Janko Gravner and Kyle Johnson Coupled Map Lattices as Mathematics Department University of California Davis, California 95616, USA Musical Instruments {gravner, kyljohnson}@math.ucdavis.edu

Abstract: We use one-dimensional coupled map lattices (CMLs) to generate sounds that reflect their spatial organization and temporal evolution from a random initial configuration corresponding to uncorrelated noise. In many instances, the process approaches an equilibrium, which generates a sustained tone. The pitch of this tone is proportional to the lattice size, so the CML behaves like an instrument that could be tuned. Among exceptional cases, we provide an example with a metastable strange attractor, which produces an evolving sound reminiscent of drone music.

A common mathematical model of a complex for sound generation this multidimensional array feedback system is iteration of a nonlinear map of needs to be reduced to a one-dimensional sequence one or more variables. The resulting unpredictability of numbers. One way to achieve this is to simply and diversity of behavior naturally invite possible produce one signal per time step, as a single function connections to music, where such models have of all variables of the system (Stefanakis, Abel, and indeed found uses, primarily to generate note Bergner 2015). There might be a canonical selection patterns for use in compositions (e.g., Battey 2004). of such a function in special cases, but in general the In this article we instead adopt this approach choice is necessarily arbitrary. Moreover, although to produce a single sound whose characteristics this method does reveal temporal evolution (at least reflect the evolution induced by a chosen recursive if the sampling variable is carefully chosen), any rule. spatial organization is not captured. To remedy this, We focus on coupled map lattices, a class of we seek to include every variable in our output iterative systems built from a large number of iden- sequence. tical functions of a single variable linearly coupled To investigate spatial organization, we must have to nearest neighbors in a network. As has been space, and a particularly useful role is played by known since their inception, these are particu- the space Zd of integer vectors with d coordinates, larly convenient models to study self-organization: the d-dimensional integer lattice. An evolving From a disordered initialization, many instances configuration attaches a variable, in our case a exhibit rapid convergence to a steady state with number, to each point in the lattice, and uses a recognizable spatial patterns (Kaneko 1993). Our rule to modify the variables over time. We will main goal, then, is to represent these patterns as assume that d = 2, as the same argument applies sounds. equally in higher dimensions. Then there is no Note that, in this article, we use the term “space” natural way to “cut the configuration into one- to refer to the abstract space in which the evolving dimensional ribbons” to generate a sound stream array of variables is arranged, not to the physical that would incorporate every variable, not even space of a listening environment or of a musical at a single time. In short, the problem is that the instrument. time (for the sound sequence) is a priori linearly ordered, whereas the two-dimensional lattice is not. Nevertheless, there have been some attempts Spatial Organization and Sound Synthesis based on two-dimensional cellular automata such as the Game of Life and voter models (Burraston Suppose we have a system that evolves in time, and and Edmonds 2005; Miranda 2007; Serquera and we want to use it to generate a sound stream. Such Miranda 2011, 2014). a system possibly has a high-dimensional set of We conclude that the most promising approaches variables, together with one-dimensional time, and are processes on Z, i.e., those whose configuration is a finite array of numbers arranged on an integer − Computer Music Journal, 42:2, pp. 22–34, Summer 2018 interval, say [0, n 1]. The number n is the lattice doi:10.1162/COMJ a 00458 size. Referring to the space, we call such processes c 2018 Massachusetts Institute of Technology. one-dimensional, although the dimension of the

