Generation of 1/F Noise from a Broken-Symmetry Model for the Arbitrary Absolute Pitch of Musical Melodies Martin Grant, and Niloufar Faghihi

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Generation of 1/f noise from a broken-symmetry model for the arbitrary absolute pitch of musical melodies Martin Grant, and Niloufar Faghihi

Citation: The Journal of the Acoustical Society of America 142, EL490 (2017); View online: https://doi.org/10.1121/1.5011150 View Table of Contents: http://asa.scitation.org/toc/jas/142/5 Published by the Acoustical Society of America Martin Grant and Niloufar Faghihi: JASA Express Letters https://doi.org/10.1121/1.5011150 Published Online 17 November 2017 Generation of 1/f noise from a broken-symmetry model for the arbitrary absolute pitch of musical melodies

Martin Grant and Niloufar Faghihi Physics Department, McGill University, Rutherford building, 3600 rue University, Montreal, Quebec H3A 2T8, Canada [email protected], [email protected]

Abstract: A model is presented to generate power spectrum noise with intensity proportional to 1=f as a function of frequency f . The model arises from a broken-symmetry variable, which corresponds to absolute pitch, where ﬂuctuations occur in an attempt to restore that symmetry, inﬂuenced by interactions in the creation of musical melodies. VC 2017 Acoustical Society of America [DMC] Date Received: August 30, 2017 Date Accepted: October 30, 2017

Correlations of melodies in music follow a power spectrum where the intensity of that spectrum varies in inverse proportion to the frequency f called 1=f noise.1 This spectrum is followed by musical rhythms,2 pitch,1,3 and consonance fluctuations in music.4 The power spectrum is related to autocorrelation functions,5,6 and hence the 1=f power spectrum signifies a correlation in, for example, the fluctuating pitch on a time scale over which the power spectrum is 1=f .7 The 1=f power spectrum in music produces a psychologically pleasing balance2,8,9 between the predictable sound intensity of a random walk 1=f 2 and random white noise 1=f 0. In this paper we show that this noise spectrum can be recovered if, first, we consider pitch as a broken-symmetry variable, which we believe to be a necessary feature of a mathematical theory of musical melodies, and if, second, we include interactions between musicians,10–13 which we find to be a sufficient feature. An understanding of music is rooted in the way the brain works.14,15 Imagine watching a film of a jazz quartet, where the film shows the progress of the musical score. As a musician begins playing, and one sees the score, the performance is not that different from watching a mathematician writing complex equations at a black- board. As the other musicians begin, it would be as if our mathematician is joined by three others, each writing varied but related theorems. This is remarkable, but the most remarkable part is the audience, perhaps humming along or tapping their fingers in time as they themselves interact with the musicians16 and even predicting the onset times of upcoming musical events.17,18 As the equations multiplied on the blackboards, the people listening to the mathematicians experience intense feelings of joy, followed by feelings of loss, and so on, as they followed the tracings of algebra. Listeners’ intense reaction to, and involvement with music shows the performance is “inside” the audience’s minds. To understand music, then, is to understand some aspects of the neuroscience of the brain;19–21 music provides a concrete way of addressing how, for example, decisions are made in the context of the creation of musical melodies. Music is particularly well suited to mathematical analysis as it is straightfor- wardly quantifiable in terms of time and pitch.22,23 While it is well known that musical melodies obey 1=f noise, there are few theories giving rise to this,24 and most demon- strations of artificial music derived from a 1=f power spectrum make use of, for example, filtered white noise6 for which no potential neurological significance can be attached. We also note that the phenomena of 1=f noise, although well docu- mented,25–33 is in seeming contradiction to the notions of time signatures, key signatures, and so forth. We will not address that in this paper, except in the sense of attempting to find the simplest model consistent with musical melodies which gives rise to 1=f noise. Crucially, we make use of the fact that a musical melody can be recognized without regard to the key of that melody, due to the arbitrariness of perceived absolute pitch: But when a melody is played in a particular key, that perfect symmetry is broken. The mathematical properties of broken-symmetry variables34–39 are well known. When a free energy has a continuously broken symmetry, a broken-symmetry variable h arises, whose fluctuations act to restore that symmetry. The modes of a broken-symmetry variable have no energy gap to propagate, and hence the correlations

