Sonic Explorations of Gumowski-­Mira Maps

Sonic Explorations of Gumowski­Mira Maps Timothy S. H. Tan Institute of Sonology The Royal Conservatoire, The Hague WLPEUHWDQ#JPDLOFRP ABSTRACT This paper studies the use of Gumowski­Mira maps for sonic arts. Gumowski­Mira maps are a set of chaotic sys­ tems that produce many organic orbits that resemble cells, flowers and other life forms. This has prompted mathema­ ticians and eventually artists to study them. These maps carry a potential for use in the sonic arts, but until now such use is non­existent. The paper describes two ways of using Gumowski­Mira maps: for synthesis and spatializa­ tion. The synthesis approach, which runs in real­time, takes the dynamical system output as the real and imagi­ nary input to an inverse Fourier transformation, thus di­ rectly sonifying the algorithm. The spatialization ap­ proach uses the shapes of Gumowski­Mira maps as shapes across the acoustic space, using the first 128 iterations of each map as audio particles. The shapes can change based on the maps’ initial parameters. The maps are explored in live performance using Leap Motion and Cycling ’74’s MIRA for iPad as control interfaces of audio processing in SuperCollider. Examples are given in two works, Cells #1 and #2. 1.INTRODUCTION The Gumowski­Mira map (henceforth called GM map) was named after I. Gumowski and C. Mira, who were stud­ ying various chaotic maps from the 1960s [1]. Around 1970, while at CERN, Gumowski was studying chaotic in­ stability in accelerators and storage rings, and collaborated with the Toulouse Research Group led by Mira [1, pp. 129­ 131]. What is now known as the GM map was based on a dissipative perturbation of one such map studied by Gumowski and then with Mira [1, pp. 121­124]. It com­ prises the following equations: ‫ݔ‬௡ାଵ ൌ ‫ݕ‬௡ ൅ ߙǤ ‫ݕ‬௡ Ǥ ሺͳ െ ߪǤ ‫ݕ‬௡ଶ ሻ ൅ ‫ܨ‬ሺ‫ݔ‬௡ ሻ ‫ݕ‬௡ାଵ ൌ െ‫ݔ‬௡ ൅ ‫ܨ‬ሺ‫ݔ‬௡ାଵ ሻ (1) There are a few variants of GM maps. According to [1, pp. 180­184], two different F(x) exist: ଶ ‫ܨ‬ሺ‫ݔ‬ሻ ൌ ߤǤ ‫ ݔ‬൅ ሺͳ െ ߤሻǤ ‫ ݔ‬Ǥ ݁ ‫ܨ‬ሺ‫ݔ‬ሻ ൌ ߤǤ ‫ ݔ‬൅ ଶǤሺଵିఓሻǤ௫ మ ଵା௫ మ భషೣమ ర (2) (3) Copyright: © 2017 Timothy S. H. Tan and PerMagnus Lindborg. This is an open­access article distributed under the terms of the Creative Commons Attribution License 3.0 Unported, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. H EA RING THE S E LF PerMagnus Lindborg School of Art, Design and Media Nanyang Technological University SHUPDJQXV#QWXHGXVJ The equation (1) also appears in a slightly different form in [1, pp. 179], with F(x) is similar to (3): ‫ݔ‬௡ାଵ ൌ ‫ݕ‬௡ ൅ ߙǤ ‫ݕ‬௡ Ǥ ሺͳ െ ߪǤ ሺ‫ݕ‬௡ଶ ൅ ͵Ǥ ‫ݔ‬௡ଶ ሻሻ ൅ ‫ܨ‬ሺ‫ݔ‬௡ ሻ ‫ݕ‬௡ାଵ ൌ െ‫ݔ‬௡ ൅ ‫ܨ‬ሺ‫ݔ‬௡ାଵ ሻ (4) ‫ܨ‬ሺ‫ݔ‬ሻ ൌ ߤǤ ‫ ݔ‬൅ ଶǤሺଵିఓሻǤ௫ మ ଵା௫ ర (5) These variations can produce similar shapes, albeit with slightly different ranges of usable initial parameters. GM maps have some benefits not found in other chaotic systems. Across the phase space, GM maps can produce intricate, organic images that resemble cells, flowers and other life­forms. These images can vary a lot in shapes, and often possess different rotational or reflection symmetries. Whereas a number of mathematicians [2] and artists [3] have studied GM maps in detail, until now nobody has used these maps as sonic parameters. Besides, GM maps are still less well­known than other chaotic systems such as the logistic map, Chua circuit, Rössler attractor and double pendula. After all, chaotic systems have been used to map onto frequencies [4, 5], rhythm and note durations [4], dynamic levels (velocities) [5], timbre [6, 7], grain fre­ quencies [6] and grain lengths [7]. Even SuperCollider [8] has many chaotic UGens that generate sounds based on audio­rate calculations of chaotic systems, such as LinCongN and GbmanTrig. This article presents how the two authors have used GM maps for synthesis and spatialization, with their methodol­ ogies, results and discussions. 