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 configuration space equals n and could be very large, Typically, our initial configuration is random. To typically in our examples between 50 and 400. be precise, x0(i), i = 0, ..., n − 1 are chosen indepen- The configuration of our process at a time t dently and uniformly from the set of possible values. therefore consists of real numbers xt(0), ..., xt(n − This maximally disordered initialization is arguably 1). This process will evolve in discrete time t = the best test of the rule’s self-organizing capabilities. 0, 1, 2, ... according to a rule that will be specified Interesting sounds can be generated with special shortly, but first we reveal how we obtain the initial configurations, however, as we will see later sequence used to generate sound. Namely, we form in sections on “Sine Wave Initialization” and “Zero the unique sequence of variables that preserves Initialization.” the order of the time, and then the order of the Coupled map lattices were utilized by Kunihiko one-dimensional lattice: Kaneko in 1983 to model spatiotemporal chaos with discrete-time, discrete-space, and continuous-state x0(0), ..., x0(n − 1), x1(0), ..., x1(n − 1), ..., xt(0), ..., dynamics (for a survey, see Kaneko 1993). We will only consider three examples of the map f :the xt(n − 1), ... (1) linear map, the logistic map used by Kaneko, and the circle map. For the latter two choices, CMLs are, in a sense, coupled nonlinear oscillators. The Coupled Map Lattices linear map facilitates the exposition through its sim- plicity, and demonstrates an interesting connection Particularly appealing update rules to study self- between CMLs and the Karplus-Strong algorithm. organization are those with very simple local The logistic map is considered because of the role it interaction. We now introduce one such class of has played in the existing literature on CMLs, and i rules, which posit that the variable at a location is because it provides good examples of periodic, inter- first updated using a function, possibly one that is mittent, and chaotic behavior. The most interesting nonlinear, of only its current state; the value at the sounds, however, are generated by the circle map, next time step is computed by a weighted average which exhibits a number of interesting evolutions. of its value and those of its nearest neighbors in Examples we provide include chaotic, periodic, and the one-dimensional lattice. Thus the second step almost periodic behavior; synchronization; inter- is linear and any perturbation spreads at a speed acting particles; and metastable strange attractors. of at most one, which is often called the “light Often, the mixture of noise and order in the CML x speed.” That is, t is given recursively, for each time is reminiscent of chimera states in neural network t = ... 0, 1, 2, ,bytherule dynamics (Hizanidis et al. 2016); see, for example, the CML given later in the section on “Evolution x + (i) = (1 − ) f (x (i)) + [ f (x (i + 1)) + f (x (i − 1))]. t 1 t 2 t t of CML with the Circle Map: Metastable Strange (2) Attractor.” Here, ∈ [0, 1] is the coupling strength that deter- There are several natural ways in which Equa- mines the influence of spatial neighbors, and f is a tion 2 can be generalized. For example, one may given function. We assume periodic boundary,that change the neighborhood of i from nearest neighbor is, for i ∈ [0, n − 1], its neighbors i − 1andi + 1are i ± 1 to a larger range r: i ± 1, ..., i ± r. This does computed modulo n. Such a rule is called a one- not seem to make a qualitative difference. Another dimensional coupled map lattice (CML). We will possibility is to forgo translation invariance and to restrict the variable values xt(i) to a suitable interval, assign either a different or a different f to different either [−1, 1] or [0, 1]. In the latter case, whenever lattice points i. A random assignment of this kind the updated variable is outside the interval we is an analogue of spin-glasses (Stein and Newman replace it with its fractional part; equivalently, we 2013) and is an intriguing direction that is largely could assume that individual variables are points on beyond the scope of this article, although we do a (continuous) circle. provide one example (see Equation 3 later in this

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 article). Finally, generalizations to other lattices are would require potentially sophisticated solvers mathematically interesting, but, as discussed, it is (Dzjaparidze 2015). unclear whether this could be done without losing There is also a theoretical advantage to CMLs: the natural correspondence to digital audio signals. No existence and uniqueness results are needed.

