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, highly contrived, or as yet unknown. |
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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 ¢. The optimal octave stretch for 31edo in the 13-limit is 0.502314 ¢, making 328ed1536 very close to optimal for 31edo.
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) | |