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