Time-frequency Analysis of Musical Rhythm
Cheng Xiao-wen, Jarod V. Hart, James S. Walker · 2007
W e shall use the mathematical techniques of Gabor transforms and continuous wavelet transforms to analyze the rhythmic structure of music and its interaction with melodic structure. This analysis reveals the hierarchical structure of rhythm. Hierarchical structure is common to rhythmic performances throughout the world’s music. The work described here is interdisciplinary and experimental. We use mathematics to aid in the understanding of the structure of music, and have developed mathematical tools that (while not completely finished) have shown themselves to be useful for this musical analysis. We aim to explore ideas with this paper, to provoke thought, not to present completely finished work. The paper is organized as follows. We first summarize the mathematical method of Gabor transforms (also known as short-time Fourier transforms, or spectrograms). Spectrograms provide a tool for visualizing the patterns of time-frequency structures within a musical passage. We then review the method of percussion scalograms, a new technique for analyzing rhythm introduced in [34]. After that, we show how percussion scalograms are used to analyze percussion passages and rhythm. We carry out four analyses of percussion passages