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 and highly contrived. |
| ← 122ed48 | 123ed48 | 124ed48 → |
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) | |