EL490 J. Acoust. Soc. Am. 142 (5), November 2017 VC 2017 Acoustical Society of America Martin Grant and Niloufar Faghihi: JASA Express Letters https://doi.org/10.1121/1.5011150 Published Online 17 November 2017 of h are power-laws. In particular, in Fourier space, as a function of wavenumber q,it can be shown that a broken-symmetry variable, often called a Goldstone variable, satisfies hjh^ðqÞj2i/1=q2 for small q, where h^ is the Fourier transform of h, defined below, and the brackets denote an average. This serves to minimize an energy function proportional to the integral of ðrhÞ2; where the broken symmetry is explicit in the invariance of h under the transformation h ! h þ const. We will review the mathematics of broken-symmetry variables below, but for the purposes of this paper, the main insight will be that those mathematics can be applied to music. A necessary feature of a theory of music is the imperfect detection of absolute pitch. A musical melody is recognized without regard to the key in which it is per- formed. Having a specific pitch, then is the same as having a specific value for a broken-symmetry variable. Hence we identify absolute pitch, in essence the key of a melody, as a broken-symmetry variable, and so the highness h of a pitch in a melody will behave with the correlations of a broken-symmetry variable. First imagine a solitary musician sitting by the piano, playing notes at random, close to a previous pitch. Following the energy function above, the decision on what note to follow a given note is given by the constraint enforced by gradients of h above, which favours nearby notes; the concept of nearby is used in the most simple sense here, and more refined descriptions exist,40 and we note that music researchers recognize that musical pitch has at least two dimensions, height and chroma;41 we trust that our simple treatment may give rise to more refined analyses. In any case, if the highness of the broken-symmetry variable pitch is called h, and the space in which the notes are played is one-dimensional time t, we have ð df hhtðÞhðÞ0 i¼ eif :tGf^ ðÞ; (1) ðÞ2p where we have introduced the frequency f replacing the Fourier wavenumber q above, and G^ ðf Þ/1=f 2. While this is a power law, it is the frequency spectrum of a random walk, not 1=f noise. Hence, recovering 1=f noise will require more than the imperfect detection of pitch: we will introduce another element, interactions, which will act to temper these variations in frequency. As such, let us insist that the solitary musician listens to other musicians, all of whom are also attempting to play melodies in the same manner as our first musician. Further, assume the musicians temper each others’ choices of notes such that notes chosen are constrained not only to be close to a previous note, but to be close to those played by the other musicians. Now the dimension of space is two-dimensional and the frequency spectrum changes: one dimension is time t and the second dimension is the direction of interaction with other musicians called x, where we can conveniently measure the strength of the interactions in units of t and x. This model provides a minimal interaction that one musician can have with others. As shown below, we immediately obtain the correlation function Pðx; f Þ¼efx=f : (2) Hence, considering the notes on the x ¼ 0 axis corresponding to a chosen musician, we have Pð0; f Þ¼1=f : (3) This is our main result, recovering 1=f noise. It arises from one necessary condition, imperfect absolute pitch detection identified as a broken-symmetry variable, and one sufficient and we believe plausible condition, interactions, which has been identified and noted in the literature.10–13 There are two further implications of our analysis which are testable. First, the exponentially damped correlations of Eq. (7) correspond to correlations across nonzero x and frequency. Second, for t ¼ 0, which corresponds to any fixed time, we find that there are power-law correlations in the Fourier transform of x, qx following 1=jqxj. We now briefly review the relevant mathematics of broken symmetry variables, and show how readily this mechanism gives rise to a 1/f spectrum. As a well-known example, consider surfaces of coexisting phases. The broken symmetry is translational invariance in the direction normal to the interface, the broken-symmetry variable is the local height of the interface, and the dynamical modes are called capillary waves or ripplons. Consider a surface in the d-dimensional x* plane of a (d þ 1Þ-dimensional system ~r ¼ð~x; yÞ. The position of the surface, assuming small fluctuations and therefore no droplets or overhangs, is y ¼ hð~xÞ, where h is the local height of the surface. The probability of being in a state hð~xÞ is proportional to eFðhÞ, where F is the free

