328ed1536
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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. |
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328 equal divisions of the 1536th harmonic (abbreviated 328ed1536) is a nonoctave tuning system that divides the interval of 1536/1 into 328 equal parts of about 38.7 ¢ each. Each step represents a frequency ratio of 15361/328, or the 328th root of 1536.
Theory
The 1536th harmonic is impossibly wide for a useful equivalence, so 328ed1536 is better thought of as a stretched version of 31edo. Indeed, tuning the 1536/1 ratio just instead of 2/1 results in octaves being stretched by about 0.79 ¢. While 31edo's approximation of the 13th harmonic is of limited accuracy, its optimal 13-limit octave stretch is 0.502314 ¢; 328ed1536 is very close, with a difference of only 1/79 ¢.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.5 | -4.4 | +1.0 | +1.9 | -3.9 | +0.3 | +1.5 | -8.8 | +2.4 | -7.7 | -3.4 |
| Relative (%) | +1.3 | -11.4 | +2.5 | +5.0 | -10.1 | +0.7 | +3.8 | -22.8 | +6.2 | -19.9 | -8.9 | |
| 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 (¢) | +12.9 | +0.8 | -2.5 | +2.0 | +13.2 | -8.3 | +14.2 | +2.9 | -4.1 | -7.2 | -6.7 | -2.9 |
| Relative (%) | +33.3 | +2.0 | -6.4 | +5.1 | +34.0 | -21.5 | +36.8 | +7.5 | -10.6 | -18.6 | -17.3 | -7.6 | |
| 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) | |