[music-dsp] Osciallator harmonics
Tom Duff
td at pixar.com
Fri Apr 26 12:38:58 EDT 2002
From: "Steve Harris" <S.W.Harris at ecs.soton.ac.uk>
> A question for you, if a square wave is
> sin(x) + sin(3x)/3 + sin(5x)/5 ...
>
> and a saw is
> sin(x) + sin(2x)/2 + sin(3x)/3 ...
>
> and a triangle is
> sin(x) + sin(3x)/9 + sin(5x)/25 ...
>
> what is the obvious missing wave in the set:
> sin(x) + sin(2x)/4 + sin(3x)/9 ...
There's a mistake here. A triangle is
sin(x) - sin(3x)/9 + sin(5x)/25 - ...
with 180 degree phase shift at each term.
So a better candidate for the missing
wave in the set is
sin(x) + cos(2x)/4 - sin(3x)/9 - cos(4x)/16 + ...
which has the same odd terms as the triangle,
and has a uniform 90 degree phase shift from
term to term.
This is, I believe, a section of a parabola, repeated
periodically.
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