Klarinet Archive - Posting 000363.txt from 2000/08

From: "Keith" <100012.1302@-----.com>
Subj: [kl] RE: klarinet Digest 14 Aug 2000 20:15:00 -0000 Issue 2468
Date: Mon, 14 Aug 2000 23:41:00 -0400

> =A0=A0 <><> Tony=A0Wakefield wrote:
> Is this a question of psychology, ( the source of Will. Wright`s theory)
> or is there some real scientific basis
>
>
I'm another real scientist, and agree with nearly all of Bill's comments.
Just the usual scientific nitpicking...

> It's a mathematical fact -- you can prove it with pencil and paper
> and something called Fourier analysis -- that so long as we limit the
> discussion to sine waves and their partials, every wave (no matter how
> complex or irregular it may appear when graphed) can be described
> accurately as the sum of a collection of sine waves.

To be precise, you have to add phase (as you did later in the argument) or
allow a mixture of sine and cosine waves. The theorem applies, rigorously
and absolutely, to periodic functions, ie to those that repeat themselves
exactly in time - i.e. a steady note. It needs modification to apply to
transients such as tongued attacks (or simply a note stopping), and the
modifications are things such as exponentials. The modifications may apply
equally to all or may themselves be frequency dependent.

> The assertion is that 'musical sounds' are, in fact, close enough
> to sine waves that this analysis applies. Experiment bears out this
> assertion (but see "wavering" below). You may wish to call it
> 'psychology' in the sense that our minds create something, but this
> assertion is verifiable and repeatable -- which is the essence of
> 'science'.

If it is a truly _periodic_ function then it isn't an assertion but a proved
theorem. What our ears/brains make of it is another matter; for example
perceived pitch is based on frequency but frequency is not sufficient to
describe pitch. For example, a sine wave of truly constant frequency is
perceived to go flatter as the volume increases - weird! Maybe non-linearity
in the ear sensors, maybe something more subtle in the brain. A physiologist
might know.

>
> Percussion is the major exception to all of this sine wave stuff, of
course.
>

Two main differences; percussion is damped not periodic, and the overtones
are not usually harmonic (i.e. wavelengths in ratios of 1:2:3 etc). But they
_can_ be harmonic. Try getting a long aluminum rod (say 6 foot long and 2
inches diameter) and hit it with a hammer on its END. Beautiful pure sound
which rings for ages (and, if you think about it, harmonic overtones). A
tympanum doesn't have harmonic overtones - they are described by Bessel
functions - but if you force a node by clamping the skin radially from a
point on the edge to the center, then the overtones _do_ become harmonic. My
first head of department (when I was a junior assistant prof) invented this,
and got me to supervise a project student to demonstrate it - a lot of fun!

> The next step in the argument is that, mathematically, each of the
> 'component' sine waves (that make up a more complex sound wave) can be
> described completely with just two numbers. One number describes the
> amplitude, and the other number describes the phase. It turns out that
> -- again, so long as we're discussing only sine waves -- all possible
> elaborations on the fundamental tone A=440 can be described completely
> and accurately with a collection of 50 two-number 'components'. If you
> move an octave higher to A@-----. If
> you move an octave lower to A=220, then you need 100 components
> in order to cover all the possibilities.

You've left off something here, which is that you have to define a cutoff
frequency, which is presumably that of a youngster's ears (not oldies like
Mark or me...). There may be other cutoffs due to the means of stimulating
and resonating the vibration, but in principle the Fourier series goes on
for ever. But given the cutoff, you are quite right. There IS less overtone
hearable from a high pitch than a low pitch. Then we are into perception. I
think an easy conclusion is that it is harder to distinguish instruments
from one another as the pitch gets high - the information presented to our
brains becomes similar. Whether this makes it harder to produce a good high
tone ... I dunno. You could also argue that as you can't hear the complex
set of overtones anyway, it should be easier as anything will do!

> My apologies if anyone didn't want to hear all of the above.

hey, it's good for them :-)

Keith Bowen

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