2.GM MAPS FOR SYNTHESIS Lindborg developed a program in Max [9] for concurrent real­time synthesis and visualization of a GM map. The state of the GM map is updated for each audio sample (Fig­ ure 1) and its output (x, y) is mapped to the real and imag­ inary input of an inverse Fourier transform (Figure 2). Used as an experimental audio synthesizer, the output has been employed in two sonic artworks in the past year [10] [11]. As pointed out by Dean et al. [12], the concurrent audio visual output of the system correspond fully on a synthesis level but this does not necessarily mean that they corre­ spond perceptually. Future work aims to evaluate experi­ mentally the extent to which auditory and visual modalities of the outputs correspond. 195 acoustic space. They also easily broach the “conflict / co­ existence” behavioral relationship addressed by Smalley [21]: Tiny changes to initial parameters create small errors that accumulate and later cause great deviations to the or­ bits. These orbits can conflict and even tangle each other. Figure 1. Max implementation of the dynamical system feedback. Figure 2. Max implementation of the GM map’s two­di­ mensional output as input to an inverse Fourier transform. 3.GM MAPS FOR SPATIALIZATION To date, chaos has been applied to spatialization in [13] and fractals in [14], but for algorithmic spatialization, there has been some progress. Some have implemented algorith­ mic spatialization with OpenMusic, as in [14]–[16], whereas Schacher et al. have composed with swarm algo­ rithms and flocking, both controlling spatialization among other parameters [17]. Lindborg has utilized particle colli­ sions registered in bubble chamber images [18] as well as the collision of particles [19] for algorithmic spatialization. Using GM maps for spatialization can prove interesting and innovative. The numerous patterns of GM maps tend to be highly distinctive and thus can provide equally dis­ tinctive spatial shapes across the acoustic space. The shapes also change unpredictably and very quickly even if there are tiny changes to the initial parameters. For Tan, this meant a vast choreography of spatial shapes. In fact, Deleuze and Guattari, while discussing philosophy, had even reflected on the speed at which these shapes change: Chaos is not so much defined by its disorder as by the infinite speed with which every form taking shape in it vanishes. It is a void that is not a nothingness but a virtual, con­ taining all possible particles and drawing out all possible forms, which spring up only to disappear immediately, without consistency or reference, without consequence. Chaos is an infinite speed of birth and disappearance [20]. Their last point makes spatializing chaotic maps like GM maps a highly efficient tool in building up tension and in­ tensity, furthermore engaging listeners through a given 196 Moreover, using chaotic maps for spatialization is more robust than for other parameters. When other parameters rely on streams of numerical outputs, the heard results of­ ten correspond poorly with the original map used. For in­ stance, the timbres of the gingerbread man map in GbmanTrig from SuperCollider may sound interesting, but listeners cannot identify its timbre specifically with GbmanTrig. Besides, scaling, rotation, translation and skewing of these maps easily preserves their visual percep­ tion, but distorts their listening experience when heard as streams of output. On the other hand, spatialization of cha­ otic maps rely on graphical representations of these maps, thereby preserving the shapes across the acoustic space. Because GM maps can create captivating visual shapes, Tan has focused on recreating these shapes through spati­ alization of GM maps in the acoustic space. 3.1 Implementation When chaotic maps exhibit their sensitivity to initial con­ ditions, they imply the need to compare between close neighboring values of initial parameters. Plotting the di­ vergent orbits one iteration at a time inadequately provides a fuller picture of these maps, and worse still when the pre­ vious iterations disappear as the sound travels along the orbit. This makes the implication of comparison too long and weak, thereby poorly demonstrating sensitivity to ini­ tial conditions. Instead, Tan uses all the iterations of each GM map as simultaneous audio particles for the acoustic space, similar to Fonseca’s audio particle system in his “Sound Particles” software [22] [23]. These iterations are updated instantaneously whenever the initial parameters