Approximate GCD's. Are these almost periodic functions?

Hello everybody,

I've been working on a theory that the waves of more complex intervals can be defined as almost periodic functions of the waves of their JI counterpart. The reason I suspect this is possible is because the former waves contain maxima that very nearly equal the GCD's (frequencies) of the latter. Moreover, the two waves seem to bear some type of 'family resemblance', their general appearance seeming to have much in common.

For example, take a graphic comparison between the Pythagorean major third sin(2pi81t) + sin(2pi64t) and the JI major third sin(2pi80t) + sin(2pi64t). The GCD of the latter, 16, can be clearly seen as the time between two of its largest peaks. These peaks, and the curvature in between them, also appear in the Pythagorean wave. By solving for the time between these analogous peaks and inverting we obtain the value 145/9 = 16.111...where it is clear that this does indeed approximate the GCD 16. Encouraged by this result I deduced the following method:

Let a/b be our larger interval and take sin(2piat) + sin(2pibt). We wish to find numbers p/q, where (p, q) are small whole-numbers, which will correspond to our 'JI' interval. Now the times at which the extrema occur for this wave will be simultaneous with that of the wave envelope [+ -]2cos(2pi(a - b)t. Setting this equal to our original wave gives

2sin(2pi(a + b)t)cos(2pi(a - b)t = [+ -]2cos(2pi(a - b)t
or
sin(2pi(a + b)t = [+ -]1.

Solving for t we obtain

t = (2k + 1)/2(a + b), k = 0,1,2,...

Now the time of the first peak will always be at k = 0 giving t = 1/2(a + b). The time of the 'approximate period' will occur at a number k = K which seems to be characteristic for each 'type' of interval (which I will get to). Thus, the difference between these times will be T = K/(a + b) and the 'approx GCD' will be (a + b)/K.

Finally, dividing our original component frequencies (a, b) by this frequency and taking them in 'quotient plus remainder' form we obtain

aK/(a + b) = p + r/(a + b),
bK/(a + b) = q - r/(a + b),
where r = aq - pb.

Adding these two values together we have K = (p + q) and our approximate GCD = (a + b)/(p + q).

The implication is that the original wave might be defined as an almost periodic function with respect to the wave, at least in the first cycle of every 'real GCD' period i.e. 81/64 have GCD 1. Perhaps with respect to the wave

sin(2pi(p*(a + b)/(p + q))t) + sin(2pi(q*(a + b)/(p + q))t)

or something along those lines? My knowledge of APF's is scantly to say the least and any help or advice would be greatly appreciated.

-Rick

I forgot to mention that for (a, b) = (81, 64) then K = 9 and (p, q) = (5, 4). For a second example, (a, b) = (32, 27) gives K = 11 and (p, q) = (6, 5).

Also, the existence of these approx GCD's might give some type of physical explanation for virtual pitch.

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

> For example, take a graphic comparison between the Pythagorean major third sin(2pi81t) + sin(2pi64t) and the JI major third sin(2pi80t) + sin(2pi64t).

These functions are both periodic with period 1. The second function actually has a period 1/16, though, because of the common factor of 16 between 64 and 80.

So far I can't figure out what you think this stuff adds that elementary number theory already does for you.

Rick is on a crusade to get the word "virtual pitch" changed because
it's a actually a "real" pitch because pseudoperiodicities are real. I
think the overall goal now is to find a precise mathematical way to
get from a complex interval like 501/400 to 5/4, or even from 81/64 to
5/4.

Here's a possible solution: convolve the waveform with a sawtooth
wave, or some form of impulse train with a rolloff. Make it be
exponentially damped. We'll call this filtering function c(wt)
(pretend the w is an authoritative-sounding greek omega sign). Its
frequency spectrum should look like a feedback comb filter with some
kind of rolloff, and of a Q that you will basically be picking with
however much damping you choose.

