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.

[From The
Music Page]