--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, rick ballan <rick_ballan@> wrote:
> >
> > Hello everyone,
> >
> > My name's Rick and I'm a new member. Forgive me for starting a new thread
but I haven't got my 'daily digest' yet and couldn't find an answer in your
archives.
> >
> > I just wanted to run an idea by someone who understands Gaussian integer
maths better than I (which wouldn't be difficult). I've had this vague thought
that
> > the concept of gcd frequency, which helps to explain basic tonality, might
be generalised to the (sometimes)
> > multiple gcd's of Gaussian integers (or some other Ring). The first
> > task is to see how, if at all, this might translate into waves. The
> > second is to see if new harmonies can be generated.
> >
> > In fact the first task seems quite credible and straightforward. For eg,
> > take e^i(a + bi)t, where it is observed that this is more than the
> > Euler identity since the frequency is now complex. We then have i(a +
> > bi) = ai - b = - (b - ai). Therefore, this suggests a wave of the form
> > e^-bt
> > . e^iat = e^-bt (cos at +isin at), which appears to be a sine wave
> > under the force of damping, or if the coefficient of b is positive, a
> > forced oscillation.
> >
> > Given another wave of complex frequency (c + id), the addition of the two
would be:
> > e^-bt
> > (cos at +isin at) + e^-dt (cos ct +isin ct), the sum of two waves with
> > not necessarily equal damping/forcing. Now since periodicity is
> > independent of amplitude, then the gcd of the waves should be just the
> > usual integer result where a/c = p/q, p and q are relatively prime and
resultant
> > frequency is g = a/p2pi = c/q2pi. But here's the rub: (a + bi)/(c +
> > di) = ((ac + bd) - i(ad + bc)) / c^2 + d^2 often gives rise to more
> > than one gcd. If we can think of how these others could "fit in" somehow, a
> > whole new area of tonality might open up. Or if it doesn't fit in at
> > all and is complete nonsense, then this too might be something.
> >
> > Thanks
>
> Rick,
>
> This looks really interesting. I am into algebra (groups, rings), and am
learning more about complex analysis. I would be fun to start
> discussing these matters here, with respect to Fourier analysis,
> and so forth.
>
> I know the Euclidean algorithm comes up a lot, and of course, gcd, lcm, and
the like, in these discussions, you might want to check out the work by people
on this newsgroup (at the Home page). There isn't as much using complex analysis
but that would be a great area to pursue, seeing its application to
acoutics/physics/soundwaves.
>
> Wish I knew more about it...
>
> PGH
>
Hi Paul,

Thanks for getting back. At this stage I don't know much about complex numbers
either so you're not alone there. Unlike reals, I can't 'picture' how one
complex number of cycles/periods would divide into another and, consequently,
how we could interpret the gcd's in terms of waves. Another doubt is that if
they give damped/forced oscillations as I mentioned above, then periodicity
(harmony) would be independent of amplitude i.e. the e^-bt part. So even if we
could find some physical interpretation for the other gcd's, they mightn't
effect the harmony. OTOH,as you said the Euclidean algorithm comes up a lot in
discussions of harmony and there just might be something I/we haven't thought
of. For eg, I once Googled 'negative frequency' and found it has a physical
interpretation (which might come into complex frequencies??).

(Excuse my ignorance but how do you get back to the homepage?)

Wish I knew more about it too...

Rick