Convolve the waveform with all of these for all w, and then get the
total energy for each resulting waveform, and then plot the resulting
energy(w). If you pick your rolloff and damping properly, you'll note
that this will basically give you what you're looking for: 81/64 will
generate a huge spike two octaves below the 64 as if it were really
5/4. You will also see some other huge spikes, and this resulting
curve basically shows you all of the "possible fundamentals" that the
brain could pick for any given chord, as well as the probability that
it will pick each one. (Normalize it so that the sum is 1 and you have
a brand new type of harmonic entropy to work with, and for chords to
boot. Voila!)

The damping refers to the tendency of the periodicity mechanism to
stop tracking the same frequency forever and instead resonate at close
pseudo-periodicities (which you are noticing yourself visually), and
the rolloff refers to the fact that higher harmonics generate much
weaker fundamentals than lower ones.

In complete layman's terms: come up with a filterbank of low-passed
comb filters (an infinitely dense one, in fact), set the incoming
waveform into all of them and see what sticks. I think this is an
excellent model for how the brain does what it does, although I'm
unsure as to the exact coefficients for the rolloff and the damping.
I'm pretty sure the damping changes with the situation, analogous to
how the free "s" parameter can change in Paul's HE model.

I call it "Harmonic Profiling," and I'll have some MATLAB code for you
all to play around with as soon as I stop going through this
post-college-graduate financial crisis that I'm going through.

-Mike

On Thu, Apr 29, 2010 at 2:57 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> > For example, take a graphic comparison between the Pythagorean major third sin(2pi81t) + sin(2pi64t) and the JI major third sin(2pi80t) + sin(2pi64t).
>
> These functions are both periodic with period 1. The second function actually has a period 1/16, though, because of the common factor of 16 between 64 and 80.
>
> So far I can't figure out what you think this stuff adds that elementary number theory already does for you.

Another note: another lovely feature of this model is that the
constituent tones themselves actually will show up as spikes on the
curve. This is analogous to the fact that if you play a perfect fifth
(with sine waves), you will not only hear a faint VF popping up an
octave below the root, you'll also hear the chromata of the tones
individually as well. When you add enough overtones, the lower comb
filter will start to resonate strongly enough that it overpowers the
individual tones, and you get "fusion." This also, handily enough,
manifests itself on the graph as the lower "spike" finally growing
larger than the spikes of the individual tones.

Getting this to work properly on the graph is completely dependent on
picking the right amount of rolloff. I'm on a different computer now
and I don't remember exactly what I picked, but I'm pretty sure it was
much higher than 1/N, closer to 1/N^2 I believe. Ironic that the
rolloff of the human voice is roughly 1/N^2 as well (props to Carl
Lumma there).

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Rick is on a crusade to get the word "virtual pitch" changed because
> it's a actually a "real" pitch because pseudoperiodicities are real. I
> think the overall goal now is to find a precise mathematical way to
> get from a complex interval like 501/400 to 5/4, or even from 81/64 to
> 5/4.

Why do you think number theory doesn't do that? The convergents to
501/400 are 1, 4/3, 5/4, 124/99 and the semiconvergents are more numerous than I care to list. For 81/64, the convergents are 1, 4/3, 5/4, 19/15 and the semiconvergents 1, 4/3, 5/4, 9/7, 14/11, 19/15. What
s missing?

> Why do you think number theory doesn't do that? The convergents to
> 501/400 are 1, 4/3, 5/4, 124/99 and the semiconvergents are more numerous than I care to list. For 81/64, the convergents are 1, 4/3, 5/4, 19/15 and the semiconvergents 1, 4/3, 5/4, 9/7, 14/11, 19/15. What
> s missing?

I can't speak for Rick, maybe that's what he's looking for. For my
purposes, what's missing is the fact that of all of those options
listed, 81/64 will most likely be perceived as 5/4 - not 1, or 4/3, or
9/7, or 14/11, or 19/15.

Rick is looking for a fundamentally mathematical reason why. I
personally don't think that there is a fundamental reason and that it
has to do more with how certain constants of how the auditory system
is set up. Hence my opting for the approach of trying to model
different parts of the auditory system and to "tweak" it until I
(hopefully) get it right.

I think my approach is sound though, although that damned rolloff
still eludes me.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I think my approach is sound though, although that damned rolloff
> still eludes me.

