These Approx GCD waves are phase modulations.

I think the solution was staring me in the face all along and its very simple.

Recall that from t = (2k + 1)/2(a + b) the peaks occur between the times k = 0 and k = K, the last being characteristic of each interval 'type'. This gave the period T = K/(a + b). Multiplying this with our original frequencies and taking it in quotient plus remainder form gave
aK/(a + b) = p + (r/(a + b),
bK/(a + b) = q - (r/(a + b),
where r = aq - pb, and adding gave K = (p + b). Our ~ GCD = (a + b)/(p + q).

Now, we multiply the (a + b)/(p + q) back out we get
a = p[(a + b)/(p + q)] + r/(p + q),
b = q[(a + b)/(p + q)] - r/(p + q).
The original wave can now be written

f(t) = sin[2pi at) + sin[2pi bt) =
sin[2pi(p[(a + b)/(p + q)])t + rt/(p + q)] +
sin[2pi(q[(a + b)/(p + q)])t - rt/(p + q)].

If we let r = 0 momentarily, then f(t) reduces to a periodic function with the exact frequency (a + b)/(p + b). This is easily proved since this is by definition the exact GCD between p[(a + b)/(p + q)] and q[(a + b)/(p + q)] i.e. since p and q are whole and in lowest form then they are the "p'th" and "q'th" harmonics to this fundamental. Thus, the original function can be regarded as a periodic function but with a modulated phase in both components, + rt/(p + q) for the first and - rt/(p + q) for the second.

> I think the solution was staring me in the face all along and its very simple.
>
> Recall that from t = (2k + 1)/2(a + b) the peaks occur between the times k = 0 and k = K, the last being characteristic of each interval 'type'. This gave the period T = K/(a + b). Multiplying this with our original frequencies and taking it in quotient plus remainder form gave
> aK/(a + b) = p + (r/(a + b),
> bK/(a + b) = q - (r/(a + b),
> where r = aq - pb, and adding gave K = (p + b). Our ~ GCD = (a + b)/(p + q).
>
> Now, we multiply the (a + b)/(p + q) back out we get
> a = p[(a + b)/(p + q)] + r/(p + q),
> b = q[(a + b)/(p + q)] - r/(p + q).
> The original wave can now be written
>
> f(t) = sin[2pi at) + sin[2pi bt) =
> sin[2pi(p[(a + b)/(p + q)])t + rt/(p + q)] +
> sin[2pi(q[(a + b)/(p + q)])t - rt/(p + q)].
>
> If we let r = 0 momentarily, then f(t) reduces to a periodic function with the exact frequency (a + b)/(p + b). This is easily proved since this is by definition the exact GCD between p[(a + b)/(p + q)] and q[(a + b)/(p + q)] i.e. since p and q are whole and in lowest form then they are the "p'th" and "q'th" harmonics to this fundamental. Thus, the original function can be regarded as a periodic function but with a modulated phase in both components, + rt/(p + q) for the first and - rt/(p + q) for the second.

Rick, it's extremely confusing to follow this when you don't define
your variables. I don't remember what k, K, a, b, p, q, or r meant
anymore, and I don't feel like flipping all the way back through the
thread to hunt for whichever message you defined them in. It would be
easier to follow if you gave a quick note before writing the
equations.

But you're right - each successive period of the wave can be viewed as
a phase-shifted previous version of the wave.

However, I have to say - this property of "duality" that you're seeing
here, in which you could interpret the wave either as a more complex
periodic function or a simpler periodic function of shifting phase -
is quite common in psychoacoustics. Take the "beating/ring modulator"
duality as well. If you add 440 Hz and 442 Hz, you're going to get 441
Hz with 2 Hz "beating." If you take a 441 Hz sine wave and multiply it
by a 1 Hz sine wave, you get the same thing.

So why does the brain tend to interpret the waveform one way sometimes
and a different way other times? That's the real question you should
be focusing on.

-Mike

Ok, sorry Mike. It is a bit hard keeping track. You know the theory so I'll just go through the steps.

1. Take a complex wave like
f(t) = sin(2pi at) + sin(2pi bt),
where a > b and (a, b) are either large coprime numbers (81, 64 say) or a is an irrational interval from b (for eg a = b*2^(1/3)).