Key Features of CML Synthesis Previous Work In our experiments, most CML dynamics rapidly Of all existing sound-synthesis techniques, our converge to a stable equilibrium, which results approach most closely resembles the linear automata in a steady tone. In almost all cases, the pitch of synthesis (LASy) of Jacques Chareyron (1990). There the tone is inversely proportional to the lattice are two main differences between LASy and CML size. The proportionality constant depends on the synthesis. The first is that the former uses cellular sampling rate and the temporal behavior of the automata with a large set of discrete variable CML. Although we do not have a formal definition, values, whereas the latter uses coupled map lattices, we refer to these cases as regular. Two exceptions which have continuous variables. In this sense, our are discussed in some detail later in this article: a approach is a natural extension of LASy. The second CML with a logistic map (Equation 4) with a highly difference is that the LASy method focuses on linear chaotic behavior (producing white noise), and a cellular automata, mentioning other possibilities CML determined by the parameters in Equation 11 only in passing, whereas we view linear rules as a with a highly organized behavior (resulting in a high starting point into more general CMLs. Notice again frequency tone). Both of these require very specific that both of these schemes use one-dimensional parameter settings. lattices; higher-dimensional cases require mapping In the regular cases, the timbre of a CML’s tone the output of the system in a more creative way. For is determined by the chosen parameters and the example, histogram mapping synthesis (Serquera initial configuration. The random initialization and Miranda 2011, 2014) uses densities of variables results in slight variations between different runs of in evolving cellular automata dynamics to control the dynamics, akin to physical instruments, which the parameters of a synthesizer. Another approach, also produce subtly different tones even when the Chaosynth, uses a cellular automaton inspired by same pitch is played with the same loudness. The neural reverberatory circuits to control the pitch effect of the parameters on the timbre of a CML is and duration of sinusoidal grains in a granular dependent on the definition of the map, but some synthesizer (Miranda 1995). general observations can be made. The degree to A recent article by Stefanakis, Abel, and Bergner which the map f itself is chaotic has the expected (2015) uses ordinary differential equations (ODEs) effect. Namely, the nonlinearity parameter (a in the for digital synthesis. This method bears some logistic map, defined later in Equation 4, and K in similarity to ours in that it uses dynamical systems. the circle map, defined in Equation 5) governs how It generates only one signal per time step, as a simple the sound is: Large nonlinearity results in function of all variables, to generate a time series chaotic noise, and other values generate tones with that produces the sound, however. By contrast, each varied degrees of complexity. The coupling strength of our variables generates a signal in the lattice order, has an unpredictable effect on evolution, and often so the resulting sound reflects spatial organization needs fine tuning. as well as temporal evolution. Furthermore, discrete space and time have the practical advantage that computations are simple and very fast. By contrast, Outline and Highlights coupled ODEs (which evolve in discrete space and continuous time) or partial differential equations Our main purpose is to use CMLs as instruments (in which both space and time are continuous) in the following fashion: Generate a random initial

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 configuration, choose the dynamic parameters, Spatial simplicity alone does not suffice for a and observe the resulting sound. Different choices pitched tone, however. As we will see in the section produce starkly different sounds, from pitched on “Zero Initialization,” there is an example with tones, to noise, to something in between. In the nearly constant equilibrium whose values change following sections, we concentrate on examples, chaotically and hence result in noise, even with both to illustrate the regular dynamics and to constant initialization. On the other hand, it is easy provide interesting unusual cases. First, however, to generate simple temporal behavior (namely, by we give an overview of our approach in the following choosing f to be the identity map with = 0), which section. We follow with a brief exposition on the leaves the spatial chaos invariant and, hence, also basic case of the linear map, and then provide three produces noise. examples with the logistic map that demonstrate We reiterate that two sounds, generated with how CMLs generate sound that range between the same map, parameters, and lattice size, but order and noise. Perhaps the most interesting are different random initial samples, will both sound the CMLs that use the circle map f . We provide similar, as they are generated by the same process, four examples: The first demonstrates that CMLs but they are subtly distinct from one another. This are capable of producing rich and varied textures balance of similarity and variation is analogous to thanks to slow convergence to the steady state, the the way sounds produced by a physical instrument second is an example of a nonunique equilibrium will have a recognizable timbre, but will reveal with unusual lack of dependence on the lattice size, subtle differences when sounding the same note. the third produces slowly changing timbre, and Another parallel is that the dominant frequency, the last CML generates a frequency that is lower and therefore the tuning procedure, cannot be than expected. Finally, we give a few examples with determined a priori for all parameter choices, but special initializations. We demonstrate the use of only after running the particular dynamics—by CMLs as musical instruments by tuning them to “playing the instrument,” as it were. We elaborate play a few melodies, some well-known and some on this point next. computer-generated. Three example melodies are With the exception of a few special cases, the described in the Appendix, including a description pitch of the tone produced by a CML is inversely of the parameters used. proportional to the lattice size n. Given a CML with a lattice of size n and a sample rate of s samples per second, the fundamental frequency of the tone Interpreting a CML Evolution as a Sound produced by the CML synthesizer will be s/(nT) Hz for some T ∈{1, 2, 3, ...}. If a CML has a steady As explained in the section “Spatial Organization state that is periodic (or nearly periodic) with a and Sound Synthesis,” we interpret each lattice sufficiently small temporal period, then T will be variable in a CML evolution as a digital audio signal, determined by the temporal period of the CML. and order these signals as in Equation 1. We have This is because the sequence of samples generated chosen a sample rate of 44.1 kHz (the compact disc by the consecutive lattice points will be periodic (or standard) in all our examples. After a short burst nearly so) with period nT. and so they will produce of white noise (due to the initial randomness), the a tone of frequency s/(nT). If the temporal period is resulting audio stream is a sound with evolving too large, s/(nT) will be below the range of human timbre. Because the dynamics tend to rapidly evolve hearing and we will perceive the strongest upper into a stable or metastable steady state, the transient partial of this frequency as the fundamental. In this period is almost imperceptible and the result is close case, the dominating frequency tends to be s/n. to a single sound. The nature of the sound depends The only exceptions to the linear dependence on upon the behavior of the CML. If the equilibrium is n that we are aware of involve rules that produce spatially simple, dominated by temporarily periodic a very close approximation to white noise and regions, our synthesizer produces a pitched tone. rare spatially periodic examples. Even chaotic