change. Both methods easily highlight the comparisons and preserve the picture of the maps, and thus the maps can continue to demonstrate sensitivity to initial condi­ tions. One challenge is the need to balance between providing an adequate image of the GM map and minimizing CPU processing usage. As such, only 128 iterations per map are used as audio particles, and only three GM maps are used. Too many particles can overload the CPU and disrupt real­ life performances involving GM maps. Additionally, whenever the GM maps change their shapes, the maps’ particles skip directly to their new posi­ tions, without the need to consider the Doppler Effect. Un­ like a particle system in which a particle can have a life of its own, here the audio particles live and die together whenever their respective maps are switched on or off. A visualizer of the particle systems ensures that the per­ former can see where the particles are; such a feature also doubles as visuals for the audience to observe the particles’ positions and orbits. The size of the speaker space, which 2017 IC M C / EM W covers the audiences in a concert hall, is similar to the size of the visualizer, so as to maintain close correspondence between audio and visuals. Audio particles can exceed the boundaries of the visualizer and sound behind the speakers (often more softly), so as not to overcrowd within the speaker space. This allows better localization of the parti­ cles (Figure 3). Each GM map can be switched on or off. Usually only one GM map starts the work, then is joined by the second, and finally the third. The range of iterations is confined within the first 128 iterations. This can be narrowed, so as to scrutinize the movements of that number of particles. With modulo op­ eration, the iteration index can be filtered with divisors of 1­7 and remainders of 0­6. At times where the GM maps often look similar, this modulo filter reveals that the same iteration index of each map occupies very different places (Figure 4). As such, the modulo filter is often used to play with the spatiality of the GM maps. Figure 3. The visualizer of the particles’ positions based on the three GM maps. Dashed square lines at the center indicate the size of the speaker space, which in real life sur­ rounds the audiences in a concert hall. Not too many particles are allowed to crowd within the speaker space, so as to make the spatial shape of the GM maps clearer. Other than using three GM maps with a max­ imum of 128 audio particles per map, second­order 2D Ambisonics is used. The spatial image of the GM maps can be zoomed in and out, so as to reveal the intricate inner shapes of the maps. The speaker setup, which surrounds the audiences, is octophonic. This is used along with the visualizer mentioned just above. Figure 4. An instance of three GM maps played together in Cells #2. Left: without modulo filter; right: with modulo filter of divisor 4 and remainder 2. The GM maps are performed with MIRA on iPad (with Max 7) by Cycling ’74 [9], as well as Leap Motion [24] (with Processing 3 [25]). These two interfaces enable the performer to play with speed: MIRA for slow changes, and Leap Motion for fast changes. Leap Motion communicates with MIRA via Open Sound Control (OSC), and likewise for MIRA with SuperCollider. In order to differentiate which particles belong to which GM map, each GM map has a different synth and thus tim­ bre. This is to prevent all the audio particles from different GM maps from blending together as if they are generated from the same map. Normally one map contains some ad­ ditive synthesis UGens, another some ChaosGens, and the last granular synthesis. In turn, their particles have varying parameters based not on their iteration index, but their po­ sitions in space. The particles’ positions can affect the par­ ticles’ frequencies and other audio effects. This is im­ portant in varying the sounds of each particle, because if the same sound is applied to every particle in a particle system, it sounds almost similar to that sound used in one particle whose size covers a far larger space (e.g. directly from one speaker), thereby making the particle system use­ less. For the