--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
>
> Hello. I've been wondering....
>
> When Banade lists a series of 'heterodyne' or comb tones elements
> such as "3P, 2P+Q...", is the "3P" component by itelf a combination
> (or 'multiplication') tone (say, 3x600 cps),


You can think of the 3P as P+P+P -- it's simply a cubic combinational
tone like just like P+P+Q (=2P+Q).

> which his syntax
> implies, or is it always, in turn, part of a difference tone set
(3P-
> 2q)?

That's a combinational tone, but not a set of combinational tones, so
I'm not clear on what you're asking.

> And if multiplication is going on, so to speak, with the
> 3P 'multiplication' tone, then do two tones ever
> produce 'multiplication' tones between them (pxQ), just as they
> produce difference tones between them (p-q)?

There can be no such thing as a multiplication tone, because its
frequency would depend on your units of measurement. But nothing in
nature depends on your units of measurement. To see what I mean, take
any two tones, determine what the 'multiplication tone' would be in
two different sets of units (say, cycles per second and cycles per
minute), and then convert back into a common set of units. Your two
results won't agree! This is a sign that such a phenomenon is
impossible in the physical world. A scientist would immediately
recognize this through 'dimensional analysis': If you have two
frequencies P Hz and Q Hz, then their product will be P*Q Hz^2. This
is not a frequency but a *square frequency* -- whatever that is, it
sure isn't a number that can be interpreted to mean cycles per second.

> I assume that the way intervals 'stack' up in a multiplicative
> fashion to create chords (5/4 x 6/5 = 3/2) is unrelated to hearing
> mechanisms and brain processing, and is a simple result of the
> logarithmic nature of pitch?

It seems to be a simple result of arithmetic, actually, having
nothing to do with the nature of pitch.