Chord complexity: Difference between revisions
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For dyads, note that we have the following property for Weil height: | For dyads, note that we have the following property for Weil height: | ||
<math>\log \max(n,d) = \frac{1}{2}\log(n\cdot d) + \frac{1}{2} \log | <math>\log \max(n,d) = \frac{1}{2}\log(n\cdot d) + \frac{1}{2} |\log (n/d)|</math> | ||
The first term on the right hand side is the Tenney height, and the second term is the span. As a result, we can see that the Weil height is equal to the Tenney height plus the span, so that it can already be viewed as an alteration of the Tenney height with even greater emphasis placed on small intervals. | The first term on the right hand side is the Tenney height, and the second term is the span. As a result, we can see that the Weil height is equal to the Tenney height plus the span, so that it can already be viewed as an alteration of the Tenney height with even greater emphasis placed on small intervals. | ||
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Note this new free parameter <math>k</math> is still independent of the free parameter <math>s</math>, which chooses how chords of different sizes scale relative to one another (for which <math>s=1</math> is still a decent default value). | Note this new free parameter <math>k</math> is still independent of the free parameter <math>s</math>, which chooses how chords of different sizes scale relative to one another (for which <math>s=1</math> is still a decent default value). | ||
=== Derivation for Dyads === | === Derivation for Dyads === | ||