> Pretty sure my original post on this subject is weak (or worse), but
> I think this article might help. One should be able to construct an
> arbitrary periodic function using a non-trigonmetric basis. In other
> words, you *can* "create a sine by adding signals."
>
> http://ieeexplore.ieee.org/iel5/31/23557/01085543.pdf
> arnumber=1085543
>
> Further, seems to me the ear can be trained to listen for non-sine
> basis functions (aka "overtones"). Take for example, the sound of
> two or more oboes. Pretty sure one could separate the parts by
> pitch, although the parts aren't themselves pure sines. Muting one
> of the oboe parts would be filtering that "frequency", wouldn't it
>
> I guess I am agreeing with the original poster, and the answer as to
> why we analyse signals in terms of trigonometric basis functions is
> just mathematical (and design ) convenience.
>
> Doug

There have been a lot of approaches to replace the Fourier
synthesis/analysis by another orthogonal basis of functions/waveforms (e.g.
Walsh functions/Hadamard transform or the Haar functions) especially because
rectangle based functions (like Walsh) can be generated much easier in the
digital world. But after all the Fourier version is the most "natural" one
as it conforms with the behaviour of the human sense of hearing. And it's
much easier to understand and to handle compared to other synthesis forms.
The other synthesis forms are mathematically correct but much more difficult
to handle and to understand compared to the simple overtone principle of the
Fourier synthesis that follows the human sense of hearing.

Just my point of view ...

Dieter Doepfer