/ Digital Synthesis of Complex Spectra by means of Multiplication of Non-linear Distorted Sine Waves
ï~~ - 6 - - 7 - The index functions (Fl) is intended primarly to describe the variation of timbre with time. However the loudness is also a function of the index and care must be taken in the appreciation of the amplitude value (P5) that will often have to be variable and not fixed as in figure 5. Some automatic correction could be calculated by the means of a normalization function as discussed by Moorer (3). However it requires storage for a new function and computation time, it complicates the instrument and it is only approximate. Moreover this variation of loudness often sounds natural and the simplest way to use this instrument is to use if necessary the PS input as a post compensation of loudness. The transfer function (F3) generates the distortion and consequently it determines the harmonic content of the output as a function of the input. In general the timbre becomes richer as the index increases, but this is highly dependent on the choise of this transfer function. However some general remarks can be made: in the standard use of the non linear distorsion the do term (the constant of the polynomial) should be set to zero to avoid a click at the beginning and end of a note especially with softer (small index value) notes; the odd and even harmonics are independent and the maximum number of harmonics is equal to the degree of the polynomial (but the higher harmonics may be insignificant). When the order is small one can use the direct equalities (3) to predict this evolution. With larger values one needs the aid of a computer program displaying the evolution of the spectrum as a function of the index value. THE CHOICE OF A TRANSFERT FUNCTION. The characteristics of the non linear distortion are entirely determined by the coefficients of the polynomial. But there are many ways to calculate these coefficients. 1.- Evaluation from a steady-state spectrum. Relation 4 allows us to calculate the 0 coefficients (the polynomial) starting from the H ones (the spectrum) given an index value XO. We have seen that the distortion of a cosine wave gives only cosine components, the amplitude of which are real but can be positive or negative. So different distortions, hence different timbre variations, can be used which verify the initial conditions, only by changing the signs of some components. An abruptly limited spectrum or a rich spectrum (more than 20 harmonics) often gives rise to irregularities in the evolution of the sound with the index, which is akin to the Gibbs phenomenon. 2.- Evaluation from a continuous transfer functior In this case, the general form of the transfer function is known, and the problem is to get a polynomial approximation of it, in order to obtain a band limited spectrum and avoid foldover. A limited development of the function approximates it very well around the origin (small values of the index) but usually not in other areas. One can also take the complete transfer function, calculate the spectrum for a specific index value and then use the first procedure with the first N harmonics. This approximates well the spectrum for that value of the index, but abrupt bandwidth limitation causes the ripple effect previously described and it may prove useful to attenuate little by little the last partials values to get smoother evolutions with almost identical spectra (this is in effect a kind of windowing). Other classical algorithms can be used, such as the least mean square approximation. 3.- Direct evaluation. An experimental and/or intuitive choice of the coefficients of the polynomial is also a good strategy: there is a strong but subtle connection between the regularity of variation of the coefficients (and their sign) and the homogenity of the resulting timbre. Choosing to affect the same sign to all odd order coefficients, and doing so for even order coefficients (though this sign may be different of the other one) produces a very steep transfer function, which may produce too brassy sounds. A good practise can be to alternate the sign of successive odd order coefficients (say Al, AS, A9 positive, and A3, A7, All negative) and do the same for even order coefficients. SOME EXAMPLES. Examples 1 to 4 use the previously described Music V instrument, except that "amplitude" input (for correction of loudness) comes from an oscillator scanning once the compensation function F4. Example 1. Brilliant sounds. Many non linear distortion create impulse like waveforms (figure 6). In this case, the timbre becomes richer as the index increases.