76ed80
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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. |
| ← 75ed80 | 76ed80 | 77ed80 → |
76 equal divisions of the 80th harmonic (abbreviated 76ed80) is a nonoctave tuning system that divides the interval of 80/1 into 76 equal parts of about 99.8 ¢ each. Each step represents a frequency ratio of 801/76, or the 76th root of 80.
Theory
The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous 12edo. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being compressed by about 2.16 ¢. The local zeta peak around 12 is located at 12.023183, which has a step size of 99.807 ¢ and an octave of 1197.686 ¢, making 76ed80 extremely close to optimal for 12edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.2 | -5.4 | -4.3 | +8.6 | -7.5 | +25.1 | -6.5 | -10.8 | +6.5 | +41.1 | -9.7 |
| Relative (%) | -2.2 | -5.4 | -4.3 | +8.7 | -7.6 | +25.1 | -6.5 | -10.8 | +6.5 | +41.2 | -9.7 | |
| Steps (reduced) |
12 (12) |
19 (19) |
24 (24) |
28 (28) |
31 (31) |
34 (34) |
36 (36) |
38 (38) |
40 (40) |
42 (42) |
43 (43) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -48.5 | +22.9 | +3.3 | -8.6 | -13.8 | -12.9 | -6.7 | +4.3 | +19.7 | +39.0 | -38.0 | -11.9 |
| Relative (%) | -48.5 | +22.9 | +3.3 | -8.7 | -13.8 | -12.9 | -6.7 | +4.3 | +19.7 | +39.0 | -38.1 | -11.9 | |
| Steps (reduced) |
44 (44) |
46 (46) |
47 (47) |
48 (48) |
49 (49) |
50 (50) |
51 (51) |
52 (52) |
53 (53) |
54 (54) |
54 (54) |
55 (55) | |