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User:Sintel/Expected Dirichlet coefficient for temperaments: Difference between revisions

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In the context of [[regular temperament theory]], a natural question is how well a given temperament approximates [[just intonation]] relative to its [[complexity]].
In the context of [[regular temperament theory]], a natural question is how well a given temperament approximates [[just intonation]] relative to its [[complexity]].
The '''Dirichlet coefficient''' gives a quantitative way to measure this.
The '''Dirichlet coefficient''' gives a quantitative way to measure this. This is the same as the "[[TE logflat badness|badness]]" used on the wiki, though the derivation here is given for the regular Euclidean norm for clarity.


Given a target vector <math>y \in \mathbb{R}^n</math>, such as the [[JIP | vector of log-primes]] in some ''p''-limit, and a rank-''k'' temperament ''X'', the Dirichlet coefficient is defined as:
Given a target vector <math>y \in \mathbb{R}^n</math>, such as the [[JIP | vector of log-primes]] in some ''p''-limit, and a rank-''k'' temperament ''X'', the Dirichlet coefficient is defined as:
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:<math>
:<math>
\begin{aligned}
\begin{aligned}
\mathrm{Vol}(d(X,y) < r)
\mathrm{Vol}(d(X,y) \le r)
&\approx \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \cdot \frac{r^{n-k}}{\frac{n-k}{2}} \\[10pt]
&\approx \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \cdot \frac{r^{n-k}}{\frac{n-k}{2}} \\[10pt]
&= \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}\, r^{n-k}
&= \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}\, r^{n-k}
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<references>
<references>
<ref name="Schmidt1967">Wolfgang M. Schmidt. ''On Heights of Algebraic Subspaces and Diophantine Approximations''. Annals of Mathematics, Vol. 85, No. 3 (1967), pp. 430-472, theorem 15 [https://doi.org/10.2307/1970352 doi:10.2307/1970352]</ref>
<ref name="Schmidt1967">Wolfgang M. Schmidt. ''On Heights of Algebraic Subspaces and Diophantine Approximations''. Annals of Mathematics, Vol. 85, No. 3 (1967), pp. 430-472, theorem 15 [https://doi.org/10.2307/1970352 doi:10.2307/1970352]</ref>
<ref name="Schmidt1968">Wolfgang M. Schmidt. ''Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height''. Duke Mathematical Journal Vol. 35 No. 2, pp. 327-339 (1968), theorem 1 [https://doi.org/10.2307/1970352 doi:10.2307/1970352]</ref>
<ref name="Schmidt1968">Wolfgang M. Schmidt. ''Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height''. Duke Mathematical Journal Vol. 35 No. 2, pp. 327-339 (1968), theorem 1 [https://doi.org/10.1215/S0012-7094-68-03532-1 doi:10.1215/S0012-7094-68-03532-1]</ref>
<ref name="Schmidt1998">Wolfgang M. Schmidt. ''The distribution of sub-lattices of Z<sup>m</sup>''. Monatshefte für Mathematik Vol. 125 No. 1, pp 37–81 (1998) [https://doi.org/10.1007/BF01489457 doi:10.1007/BF01489457]</ref>
<ref name="Schmidt1998">Wolfgang M. Schmidt. ''The distribution of sub-lattices of Z<sup>m</sup>''. Monatshefte für Mathematik Vol. 125 No. 1, pp 37–81 (1998) [https://doi.org/10.1007/BF01489457 doi:10.1007/BF01489457]</ref>
</references>
</references>
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