J. Acoust. Soc. Am. 142 (5), November 2017 Martin Grant and Niloufar Faghihi EL491 Martin Grant and Niloufar Faghihi: JASA Express Letters https://doi.org/10.1121/1.5011150 Published Online 17 November 2017 energy in units of the temperature. If the free energy does not depend on the absolute position of the interface, which corresponds to translational invariance of space in the direction normal to the interface, it is invariant under h ! h þ const. Hence it can only depend on h through gradients. To lowest order in gradients, we have ð F / d~x ðrhÞ2: (4)

Note that the free energy is minimized by minimizing variations in h, any constant h minimizes F. This form, being quadratic, can be diagonalized and solved as a product of independent Gaussians by using Fourierð transforms, h^ð~qÞ¼ d~xei~q:~x hð~xÞ; (5) Ð d i~q~x ^ where we can write hð~xÞ¼Ð ½d~q=ð2pÞ e hð~qÞ, since the Dirac delta function ensures completeness, dð~x x~0 Þ¼ ½d~q=ð2pÞd ei~qð~x ~x0 Þ. The solution is

hh^ð~qÞh^ðq~0 Þi¼ð2pÞd dð~q þ~q0ÞG^ ðqÞ; (6) where G^ ðqÞ/1=q2: This is the standard result for a broken-symmetry Goldstone variable.34 As the integrals are Gaussian, all moments can be obtained from the second moment. In real space, the correlation function hhð~xÞhð0Þi is the Fourier transform of G^ ðqÞ. Translational and rotational invariance of space in the plane parallel to the interface gives rise to the delta 0 function dð~q þ~q Þ, and implies that hhð~xÞhð0Þi ¼ hhð~x þ ~x0Þhð~x0Þi, where ~x0 corresponds to the arbitrary origin, and can be ignored. This general result can be obtained in other ways, and the noise source need not be thermal which is important for our present investigation; the derivation of the Kardar-Parisi-Zhang equation35–37 provides an example. If the rate in change of time s of a broken-symmetry variable has the invariance h ! h þ const, and we endeavour to find the rate of change of h via a gradient expansion in h, and to lowest order in time derivatives, we obtain @h=@s /r2h þ l, where l is a noise. By the central-limit theorem, the only nontrivial moment of the noise satisfies hlð~x; sÞlð~x0; s0Þi / dð~x ~x0Þdðs s0Þ. In steady state, one again recovers the same probability and free energy, and the same results, including G^ / 1=q2. ~ Finally, let us define a convenient partial Fourier transform using ~q ¼ðqx; kÞ, ~ such that qx is one-dimensional, and k is ðd 1Þ-dimensional. Namely, let ð dqx Px;~k ¼ eiqxxGq^ ;~k : (7) 2p x

2 2 2 ~ jkjx ~ Using G^ ¼ 1=q ¼ 1=ðqx þ k Þ gives Pðx; kÞ¼e =jkj so that Pð0; kÞ¼1=jkj. Note that the x ¼ 0 axis is equivalent to any constant x axis from translational invariance in the plane parallel to the interface.

FIG. 1. The power P spectrum of the solid-on-solid model showing 1=f noise as a function of logðPÞ versus log ðf Þ.

