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 and highly contrived. |
| ← 56ed2560 | 57ed2560 | 58ed2560 → |
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) | |