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Rhythmic Complexity in Jazz: An Information Theory Approach

Abstract
Many techniques of quantifying rhythmic complexity have been explored, including methods based on the concept of entropy. Roughly speaking, entropy measures a rhythm’s unpredictability. The primary goals of this study were to answer two questions: 1) Does rhythmic entropy correspond to perceived rhythmic complexity? and 2) Does entropy of a jazz solo depend on soloist? Additionally, I used entropy to study the relationship between sheet music and jazz versions of songs from the American songbook, and I used the concept of mutual information to study soloist-accompanist interactions in the music of Charlie Parker. I asked fifteen UMass music majors to rate short, eighth-note based jazz rhythms for complexity. Entropies were calculated by constructing distributions based on the inter-onset intervals (IOI’s) between notes. Using a mixed effects multiple regression model, I found, as expected, that higher entropy resulted in higher complexity ratings. Other factors did, too, namely: number of notes, syncopation, lack of periodicity, and the effects of each complexity rating on the following one. It is possible that entropy was mediated by lack of periodicity. I then transcribed (or compiled and checked) a corpus of 88 solos by Armstrong, Hawkins, Young, Christian, and Parker, and calculated entropies based on the IOI’s between stress-accented notes. I used the technique of estimated marginal means with number of distinct IOI’s and number of accents as covariates to show that entropy depends significantly on soloist: solos by Lester Young were significantly more entropic than those by Armstrong, Christian, and Parker. Stress accent density and contour accent density were used to explain the unexpected lack of differentiation between Parker and Hawkins in terms of entropy. I demonstrated that jazz renditions of popular songs had higher entropy than their sheet music counterparts. Finally, I used mutual information to show that interrelationships between Parker and his accompanists were stronger than those between Parker and a Charleston comping rhythm. This work demonstrates the utility of entropy-based methods in predicting a listener’s perceived complexity, in characterizing a soloist’s oeuvre, and in describing embellished versions of songs. It also demonstrates the utility of mutual information in describing soloist/accompanist interactions.
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