If you can come up with an alternative to harmonic entropy it will be interesting.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
>
> > For example, take a graphic comparison between the Pythagorean major third sin(2pi81t) + sin(2pi64t) and the JI major third sin(2pi80t) + sin(2pi64t).
>
> These functions are both periodic with period 1. The second function actually has a period 1/16, though, because of the common factor of 16 between 64 and 80.

Gene, only the second sentence here is correct. If we take 80/64 the frequency is its GCD 16 and period 1/16, not 1. 64 and 80 equal the 4th and 5th harmonics to this fundamental, respectively. Of course I know this. This is my starting point.
>
> So far I can't figure out what you think this stuff adds that elementary number theory already does for you.
>
Well if we detune the 80 to 81, say, then the GCD jumps down to 1 which is nowhere near 16. But the wave itself doesn't change all that dramatically. What I've shown in this post is that these waves do indeed contain a frequency that is close to the GCD of its corresponding epimoric (or JI) interval. It's formula is ~ GCD = (a + b)/(p + q). So for this example, the JI interval is p/q = 5/4, ~ GCD = (81 + 64)/(5 + 4) = 145/9 = 16.111... Therefore, 64 becomes a slightly detuned 4th harmonic to this fundamental and 81 a slightly detuned 5th harmonic. And the formula is completely general. If we take a as an irrational interval from b then the formula still holds. For example, if a = 64*(2^(1/3)) then ~ GCD = 64*[(2^(1/3)) + 1]/9 = 16.0705...

Being able to map any tuning system back onto its JI counterpart is hardly 'elementary number theory'. Nor is a mathematics that assigns a many-to-one intervallic range to each interval class something that is irrelevant to questions of tuning. If we form a ring with these classes, then it is quite conceivable that ~2 + ~2 = ~5 in this number system, if you take my meaning.

Please put some time in to understand the problem before answering.

Thanks

-Rick

--- In tuning-math@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > > For example, take a graphic comparison between the Pythagorean major third sin(2pi81t) + sin(2pi64t) and the JI major third sin(2pi80t) + sin(2pi64t).
> >
> > These functions are both periodic with period 1. The second function actually has a period 1/16, though, because of the common factor of 16 between 64 and 80.
>
> Gene, only the second sentence here is correct. If we take 80/64 the frequency is its GCD 16 and period 1/16, not 1. 64 and 80 equal the 4th and 5th harmonics to this fundamental, respectively. Of course I know this. This is my starting point.

I said the period is 1/16 and you seem to be agreeing, so I don't know what you think is incorrect. If this is your starting point, where do the almost periodic functions come in?

> > So far I can't figure out what you think this stuff adds that elementary number theory already does for you.
> >
> Well if we detune the 80 to 81, say, then the GCD jumps down to 1 which is nowhere near 16. But the wave itself doesn't change all that dramatically. What I've shown in this post is that these waves do indeed contain a frequency that is close to the GCD of its corresponding epimoric (or JI) interval. It's formula is ~ GCD = (a + b)/(p + q). So for this example, the JI interval is p/q = 5/4, ~ GCD = (81 + 64)/(5 + 4) = 145/9 = 16.111...

That's the mediant between 5/4 and 81/64, but it nowhere appears as a frequency.

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Rick is on a crusade to get the word "virtual pitch" changed because
> it's a actually a "real" pitch because pseudoperiodicities are real. I
> think the overall goal now is to find a precise mathematical way to
> get from a complex interval like 501/400 to 5/4, or even from 81/64 to
> 5/4.
>
Mike, I'm not really on a crusade to change virtual pitch. The impression I got was that VPT was some type of guessing game done by the brain that prevented people from looking into this question of ~ GCD = tonic within the waves more seriously, which made me frustrated. But it was more of a misunderstanding really. For the record, I'm quite happy now to adopt the accepted term. (Btw, you recently mentioned that the minor triad 10:12:15 has VP around 5 which shows that the 3/2 interval 15/10 overrides the other two GCD's. You then said that what I was saying wouldn't explain musical harmony. However, this is exactly the type of thing that my theory does seem to explain. Minor triad as 16:19:24 for eg maps onto 5:6, 4:5, while the 16:24 = 2/3 directly. This predicts a GCD of 8 which is 5 x 8/5).

