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Page 390 ï~~Synthesis from attractors Gordon Monro School of Mathematics and Statistics University of Sydney NSW 2006 Australia monro-gOmaths. su. oz. au Abstract This paper describes a method of synthesis of sustained sounds, derived from ideas in nonlinear dynamics, and making use of geometric shapes called attractors. An attractor can be either a simple geometric construction, or derived from a natural sound. The synthesis involves following the attractor and projecting the trajectory down to one dimension. Various sorts of timbral modification and cross-synthesis can be obtained. The method also produces multi-channel synthesis in a natural fashion. 1 Attractors Strictly speaking, an attractor is a set of points in n-dimensional space to which the solution of a system of differential equations converges. Examples are the well-known Lorenz and Rossler attractors of chaos theory, which are sets of points in 3-dimensional space [Holden 1986]. Attractors for experimental systems can be determined by making a sequence of measurements in time. (This is further discussed below.) For the purposes of this paper, an attractor is taken to be a set of points in n-dimensional space, where all (or most) of the points have designated successor points. 2 Artificial attractors A simple example in 2-dimensional space is the square shown in Figure 1. Here the. small circles represent the points making up the attractor, and the arrows indicate successor points. (Attractors used in sound synthesis normally contain more than 16 points.) To synthesize a sound from this attractor, we first construct a sequence of points: the point Pi is some point on the square; P2 is the successor point of pl, and so forth. To get a sequence of numerical values (sound samples), we project the points P1,P2... down to a line. If we project down to the line x = y we obtain the triangle wave shown in Figure 2a; projection down to the x-axis gives the waveform shown in Figure 2b. Changing the direction of projection amounts to cross-fading between two out-of-phase triangle waves. Rapid rotation of the direction of projection produces the waveform shown in Figure 3. Such rapid rotation amounts to a combination of amplitude modulation and comb filtering. It is also possible to modulate the lag, giving frequency modulation effects. Similar manipulations using 3-dimensional attractors can produce a large variety of waveforms.. ---}.. -.0-- - -+.4 -I I I I0 " -4--- *.4-- -- - * -- Figure 1: An artificial attractor Figure 2a: Sound obtained by projection Figure 2b: Effect of a different projection 4P.07 390 ICMC Proceedings 1993
Page 391 ï~~Figure 3: Effect of rapid rotation 3 Obtaining attractors from natural sounds Given a sequence of sound samples S.,,2,... we can obtain an attractor (in so-called pseudophase-space) as follows. Choose the dimension of the space: in this example I choose the dimension 4. Then choose a lag k. The first point of the attractor is p1 = (Si, 8k+1, 82k+1, 83k+1), the second point is p2 = (S2,$Sk+2,S82k+2) $3k+2), and so on. Furthermore we designate P2 to be the successor point of Pi, and so on. In this way we obtain a sequence of points in 4-dimensional space; each point has a successor except the last. If we carry out this procedure with a sine wave, we simply get an ellipse; with a natural sound the results can be like Figure 4 (which is a 2-dimensional attractor obtained from a bagpipe note). This method of constructing a higher-dimensional figure from a sequence of numerical values was given a theoretical basis in [Takens, 1981] and is widely used. An appropriate lag is about 1/3 of the period of the waveform (assuming that it is close to periodic); for a technique to calculate an optimal lag, see [Buzug et al., 1990]. by Jeff Pressing and co-workers [Pressing et al., 1993]. Figure 5 shows a 2-dimensional attractor obtained from a sound synthesized by the method of fractal interpolation [Monro, 1993]. Figure 5: Fractal waveform 4 Resynthesis from natural attractors For resynthesis from natural attractors, two techniques have been tried. The simpler technique is just to start with a short sample loop, and form an attractor in n-dimensional space, as in Section 3. If we declare the successor of the last point to be the first point, a closed trajectory is obtained. This can be projected on to different directions, rotated, and so forth, as indicated in Section 2. A wide variety of timbres can thus be derived from the original natural sound, at a fairly low computational cost. A second technique has been employed with longish samples where a closed trajectory is not obtained. First an attractor is constructed as in Section 3; this time not every point has a successor. We then proceed as follows [Casdagli 1989]. (It is assumed that we are working in n-dimensional space, where n has to be chosen large enough to avoid self-intersections of the attractor.) We construct a sequence of points in n-dimensional space; the sound will be obtained by projecting this sequence on to a line, as in Section 2. Start with a point p1 near the attractor. Find several (at least n +1) nearby attractor points, restricting ourselves to points that have successors. Use the successors of these nearby points to construct an "n-dimensional subspace of best fit" (analogous to a line of best fit but in 2n-dimensional space rather than 2-dimensional space). Use this to obtain the next point P2 -Then find points on the attractor near to P2, use Figure 4: Bagpipe note Colouring a picture like Figure 4 according to the number of times each pixel is visited by the trajectory can produce striking images. Images have also been obtained from 3-dimensional attractors, using colour to indicate the third dimension (depth). This method of visualization of sound, which as far as I know has not been previously introduced, has been further developed ICMC Proceedings 1993 391 4P.07
Page 392 ï~~these to compute p3, and so on. The method of singular value decomposition was used for these calculations. This technique is computationally intensive, but it can produce indefinitely sustained sounds. It differs from the previous technique in that the successive points obtained are not necessarily on the attractor, merely close to it. The trajectory can be varied by weighting points differentially and also by using variable scaling on the n coordinate axes. 5 Extensions It is possible to combine more than one sound. For example a 2-dimensional artificial attractor was combined with a 3-dimensional attractor derived from a bassoon note to make a 5 -dimensional attractor. The techniques of Section 2 can then be applied, yielding cross-syntheses. The method of this paper lends itself very easily to stereo or multichannel synthesis. For stereo, simply choose two different directions of projection, one for each channel. This is a case where the multichannel sound arises directly from the synthesis method, rather than being obtained from a monophonic sound by some separate panning process. The method is suited to synthesis of sustained sounds, and the natural sounds used were sustained portions of instrumental sounds. However, since we can choose the starting point on an attractor, it is simple to construct a sequence of points from an attack, and graft this seamlessly on to the beginning of the sound obtained by the methods described above. References [Buzug et al., 1990] Th. Buzug, T. Reimers and G. Pfister. Optimal reconstruction of strange attractors from purely geometrical arguments. Europhysics Letters, 13(7): pp.605-610, 1990. [Casdagli, 1989] Martin Casdagli. Nonlinear prediction of chaotic time series. Physica D, 35: pp.335-356, 1989. [Holden, 1986] Arun V. Holden (Ed.): Chaos. Manchester University Press, (Series Nonlinear science: theory and applications, ed. A.V. Holden), 1986. [Monro, 1993] Gordon Monro. Fractal interpolation waveforms. Manuscript accepted for publication. [Pressing et al., 1993] Jeff Pressing, Chris Scallan and Neil Dicker, Proceedings of the International Computer Music Conference, Tokyo, 1993. [Takens, 1981] F. Takens, Detecting strange attractors in fluid turbulence. In D. Rand and L.-S. Young (Eds): Dynamical Systems and Turbulence, series Lecture Notes in Mathematics, vol 898, Springer-Verlag, Berlin, 1981. 4P.07 392 ICMC Proceedings 1993