204ed96
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited and highly contrived. |
| ← 203ed96 | 204ed96 | 205ed96 → |
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 is too wide to be a useful 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 31 is located at 30.978382, which has a step size of 38.737 ¢ and an octave of 1200.837 ¢ (which is stretched by 0.837 ¢), making 204ed96 extremely close to optimal for 31edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.8 | -3.9 | +1.6 | +2.6 | -3.1 | +1.1 | +2.4 | -7.9 | +3.4 | -6.7 | -2.4 |
| Relative (%) | +2.0 | -10.2 | +4.1 | +6.7 | -8.1 | +2.9 | +6.1 | -20.3 | +8.8 | -17.2 | -6.1 | |
| Steps (reduced) |
31 (31) |
49 (49) |
62 (62) |
72 (72) |
80 (80) |
87 (87) |
93 (93) |
98 (98) |
103 (103) |
107 (107) |
111 (111) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +14.0 | +1.9 | -1.3 | +3.1 | +14.4 | -7.1 | +15.5 | +4.2 | -2.8 | -5.9 | -5.4 | -1.6 |
| Relative (%) | +36.2 | +4.9 | -3.4 | +8.1 | +37.2 | -18.3 | +40.1 | +10.8 | -7.3 | -15.2 | -13.8 | -4.1 | |
| Steps (reduced) |
115 (115) |
118 (118) |
121 (121) |
124 (124) |
127 (127) |
129 (129) |
132 (132) |
134 (134) |
136 (136) |
138 (138) |
140 (140) |
142 (142) | |