Abstract

A new application of the well-known process of frequency modulation is shown to result in a surprising control of audio spectra. The technique provides a means of great simplicity to control the spectral components and their evolution in time. Such dynamic spectra are diverse in their subjective impressions and include sounds both known and unknown. . INTRODUCTION Of interest in both acoustical re-search and electronic music is the synthesis of natural sound. For the researcher, it is the ultimate test of acoustical theory, while for the composer of electronic music it is an extraordinarily rich point of departure in the domain of timbre, or tone quality. The synthesis of natural sounds has been elusive; however, recent research in computer analysis and synthesis of some tones of musical instruments has yielded an insight which may prove to have general relevance in all natural sounds: the character of the temporal evolution of the spectral components is of critical importance in the determination of timbre. 1 In natural sounds the amplitudes of the frequency components of the spectrum are time-variant, or dynamic. The energy of the components often evolves in complicated ways, in particular, during the attack and decay portions of the sound. The temporal evolution of the spectrum is in some cases easily followed as with bells, whereas in other cases not, because the evolution occurs in a very short time period, but it is nevertheless perceived and is an important cue in the recognition of timbre. Many natural sounds seem to have characteristic spectral evolutions that, in addition to providing their "signature," are largely responsible for what we judge to be their lively quality. In contrast, it is largely the fixed proportion spectrum of most synthesized sounds that so readily imparts to the listener the electronic cue and lifeless quality. The special application of the equation for frequency modulation, described below, allows the production of complex spectra with very great simplicity. The fact that the temporal evolution of the frequency components of the spectrum can be easily controlled is perhaps the most striking attribute of the technique, for dynamic spectra are achieved only with considerable difficulty using current techniques of synthesis. At the end of this paper some simulations of brass, woodwind, and percussive sounds are given. The importance of these simulations is as much in their elegance and simplicity as it is in their accuracy. This frequency modulation technique, although not a physical model for natural sound, is shown to be a very powerful perceptual model for at least some. FREQUENCY MODULATION Frequency modulation (FM) is well understood as applied in radio transmission, but the relevant equations have not been applied in any significant way to the generation of audio spectra where both the carrier and the modulating frequencies are in the audio band and the side frequencies form the spectrum directly. In FM, the instantaneous frequency of a carrier wave is varied according to a modulating wave, such that the rate at which the carrier varies is the frequency of the modulating wave, or modulating frequency. The amount the carrier varies around its average, or peak frequency deviation, is proportional to the amplitude of the modulating wave. The parameters of a frequency-modulated signal are c = carrier frequency or average frequency m = modulating frequency d = peak deviation. The equation for a frequency-modulated wave of peak amplitude A where both the carrier and modulating waves are sinusoids is where e = the instantaneous amplitude of the modulated carrier ! = the carrier frequency in rad/s ! = the modulating frequency in rad/s I = d/m = the modulation index, the ratio of the peak deviation to the modulating frequency. It is obvious that when I = 0 the frequency deviation must also be zero and there is no modulation. When I is greater than zero, however, frequencies occur above and below the carrier frequency at intervals of the modulating frequency. The number of side frequencies that occur is related to the modulation index in such a way that as I increases from zero, energy is "stolen" from the carrier and distributed among an increasing number of side