Relative (%) | -0.6 | +0.6 | -1.3 | -21.4 | +0.0 | -12.0 | -1.9 | +1.3 | -22.0 | +14.1 | -0.6 | |
| Steps (reduced) |
41 (41) |
65 (65) |
82 (82) |
95 (95) |
106 (0) |
115 (9) |
123 (17) |
130 (24) |
136 (30) |
142 (36) |
147 (41) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.6 | -3.7 | -6.1 | -0.7 | +11.4 | +0.2 | -5.6 | -6.6 | -3.3 | +3.9 | -14.5 | -0.4 |
| Relative (%) | +25.8 | -12.6 | -20.8 | -2.6 | +38.8 | +0.6 | -19.2 | -22.7 | -11.3 | +13.5 | -49.5 | -1.3 | |
| Steps (reduced) |
152 (46) |
156 (50) |
160 (54) |
164 (58) |
168 (62) |
171 (65) |
174 (68) |
177 (71) |
180 (74) |
183 (77) |
185 (79) |
188 (82) | |
Subsets and supersets
Since 106 factors into primes as 2 × 53, 106ed6 contains 2ed6 and 53ed6 as subset ed6's.