EL492 J. Acoust. Soc. Am. 142 (5), November 2017 Martin Grant and Niloufar Faghihi Martin Grant and Niloufar Faghihi: JASA Express Letters https://doi.org/10.1121/1.5011150 Published Online 17 November 2017 As an example of how readily a 1=f spectrum can be obtained, for small f corresponding to large time scales, we show the results of a Monte Carlo simulation of the equilibrated solid-on-solid model,36,42 which is known to be a good model of capillary wave fluctuations. Each lattice site of a two-dimensional square grid can have any integer height hi, where i ¼ 1; 2; …; N runs overP all sites of the square lattice of N sites. The energy of interaction is proportional to ijhi hnnj, where nn denotes the nearest neighbours of i. In Fig. 1 we show the power spectrum of the equilibrated solid-on- solid model with N ¼ 1024 1024 at noise intensity T ¼ 4, in units of the interaction constant. The flattening of the spectrum at very low frequencies corresponds to finite- size effects. This is the averaged power spectrum of one “row” of the model, corresponding to the time axis of the evolving melody. Acknowledgments We thank Dan Levitin, Charles Gale, and Caroline Palmer for useful discussions. This work was supported by the Natural Sciences and Engineering Research Council of Canada and by le Fonds de recherche du Quebec—Nature et technologies. References and links 1B. Manaris, J. Romero, P. MacHado, D. Krehbiel, T. Hirzel, W. Pharr, and R. B. Davis, “Zipf’s law, music classification, and aesthetics,” Comput. Music J. 29, 55–69 (2005). 2D. J. Levitin, P. Chordia, and V. Menon, “Musical rhythm spectra from Bach to Joplin obey a 1/f power law,” Proc. Natl. Acad. Sci. U.S.A. 109, 3716–3720 (2012). 3L. Liu, J. Wei, H. Zhang, J. Xin, and J. Huang, “A statistical physics view of pitch fluctuations in the classical music from Bach to Chopin: Evidence for scaling,” PLoS One 8, e58710 (2013). 4D. Wu, K. M. Kendrick, D. J. Levitin, C. Li, and D. Yao, “Bach is the father of harmony: Revealed by a 1/f fluctuation analysis across musical genres,” PLoS One 10, e0142431 (2015). 5R. F. Voss and J. Clarke, “1/f noise in music and speech,” Nature 258, 317–318 (1975). 6M. Gardner, Fractal Music, Hypercards and More: Mathematical Recreations from Scientific American (W. H. Freeman and Company, NewYork, 1991). 7R. F. Voss and J. Clarke, “1/f noise in music: Music from 1/f noise,” J. Acoust. Soc. Am. 63, 258–263 (1978). 8D. J. Levitin, “Why music moves us,” Nature 464, 834–835 (2010). 9D. J. Levitin, This is Your Brain on Music: The Science of a Human Obsession (Dutton Adult, Boston, MA, 2006). 10D. L. Gilden, T. Thornton, and M. Mallon, “1/f noise in human cognition,” Science 267, 1837–1839 (1995). 11H. Hennig, R. Fleischmann, and T. Geisel, “Musical rhythms: The science of being slightly off,” Phys. Today 65(7), 64–65 (2012). 12H. Hennig, “Synchronization in human musical rhythms and mutually interacting complex systems,” Proc. Natl. Acad. Sci. 111, 12974–12979 (2014). 13M. H. Ruiz, S. Hong, H. Hennig, E. Altenmller, and A. A. Kuhn, “Long-range correlation properties in timing of skilled piano performance: The influence of auditory feedback and deep brain stimulation,” Front. Psychol. 5, 1030 (2014). 14D. Wu, C.-Y. Li, and D.-Z. Yao, “Scale-free music of the brain,” PLoS One 4, e5915 (2009). 15R. N. Shepard, “Perceptual-cognitive universals as reflections of the world,” Psychon. Bull. Rev. 1, 2–28 (1994). 16B. H. Repp, “Sensorimotor synchronization: A review of the tapping literature,” Psychon. Bull. Rev. 12, 969–992 (2005). 17S. K. Rankin, P. W. Fink, and E. W. Large, “Fractal tempo fluctuation and pulse prediction,” Music Percept. 26, 401–413 (2009). 18S. K. Rankin, P. W. Fink, and E. W. Large, “Fractal structure enables temporal prediction in music,” J. Acoust. Soc. Am. 136, EL256–EL262 (2014). 19E. W. Large, “Resonating to musical rhythm: Theory and experiment,” in The Psychology of Time, edited by S. Grondin (Emerald Group Publishing, Bingley, UK, 2008) Chap. 6, pp. 189–231. 20E. W. Large and J. S. Snyder, “Pulse and meter as neural resonance,” Ann. N.Y. Acad. Sci. 1169, 46–57 (2009). 21C. Palmer, “Music performance,” Ann. Rev. Psychol. 48, 115–138 (1997). 22P. Campbell, “The music of digital computers,” Nature 324, 523–528 (1986). 23M. Schroeder, “In there such a thing as fractal music?,” Nature 325, 765–766 (2002). 24Yu. L. Klimontovich and J.-P. Boon, “Natural flickr noise (‘1/f noise’) in music,” Europhys. Lett. 3, 395–399 (1987). This paper shows that flicker noise, which is 1/f noise, can give rise to a 1/f spectrum in music, but does not provide a further rationale for the assumption of 1/f noise. 25B. Kaulakys and T. Meskauskas, “Modeling 1/f noise,” Phys. Rev. E 58, 7013–7019 (1998). 26B. Kaulakys, “Autoregressive model of 1/f noise,” Phys. Lett. A 257, 37–42 (1999). 27B. Kaulakys, “The intrinsic origin of 1/f noise,” Mol. Reliab. 40, 1787–1790 (2000). 28B. Kaulakys and J. Ruseckas, “Stochastic nonlinear differential equation generating 1/f noise,” Phys. Rev. E 70, 020101(R) (2004). 29B. Kaulakys, J. Ruseckas, V. Gonits, and M. Alaburda, “Nonlinear stochastic models of 1/f noise and power-law distributions,” Physica A 365, 217–221 (2006).

