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Music: Broken Symmetry, Geometry, and Complexity

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Music: Broken Symmetry, Geometry, and Complexity

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Music:BrokenSymmetry, Geometry,andComplexity Gary W. Don, Karyn K. Muir, Gordon B. Volk, James S. Walker
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T cfhamluenuunddsrdiiepacnliatgmth:cieahoPn,snytataanhlbsldaeogatnmonwgrdaeeteoaahnvnneedmrmhtaraaoitrticnhcmheaeosl,mhngifiasrrctteoiotqcurhsupyee,taonhirncneydy-,-ory in musical scores [7, 47, 56, 15]. This article is part of a special issue on the theme of math-ematics, creativity, and the arts. We shall explore some of the ways that mathematics can aid in creativity and understanding artistic expression in the realm of the musical arts. In particular, we hope to provide some intriguing new insights on such questions as: Does Louis Armstrong’s voice sound like his trumpet? What do Ludwig van Beethoven, Ben-ny Goodman, and Jimi Hendrix have in common? How does the brain fool us sometimes when listening to music? And how have composers used such illusions? How can mathematics help us create new music? Gary W. Don is professor of music at the University of Wisconsin-Eau Claire. His email address is dongw@ uwec.edu . Karyn K. Muir is a mathematics major at the State Uni-versity of New York at Geneseo. Her email address is kkm5@geneseo.edu . Gordon B. Volk is a mathematics major at the Universi-ty of Wisconsin-Eau Claire. His email address is volkgb@ uwec.edu . James S. Walker is professor of mathematics at the Uni-versity of Wisconsin-Eau Claire. His email address is walkerjs@uwec.edu .
Melody contains both pitch and rhythm. Is it possible to objectively describe their con-nection? Is it possible to objectively describe the com-plexity of musical rhythm? In discussing these and other questions, we shall outline the mathematical methods we use and provide some illustrative examples from a wide variety of music. The paper is organized as follows. We first sum-marize the mathematical method of Gabor trans-forms (also known as short-time Fourier trans-forms, or spectrograms). This summary empha-sizes the use of a discrete Gabor frame to perform the analysis. The section that follows illustrates the value of spectrograms in providing objec-tive descriptions of musical performance and the geometric time-frequency structure of recorded musical sound. Our examples cover a wide range of musical genres and interpretation styles, in-cluding: Pavarotti singing an aria by Puccini [17], the 1982 Atlanta Symphony Orchestra recording of Copland’s Appalachian Spring symphony [5], the 1950 Louis Armstrong recording of “La Vie en Rose” [64], the 1970 rock music introduction to “Layla” by Duane Allman and Eric Clapton [63], the 1968 Beatles’ song “Blackbird” [11], and the Re-naissance motet, “Non vos relinquam orphanos”, by William Byrd [8]. We then discuss signal syn-thesis using dual Gabor frames, and illustrate how this synthesis can be used for processing recorded sound and creating new music. Then we turn to the method of continuous wavelet trans-forms and show how they can be used together with spectrograms for two applications: (1) zoom-ing in on spectrograms to provide more detailed views and (2) producing objective time-frequency
Notices of the AMS Volume 57 , Number 1
portraits of melody and rhythm. The musical il-lustrations for these two applications are from a 1983 Aldo Ciccolini performance of Erik Satie’s “Gymnopédie I” [81] and a 1961 Dave Brubeck jazz recording “Unsquare Dance” [94]. We conclude the paper with a quantitative, objective description of the complexity of rhythmic styles, combining ideas from music and information theory. Discrete Gabor Transforms: Signal Analysis We briefly review the widely employed method of Gabor transforms [53], also known as short-time Fourier transforms, or spectrograms, or sono-grams. The first comprehensive effort in employing spectrograms in musical analysis was Robert Co-gan’s masterpiece, New Images of Musical Sound [27] — a book that still deserves close study. A more recent contribution is [62]. In [37, 38], Dörfler de-scribes the fundamental mathematical aspects of using Gabor transforms for musical analysis. Other sources for theory and applications of short-time Fourier transforms include [3, 76, 19, 83, 65]. There is also considerable mathematical background in [50, 51, 55], with musical applications in [40]. Us-ing sonograms or spectrograms for analyzing the music of birdsong is described in [61, 80, 67]. The theory of Gabor transforms is discussed in com-plete detail in [50, 51, 55] from the standpoint of function space theory. Our focus here, however, will be on its discrete aspects, as we are going to be processing digital recordings. The sound signals that we analyze are all dig-ital, hence discrete, so we assume that a sound signal has the form { f t k } , for uniformly spaced values t k k Ñ t in a finite interval 0  T  . A Gabor transform of f , with window function w , is defined as follows. First, multiply { f t k } by a sequence of M shifted window functions { w t k τ } 0 , produc-ing time-localized subsignals, { f t k w t k τ } M 0 . Uniformly spaced time values, { τ t j ℓ } M 0 , are used for the shifts ( j being a positive integer greater than 1). The windows { w t k τ } M 0 are all compactly supported and overlap each other; see Figure 1. The value of M is determined by the minimum number of windows needed to cover 0  T  , as illustrated in Figure 1(b). Second, because w is compactly supported, we treat each subsignal { f t k w t k τ } as a finite sequence and apply an FFT F to it. This yields the Gabor transform of { f t k } : (1) {F { f t k w t k τ }} M 0 We shall describe (1) more explicitly in a moment (see Remark 1 below). For now, note that because the values t k belong to the finite interval 0  T  , we always extend our signal values beyond the interval’s endpoints by appending zeros; hence the full supports of all windows are included.
(a) (b) (c) Figure 1. (a) Signal. (b) Succession of window functions. (c) Signal multiplied by middle window in (b); an FFT can now be applied to this windowed signal. The Gabor transform that we employ uses a Blackman window defined by 0 42 0 5 cos 2 π t λ w t  0 08 cos 4 π t λ for | t | ≤ λ 2 0 for | t | > λ 2 for a positive parameter λ equaling the width of the window where the FFT is performed. In Figure 1(b) we show a succession of these Black-man windows. Further background on why we use Blackman windows can be found in [20]. The efficacy of these Gabor transforms is shown by how well they produce time-frequency portraits that accord well with our auditory perception, which is described in the vast literature on Ga-bor transforms that we briefly summarized above. In this paper we shall provide many additional examples illustrating their efficacy. Remark 1. To see how spectrograms display the frequency content of recorded sound, it helps to write the FFT F in (1) in a more explicit form. The FFT that we use is given, for even N , by the following mapping of a sequence of real numbers { a m } m N  2 1 : m − N  2 } F 1 N  2 1 (2) { a m -n A ν X a m e i 2 π mν N o N m − N  2 where ν is any integer. In applying F in (1), we make use of the fact that each Blackman window w t k τ is centered on τ jℓ Ñ t and is 0 for t k outside of its support, which runs from t k jℓ N 2 Ñ t to t k jℓ N 2 Ñ t and is 0 at t k jℓ ± N 2 Ñ t . So, for a given windowing spec-ified by , the FFT F in (2) is applied to the vector a m Nm2 −− N 1 2 defined by f t k w k jℓ Ñ t  kjj N 2 N 12 . In (2), the variable ν corresponds to frequencies for the discrete complex exponentials e i 2 π mν N used in defining the FFT F . For real-valued data, such as recorded sound, the FFT values A ν satisfy the symmetry condition A ν A ν , where A ν is the complex conjugate of A ν . Hence, no significant information is gained with negative frequencies. Moreover, when the Gabor transform is displayed, the values of the Gabor transform are plotted as
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