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 sounds that do not produce a tone having a clear, a is in [−1, 1] (values of a outside of this range can definite musical pitch do have an audible change be used, but then we have to use a fractional part of pitch as a result of a change in the lattice size. operation to map back into the interval [−1, 1]). Additionally, the discontinuity between temporal If we start from a random initialization and take updates provides a periodic signal with frequency ∈ (0, 1], this gives us a CML which is, effectively, inversely proportional to the lattice size. These an implementation of the Karplus-Strong algorithm two contributions are fundamentally different. For for plucked-string synthesis (Karplus and Strong example, if n is large, the second effect amounts to a 1983). Indeed, the transition rule of our CML takes sequence of “pops.” To reiterate, the inverse of n will a weighted average of adjacent samples in the sound be present in the frequency domain, except when stream generated by Equation 1, and thus acts as a the lattice configuration is spatially periodic and low-pass filter on the signal. temporarily constant. The latter level of regularity This linear case is particularly attractive for is very difficult to achieve with a CML starting from the following reason: When it starts from ran- a random initialization. We do, however, provide domness, it emulates a plucked string of length an example in which this happens at least with a proportional to the length of lattice. In fact, this positive probability (this will be discussed in greater case has appeared in the literature in the guise detail in the section “Evolution of CML with the of a linear cellular automaton with a large num- Circle Map: Annihilating Diffusions”). Trivially, we ber of variable values (Chareyron 1990). A few can also achieve this by the identity map f , = 0, examples generated using a = 0.99 and = 0.5 and with a periodic initial configuration. are illustrated visually in Figure 1 and can be The diffusion caused by the local averaging acts, heard as Sound Examples 1–3. [Editor’s note: All in the frequency domain, as a low-pass filter on the sound examples can be heard on the CMJ Web site signal after the possibly chaotic map f has been at www.mitpressjournals.org/doi/suppl/10.1162 applied to each lattice point. This filtering operation /COMJ a 00458.] mitigates the extent to which higher partial tones In Figure 1a (and in later figures), we exhibit are present in the sound produced by the CML and space–time pictures of the corresponding CML thus helps to reinforce the fundamental frequency evolutions. As is standard in the CML literature, the proportional to the inverse of n. The strong and entire spatial configurations xt(0), ..., xt(n − 1) are predictable relationship between the lattice size and depicted horizontally, with the configuration at time the frequency of the sound enables CMLs to be used t + 1 below the one at time t. Thus, the initial state as a natural method of digital synthesis. at time t = 0 is at the top, and time runs downwards. For example, a tone generated by a CML with The variable at each lattice point is linearly mapped temporal period 2 and lattice size 50 at 44.1 kHz to a shade of gray, so that white corresponds to the will have a fundamental frequency of 441 Hz, which right endpoint of the state space and black to the is about 4 cents sharper than the middle A tuned left endpoint. For example, in Figure 1a the variable at 440 Hz. Notice that with the standard 44.1-kHz is in [−1, 1], so the lighter and darker shades depict sample rate, we cannot get closer to the frequency values close to 1 and −1, respectively. 440 Hz. The relationship between n and frequency Figure 1b displays a spectrogram of the sound allows us to easily control the pitch generated by a generated by the same CML, with lighter shades CML, but it also causes the precision of our tuning of gray corresponding to larger amplitudes, using to be limited by the sample rate. the same time period as the space–time pictures. In this (and the following figures), time runs hor- izontally from left to right, as is standard for A Simple Example: The Linear Case spectrograms. We also provide a preliminary example of the dis- Consider a linear map f :[−1, 1] → [−1, 1], defined ordered version of the CML rule given by Equation 2, by f (x) = ax for all x ∈ [−1, 1]. Here, the coefficient namely,

