204ed96
204 equal divisions of the 96th harmonic (abbreviated 204ed96) is a nonoctave tuning system that divides the interval of 96/1 into 204 equal parts of about 38.7 ¢ each. Each step represents a frequency ratio of 961/204, or the 204th root of 96.
Theory
The 96th harmonic would be extremely wide for an equivalence, so 204ed96 is better thought of as a stretched version of 31edo. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being stretched by about 0.79 ¢. The local zeta peak around 12 is located at 30.978382, which has a step size of 38.737 ¢ and an octave of 1200.837 ¢ (which is compressed by 2.31 ¢), making 204ed96 extremely close to optimal for 31edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.0 | +4.0 | -4.0 | -9.1 | +2.0 | +7.4 | -6.0 | +7.9 | -11.1 | -12.7 | +0.0 |
| Relative (%) | -3.6 | +7.3 | -7.3 | -16.7 | +3.6 | +13.6 | -10.9 | +14.6 | -20.4 | -23.4 | +0.0 | |
| Steps (reduced) |
22 (22) |
35 (35) |
44 (44) |
51 (51) |
57 (57) |
62 (62) |
66 (66) |
70 (70) |
73 (73) |
76 (76) |
79 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +24.8 | +5.4 | -5.1 | -7.9 | -4.0 | +6.0 | +21.3 | -13.1 | +11.4 | -14.7 | +17.2 | -2.0 |
| Relative (%) | +45.5 | +9.9 | -9.4 | -14.6 | -7.3 | +10.9 | +39.1 | -24.0 | +20.9 | -27.0 | +31.7 | -3.6 | |
| Steps (reduced) |
82 (3) |
84 (5) |
86 (7) |
88 (9) |
90 (11) |
92 (13) |
94 (15) |
95 (16) |
97 (18) |
98 (19) |
100 (21) |
101 (22) | |
Subsets and supersets
79ed12 is the 22nd prime ed12, so it does not contain any nontrivial subset ed12's.