visualizer, each GM map is assigned a color, as a visual aid to differentiate the GM maps. Leap Motion is very sensitive to the performer’s hand positions. This fits well for performing chaotic works, since chaotic systems are also sensitive to initial condi­ tions, and accuracy is not so important in this context. This enables the resultant shapes to transform quickly, but not slowly, because it is difficult to control hand positions un­ der Leap Motion’s high sensitivity. The hand parameters are chosen and mapped based on ease and comfort. Only the ranges of Į= [­0.25, 0.25], ı= [­0.25, 0.25], ȝ = [­1.0, 1.0], x0 = [­2.0, 2.0] and y0 = [­2.0, 2.0] are used as the initial parameters, as in these ranges, generally the maps are conservative, do not enter infinity too soon, and do not settle into a stable attractor at the origin within the first 128 iterations. The last feature tends to create an unwanted cen­ tral ringing tone that overwhelms other audio particles. The form is based on the rate of change of the initial pa­ rameters. The music starts slowly, exploring and scrutiniz­ ing the different spatial shapes of each map, before speed­ ing up from the middle. Often the shapes are repeated at the start, so that listeners can realize that they are not lis­ tening to a random system, but a chaotic system of unpre­ dictable, but specific shapes. In contrast, MIRA allows the performer to play with one or two parameters at a time on the iPad. This mode allows better accuracy and slower changes than the Leap Motion mode. One can switch between the MIRA mode and the Leap Motion mode, in which only one of the two can affect the parameters. For both cases, MIRA will track the changes in the parameters as a visual feedback to the per­ former. (MIRA also tracks the parameters changed by Leap Motion through OSC.) H EA RING THE S E LF 197 Figure 5. Three GM maps for Cells #1. Top: (1) with (2) for F(x), middle: (1) with (3), bottom: (4) with (5). Į = ­0.03125, ı = ­0.125, ȝ = ­0.765625, x0 and y0 = 0.25, and n = 128. Ranges of both axes are [­12.5, 12.5]. 198 Figure 6. Three GM maps for Cells #2, using (1) with (3) for F(x), Į = ­0.03125, ı = ­0.125, x0 and y0 = 0.25, and n = 128, but ȝtop = ­0.765625, ȝmiddle = ȝtop + 2­12 and ȝbottom = ȝtop + 2­11. Ranges of both axes are [­12.5, 12.5]. 2017 IC M C / EM W 3.2 &HOOVand&HOOV Tan composed Cells #1 (2016, rev. 2017) and Cells #2 (2017) with Leap Motion and Max with MIRA for perfor­ mance interface, and SuperCollider 3.8.0 for audio pro­ cessing. Both works are to be performed live for a two­ dimensional 8.1 speaker setup. Usually visuals of the GM maps are projected to a large screen in front of the audi­ ences, because GM maps remain unknown to many. Cells #1 involves three GM maps, formed by coupling (2) and (3) with (1), and (5) with (4), and using the first 128 iterations (Figure 5). Tan wants to study how the shapes will look and sound if the same set of initial parameters to these three maps is applied. Cells #2 also involves three GM maps. This time, all three use the same equation (1), with (3) for F(x), but the first and the second had ȝ that differs by 2­12, and likewise for the second and third (Fig­ ure 6). In both works, all these maps exist alongside each other and contest for spatial prominence and likeness and differences on their orbits in the same acoustic space. Both works are meant for an octaphonic speaker setup with sec­ ond­order Ambisonics, for greater spatial clarity. 3.3 Artistic Results The spatial shapes can generally be identified, especially by applying the modulo filter, but more work is needed to make spatial shapes even clearer. The modulo filter and the limit on the range of iterations work best for Cells #2, es­ pecially when the maps often overlap each other but the particles with the same iteration indices are at different or even opposite places. Timbre­wise, the chosen timbres for each GM map remain distinct and do not blend, and this is required. One listener had even highlighted the timbral va­ riety of Cells #2. Additionally, some listeners had found the performance with MIRA and Leap Motion very engag­ ing, and noted the close correspondence between the sounds and visuals. Meanwhile, Cells #1, originally com­ posed in 2016, is now being revised to be on par with Cells #2. 