Example E05.chebychev.pd (Figure 5.12) demonstrates how you can use waveshaping to generate pure harmonics. We'll limit ourselves to a specific example here in which we would like to generate the pure fifth harmonic,

by waveshaping a sinusoid

We need to find a suitable transfer function . First we recall the formula for the waveshaping function (Page ), which gives first, third and fifth harmonics:

Next we add a suitable multiple of to cancel the third harmonic:

and then a multiple of to cancel the first harmonic:

So for our waveshaping function we choose

This procedure allows us to isolate any desired harmonic; the resulting functions are known as

To incorporate this in a waveshaping instrument, we simply build a patch
that works as in Figure 5.5, computing the expression

where is a suitable

By suitably combining Chebychev polynomials we can fix any desired superposition of components in the output waveform (again, as long as the waveshaping index is one). But the real promise of waveshaping--that by simply changing the index we can manufacture spectra that evolve in interesting but controllable ways--is not addressed, at least directly, in the Chebychev picture.