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: | ||