J. Acoust. Soc. Am. 142 (5), November 2017 Martin Grant and Niloufar Faghihi EL493 Martin Grant and Niloufar Faghihi: JASA Express Letters https://doi.org/10.1121/1.5011150 Published Online 17 November 2017 30B. Kaulakys and M. Alaburda, “Modeling scaled processes and 1/fb noise using nonlinear stochastic differential equations,” Stat. Mech.: Theor. Exp. 2009, P02051 (2009). 31A. Balandin, “Low-frequency 1/f noise in grapheme devices,” Nat. Nanotechnol. 8, 549–555 (2013). 32M. Baiesi and C. Maes, “Realistic time correlations in sandpiles,” Europhys. Lett. 75, 413–419 (2006). 33J. Ruseckas, R. Kazakevicius, and B. Kaulakys, “1/f noise from point process and time-subordinated Langevin equations,” J. Stat. Mech.: Theor. Exp. 2016, 054022 (2016). 34D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, Monograph 47 of Frontiers in Physics (W. A. Benjamin, San Francisco, CA, 1975). 35M. Kardar, G. Parisi, and Y.-C. Zhang, “Dynamic scaling of growing interfaces,” Phys. Rev. Lett. 56, 889–892 (1986). 36B. Grossmann, H. Guo, and M. Grant, “Kinetic roughening of interfaces in driven systems,” Phys. Rev. A 43, 1727–1743 (1991). 37B. Schmittmann and R. Zia, “Driven diffusive systems. an introduction and recent developments,” Phys. Rep. 301, 45–64 (1998). 38P. Bak, C. Tang, and K. Wiesenfeld, “Self-organized criticality: An explanation of the 1/f noise,” Phys. Rev. Lett. 59, 381–384 (1987). 39P. Bak, How Nature Works: The Science of Self-Organized Criticality (Springer-Verlag Telos, Berlin, 1996). 40R. N. Shepard, “Geometrical approximations to the structure of musical pitch,” Psychol. Rev. 89, 305–333 (1982). 41J. D. Warren, S. Uppenkamp, R. D. Patterson, and T. D. Grifﬁths, “Separating pitch chroma and pitch height in the human brain,” Proc. Natl. Acad. Sci. 100, 10038–10042 (2003). 42R. H. Swendsen, “Roughening transition in the solid-on-solid model,” Phys. Rev. B 15, 689–692 (1977).

EL494 J. Acoust. Soc. Am. 142 (5), November 2017 Martin Grant and Niloufar Faghihi