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 Figure 1. The evolution of time domain (a) and in the and space on the the linear coupled map frequency domain (b). The horizontal axis, as is lattice (CML) with lattice time-domain typicalintheCML size n = 100, a = 0.99, and representation used here literature. = 0.5, from a random places time on the vertical initial configuration, in the axis, oriented downwards,

i We note that when a = 2and = 0, each lattice xt+1(i) = (1 − i) f (xt(i)) + [ f (xt(i + 1)) + f (xt(i − 1))], 2 point behaves chaotically and independently of (3) others, resulting in a close approximation to white where i are random, in our case chosen inde- noise. This is a simple instance in which the lattice pendently and uniformly on [0, 1]. This produces size has no effect on the sound produced by the a sound with more pronounced upper partials CML. that correspond to enhanced horizontal lines in the spectrogram in Figure 2; refer also to Sound Examples 4–6. The Circle Map

In this case, the variables have values in the interval The Logistic Map [0, 1] and K We use the following version of the logistic map: f (x) = x + ω − sin(2πx)mod1. (5) 2π f (x) = 1 − ax2. (4) Here, ω represents a phase shift and K the strength of nonlinearity. Taking the fractional part mod 1 is We will assume that the nonlinearity parameter a needed because without it f does not necessarily is in the interval [1, 2], so that f maps the interval map [0, 1] into [0, 1]. In the following discussion, we [−1, 1] into itself. The choice of Equation 4 was examine the behavior in the case of four different pa- made by Kaneko in his pioneering work on CMLs rameter settings with random initial configurations (Kaneko 1993). Other versions of the logistic map and explore special initializations. are equally natural and produce similar results The Kuramoto model is an ODE that has been (e.g., f (x) = ax(1 − x)). widely studied as a prototype for synchronization, Values of a near 1 result in a sound that rapidly whereby the evolution leads to nearly equal variable converges to a steady tone. As a increases, the CML values across space in spite of the possibly chaotic behaves more chaotically, resulting in sounds that temporal evolution (for a comprehensive survey, become closer to noise than to a pitched tone. See see Acebron´ et al. (2005); for examples, see our Figure 3 for time domain and spectrogram plots of discussions of the metastable strange attractor logistic map CMLs with a few different parameter and near-periodicity later in this article). With the values. (Refer to Sound Examples 7–15.) circle map f as in Equation 5, the CML could be

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 Figure 2. The evolution of and uniformly on [0, 1], the disordered CML given from a random initial by Equation 3, with lattice configuration, in the time size n = 100, a = 0.99, and domain (a) and in the i chosen independently frequency domain (b).

interpreted as a version of the Kuramoto model Note that, when negative, i − k is interpreted as i − with discrete time and local coupling. Thus it is no k+ n because of the periodic boundary conditions. surprise that we observe synchronization in several This quantity compares a configuration with its of our examples (see our remarks after Equations 6 spatial shift by k, and leads to two other statistics: and 13), although conditions on the parameter values ρt = max cross corrt(k), (9) that result in this scenario are rather unclear. k

τt = min{k : cross corrt(k) = ρt}. (10)

Evolution of CML with the Circle Map: Metastable The quantity ρt measures the extent to which xt is a Strange Attractor translation of xt−1,andτt is the optimal translation, that is, one with largest cross-correlation. See First, we consider the circle map with parameters Figure 4 for the resulting evolution for three lattices sizes, up to time 107. By this time, the dynamics with ω = 5/9, K = 1, = 0.3. (6) n = 70 reached a stable synchronized configuration, with δt very close to 0. Therefore, at any fixed Owing to the complexity and longevity of its time, all the variables are nearly equal. Their evolution to the steady state, we need to develop approximately common value, however, changes some statistical tools to understand this CML. chaotically over time, owing to the nonlinearity of Figure 4 depicts the behavior of three statistics. The f . The resulting sound evolves into chaotic taps simplest is the unscaled variance because of the discontinuities between iterates. A   tone with frequency proportional to 1/n is also heard − − 2 n 1 1 n 1 (see the discussion in the section on “Interpreting δ = x (i)2 − x (i) ,(7) t t n t a CML Evolution as a Sound”). We conjecture that i=0 i=0 this is the fate of the CML with parameters given by Equation 6 for every lattice size. The waiting time for which simply measures how far x is from a constant. t n ≥ 100 is so large, however, that synchronicity is in A more involved tool is the cross-correlation (Stoica practice never reached. Instead, for n between about and Moses 2005), defined for t = 1, 2, ...by 100 and 300 we see the dynamics largely oscillating n−1 between two metastable polarity configurations, cross corrt(k) = xt−1(i − k) xt(i). (8) characterized by waves traveling in two opposite i=0 directions, with brief periods of polarity reversal

