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Page 327 ï~~Geometric Sound Transformations Jon Drummond JRCASE CSIRO-Macquarie University NSW 2109 Australia j ond~macadam. mpce. mq. edu. au Gordon Monro Mathematics and Statistics University of Sydney NSW 2006 Australia monrog.maths, su. oz. au Abstract This paper describes a family of geometrically based methods for sound modification, inspired by techniques used for image manipulation. The methods can be seen as higherdimensional analogues of wave-shaping, with the advantage of an intuitive visual interface for the operations being performed. All the methods described here operate in the time domain. 1 Introduction This paper describes a family of geometrically based methods for sound modification, inspired by techniques used for image manipulation. The methods can be seen as higher-dimensional analogues of wave-shaping, with the advantage of an intuitive visual interface for the operations being performed. All the methods described here operate in the time domain. The starting point is a previously developed procedure for representing a monophonic sound as a two-dimensional (or higher-dimensional) image. This is the method of lagged embedding ([Monro 1993], [Pressing et al 1993]), where a sequence of samples 1,8s2,... together with the choice of a lag k generates the sequence of points in the planeP1 = (s1,S1+k), P2 = (82,82+k), etc. 2 Image Warping Our first group of procedures is derived from image warping: by distorting the plane containing the image derived from the sound and then reconstructing the sound, we obtain a modified sound. A warping can be written as a function f: It x JR --f R x JR (where JR is the real line). Such a function has two components f, (x, y) and f2(z,y), where f(z,y) = (fi(z,y),f2(x,y)). The component f, gives the warping in the x direction of each point and the component f2 gives the warping in the y direction. We have used a grid warping technique, which generalizes readily to three or more dimensions. For the two-dimensional warping above, we can reconstruct the sound by projecting onto the x axis; the sequence of output samples is then fl(sl,sl+k), fl(s2, s2+k), etc. Since f, is a nonlinear function the result is not a straightforward transformation of the input sound; we can consider the technique as a higher-dimensional version of wave-shaping. Projecting onto the y axis in general produces a different waveform. The illustration shows a warping of the plane, together with the result of this warping applied to an ellipse. (The ellipse was derived from a sine wave.) Warping of the plane ICMC Proceedings 1994 327 Audio Signal Processing
Page 328 ï~~The waveform obtained by projecting the warped ellipse onto the x axis is shown below. Twisted waveform Waveform from warped ellipse Other geometric operations include, for example, constructing a three-dimensional image of a sound and then clipping to a spherical volume; a variety of "soft clippings" can be obtained. Much more radical geometric transformations are also possible, for example the transformation of inverting the waveform in a sphere. 3 Dynamic transformations The warping procedures described above are "static", in that the warping function does not change with time. Another class of procedures arises from explicitly including time in the representation. We can create a three-dimensional representation of a sound with two space dimensions and one time dimension: the result is similar to a helix. We can then apply geometric transformations to the whole three-dimensional image. This allows new forms of modulation to be created. One we have investigated is twisting the whole helix (like tightening up a spring). This functions like a generalized amplitude modulation, with selective cancellation of harmonics. The illustration shows (a) a two-dimensional plot of a simple waveform (with just the fundamental and second harmonic present), and (b) a similar plot of the same waveform "twisted" in the manner just described. All of these methods can be extended to "artificially constructed" sequences of points not deriving from a natural sound, thus giving a family of sound synthesis methods, and also to multichannel sound tracks. These latter allow operations where one sound subtly influences another. Representation of an audio signal in two or three dimensions in the manner indicated above allows for a very intuitive and powerful analysis of the signal. Many subtle variations are easily revealed using this representation such as the shape of envelopes, similarities and differences between different timbres, and so forth. Such analyses and comparisons can be used to provide a basis for further modifications to the signal. These can be made directly in the display, graphically or algorithmically. For display we use the Xli window system; colour is used in our images either to help with perception of the third dimension or to indicate the passage of time in a sound sample. 4 References [Monro, 1993] Gordon Monro, Synthesis from attractors. Proc. Internat. Computer Music Conf., Tokyo: pp. 390-392, 1993. [Pressing et al, 1993] Jeff Pressing, Chris Scallan and Neil Dicker, Visualization and predictive modelling of musical signals using embedding techniques Proc. Internat. Computer Music Conf., Tokyo: pp. 110-113, 1993. Original waveform Audio Signal Processing 328 ICMC Proceedings 1994