2. It's extrema occur when this wave meets its wave envelope +/-2cos(pi(a - b)t. We get 2sin(pi(a + b)t)cos(pi(a - b)t = +/-2cos(pi(a - b)t or sin(pi(a + b)t = +/-1. Solving for time gives
t = (2k + 1)/2(a + b) where k = 0,1,2,3,...

3. The times corresponding to the approx GCD occurs at k = 0 and k = K where K is an integer that is *given* for each interval. (For eg, major thirds always give 9, minor thirds always 11. This I'm still trying to prove).

4. The difference between these two times is T = K/(a + b) and 1/T = ~GCD = (a + b)/K.

5. Dividing our original a and b by this 'fundamental' frequency and taking it in quotient + remainder form gives
aK/(a + b) = p + (r/(a + b),
bK/(a + b) = q - (r/(a + b),
where r = aq - pb and p and q are whole numbers. We see that (a, b) are approximately the (p, q)th harmonics to this 'fundamental'. If we add these two values above we get K = (p + q) and substituting back into 4. gives ~GCD = (a + b)/(p + q).

This brings us up to date:
6. Now, we multiply the (a + b)/(p + q) back out we get
> > a = p[(a + b)/(p + q)] + r/(p + q),
> > b = q[(a + b)/(p + q)] - r/(p + q).
> > The original wave can now be written
> >
> > f(t) = sin[2pi at) + sin[2pi bt) =
> > sin[2pi(p[(a + b)/(p + q)])t + rt/(p + q)] +
> > sin[2pi(q[(a + b)/(p + q)])t - rt/(p + q)].
> >
> > If we let r = 0 momentarily, then f(t) reduces to a periodic function with the exact frequency (a + b)/(p + b). This is easily proved since this is by definition the exact GCD between p[(a + b)/(p + q)] and q[(a + b)/(p + q)] i.e. since p and q are whole and in lowest form then they are the "p'th" and "q'th" harmonics to this fundamental. Thus, the original function can be regarded as a periodic function but with a modulated phase in both components, + rt/(p + q) for the first and - rt/(p + q) for the second.
>
> Rick, it's extremely confusing to follow this when you don't define
> your variables. I don't remember what k, K, a, b, p, q, or r meant
> anymore, and I don't feel like flipping all the way back through the
> thread to hunt for whichever message you defined them in. It would be
> easier to follow if you gave a quick note before writing the
> equations.
>
But you're right - each successive period of the wave can be viewed as a phase-shifted previous version of the wave.
>
However, I have to say - this property of "duality" that you're seeing
here, in which you could interpret the wave either as a more complex
periodic function or a simpler periodic function of shifting phase -
is quite common in psychoacoustics. Take the "beating/ring modulator"
duality as well.

I have no doubt. I looked at AM, FM and PM many times trying to find a solution to this problem. But for me at least, the important thing here is the *emphasis* placed on the transference from GCD's, how they behave in JI, to these approx GCD's, and not on the fact that it uses PM per se. We now have a fundamental "1" for a whole world of possibilities. Since we also have a rigorous way of defining how far an interval is from the JI interval (i.e. by the remainder), then it might lead to something like harmonic entropy. These are musical questions that have been plaguing me for years. So while I might read a bit of this problem here and that problem there, I feel very comfortable with the way it's all coming together.

If you add 440 Hz and 442 Hz, you're going to get 441 Hz with 2 Hz "beating." If you take a 441 Hz sine wave and multiply it by a 1 Hz sine wave, you get the same thing.

I understand the first sentence here Mike, 1/2(442+/-440), but not the second. You mean that it wouldn't make a difference whether we multiplied 441Hz by 1Hz or 2Hz? I've never heard the term "ring modulator" before.
>
So why does the brain tend to interpret the waveform one way sometimes and a different way other times? That's the real question you should be focusing on.
>
> -Mike

I've just never experienced that.

Ah, it just occurred to me what I wanted to say Mike. It is well known that two sines can be rewritten as an amplitude modulation because this is a basic trig identity sinA + sinB = 2sin((A + B)/2)cos((A - B)/2). In contrast, when we read about phase or frequency modulation, we get the impression that they are some type of *alteration* of the initial wave, techniques used in signal processing and media communication etc. But now we see that the original wave itself can be *rewritten* as the sum of two other periodic waves with an equal and opposite phase modulation in each component. They are equivalent; that's the difference. This itself is not unlike an (albeit more complicated) trig identity. Further, if a Fourier analysis can break a wave down into component sines, then it should in principle be able to break it down into these simpler JI components. And in doing so, basic harmonies are revealed. IOW it is not strictly true that the sine components of a FA are 'elementary' (for the word does mean that something cannot be analysed any further). At any rate, it's a world full of new possibilities.

