Re: 7-tet and almost seven equal - Chopi

Hi Andris,

Yes, Fourier's work was on heat flow,
and using fourier series to analyse that. However,
the fourier series, as you said, are also used to
analyse sounds into their partials - when you
use a program that finds the partials of a timbre,
then that is likely to be the method that it uses
to do it.

One can use fourier series to convert any kind of
a curve that isn't too wildly discontinuous
into a sequence of frequencies that adds up to
the same shape over a finite region.

The reasons may not be well understood, but
when one finds the frequencies that add up to
make a musical note, the frequencies one gets
correspond to the partials that the ear actually
hears when listening to that note, if you listen
to it carefully. At least, one can train to
hear the partials in a musical timbre;
it's something the ear is able to do.

If you make a note out of those partials then
the result sounds pretty much like the original
in a general kind of way, however it isn't exact
because of the time varying component of it
- in real life partials of a sound start, stop,
and change in pitch as the note progresses.

The fourier series if done exactly would capture
all that too, by adding in suitable low frequency
components, because it can exactly reconstitute the original
wave shape, so long as one also preserves the
phase information too. However, in practice
that doesn't really work too well and
combined approaches are needed.

Anyway, it's not so surprising that if one adds
up the partials one gets a note that sounds
somewhat like the original timbre. It is
a little surprising that the ear is able to
hear those constituent partials of a timbre
- if you look at it on an oscilloscope you
see the basic shape as a repeating pattern,
but you don't see the partials at all.
E.g. a saw tooth waveform could just as
easily be perceived as a simple shape in
its own right. Also a square wave ditto.

I suppose the question is,
why do we hear sounds as made up of sine
waves, rather than, say, square waves
(which is anohter way of analysing it).

Also, do we hear in terms of exact sine
waves, or is that just a first approximation
to the way we hear the partials of a sound?

Robert

--- In MakeMicroMusic@y..., "Robert Walker" <robertwalker@n...> wrote:

> I suppose the question is,
> why do we hear sounds as made up of sine
> waves, rather than, say, square waves
> (which is anohter way of analysing it).

simple: study the physics of the cochlea (or of any resonator).
square waves may work mathematically, but they don't make sense
*physically*. a resonant body will be excited anytime there's a *sine
wave component* present that agrees with its resonant frequency.
pretty simple differential equations behind this. the cochlea is
essentially a system of a large number of resonators, overlapping so
as to be sensitive to the full range of frequencies 20-20,000 Hz (at
best).

> Also, do we hear in terms of exact sine
> waves, or is that just a first approximation
> to the way we hear the partials of a sound?

well, i suppose it's a first approximation in the sense that what we
hear is actually time-varying . . . but in the sense that some other
periodic waveform would be more correct, no -- i remember wondering
about this pretty intensely as an undergraduate and ended up
discovering that the sine/cosine function is really unique in this
regard.

this should probably continue elsewhere, either metatuning or
tuning . . .

On 2/26/02 5:30 PM, "paulerlich" <paul@...> wrote:

> --- In MakeMicroMusic@y..., "Robert Walker" <robertwalker@n...> wrote:
>
>> I suppose the question is, why do we hear sounds as made up of sine waves,
>> rather than, say, square waves (which is anohter way of analysing it).
>>
> simple: study the physics of the cochlea (or of any resonator). square waves
> may work mathematically, but they don't make sense *physically*. a resonant
> body will be excited anytime there's a *sine wave component* present that
> agrees with its resonant frequency. pretty simple differential equations
> behind this. the cochlea is essentially a system of a large number of
> resonators, overlapping so as to be sensitive to the full range of frequencies
> 20-20,000 Hz (at best).
>

I never noticed that but it makes a lot of sense.

>> Also, do we hear in terms of exact sine waves, or is that just a first
>> approximation to the way we hear the partials of a sound?
>>
> well, i suppose it's a first approximation in the sense that what we hear is
> actually time-varying . . . but in the sense that some other periodic waveform
> would be more correct, no -- i remember wondering about this pretty intensely
> as an undergraduate and ended up discovering that the sine/cosine function is
> really unique in this regard.
>

This could prove to be a powerful observation. Especially since the
breakdown of a square wave into sine wave components is so straightforward.

I can appreciate the fact that a Fourier Transform can break even random
noise into a series of harmonics which when reconstructed can exactly
reproduce a series of digital frames... But would that imply that the FT is
actually the way the cochlea breaks sound down? That's an obscure way of
asking I know.

Marc

--- In MakeMicroMusic@y..., "Orphon Soul, Inc." <tuning@o...> wrote:

But would that imply that the FT is
> actually the way the cochlea breaks sound down? That's an obscure way of
> asking I know.

It gets at the energy spectrum pretty well--phase information is another matter.

--- In MakeMicroMusic@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> --- In MakeMicroMusic@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
>
> But would that imply that the FT is
> > actually the way the cochlea breaks sound down? That's an
obscure way of
> > asking I know.
>
> It gets at the energy spectrum pretty well--phase information is
>another matter.

i think gene is merely pointing out that the cochlea pretty much
ignores phase information -- right gene? i thought there was a
possibility for misinterpretation here, so i thought i'd try to
clarify . . .