Having said that, yes what you say here is exactly what I'm trying to prove. The fact that I can now arrive at the JI interval (p, q) directly from the complex (a, b) is a step in the right direction.

> Here's a possible solution: convolve the waveform with a sawtooth
> wave, or some form of impulse train with a rolloff. Make it be
> exponentially damped. We'll call this filtering function c(wt)
> (pretend the w is an authoritative-sounding greek omega sign). Its
> frequency spectrum should look like a feedback comb filter with some
> kind of rolloff, and of a Q that you will basically be picking with
> however much damping you choose.
>
> Convolve the waveform with all of these for all w, and then get the
> total energy for each resulting waveform, and then plot the resulting
> energy(w). If you pick your rolloff and damping properly, you'll note
> that this will basically give you what you're looking for: 81/64 will
> generate a huge spike two octaves below the 64 as if it were really
> 5/4. You will also see some other huge spikes, and this resulting
> curve basically shows you all of the "possible fundamentals" that the
> brain could pick for any given chord, as well as the probability that
> it will pick each one. (Normalize it so that the sum is 1 and you have
> a brand new type of harmonic entropy to work with, and for chords to
> boot. Voila!)
>
> The damping refers to the tendency of the periodicity mechanism to
> stop tracking the same frequency forever and instead resonate at close
> pseudo-periodicities (which you are noticing yourself visually), and
> the rolloff refers to the fact that higher harmonics generate much
> weaker fundamentals than lower ones.
>
> In complete layman's terms: come up with a filterbank of low-passed
> comb filters (an infinitely dense one, in fact), set the incoming
> waveform into all of them and see what sticks. I think this is an
> excellent model for how the brain does what it does, although I'm
> unsure as to the exact coefficients for the rolloff and the damping.
> I'm pretty sure the damping changes with the situation, analogous to
> how the free "s" parameter can change in Paul's HE model.
>
> I call it "Harmonic Profiling," and I'll have some MATLAB code for you
> all to play around with as soon as I stop going through this
> post-college-graduate financial crisis that I'm going through.
>
> -Mike
>
Ok thanks Mike, this looks great (though it will take me some time to understand). My ideas were a bit more simple, something like finding a way to prove that the a, b wave was an almost periodic function with regard to sin(2pi p*GCD t) + sin(2pi q*GCD t), where the GCD here is the exact value (16 b/w 80 and 64). But like all maths, one proof leads to another.

I look forward to seeing your MATLAB code. Hopefully I'll be able to translate it into Mathematica.

Thanks,

-Rick
>
> On Thu, Apr 29, 2010 at 2:57 AM, genewardsmith
> <genewardsmith@...> wrote:
> > --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > > For example, take a graphic comparison between the Pythagorean major third sin(2pi81t) + sin(2pi64t) and the JI major third sin(2pi80t) + sin(2pi64t).
> >
> > These functions are both periodic with period 1. The second function actually has a period 1/16, though, because of the common factor of 16 between 64 and 80.
> >
> > So far I can't figure out what you think this stuff adds that elementary number theory already does for you.
>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Why do you think number theory doesn't do that? The convergents to
> > 501/400 are 1, 4/3, 5/4, 124/99 and the semiconvergents are more numerous than I care to list. For 81/64, the convergents are 1, 4/3, 5/4, 19/15 and the semiconvergents 1, 4/3, 5/4, 9/7, 14/11, 19/15. What
> > s missing?
>
> I can't speak for Rick, maybe that's what he's looking for. For my
> purposes, what's missing is the fact that of all of those options
> listed, 81/64 will most likely be perceived as 5/4 - not 1, or 4/3, or
> 9/7, or 14/11, or 19/15.

