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## Sinusoidal waveshaping: evenness and oddness

Another interesting class of waveshaping transfer functions is the sinusoids:

which include the cosine and sine functions (got by choosing and , respectively). These functions, one being even and the other odd, give rise to even and odd harmonic spectra, which turn out to be:

The functions are the Bessel functions of the first kind, which engineers sometimes use to solve problems about vibrations or heat flow on discs. For other values of , we can expand the expression for :

so the result is a mix between the even and the odd harmonics, with controlling the relative amplitudes of the two. This is demonstrated in Patch E07.evenodd.pd, shown in Figure 5.14.

Figure 5.14: Using an additive offset to a cosine transfer function to alter the symmetry between even and odd. With no offset the symmetry is even. For odd symmetry, a quarter cycle is added to the phase. Smaller offsets give a mixture of even and odd.

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