We return again to the spectra computed on Page ,
corresponding to waveshaping functions of the form . We note
with pleasure that not only are they all in phase (so that they can
be superposed with easily predictable results) but also that the spectra
spread out as increases. Also, in a series
of the form,

a higher index of modulation will lend more relative weight to the higher power terms in the expansion; as we saw seen earlier, if the index of modulation is , the various terms are multiplied by , , , and so on.

Now suppose we wish to arrange for different terms in the above expansion
to dominate the result in a predictable way as a function of the index .
To choose the simplest possible example, suppose we wish to be the largest
term for , then for it to be overtaken by the more quickly growing
term for , which is then overtaken by the term for
and so on, so that each th term takes over at index .
To make this happen we just require that

and so choosing , we get , , , and in general,

These happen to be the coefficients of the power series for the function

where is Euler's constant.

Before plugging in as a transfer function it's wise to plan how we
will deal with signal amplitude, since grows quickly as
increases. If we're going to plug in a sinusoid of amplitude , the maximum output
will be , occurring whenever the phase is zero. A simple and natural
choice is simply to divide by to reduce the peak to one, giving:

This is realized in Example E06.exponential.pd. Resulting spectra for 0, 4, and 16 are shown in Figure 5.13. As the waveshaping index rises, progressively less energy is present in the fundamental; the energy is increasingly spread over the partials.