Volume 10, Number 2, June 2004
Copyright © 2004 Society for Music Theory
Bret Aarden and Paul T. von Hippel
Rules for Chord Doubling (and Spacing): Which Ones Do We Need?
 We developed a statistical model for discriminating composed from random triads. The model used features of the triads' spacing and doubling. (See §5.1.) For each feature, the model estimated an optimal weight. Larger weights were assigned to features that were more useful for discrimination. Positive weights were assigned to features that were more common in composed triads. Negative weights were assigned to features that were more common in random triads.
 The features may be represented as X1,X2,.... The weights may be represented as β1,β2,.... A triad's weighted features are added together to represent the probability that the triad is composed rather than random:
Prob (composed) = Λ ( β1X1 + β2X2 + ... )
 (Here Λ is the cumulative logistic distribution function.) This way of
modeling probabilities is known as logistic regression analysis.(80) When
logistic regression is used to discriminate between two types of objects, as
here, it is called a logistic discrimination model.
 In its pure form, logistic discrimination is used for observations that are independent of one another. Our composed and random triads, however, are not independent but paired. To accommodate this pairing, we used a conditional logistic model that is appropriate for matched-pair data.(81)
Brent Yorgason, Managing Editor
Updated 03 June 2004