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 Figure 3. Three examples spectrograms of the signal Examples 7–15 are of the CML with the generated by the same generated by these three logistic map (Equation 4), CML on the right. The pairs of parameters (see each with 100 iterations parameter values used are: main text for the URL to and lattice size n = 100. a = 1.1, = 0.3 (a); a = the audio files). Time-domain plots are 1.75, = 0.3 (b); and a = 2, given on the left, with = 0.5 (c). Sound

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 Figure 4. Evolution of the bottom (medium gray) CML with the circle map curve depicts δt,the (Equation 5) and middle (dark gray) ρt and parameters given by the top (light gray) τt.See Equation 6, with n = 70 text for complete (a), n = 200 (b), and n = explanation. 400 (c). In each frame, the

between them. This is most clearly seen in the Although these properties make these parameter middle frame of Figure 4, where polarity periods settings inappropriate for use in a melodic or har- are distinguished by flat lines in the translation monic context, they result in a rich evolving timbre statistic τt. The evolution is thus dominated by a that is well-suited for avant-garde compositions. kind of strange attractor that creates an evolving (Refer to Sound Examples 16–18.) sound reminiscent of drone music (particularly of popular German electronic music of the 1970s, nicely exemplified on the Tangerine Dream album Zeit) punctuated by rhythmic crackling noises Evolution of CML with the Circle Map: during reversals. For larger n, the temporal evolution Annihilating Diffusions appears chaotic in Figure 4, as the polarity is only rarely reached; however, the spatial organization is We now consider the case with parameters similar to that of the polarity configurations and thus the droning sound persists with no, or only rare, reversals. ω = 0.5, K = 1, = 1. (11)

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 Figure 5. The first 1,000 Figure 6. First 250 time CML with the circle map time steps of the evolution steps (a), followed by 500 (Equation 5) and of the CML with the circle time steps starting later parameters given by map (Equation 5) and around time step 40,000 Equation 12, with n = 400. parameters given by (b), of the evolution of the Equation 11, with n = 400.

independent of the lattice size. (Refer to Sound Examples 19 and 20.)

Evolution of CML with the Circle Map: Repelling Particles

These generate two competing equilibria, one Next is the case with parameters with horizontal stripes (with temporal period 2 ω = 0.5, K = 0.1, = 0.6. (12) and spatial period 1) and one with vertical stripes (with temporal period 1 and spatial period 2), with After a considerable time, the evolution self- diffusing boundaries between the two equilibria. See organizes into a system of repelling particles on Figure 5 for an illustration. There are also stationary a slowly evolving background. Figure 6 provides defects that generate pronounced vertical lines and an illustration. As a result, the tone produced by that interact with the boundaries in an unpredictable the CML with these parameters will have steadily fashion. The invariant state is therefore not unique: changing timbre. Because the repelling particles One or the other equilibrium is reached with equal travel slowly, a high degree of structural similarity probability. When the lattice size is small, it is very is preserved between consecutive lattice variables, likely that the stationary defects disappear and so, and so the tone evolves smoothly. After an initial with probability close to 1/2, the generated sound period during which the particles are created and is a square wave with frequency equal to half the destroyed, their number stabilizes, which ensures sample rate; this sound is very high-pitched and an upper bound on the difference between two

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 Figure 7. The first 250 time CML with the circle map steps (a), followed by 500 (Equation 5) and time steps starting later parameters given by around time step 40,000 Equation 13, with n = 400. (b), of the evolution of the

by Equation 6. In addition to being nearly periodic spatially, these dynamics are also close to being temporally periodic, with a temporal period of 5. As a result, the fundamental frequency F of a tone produced at a sample rate s with a lattice of size n is given by F = s/(5n). The fifth upper partial (frequency s/n) is particularly strong and can be heard distinctly above the fundamental. (Refer to Sound Examples 23 and 24.)

