/ Digital Synthesis of Complex Spectra by means of Multiplication of Non-linear Distorted Sine Waves
ï~~ ACKNOWLEDGMENTS DIGITAL SYNTHESIS OF COMPLEX SPECTRA BY MEANS OF The author is endebted to the whole computer music community, and especially to J.C. RISSET, whose concern about timbre problems has been inspiring (and for a constant dialogue with him has proved to be very helpful), to Andy MOORER and Marc LE BRUN (who has done a parallel work on waveshaping - the two works are in fact complementary - indepen dently in Stanford University) for their comments and reviewing. This work has been done in the Laboratoire de Mdcanique et d'Acoustique of C.N.R.S. in Marseille (France). Â~ Compiler's note: This paper has been reprinted in these Proceedings with the kind permission of the Audio Engineering Society. This preprint has been reproduced from the author's advance manuscript, without editing, corretions or consideration by the Review Board. For this reason there may be changes should this paper be published in the Audio Engineering Society Journal. Additional preprints may be obtained by sending request and remittance to the Audio Engineering Society, Room 449,60 East 42nd Street, New York, N. Y. 10017. Â~Copyright 1978 by the Audio Engineering Society. All rights reserved. Reproduction of this preprint, orany portion thereof, is not permitted without direct permission from the publication office of the Society. MULTIPLICATION OF NON LINEAR DISTORTED SINE WAVES par DANIEL ARFIB Laboratoire de M6canique et d'Acoustique - Marseille - France INTRODUCTION. Synthesizing a sound by digital methods, for example using the Music V program (1) or special digital circuits, implies the definition of the mathematical procedure to be followed for constructing the sound, and for specifying its parameters. For example, additive synthesis consists of adding sine waves, the amplitudes of which are functions of time. Subtractive synthesis corresponds to the digital filtering of a complex waveform, wherein one changes the filter characteristics with time. However, with these two methods one must be painstaking to obtain sounds with rich musical quality and various playing possibilities. Recent research has focused on global methods of synthesis, which can be called modulation methods: a sound is defined by the evolution in time of one amplitude, one or more frequencies determining a line spectrum and one or more indexes of timbre determining the amplitude of these partials. This class of methods includes the FM synthesis technique described by Chowning (2) where the time variables are the amplitude, the carrier and modulation frequencies and the modulation index(es), and the discrete summation formula method described by Moorer (3,4) where the variables are the amplitude, the starting frequency, the frequency difference between partials and one index controlling the ratio of the successive partials. The power of such methods depends upon their ability to generate a palatte of related but distinct sounds. Similarly, in the syntax of the Music V program, there are two levels the instrument (INS definition) which can define a class of sounds, and the notes parameters (NOT), which can define one specific sound inside this class. In this sense, a global method is powerful if one can define differentiated instruments whith enough possible articulations in notes. Non linear distortion of sine waves (also called wave shaping) is a global method, which can be introduced as follows: the distortion of a sine wave by a "bad" amplifier produces a number of harmonics, the amplitude of which depend upon the amplitude of the input and the non linearity of the amplifier. This technique has been already used in computer
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