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Music: Broken Symmetry, Geometry, and Complexity Gary W Music: Broken Symmetry, Geometry, and Complexity Gary W. Don, Karyn K. Muir, Gordon B. Volk, James S. Walker he relation between mathematics and Melody contains both pitch and rhythm. Is • music has a long and rich history, in- it possible to objectively describe their con- cluding: Pythagorean harmonic theory, nection? fundamentals and overtones, frequency Is it possible to objectively describe the com- • Tand pitch, and mathematical group the- plexity of musical rhythm? ory in musical scores [7, 47, 56, 15]. This article In discussing these and other questions, we shall is part of a special issue on the theme of math- outline the mathematical methods we use and ematics, creativity, and the arts. We shall explore provide some illustrative examples from a wide some of the ways that mathematics can aid in variety of music. creativity and understanding artistic expression The paper is organized as follows. We first sum- in the realm of the musical arts. In particular, we marize the mathematical method of Gabor trans- hope to provide some intriguing new insights on forms (also known as short-time Fourier trans- such questions as: forms, or spectrograms). This summary empha- sizes the use of a discrete Gabor frame to perform Does Louis Armstrong’s voice sound like his • the analysis. The section that follows illustrates trumpet? the value of spectrograms in providing objec- What do Ludwig van Beethoven, Ben- • tive descriptions of musical performance and the ny Goodman, and Jimi Hendrix have in geometric time-frequency structure of recorded common? musical sound. Our examples cover a wide range How does the brain fool us sometimes • of musical genres and interpretation styles, in- when listening to music? And how have cluding: Pavarotti singing an aria by Puccini [17], composers used such illusions? the 1982 Atlanta Symphony Orchestra recording How can mathematics help us create new of Copland’s Appalachian Spring symphony [5], • music? the 1950 Louis Armstrong recording of “La Vie en Rose” [64], the 1970 rock music introduction to Gary W. Don is professor of music at the University “Layla” by Duane Allman and Eric Clapton [63], the of Wisconsin-Eau Claire. His email address is dongw@ 1968 Beatles’ song “Blackbird” [11], and the Re- uwec.edu. naissance motet, “Non vos relinquam orphanos”, Karyn K. Muir is a mathematics major at the State Uni- by William Byrd [8]. We then discuss signal syn- versity of New York at Geneseo. Her email address is thesis using dual Gabor frames, and illustrate [email protected]. how this synthesis can be used for processing Gordon B. Volk is a mathematics major at the Universi- recorded sound and creating new music. Then we ty of Wisconsin-Eau Claire. His email address is volkgb@ turn to the method of continuous wavelet trans- uwec.edu. forms and show how they can be used together James S. Walker is professor of mathematics at the Uni- with spectrograms for two applications: (1) zoom- versity of Wisconsin-Eau Claire. His email address is ing in on spectrograms to provide more detailed [email protected]. views and (2) producing objective time-frequency 30 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 (a) (b) (c) of the complexity of rhythmic styles, combining Figure 1. (a) Signal. (b) Succession of window ideas from music and information theory. functions. (c) Signal multiplied by middle window in (b); an FFT can now be applied to Discrete Gabor Transforms: Signal this windowed signal. Analysis We briefly review the widely employed method of Gabor transforms [53], also known as short-time The Gabor transform that we employ uses a Fourier transforms, or spectrograms, or sono- Blackman window defined by grams. The first comprehensive effort in employing 0.42 0.5 cos(2πt/λ) spectrograms in musical analysis was Robert Co- + + gan’s masterpiece, New Images of Musical Sound w(t) 0.08 cos(4πt/λ) for t λ/2 = | |≤ [27] — a book that still deserves close study. A more 0 for t >λ/2 | | recent contribution is [62]. In [37, 38], Dörfler de- for a positive parameter λ equaling the width scribes the fundamental mathematical aspects of of the window where the FFT is performed. In using Gabor transforms for musical analysis. Other Figure 1(b) we show a succession of these Black- sources for theory and applications of short-time man windows. Further background on why we use Fourier transforms include [3, 76, 19, 83, 65]. There Blackman windows can be