Sine Wave Initialization

Interesting results can be obtained by setting the initial configuration to

= 1 π / + = ... − xi(0) 2 (sin(2 mi n) 1), i 0, , n 1, (14) where m is a positive integer parameter. Note that with identity map f and = 0, this initial configuration would result in a sine wave with frequency sm/n. In contrast, with a random initial configuration, which approximates a range of frequencies at equal strength, this initialization uses a single frequency. timbres at different times. This can be heard in Usig the circle map f (of the form in Equation 5), Sound Examples 21 and 22. and a sufficiently large lattice size (n greater than about 10,000), this setting creates a rhythmically pulsing tone. With a given sample rate s,andmand Evolution of CML with the Circle Map: n, the tone has frequency f = sm/n. The rhythm is Near-Periodicity determined by the dynamic parameters ω, K,and. As a rule of thumb, if ω = p/q is a rational number Finally, we consider the case with parameters represented by a reduced fraction, then the rhythm ω = 0.2, K = 0.2, = 0.12. (13) can be interpreted as having q beats per measure, with tempo s/n. Because of rounding errors, the This selection leads, after a few thousand itera- pulsing gradually devolves into noise. This “descent tions, to a stable, nearly periodic equilibrium with into chaos” is particularly striking when m= 100 a spatial period of about 80; see Figure 7. We note and f is the circle map with parameters given by that requires some finetuning here: A significantly Equation 6. Sound Examples 25 and 26 use the larger value for leads to a chaotic state, whereas a parameters provided in Table 1. significantly smaller leads to synchronization. Un- like in the CML determined by Equation 6, however, the approach to synchronicity is gradual. The sound Zero Initialization produced by these parameters has a timbre that varies somewhat more than the repelling-particle If the system is synchronized initially, as when case given by Equation 12, but considerably less xi(0) = 0 for all i, then it remains synchronized than the metastable strange-attractor case given forever. Therefore, the system reduces to the

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 Table 1. Parameter Values for Sine the resulting sound interpolates between a pitched Wave Initializations tone and noise. By far, the most versatile is the circle map, for ω nm K which we have provided four particularly interesting Example 1 10,000 10 5/9 1 0.3 choices for its three parameters. We have given two Example 2 10,000 10 3/7 1 0.3 examples with embedded particle systems, one Example 3 200 1 5/9 1 0.3 with annihilating diffusions with two competing Example 4 10,000 100 5/9 1 0.3 equilibria and another with repelling particles that produces a steadily changing timbre. Another circle map case exhibits nearly periodic behavior with a iteration of the map f , and the coupling strength resulting combination of frequencies. Our leading has no influence. This allows us to provide a circle map example converges to its equilibrium simple example of a sound generated by CML that very slowly, providing an opportunity to hear has spatial simplicity but still produces noise. This a sonic representation of a metastable strange is illustrated by the circle map with parameters attractor, which turns out to resemble drone music. ω = / = . 5 9andK 2 5. The generated sounds make The same parameter choice also produces, when it clear that the temporal chaos “drowns out” the initialized by a sine wave, a melody that gradually spatial order. This is especially pronounced with devolves into noise. smaller lattice sizes, as shown by Sound Examples Our results open several avenues for further 29 and 30. exploration. Clearly, one could look for new features using different functions f from those that were investigated here. Perhaps a greater priority is to Conclusions and Future Directions fully understand the potential of the circle map with its three parameters , ω,andK. Indeed, We have explored the use of complex nonlinear our most interesting examples were obtained by dynamics as a sound generator. Specifically, we little more than educated guessing. An “atlas” of used one-dimensional coupled map lattices, which parameter space with a catalog of behaviors would combine the iteration of a nonlinear map with be expedient if one wanted to contemplate possible linear local averaging. We presented a method of applications to, say, sound effects in computer synthesizing sound from the resulting dynamics games. Particularly the role of the coupling strength by using variables at all spatial locations in the is mysterious and counterintuitive, and it deserves lattice order (and also in temporal order). In a additional investigation. Furthermore, we only typical case, rapid convergence to an equilibrium touched on the effects of spatial heterogeneity: produces a steady tone. The spatial organization Either or a parameter in the function f could vary of the steady state determines a characteristic from location to location (e.g., by a random choice). timbre of this tone, with subtle variations due to Temporal heterogeneity is another possibility: The different initial conditions. Moreover, the size (i.e., update rule may change abruptly or gradually—for the length) of the network determines the pitch. example, to create a tone decay. Finally, adding From this perspective, such a mathematical object random noise to the update rule may test the is an abstract form of a musical instrument. stability of some of the observed behaviors. We have limited our exploration to several case studies with a linear, logistic, or circle map. The linear case emulates the sound of a plucked Acknowledgments string, with detectable alteration when the coupling strengths are chosen at random. The behavior We thank the reviewers and the editor for their induced by our first nonlinear function, the logistic constructive comments, which significantly im- map, is as expected: As the nonlinearity increases, proved the presentation. Both authors were partially

