57ed2560
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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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57 equal divisions of the 2560th harmonic (abbreviated 57ed2560) is a nonoctave tuning system that divides the interval of 2560/1 into 57 equal parts of about 238 ¢ each. Each step represents a frequency ratio of 25601/57, or the 57th root of 2560.
Theory
The 2560th harmonic is far too wide to be a useful equivalence, so 57ed2560 is better thought of as a compressed version of 5edo. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being compressed by about 8.237 ¢. It is almost exactly equal to the local zeta peak around 5, with an octave only 0.000612 ¢ off from the ideal size.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -8 | +5 | -16 | +74 | -3 | -32 | -25 | +10 | +66 | -99 | -12 |
| Relative (%) | -3.4 | +2.1 | -6.9 | +31.0 | -1.4 | -13.4 | -10.3 | +4.1 | +27.6 | -41.6 | -4.8 | |
| Steps (reduced) |
5 (5) |
8 (8) |
10 (10) |
12 (12) |
13 (13) |
14 (14) |
15 (15) |
16 (16) |
17 (17) |
17 (17) |
18 (18) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +88 | -40 | +79 | -33 | +101 | +2 | -92 | +58 | -27 | -107 | +54 | -20 |
| Relative (%) | +37.0 | -16.8 | +33.1 | -13.8 | +42.2 | +0.7 | -38.6 | +24.1 | -11.3 | -45.1 | +22.6 | -8.3 | |
| Steps (reduced) |
19 (19) |
19 (19) |
20 (20) |
20 (20) |
21 (21) |
21 (21) |
21 (21) |
22 (22) |
22 (22) |
22 (22) |
23 (23) |
23 (23) | |