3.4 Discussion Chaotic maps, such as GM maps, can be a powerful tool for spatialization. However, it can be difficult trying to rep­ licate the success of visualizing GM maps (or other chaotic maps) across the audio spatial realm, when creative moti­ vations conflict with technical constraints. Attempts to produce complete shapes of GM maps au­ rally with a lot of audio particles will cause overcrowding inside the listening space, thus blurring the choreography of the spatial shapes. As such, just the first 128 iterations for one map are already adequate enough as audio parti­ cles. Besides, having a lot of particles enhances and does not blur the visual experience, but blurs and distorts the listening experience. While visual particles do not diffuse light themselves and blur their own shapes, audio particles can at best only represent the ideal visual particle. Breg­ man elaborates that: H EA RING THE S E LF This way of using sound has the effect, how­ ever, of making acoustic events transparent; they do not occlude energy from what lies behind them. The auditory world is like the visual world would be if all objects were very, very transparent and glowed in sput­ ters and starts by their own light, as well as reflecting the light of their neighbors. This would be hard world for the visual system to deal with [26]. Despite efforts to reproduce the shapes of the GM maps as faithfully as possible, the acoustic space has some re­ strictions. One implication of Bregman’s is that while anyone can replicate the directionality, scale and distribu­ tion of particles across the space, the sharpness in the edges and corners remains poorly defined. This can easily blur the shapes’ intricate designs and corners, as well as cover the holes in between the particles. As such, having thou­ sands of audio particles like of visual particles do not en­ hance the listening experience similarly to the visual expe­ rience. By this approach, one can increase the number of speakers for better spatial definition and clarity, though perhaps not so cost­effectively. One drawback of performing with real­time audio parti­ cle systems is that as the number of particles increase, both the demand for audio processing power and the chance of a breakdown also increase. Whereas visual implementa­ tions of GM maps tend to involve hundreds or even thou­ sands of particles, doing likewise for audio easily over­ loads the CPU and thus handicaps real­life performances of the works. Currently Fonseca’s “Sound Particles” soft­ ware still cannot perform real­time rendering and play­ back. Cells #2 went smoothly during rehearsals and sound­ check, but halfway through the actual performance at the Institute of Sonology, crashed just after the three GM maps have been introduced. This was likely due to the audio in­ terface (Focusrite Scarlett 18i20) unable to handle Super­ Collider’s immense processing power, thereby causing the audio interface to disconnect from SuperCollider. The per­ formance had to be aborted. Work is underway reducing CPU usage in one laptop, perhaps by splitting the pro­ cessing power into more than one laptop and inviting one extra performer, for both Cells. While tempo is used to build form, listeners often per­ ceive fast changes in spatial shapes as random instead of being chaotic. These intricate shapes are hence no longer perceived as specific to GM maps, even if they indeed are. As such, the tempo will be reduced across both Cells. Cer­ tain shapes may be paused, so that the listeners can per­ ceive these shapes fully. Another problem is the need to adjust the amplitudes of all the audio particles, since audio particles can cancel each other a bit, possibly due to phasing. This occurs especially with larger number of audio particles. As such, the sum of the audio particles can sound less loudly than the ideal to­ tal, and needs correction. 199 4.CONCLUSION Using GM maps for sonification remains a promising sub­ ject to explore. GM maps’ captivating shapes have in­ spired both authors to sonify them. 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