do combination tones ever include multiplication alone?

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

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)?

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?

thanks, Kelly

--- 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.