Rick here. Yes, the point is that this was deduced *from* the wave directly. It doesn't converge or anything like that (though they may play some part in a possible proof). The solution of the wave period to 64:81 IS a 5/4 and nothing else.
>
> Rick is looking for a fundamentally mathematical reason why. I
> personally don't think that there is a fundamental reason and that it
> has to do more with how certain constants of how the auditory system
> is set up. Hence my opting for the approach of trying to model
> different parts of the auditory system and to "tweak" it until I
> (hopefully) get it right.
>
> I think my approach is sound though, although that damned rolloff
> still eludes me.
>
> -Mike
>
As I said Mike your model looks very promising in its own right. When I was young I spent years converting what I knew about harmony into numbers and looking for laws which explained things. I came to the conclusion that GCD's, not combination tones, were the key to tonality but that these could be detuned, compounded, mixed. I think that we're both arriving at the same conclusions from different points of view. (oh, I also got interested in general physics).

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > >
> > >
> > >
> > > --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > >
> > > > For example, take a graphic comparison between the Pythagorean major third sin(2pi81t) + sin(2pi64t) and the JI major third sin(2pi80t) + sin(2pi64t).
> > >
> > > These functions are both periodic with period 1. The second function actually has a period 1/16, though, because of the common factor of 16 between 64 and 80.
> >
> > Gene, only the second sentence here is correct. If we take 80/64 the frequency is its GCD 16 and period 1/16, not 1. 64 and 80 equal the 4th and 5th harmonics to this fundamental, respectively. Of course I know this. This is my starting point.
>
> I said the period is 1/16 and you seem to be agreeing, so I don't know what you think is incorrect. If this is your starting point, where do the almost periodic functions come in?

Just didn't understand what you meant by "These functions are both periodic with period 1" because the 80 to 64 doesn't.
>
> > > So far I can't figure out what you think this stuff adds that elementary number theory already does for you.
> > >
> > Well if we detune the 80 to 81, say, then the GCD jumps down to 1 which is nowhere near 16. But the wave itself doesn't change all that dramatically. What I've shown in this post is that these waves do indeed contain a frequency that is close to the GCD of its corresponding epimoric (or JI) interval. It's formula is ~ GCD = (a + b)/(p + q). So for this example, the JI interval is p/q = 5/4, ~ GCD = (81 + 64)/(5 + 4) = 145/9 = 16.111...
>
> That's the mediant between 5/4 and 81/64, but it nowhere appears as a frequency.
>
Not a periodic frequency in the strict sense, which is why it's almost periodic. The important point to keep in mind is that this mediant between 81/64 and 5/4, or more generally between the complex interval a/b and its JI counterpart p/q, represents a solution to the time between the wave maxima and was not designed by myself i.e.
t = (2k + 1)/2(a + b) gives k = (p + q). The first peak is at t = 1/2(a + b) which we subtract to get T = (p + q)/(a + b). Substituting back into the time equation we see that maxima occurs at all t = [1/2(a + b)] + NT, N = 0,1,2,...Interestingly, these mediants now serve as almost GCD's to the original frequencies. They also now seem to be a physical feature of the wave instead of just a tool for finding approx ratios.

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

> > > > > For example, take a graphic comparison between the Pythagorean major third sin(2pi81t) + sin(2pi64t) and the JI major third sin(2pi80t) + sin(2pi64t).

> Just didn't understand what you meant by "These functions are both periodic with period 1" because the 80 to 64 doesn't.

A function f is periodic with period 1 if f(t+1)=f(t). Sometimes people exclude constant functions from that definition, but clearly it applies to sin(2pi81t) + sin(2pi64t).

Also I'd add to this that another very basic mathematical way to
figure out "how periodic" a function is, or where it's "almost
periodic" is to just autocorrelate it with itself.