Whether this technique is already known in some circles is beside the point. Questions of 'originality' never much interested me because everything 'original' is really just seeing old material in a slightly different or new way.

Rick

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I think the solution was staring me in the face all along and its very simple.
> >
> > Recall that from t = (2k + 1)/2(a + b) the peaks occur between the times k = 0 and k = K, the last being characteristic of each interval 'type'. This gave the period T = K/(a + b). Multiplying this with our original frequencies and taking it in quotient plus remainder form gave
> > aK/(a + b) = p + (r/(a + b),
> > bK/(a + b) = q - (r/(a + b),
> > where r = aq - pb, and adding gave K = (p + b). Our ~ GCD = (a + b)/(p + q).
> >
> > Now, we multiply the (a + b)/(p + q) back out we get
> > a = p[(a + b)/(p + q)] + r/(p + q),
> > b = q[(a + b)/(p + q)] - r/(p + q).
> > The original wave can now be written
> >
> > f(t) = sin[2pi at) + sin[2pi bt) =
> > sin[2pi(p[(a + b)/(p + q)])t + rt/(p + q)] +
> > sin[2pi(q[(a + b)/(p + q)])t - rt/(p + q)].
> >
> > If we let r = 0 momentarily, then f(t) reduces to a periodic function with the exact frequency (a + b)/(p + b). This is easily proved since this is by definition the exact GCD between p[(a + b)/(p + q)] and q[(a + b)/(p + q)] i.e. since p and q are whole and in lowest form then they are the "p'th" and "q'th" harmonics to this fundamental. Thus, the original function can be regarded as a periodic function but with a modulated phase in both components, + rt/(p + q) for the first and - rt/(p + q) for the second.
>
> Rick, it's extremely confusing to follow this when you don't define
> your variables. I don't remember what k, K, a, b, p, q, or r meant
> anymore, and I don't feel like flipping all the way back through the
> thread to hunt for whichever message you defined them in. It would be
> easier to follow if you gave a quick note before writing the
> equations.
>
> But you're right - each successive period of the wave can be viewed as
> a phase-shifted previous version of the wave.
>
> However, I have to say - this property of "duality" that you're seeing
> here, in which you could interpret the wave either as a more complex
> periodic function or a simpler periodic function of shifting phase -
> is quite common in psychoacoustics. Take the "beating/ring modulator"
> duality as well. If you add 440 Hz and 442 Hz, you're going to get 441
> Hz with 2 Hz "beating." If you take a 441 Hz sine wave and multiply it
> by a 1 Hz sine wave, you get the same thing.
>
> So why does the brain tend to interpret the waveform one way sometimes
> and a different way other times? That's the real question you should
> be focusing on.
>
> -Mike
>

> This itself is not unlike an (albeit more complicated) trig identity. Further, if a Fourier analysis can break a wave down into component sines, then it should in principle be able to break it down into these simpler JI components. And in doing so, basic harmonies are revealed. IOW it is not strictly true that the sine components of a FA are 'elementary' (for the word does mean that something cannot be analysed any further). At any rate, it's a world full of new possibilities.

OK, but you're still using fourier analysis here. Also, you could take
the 5/4 and use the same identity and say that sin(5t) + sin(4t) ends
up being a phase-modulated version of an even more complex wave.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > This itself is not unlike an (albeit more complicated) trig identity. Further, if a Fourier analysis can break a wave down into component sines, then it should in principle be able to break it down into these simpler JI components. And in doing so, basic harmonies are revealed. IOW it is not strictly true that the sine components of a FA are 'elementary' (for the word does mean that something cannot be analysed any further). At any rate, it's a world full of new possibilities.
>
> OK, but you're still using fourier analysis here. Also, you could take
> the 5/4 and use the same identity and say that sin(5t) + sin(4t) ends
> up being a phase-modulated version of an even more complex wave.
>
> -Mike
>
Yes, from what I can tell the relationship of (a, b) to (p, q) seems to be many-to-one. I never said that the wave wasn't Fourier, just that there seems to be something else going on along side it.