found in [20]. is also considerable mathematical background in The efficacy of these Gabor transforms is shown [50, 51, 55], with musical applications in [40]. Us- by how well they produce time-frequency portraits ing sonograms or spectrograms for analyzing the that accord well with our auditory perception, music of birdsong is described in [61, 80, 67]. The which is described in the vast literature on Ga- theory of Gabor transforms is discussed in com- bor transforms that we briefly summarized above. plete detail in [50, 51, 55] from the standpoint of In this paper we shall provide many additional function space theory. Our focus here, however, examples illustrating their efficacy. will be on its discrete aspects, as we are going to Remark 1. To see how spectrograms display the be processing digital recordings. frequency content of recorded sound, it helps to The sound signals that we analyze are all dig- write the FFT in (1) in a more explicit form. ital, hence discrete, so we assume that a sound The FFT that weF use is given, for even N, by the signal has the form f (tk) , for uniformly spaced { } following mapping of a sequence of real numbers values tk k t in a finite interval [0,T]. A Gabor m N/2 1 = am m= N/−2 : transform of f , with window function w, is defined { } =− N/2 1 as follows. First,∆ multiply f (tk) by a sequence of 1 − { } M F i2πmν/N shifted window functions w(t τ ) , produc- (2) am !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Aν am e− , k " " 0 { } → = √N { − } = M & m 'N/2 ( ing time-localized subsignals, f (tk)w(tk τ") " 0. =− { − M } = where ν is any integer. In applying in (1), we Uniformly spaced time values, τ" tj" " 0, are F used for the shifts (j being a{ positive= } integer= make use of the fact that each Blackman window M w(tk τ") is centered on τ" j" t and is 0 for greater than 1). The windows w(tk τ") " 0 are − = { − } = tk outside of its support, which runs from tk all compactly supported and overlap each other; = (j" N/2) t to tk (j" N/2)∆ t and is 0 at see Figure 1. The value of M is determined by − = + tk (j" N/2) t. So, for a given windowing spec- the minimum number of windows needed to cover = ± ified by ", the∆ FFT in (2) is applied∆ to the vector [0,T], as illustrated in Figure 1(b). Second, because N/2 1 F w is compactly supported, we treat each subsignal (am)m −N/2 defined∆ by =− j" N/2 1 f (t )w(t τ ) as a finite sequence and apply + − k k " f (tk)w([k j"] t) . { − } k j" N/2 an FFT to it. This yields the Gabor transform of − = − F ) * f (tk) : In (2), the variable∆ ν corresponds to frequencies { } i2πmν/N M for the discrete complex exponentials e− (1) f (tk)w(tk τ") " 0. {F{ − }} = used in defining the FFT . For real-valued data, F We shall describe (1) more explicitly in a moment such as recorded sound, the FFT values Aν satisfy (see Remark 1 below). For now, note that because the symmetry condition A ν Aν∗, where Aν∗ is − = the values tk belong to the finite interval [0,T], the complex conjugate of Aν . Hence, no significant we always extend our signal values beyond the information is gained with negative frequencies. interval’s endpoints by appending zeros; hence Moreover, when the Gabor transform is displayed, the full supports of all windows are included. the values of the Gabor transform are plotted as January 2010 Notices of the AMS 31 squared magnitudes (we refer to such plots as require that the windows satisfy spectrograms). There is perfect symmetry at ν M 2 and ν, so the negative frequency values are not (6) A w (tk τ") B − ≤ − ≤ displayed in the spectrograms. " 0 '= for two positive constants A and B (the frame Gabor Frames constants). The constants A and B ensure numeri- When we discuss audio synthesis, it will be im- cal stability, including preventing overflow during portant to make use of the expression of Gabor analysis and synthesis. The inequalities in (6) transforms in terms of Gabor frames. An important obviously hold for our Blackman windows when seminal paper in this field is [92]. A comprehensive they are overlapping as shown in Figure 1(b). Us- introduction can be found in [48]. We introduce ing (4) through (6), along with the Cauchy-Schwarz Gabor frames here because they follow naturally inequality, we obtain (for K : k1 k0 1): from combining (1) and (2). If you prefer to see = − + M,N/2 1 − musical examples, then please skip ahead to the 2 2 (7) C",ν (KB) f , next section and return here when we discuss | | ≤ ' ' " 0,ν N/2 signal synthesis.
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