Paper Info
Title
Asymmetric Rhythms and Tiling Canons
Abstract
1. INTRODUCTION. If you walk into your neighborhood record store, you will be confronted with an array of different musical genres. What makes a musical style distinctive? Certainly, instrumentation is important: one does not expect to hear a trum- pet playing bluegrass or a banjo in a mariachi band. However, drums and guitars are almost ubiquitous in popular music around the world; instrumentation is clearly not the whole story. Our own personal likes and dislikes are strongly influenced by the rhythms, melodic structures, harmonies, and lyrics in the songs we hear. This article focuses on what may be the most important of these aspects: rhythm. We examine the mathematics of some rhythmic structures common in popular and folk music. Anyone who listens to rock music is familiar with the repeated drum beat—one, two, three, four—based on a 4/4 measure. Fifteen minutes listening to a Top 40 ra- dio station offers evidence enough that most rock music has this basic beat (Audio Example 1).1 But if we tune the radio to different frequencies, we may hear popular music (jazz, Latin, African) with different characteristic rhythms (Audio Example 2). Although much of this music is also based on the 4/4 measure, some instruments play repeated patterns that are not synchronized with the 4/4 beat, creating syncopation— an exciting tension between different components of the rhythm. This article is con- cerned with classifying and counting rhythms that are maximally syncopated in the sense that, even when shifted, they cannot be synchronized with the division of a mea- sure into two parts. In addition, we discuss rhythms that cannot be aligned with other even divisions of the measure. Our results have a surprising application to rhythmic canons.
Year
DOI
Venue
2006
10.2307/27642087
AMERICAN MATHEMATICAL MONTHLY
Field
DocType
Volume
Melody,Jazz,Algebra,Syncopation,Popular music,Beat (music),Lyrics,Linguistics,Rhythm,Mathematics,Harmony (Music)
Journal
113
Issue
ISSN
Citations 
10
0002-9890
0
PageRank 
References 
Authors
0.34
3
2
Name
Order
Citations
PageRank
Rachel W. Hall121.29
Paul Klingsberg2212.67