198ed304
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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. |
| ← 197ed304 | 198ed304 | 199ed304 → |
198 equal divisions of the 304th harmonic (abbreviated 198ed304) is a nonoctave tuning system that divides the interval of 304/1 into 198 equal parts of about 50 ¢ each. Each step represents a frequency ratio of 3041/198, or the 198th root of 304.
Theory
The 304th harmonic is far too wide to be a useful equivalence, so 198ed304 is better thought of as a compressed version of 24edo. Indeed, tuning the 304/1 ratio just instead of 2/1 results in octaves being stretched by about 0.301 ¢. The local zeta peak around 24 is located at 24.005742, which has a step size of 49.98804 ¢ and an octave of 1199.713 ¢ (which is compressed by 0.287 ¢), making 198ed304 extremely close to optimal for 24edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.3 | -8.8 | -8.7 | +3.6 | -13.1 | -14.3 | -13.0 | +15.6 | -0.7 | +0.3 | +15.8 |
| Relative (%) | -13.0 | -26.5 | -26.1 | +10.8 | -39.5 | -43.1 | -39.1 | +47.0 | -2.2 | +1.0 | +47.4 | |
| Steps (reduced) |
36 (36) |
57 (57) |
72 (72) |
84 (84) |
93 (93) |
101 (101) |
108 (108) |
115 (115) |
120 (120) |
125 (125) |
130 (130) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.4 | -20.0 | +10.6 | -1.2 | -6.2 | -5.2 | +1.2 | +12.4 | -22.1 | -2.7 | +20.4 | -3.3 |
| Relative (%) | +16.7 | -39.9 | +21.1 | -2.4 | -12.4 | -10.3 | +2.4 | +24.8 | -44.2 | -5.3 | +40.7 | -6.7 | |
| Steps (reduced) |
89 (89) |
91 (91) |
94 (94) |
96 (96) |
98 (98) |
100 (100) |
102 (102) |
104 (104) |
105 (105) |
107 (107) |
109 (109) |
110 (110) | |