--- 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
>
Actually Paul, there is one other problem that came up on the alternate tunings
list which I can't find a math's solution for. And that is, how can the lcm be
interpreted in terms of waves? For eg, given the freq's p and q, p > q, and p/q
= a/b which are relatively prime, then as you know this gives two other
equations, gcd = p/a = q/b and lcm = pb = qa. The first gives the period of the
wave, proved but adding whole numbered multiples of the gcd period (T = Na/p =
Nb/q), but I can't for the life of me find how we could hear the lcm. Is it the
time between smallest wave-crests or something?

Rick

PS: I did look up the messages and there's allot their.