-Mike

On Fri, Apr 30, 2010 at 10:09 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@...> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > > > > > For example, take a graphic comparison between the Pythagorean major third sin(2pi81t) + sin(2pi64t) and the JI major third sin(2pi80t) + sin(2pi64t).
>
> > Just didn't understand what you meant by "These functions are both periodic with period 1" because the 80 to 64 doesn't.
>
> A function f is periodic with period 1 if f(t+1)=f(t). Sometimes people exclude constant functions from that definition, but clearly it applies to sin(2pi81t) + sin(2pi64t).
>
>

Just one more thing for the moment. As you know, periodic functions require their component frequencies to be rational numbers, the frequency of the wave being the GCD. It follows that irrational intervals must be aperiodic (Pythagoras'proof that sqrt2 is not a rational number, for eg, means that the b5 has no GCD and produces a wave that is aperiodic). Now at that time I was exploring the idea that GCD = tonic. But this left the problem that irrationals had no GCD and therefore no tonic. It seemed also to follow that irrationals had something to do with atonality (or 'pantonality'), diminished 7 chords etc...And mixing them became like oil and water. Then how can tonality be possible on say a piano, tuned as it is to 2^(1/12)?

But now this problem is circumvented. With ~ GCD = (a + b)/(p + q) where a is an irrational interval from b, since p and q are always whole numbers, then there is no longer any conflict of interest.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > I think my approach is sound though, although that damned rolloff
> > still eludes me.
>
> If you can come up with an alternative to harmonic entropy it will be interesting.
>

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

> But now this problem is circumvented. With ~ GCD = (a + b)/(p + q) where a is an irrational interval from b, since p and q are always whole numbers, then there is no longer any conflict of interest.

That's undefined without a way of determining p. You might note, however, that irrational numbers also have continued fractions.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
>
> > But now this problem is circumvented. With ~ GCD = (a + b)/(p + q) where a is an irrational interval from b, since p and q are always whole numbers, then there is no longer any conflict of interest.
>
> That's undefined without a way of determining p. You might note, however, that irrational numbers also have continued fractions.
>
Sorry Gene, I don't understand what you mean by the first statement. Nothing I've said has been undefined. Both p and q are determined by the 'quotient plus remainder' method I gave earlier. They are necessarily whole.

The fact that both rationals and irrationals can be approximated by continued fractions is neither here nor there. The problem itself naturally led me to what seems on the surface to be 'elementary number theory'. I'll let you know if and when Fourier Transforms or wavelets come into the picture.

--- In tuning-math@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> Sorry Gene, I don't understand what you mean by the first statement. Nothing I've said has been undefined. Both p and q are determined by the 'quotient plus remainder' method I gave earlier. They are necessarily whole.

OK, take an irrational number--say, pi--and show us. How do you write it uniquely as a/p, with p an integer?

> > >
> > Sorry Gene, I don't understand what you mean by the first statement. Nothing I've said has been undefined. Both p and q are determined by the 'quotient plus remainder' method I gave earlier. They are necessarily whole.
>
> OK, take an irrational number--say, pi--and show us. How do you write it uniquely as a/p, with p an integer?
>
Well the two pairs are our starting interval a/b from which we deduce the interval p/q where only the latter needs to be integers i.e. they came from t = (2k + 1)/2(a + b) where k = 0,1,2, and k = K = (p + q). By saying that our original 'a' can be an irrational interval from 'b' I simply meant that we can multiply 'b' by this number. For pi, a = pi*b and a/b = pi.

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

> Well the two pairs are our starting interval a/b from which we deduce the interval p/q where only the latter needs to be integers i.e. they came from t = (2k + 1)/2(a + b) where k = 0,1,2, and k = K = (p + q). By saying that our original 'a' can be an irrational interval from 'b' I simply meant that we can multiply 'b' by this number. For pi, a = pi*b and a/b = pi.
>

