
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? 

[1] 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.
[2] The features may be represented as X_{1},X_{2},.... The weights may be represented as β_{1},β_{2},.... A triad's weighted features are added together to represent the probability that the triad is composed rather than random:
Prob (composed) = Λ ( β_{1}X_{1} + β_{2}X_{2} + ... )
[3] (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.
[4] 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 matchedpair data.^{(81)}
Prepared by
Brent Yorgason, Managing Editor
Updated
03 June 2004