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00458 by guest on 30 September 2021 supported by NSF grant DMS-1513340. Janko Serquera, J., and E. R. Miranda. 2011. “Cellular Automata Gravner also received partial support through Dynamic Control for Sound Design with Histogram Simons Foundation Award no. 281,309 and the Mapping Synthesis and the Multitype Voter Model.” Republic of Slovenia’s Ministry of Science program In Proceedings of the International Computer Music P1-285. Conference, pp. 139–142. Serquera, J., and E. R. Miranda. 2014. “Histogram Mapping Synthesis: A Cellular Automata-Based Technique for Flexible Sound Design.” Computer Music Journal References 38(4):38–52. Stefanakis, N., M. Abel, and A. Bergner. 2015. “Sound Acebron,´ J. A., et al. 2005. “The Kuramoto Model: A Synthesis Based on Ordinary Differential Equations.” Simple Paradigm for Synchronization Phenomena.” Computer Music Journal 39(3):25–43. Reviews of Modern Physics 77(1):137–185. Stein, D. L., and C. M. Newman. 2013. Spin Glasses Battey, B. 2004. “Musical Pattern Generation with and Complexity. Princeton, New Jersey: Princeton Variable-Coupled Iterated Map Networks.” Organised University Press. Sound 9(2):137–150. Stoica, P., and R. Moses. 2005. Spectral Analysis of Burraston, D., and E. Edmonds. 2005. “Cellular Automata Signals. Upper Saddle River, New Jersey: Prentice in Generative Electronic Music and Sonic Art: A Hall. Historical and Technical Review.” Digital Creativity 16(3):165–185. Chareyron, J. 1990. “Digital Synthesis of Self-Modifying Waveforms by Means of Linear Automata.” Computer Appendix: Parameters in Examples of Melodies Music Journal 14(4):25–41. Dzjaparidze, M. 2015. “Exploring the Creative Potential of We provide three examples of melodies: Physically Inspired Sound Synthesis.” PhD dissertation, Queen’s University Belfast. Sound Example A1: To play “In the Hall of the Hizanidis, J., et al. 2016. “Chimera-like States in Mod- Mountain King” by Edvard Grieg, from his incidental ular Neural Networks.” Scientific Reports 6:Art. music to Peer Gynt, we use the circle map with 19845. Available online at www.nature.com/articles ω = 0.5andK = 0.1 throughout the excerpt with /srep19845. Accessed February 2018. three different values of , starting with = 0.6, Kaneko, K. 1993. “The Coupled Map Lattice.” In switching a third of the way through to = 0.65, K. Kaneko, ed. Theory and Applications of Coupled and in the final third to = 0.7. Map Lattices. New York: Wiley, pp. 1–49. Sound Example A2: The opening to Ludwig Karplus, K., and A. Strong. 1983. “Digital Synthesis of van Beethoven’s Fifth Symphony is played using Plucked String and Drum Timbres.” Computer Music the logistic map with different parameter values Journal 7(3):43–55. for each note. So there are too many values to Miranda, E. R. 1995. “Granular Synthesis of Sounds by Means of a Cellular Automaton.” Leonardo 28(4):297– list. 300. Sound Example A3: Finally, the sequence of notes Miranda, E. R. 2007. “Cellular Automata Music: From for a generated melody was produced by a computer Sound Synthesis to Musical Form.” In E. R. Miranda algorithm unrelated to the topic of this article. and J. A. Biles, eds. Evolutionary Computer Music. It is played using the circle map with parameters Berlin: Springer, pp. 170–193. ω = 0.13, K = 0.9, and = 0.3.

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