Generalising gcd frequency to Gaussian Integers

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

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--- 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

--- 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

--- 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.

--- In tuning-math@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> --- 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.

I can't seem to get to a lot of the links at the top. (Resources).
I would say try to get to Gene's stuff and Graham's stuff to start.
And don't worry about the physical waves too much (crests and troughs, phase, period, etc.) at first, just the theories involving
tuning, Euclidean algorithm and such. These ideas come into play
with respect to linear tunings, multilinear algebra, and the like.

Another interesting property is the Farey sequence, and the Stern-Brocat tree. I think it would be fun to bring in complex analysis still though...

PGH