Chord complexity: Difference between revisions

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= Basics =

== Summary ==

In this article we derive a fairly simple set of expressions which evaluate what we call the "simple" chord complexity (or "otonalness") of a chord. These generalize the familiar expressions for both the Benedetti/Tenney height and the Weil height of dyads. These expressions are as follows for chord <math>x_1:x_2:\ldots:x_N</math>:

Benedetti height:

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

Weil height:

<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>.

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.

== The Psychoacoustics of a Dyad ==

Consonance and dissonance are rather tricky and elusive phenomena to model, in part because the terms don't unambiguously refer to one thing. David Huron, for instance, lists at least 14 different types of dissonance [https://archive.md/ICWLQ here], some of which are psychoacoustic, some of which depend on some kind of larger musical or "tonal" setting, and some of which are clearly dependent on learned expectations. It is thus very likely that consonance is a multidimensional quantity that cannot be represented by a single scalar value.

@@ Line 2: / Line 2: @@
 = Basics =
+== Summary ==
+In this article we derive a fairly simple set of expressions which evaluate what we call the "simple" chord complexity (or "otonalness") of a chord. These generalize the familiar expressions for both the Benedetti/Tenney height and the Weil height of dyads. These expressions are as follows for chord <math>x_1:x_2:\ldots:x_N</math>:
+Benedetti height:
+<math>\displaystyle B_s(x_1, x_2, \ldots, x_N) = \frac{(x_1 \cdot x_2 \cdot \ldots \cdot x_N)^{1/N}}{N^{1/s}}</math>
+Weil height:
+<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>.
+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.
 == The Psychoacoustics of a Dyad ==
 Consonance and dissonance are rather tricky and elusive phenomena to model, in part because the terms don't unambiguously refer to one thing. David Huron, for instance, lists at least 14 different types of dissonance [https://archive.md/ICWLQ here], some of which are psychoacoustic, some of which depend on some kind of larger musical or "tonal" setting, and some of which are clearly dependent on learned expectations. It is thus very likely that consonance is a multidimensional quantity that cannot be represented by a single scalar value.