123ed48
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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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123 equal divisions of the 48th harmonic (abbreviated 123ed48) is a nonoctave tuning system that divides the interval of 48/1 into 123 equal parts of about 54.5 ¢ each. Each step represents a frequency ratio of 481/123, or the 123rd root of 48.
Theory
The 48th harmonic is too wide to be a useful equivalence, so 123ed48 is better thought of as a compressed version of 22edo. The local zeta peak around 22 is located at 22.025147, which has the octave compressed by 1.37 ¢; the octave of 123ed48 comes extremely close (differing by only 1/10 ¢), thus minimizing relative error as much as possible.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.3 | +5.1 | -2.6 | -7.5 | +3.8 | +9.4 | -3.8 | +10.2 | -8.7 | -10.3 | +2.6 |
| Relative (%) | -2.3 | +9.4 | -4.7 | -13.7 | +7.0 | +17.2 | -7.0 | +18.7 | -16.0 | -18.9 | +4.7 | |
| Steps (reduced) |
22 (22) |
35 (35) |
44 (44) |
51 (51) |
57 (57) |
62 (62) |
66 (66) |
70 (70) |
73 (73) |
76 (76) |
79 (79) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -27.0 | +8.1 | -2.3 | -5.1 | -1.1 | +8.9 | +24.3 | -10.0 | +14.5 | -11.5 | +20.5 | +1.3 |
| Relative (%) | -49.6 | +14.9 | -4.3 | -9.4 | -2.0 | +16.4 | +44.6 | -18.4 | +26.6 | -21.2 | +37.6 | +2.3 | |
| Steps (reduced) |
81 (81) |
84 (84) |
86 (86) |
88 (88) |
90 (90) |
92 (92) |
94 (94) |
95 (95) |
97 (97) |
98 (98) |
100 (100) |
101 (101) | |