Computer Music: Musc 216
Computer Music Terms

Wave Shaping Synthesis

There is more than one form of wave shaping synthesis around. A form developed by Risset, is used quite effectively in several wave editing programs such as Cool Edit (Syntrillium Software) as an algorithm for the creation of a wide variety of distortion effects. For some reason, Cool Edit provides an input interface which allows only a form of input suitable to distortion, even while the algorithm may be used to create other effects also. But for reason of its usefulness in distortion, it is detailed in another article on this site, Distortion and Civilised Behaviour.

It will be more productive, however, to look for now at the application of Chebychev Shaping Functions to soundwaves, as developed by Arfib and LeBrun. It has been demonstrated that the Chebychev polynomials can be used to add specific harmonics to a constant sinusoid wave. A shaping function is derived from the input wave, either a sine or cosine, which consists of values between -1 and +1, and an appropriate member of the set of Chebychev polynomials. Taking T(sub k) to indicate one of these polynomials, and k to be the ordinal value of the polynomial within the set, the shaping function, w, can be described like this:

This is another way of saying that the frequency of the wave which is generated from the left hand portion is k times the frequency of the fundamental or input wave. To put it yet another way, the output is the kth harmonic.

As an example, a shaping function to generate a steady cosine wave so as to add a 2nd harmonic at 0.4 of the fundamental amplitude and a 3rd harmonic at 0.2 of the fundamental amplitude a formula like this would be used:

The results, using changing x values for the Chebychevs, can then be placed in a transfer function wavetable (or to put it in programming jargon, a lookup table). An input cosine wave then contains the harmonics that appear here as the k values.

The foregoing has been only the foundation to the theory. Arfib showed that an input of a wave having a changing frequency (as opposed to the constant frequency of the above) yields inharmonic partials and formant structures. More importantly from a musician's point of view is the effect generated when the input wave is a complex recorded sound. The effect is similar to phase shifting, as undulating harmonics are generated.

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