129ed12
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| ← 128ed12 | 129ed12 | 130ed12 → |
129 equal divisions of the 12th harmonic (abbreviated 129ed12) is a nonoctave tuning system that divides the interval of 12/1 into 129 equal parts of about 33.3 ¢ each. Each step represents a frequency ratio of 121/129, or the 129th root of 12.
Theory
129ed12 is very nearly identical to 36edo (sixth-tone tuning), but with the 12th harmonic rather than the octave being just. This stretches the octave by about 0.546 ¢. 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 ¢; 129ed12's octave is extremely close to optimal, being only 0.0418 ¢ (1/23 of a cent) off from the zeta peak.
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 | -0.6 | +1.6 | -2.2 | +15.5 | -16.1 | +0.0 |
| Relative (%) | +1.6 | -3.3 | +3.3 | +44.9 | -1.6 | -1.9 | +4.9 | -6.5 | +46.5 | -48.3 | +0.0 | |
| Steps (reduced) |
36 (36) |
57 (57) |
72 (72) |
84 (84) |
93 (93) |
101 (101) |
108 (108) |
114 (114) |
120 (120) |
124 (124) |
129 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.2 | -0.1 | +13.9 | +2.2 | -2.7 | -1.6 | +4.8 | +16.1 | -1.7 | -15.6 | +7.5 | +0.5 |
| Relative (%) | -15.5 | -0.2 | +41.6 | +6.5 | -8.2 | -4.9 | +14.4 | +48.1 | -5.2 | -46.7 | +22.6 | +1.6 | |
| Steps (reduced) |
133 (4) |
137 (8) |
141 (12) |
144 (15) |
147 (18) |
150 (21) |
153 (24) |
156 (27) |
158 (29) |
160 (31) |
163 (34) |
165 (36) | |
Subsets and supersets
Since 129 factors into primes as 3 × 43, 129ed12 contains subset ed12's 3ed12 and 43ed12.