Chord complexity: Difference between revisions

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is the "Benedetti" version. The numerator is the geometric mean, and the denominator normalizes by the size of the chord. The logarithmic "Tenney" version is as follows:

<math>\displaystyle T_s(x_1, x_2, \ldots, x_N) = \frac{1}{N} \log(x_1 \cdot x_2 \cdot \ldots \cdot x_N) - \frac{1}{s}\~~logN~~</math>

<math>\displaystyle T_s(x_1, x_2, \ldots, x_N) = \frac{1}{N} \log(x_1 \cdot x_2 \cdot \ldots \cdot x_N) - \frac{1}{s}\log(N)</math>

In both cases, the free parameter <math>s</math>, which is derived from the original expression, now only determines the way that differently-sized chords scale relative to one another. The results for some value of <math>s</math>, when comparing chords of different sizes, will closely resemble the relative scaling of chord sizes in the Dirichlet complexity of equal value <math>s</math>, but without the caveats regarding span.

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 is the "Benedetti" version. The numerator is the geometric mean, and the denominator normalizes by the size of the chord. The logarithmic "Tenney" version is as follows:
-<math>\displaystyle T_s(x_1, x_2, \ldots, x_N) = \frac{1}{N} \log(x_1 \cdot x_2 \cdot \ldots \cdot x_N) - \frac{1}{s}\logN</math>
+<math>\displaystyle T_s(x_1, x_2, \ldots, x_N) = \frac{1}{N} \log(x_1 \cdot x_2 \cdot \ldots \cdot x_N) - \frac{1}{s}\log(N)</math>
 In both cases, the free parameter <math>s</math>, which is derived from the original expression, now only determines the way that differently-sized chords scale relative to one another. The results for some value of <math>s</math>, when comparing chords of different sizes, will closely resemble the relative scaling of chord sizes in the Dirichlet complexity of equal value <math>s</math>, but without the caveats regarding span.