That still leaves your mediant completely undefined.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
>
> > Well the two pairs are our starting interval a/b from which we deduce the interval p/q where only the latter needs to be integers i.e. they came from t = (2k + 1)/2(a + b) where k = 0,1,2, and k = K = (p + q). By saying that our original 'a' can be an irrational interval from 'b' I simply meant that we can multiply 'b' by this number. For pi, a = pi*b and a/b = pi.
> >
>
> That still leaves your mediant completely undefined.
>
I think you came in at the tail end of the discussion Gene. A few months ago I acquired Mathematica which allowed me to test some of my old ideas and see graphs of waves in an instant. The first thing I noticed was that if we overlay, say, a 5/4 with an 81/64 in the same time scale (the justification of course being that they're both 'major thirds'), then their exists in the more 'complex' 81/64 wave something akin to the period (inverse GCD) of the 'simple' wave 5/4. Both had maxima at almost identical times while the 'shape' of the complex wave looked the same between the first two peaks. Generalising, I called the complex interval a/b and simple one p/q and was then able to deduce the latter from the former. The appearance of the mediant was a consequence, not a condition. By solving the period between the peaks in the complex wave I obtained T = (p + q)/(a + b), and 1/T serves as an approximate GCD to a and b. Again, in the proof I didn't assume the p/q to begin with but deduced it.

--- In tuning-math@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>Again, in the proof I didn't assume the p/q to begin with but deduced it.

To have a proof, you need precise statements and definitions. You can't just draw pictures.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >Again, in the proof I didn't assume the p/q to begin with but deduced it.
>
> To have a proof, you need precise statements and definitions. You can't just draw pictures.
>
Yes I realise that. I was just giving you the background of my original hypothesis. However, being able to rewrite what I thought were irreducible sine waves as more 'simple' sines with phase modulations is really panning out. There is a proof to be had. I just have to figure out how to say it exactly.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >Again, in the proof I didn't assume the p/q to begin with but deduced it.
>
> To have a proof, you need precise statements and definitions. You can't just draw pictures.
>
I can, however, now state the problem perhaps a bit more mathematically. Given a wave of the form

f(t) = sin(2pi at) + sin(2pi bt)

where a > b, a, b are either coprime or a is an irrational ratio from b, then there exists unique integers p and q, p > q, such that

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

IOW more 'complex' sine waves can be rewritten as the sum of 'simpler' sines with an equal and opposite phase modulation in each component. Even though I deduced it, I can't think of how to prove this yet. I do know that it works for many examples, a/b = 81/64 and p/q = 5/4 being one, or a/b = 32/27 and p/q = 6/5 being another. There are also certain features which stand out. For eg, if a = Np and b = Nq then a/b = p/q, r = 0, (a + b)/(p + q) = N and the equation reduces to the original. But beyond that I don't yet know.

If you can think of a way to prove this please let me know.

Rick

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

> f(t) = sin(2pi at) + sin(2pi bt) =
> sin[2pi(p[(a + b)/(p + q)])t + rt/(p + q)] + sin[2pi(p[(a + b)/(p + q)])t - rt/(p + q)], where r = aq - pb.

The first term is an algebraic identity, which proves it. For the second term, you need to have sin[2pi(q[(a + b)/(p + q)])t - rt/(p + q)] instead, and this would prove the whole identity. However, it would also prove that p and q are not unique and need not be integers.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
>
> > f(t) = sin(2pi at) + sin(2pi bt) =
> > sin[2pi(p[(a + b)/(p + q)])t + rt/(p + q)] + sin[2pi(p[(a + b)/(p + q)])t - rt/(p + q)], where r = aq - pb.
>
> The first term is an algebraic identity, which proves it. For the second term, you need to have sin[2pi(q[(a + b)/(p + q)])t - rt/(p + q)] instead, and this would prove the whole identity. However, it would also prove that p and q are not unique and need not be integers.
>
Thanks Gene, but could you be a little more specific please? By the first term do you mean sin[2pi(p[(a + b)/(p + q)])t + rt/(p + q)]? I also dont know what you mean by you need to have for the second term [2pi(q[(a + b)/(p + q)])t - rt/(p + q)] instead, because this is what I wrote. Did you mean substituting it somewhere?

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

> Thanks Gene, but could you be a little more specific please? By the first term do you mean sin[2pi(p[(a + b)/(p + q)])t + rt/(p + q)]? I also dont know what you mean by you need to have for the second term [2pi(q[(a + b)/(p + q)])t - rt/(p + q)] instead, because this is what I wrote. Did you mean substituting it somewhere?

It's not what you wrote. You did the algebra correctly in a recent post, so you've got it figured out in any case.