361ed448
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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. |
| ← 360ed448 | 361ed448 | 362ed448 → |
361 equal divisions of the 448th harmonic (abbreviated 361ed448) is a nonoctave tuning system that divides the interval of 448/1 into 361 equal parts of about 29.3 ¢ each. Each step represents a frequency ratio of 4481/361, or the 361st root of 448.
Theory
The 448th harmonic is far too wide to be a useful equivalence, so 361ed448 is better thought of as a stretched version of 41edo. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being stretched by about 0.338 ¢. The local zeta peak around 41 is located at 40.988078, which has a step size of 29.277 ¢ and an octave of 1200.349 ¢ (which is stretched by 0.837 ¢), making 361ed448 extremely close to optimal for 41edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.3 | +1.0 | +0.7 | -5.0 | +1.4 | -2.0 | +1.0 | +2.0 | -4.7 | +5.9 | +1.7 |
| Relative (%) | +1.2 | +3.5 | +2.3 | -17.2 | +4.6 | -6.9 | +3.5 | +7.0 | -16.1 | +20.3 | +5.8 | |
| Steps (reduced) |
41 (41) |
65 (65) |
82 (82) |
95 (95) |
106 (106) |
115 (115) |
123 (123) |
130 (130) |
136 (136) |
142 (142) |
147 (147) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +9.5 | -1.7 | -4.0 | +1.4 | +13.5 | +2.4 | -3.4 | -4.4 | -1.0 | +6.3 | -12.1 | +2.0 |
| Relative (%) | +32.5 | -5.8 | -13.7 | +4.6 | +46.1 | +8.1 | -11.6 | -14.9 | -3.4 | +21.5 | -41.4 | +6.9 | |
| Steps (reduced) |
152 (152) |
156 (156) |
160 (160) |
164 (164) |
168 (168) |
171 (171) |
174 (174) |
177 (177) |
180 (180) |
183 (183) |
185 (185) |
188 (188) | |