Chord complexity: Difference between revisions

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<math>\displaystyle W_s(x_1, x_2, \ldots, x_N) = \frac{\max(x_1, x_2, \ldots, x_N)}{N^{1/s}}</math>

The use of either the geometric mean or maximum has a pretty long "folklore" history of being used to evaluate the complexity of a chord; such expressions routinely show up in the computation of [[Harmonic Entropy]], for instance. These expressions are the same, but simply multiply the result by an extra normalizing term of <math>1/N^{1/s}</math>. This normalizing term doesn't affect the rankings for chords of the same size, but does affect how chords of different sizes scale in complexity with regard to one another. There is one free parameter <math>s</math> which can be used to adjust this scaling between chords of different sizes; we suggest setting <math>s=1</math> as a good default value. We also note that we get the usual raw geometric mean and maximum as <math>s \to \infty</math>. Strictly speaking, the original formulation of Benedetti/Tenney height for dyads is equal to the geometric mean squared; the additional step of taking the <math>N</math>th root of the product generalizes the expression to measure any number of tones in a similar way.

The use of either the geometric mean<ref>We also note, strictly speaking, that the original formulation of Benedetti height for dyads is equal to the product rather than the geometric mean, although the geometric mean ranks chords of the same size identically to the product. However, the geometric mean version, with the additional step of taking the <math>N</math>th root of the product, has shown up in many "natural" settings such as Harmonic Entropy, and can be thought of as generalizing the expression to measure any number of tones in a similar way, and the additional adjustment of dividing by <math>N^{1/s}</math> calibrates chords of different sizes relative to one another.</ref> or maximum has a pretty long "folklore" history of being used to evaluate the complexity of a chord; such expressions routinely show up in the computation of [[Harmonic Entropy]], for instance. These expressions are the same, but simply multiply the result by an extra normalizing term of <math>1/N^{1/s}</math>. This normalizing term doesn't affect the rankings for chords of the same size, but does affect how chords of different sizes scale in complexity with regard to one another. There is one free parameter <math>s</math> which can be used to adjust this scaling between chords of different sizes; we suggest setting <math>s=1</math> as a good default value. We also note that we get the usual raw geometric mean and maximum as <math>s \to \infty</math>.

In this article we derive these expressions rigorously, as a slight adjustment or "span-correction" of a slightly different metric which satisfies certain axioms regarding simple chord complexity.

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 <math>\displaystyle W_s(x_1, x_2, \ldots, x_N) = \frac{\max(x_1, x_2, \ldots, x_N)}{N^{1/s}}</math>
-The use of either the geometric mean or maximum has a pretty long "folklore" history of being used to evaluate the complexity of a chord; such expressions routinely show up in the computation of [[Harmonic Entropy]], for instance. These expressions are the same, but simply multiply the result by an extra normalizing term of <math>1/N^{1/s}</math>. This normalizing term doesn't affect the rankings for chords of the same size, but does affect how chords of different sizes scale in complexity with regard to one another. There is one free parameter <math>s</math> which can be used to adjust this scaling between chords of different sizes; we suggest setting <math>s=1</math> as a good default value. We also note that we get the usual raw geometric mean and maximum as <math>s \to \infty</math>. Strictly speaking, the original formulation of Benedetti/Tenney height for dyads is equal to the geometric mean squared; the additional step of taking the <math>N</math>th root of the product generalizes the expression to measure any number of tones in a similar way.
+The use of either the geometric mean<ref>We also note, strictly speaking, that the original formulation of Benedetti height for dyads is equal to the product rather than the geometric mean, although the geometric mean ranks chords of the same size identically to the product. However, the geometric mean version, with the additional step of taking the <math>N</math>th root of the product, has shown up in many "natural" settings such as Harmonic Entropy, and can be thought of as generalizing the expression to measure any number of tones in a similar way, and the additional adjustment of dividing by <math>N^{1/s}</math> calibrates chords of different sizes relative to one another.</ref> or maximum has a pretty long "folklore" history of being used to evaluate the complexity of a chord; such expressions routinely show up in the computation of [[Harmonic Entropy]], for instance. These expressions are the same, but simply multiply the result by an extra normalizing term of <math>1/N^{1/s}</math>. This normalizing term doesn't affect the rankings for chords of the same size, but does affect how chords of different sizes scale in complexity with regard to one another. There is one free parameter <math>s</math> which can be used to adjust this scaling between chords of different sizes; we suggest setting <math>s=1</math> as a good default value. We also note that we get the usual raw geometric mean and maximum as <math>s \to \infty</math>.
 In this article we derive these expressions rigorously, as a slight adjustment or "span-correction" of a slightly different metric which satisfies certain axioms regarding simple chord complexity.