143ed11: Difference between revisions
Contribution (talk | contribs) Created page with "143 equal divisions of the 11th harmonic (abbreviated 143ed11), is the tuning system that divides the 11th harmonic into 143 equal parts of about 29.0302 ¢ each. Each step represents a frequency ratio of <math>11^{\frac{1}{143}}</math>, or the 143rd root of 11. == Theory == === Why 143ed11 matter === 143ed11 is extremely strong in the 5.7.8.9.11.13.17.23 subgroup. All ratios in the 5.7.8.9.11.13.17.23 subgroup and 25-integer-limit are approximated in 143ed11 with..." |
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All ratios in the 5.7.8.9.11.13.17.23 subgroup and 25-integer-limit are approximated in 143ed11 with less than 2.5 ¢ error. | All ratios in the 5.7.8.9.11.13.17.23 subgroup and 25-integer-limit are approximated in 143ed11 with less than 2.5 ¢ error. | ||
According to the finite | According to the finite Euler product with sigma = 1, the maxima in the 5.7.8.9.11.13.17.23 subgroup is about 41.33601-ED2 (or 29.03038 ¢). | ||
The Tenney–Euclidean regular temperement in the 5.7.8.9.11.13.17.23 subgroup mapped with [⟨96 116 124 131 143 153 169 187]] gives 41.33650-ED2 (or 29.03005 ¢). The finite | The Tenney–Euclidean regular temperement in the 5.7.8.9.11.13.17.23 subgroup mapped with [⟨96 116 124 131 143 153 169 187]] gives 41.33650-ED2 (or 29.03005 ¢). The finite Euler product with sigma = 1/2 gives about the same result. | ||
143ed11, with its size of about 41.33627-ED2 (or 29.03020 ¢), gives a simple, very near compromise between the different methods approaching 5.7.8.9.11.13.17.23 in this area. | 143ed11, with its size of about 41.33627-ED2 (or 29.03020 ¢), gives a simple, very near compromise between the different methods approaching 5.7.8.9.11.13.17.23 in this area. | ||
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| -2.5 | | -2.5 | ||
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