[sdiy] harmonic generator & megaloopers

[jhaible]

> The only time I have seen this mentioned is Electronotes #45 Page 21
> "Transform methods in musical engineering" where Bernie talks about how
the
> Fourier series fails for gated sine waves. But he does not say what does
> happens exactly, or why fading in a sine wave vs. gating it removes the
> clicks mathematically.

Think of a gated or faded sine as a multiplication of the sine and the
envelope
(step functions for gating, ramp functions for linear fading, etc.)
Multiplication in time domain becomes convolution in frequency domain.
Fourier transform of rectangle function is a sin(x)/x function. Fourier
transform
of sine is a dirac impulse. (Actually two; the 2nd one is at negative
frequency).
So the convolution is easy: it's the sin(x)/x function itself.
Instead of a single "line" (dirac) at the frequency of the sine (f) you get
a function
with an absolute maximum at f, and with an infinite number of local maxima
with
decreasing amplitude. Distance between maxima depends on gate length.
This was for a rectangular gate function (I hope I got that right). You can
do a
similar calculation for other window functions.
As the convolution is always done with a dirac pulse (the sine wave in
frequency
domain), the harmonic contents of the window function will directly reflect
the amount of artefacts in the gated sine signal.
So a triangle (or trapezoid) will produce less "click" than a rectangle, and
I guess
a raised cosine would be even smoother.

JH.