Sine functions can be represented as Taylor series. The terms of the Taylor
series are not periodic functions however. But we can redo the Taylor
series, taking advantage of the periodic nature of the sine wave, doing it
normally on the interval [0,2pi) and then "re-centering" at each multiple of
2pi, just copying the function from the [0,2pi) interval. The result is the
sine wave expressed as a sum of increasingly curvy sawtooths, albeit all of
the same frequency.

I think the explanation has more to do with the cochlea. As I understand
it, It has different sensors for different frequencies, and the frequency
sensed increases as you travel further into the tube. The sensor hairs
probably vibrate naturally in a sine waveform. Thus, I'm guessing, the
human ear naturally decomposes a wave into its Fourier components, absorbing
the energy from the lower frequencies in sine form, and then passing off the
rest down the tube.

This leaves open the question of a synthesizer filter based on a different
resonance waveform. Any thoughts on whether that's possible, what it would
sound like

Monroe

On Thu, Jul 3, 2008 at 1:50 PM, laryn91 < caymus91@... > wrote:

> I don't believe you can create a sine by adding signals with overtones.
> You can transfer the
> energy around in the spectrum with specific phase cancellations. But every
> time you
> cancel an overtone you create or reinforce another.
>
> In nature, sine functions are prevalent everywhere. On the other hand,
> square waves are
> non-existent and must always be synthesized. It would very atypical for
> nature to miss
> something as mathematically elegant as sines in favor of something more
> convoluted.
>
>
> --- In Doepfer_a100@yahoogroups.com <Doepfer_a100%40yahoogroups.com>,
> "Monroe Eskew" <monroe.eskew@...> wrote:
> >
> > I'm curious about harmonics. I've been looking for an explanation of why
> > different waveforms have different overtones. One explanation offered is
> in
> > terms of Fourier series. Every periodic function can be expressed as an
> > infinite sum of sine waves of increasing frequency and decreasing
> amplitude.
> > If we look at the Fourier series for a given curve (like a sawtooth or
> > square wave), then we can find the overtones by looking at the terms in
> the
> > sum.
> >
> > Now I like mathematics, but I'm not satisfied by this explanation. We can
> > express a function as a Fourier series, but we can also express it in
> other
> > ways. Perhaps a sine wave can be expressed as an infinite series of
> square
> > waves. Then a sine wave should have a lot of overtones.
> >
> > Here's my guess-- Qualitatively, different waveforms have different
> sounds,
> > and this does not necessarily need to be interpreted as having overtones.
> > However FILTERS are what truly reveal overtones. But the function of a
> > filter is determined by the fact that its resonant frequency is always a
> > sine wave. If we had square wave resonance, then we'd have totally
> > different filters, with the square wave being the least affected by the
> > filter.
> >
> > Is that more or less correct
> >
> > Also, does the Fourier expression make the most sense to the human ear
> > (i.e. Does the human ear have something akin to sine wave resonance )
> >
> > Thanks,
> > Monroe
> >
> >
> > [Non-text portions of this message have been removed]
> >
>
>
>


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