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, highly contrived, or as yet unknown. |
| ← 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 0.349 ¢), 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) | |