Yahoo Tuning Groups Ultimate Backup tuning Approx GCD's with Irrational numbers

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I don't know what you mean by "the scientists". I disagree with the particular psychoacoustic explanation, not science per se. And because these frequencies are not strictly periodic then they will not be revealed by Fourier Analysis either.
>
> Well, that's not true.

Yes it is true. Stop guessing Mike. Again, given that (a, b) are coprime, sin(2pi*at) + sin(2pi*bt) = 2sin(pi(a + b)t)cos(pi(a - b)t), the maxima of this wave satisfy
2sin(pi(a + b)t)cos(pi(a - b)t) = (+ or -) 2cos(pi(a - b)t) giving
sin (pi(a + b)t) = (+ or -)1.
Solving for time gives t = (2k + 1)/2(a + b) where k = 0,1,2... The first largest maxima occurs at K = 0 and t = 1/2(a + b). The second occurs at k = (p + q), where p and q are the corresponding JI interval, and t = 2(p + q)/2(a + b). The difference is T = (p + q)/(a + b) and ~ GCD = 1/T. Since (p + q) is whole then whole numbered multiples of it will also belong to the set k. IOW this period will continue throughout the wave starting from the initial time.

This frequency isn't revealed by a Fourier Analysis. It's maxima are not all equal, nor are the values of the function in between. But this is not to say that the wave itself is not Fourier analysable.

Now the remaining problem is to find out WHY k = (p + q) in a rigorous mathematical sense. Even though I've tested many examples, including totally random ones, and they all surprisingly give the expected result, I'm still at a loss to find *the* mathematical proof for it. As I mentioned, it also works for when a is an irrational interval from b. (p and q are still whole since they are members of the integers k) But this so far is still just an observation, not a proof. So why don't you work with me to help find this proof instead of going over old ground?

>
> > Therefore I disagree with the view that FA is complete. Yet it doesn't contradict FA either. While the theory of Almost periodic functions comes close (and will probably play some part yet), I've never come across any scientist who has even touched on this particular theory, let alone solved it. IOW its probably the humble beginnings of a new *scientific* theory. Have I ever said anything different?
>
> I don't understand what you mean by this. What particular theory?
> You'll note that if let the two intervals mentioned run on long
> enough, that they do end up diverging. How close is "close enough" for them to be considered equivalent?

-Mike

What two intervals? I'm not simply 'comparing' an 81/64 with a 5/4 for eg but saying directly that the former wave *already* contains slightly detuned cycles of the latter. For this example, (a + b)/(p + q) = 145/9 = 16.111..., 64/(64.111..) = 4 - (4/145) and 81/(64.111..) = 5 + (4/145). Since the ear/brain does not perceive or need to perceive every cycle of a wave, then the theory is that sub-harmonic matching does seem to have some physical basis in wave theory.

-Rick
>
>

🔗Marcel de Velde <m.develde@...>

🔗Mike Battaglia <battaglia01@...>

> > > I don't know what you mean by "the scientists". I disagree with the particular psychoacoustic explanation, not science per se. And because these frequencies are not strictly periodic then they will not be revealed by Fourier Analysis either.
> >
> > Well, that's not true.
>
> Yes it is true. Stop guessing Mike.

Fourier Analysis isn't limited to periodic functions and it never has
been. A Fourier "Series" is limited to periodic functions. The
classical Fourier Transform works on aperiodic functions from t = -Inf
to Inf and the transform is a continuous function.

Again, given that (a, b) are coprime, sin(2pi*at) + sin(2pi*bt) =
2sin(pi(a + b)t)cos(pi(a - b)t), the maxima of this wave satisfy
> 2sin(pi(a + b)t)cos(pi(a - b)t) = (+ or -) 2cos(pi(a - b)t) giving
> sin (pi(a + b)t) = (+ or -)1.

I don't get this jump here, can you explain? Why would the maxima
satisfy that equation?

> Solving for time gives t = (2k + 1)/2(a + b) where k = 0,1,2... The first largest maxima occurs at K = 0 and t = 1/2(a + b). The second occurs at k = (p + q), where p and q are the corresponding JI interval, and t = 2(p + q)/2(a + b). The difference is T = (p + q)/(a + b) and ~ GCD = 1/T. Since (p + q) is whole then whole numbered multiples of it will also belong to the set k. IOW this period will continue throughout the wave starting from the initial time.

I don't have MATLAB on this computer to test that but I'll assume
you're right. So your goal is to find the maxima of these functions
and then show that they are "nearly" the maxima of some simpler
function, right?

> This frequency isn't revealed by a Fourier Analysis. It's maxima are not all equal, nor are the values of the function in between. But this is not to say that the wave itself is not Fourier analysable.

WHAT frequency? The frequency of the sin (pi(a + b)t) term? No, it
won't appear as an impulse in the Fourier transform of the wave, but
its existence can be clearly deduced by finding the two spikes on the
transformed curve, getting the distance between them, and dividing by
two, as you have done mathematically here.

> Now the remaining problem is to find out WHY k = (p + q) in a rigorous mathematical sense. Even though I've tested many examples, including totally random ones, and they all surprisingly give the expected result, I'm still at a loss to find *the* mathematical proof for it. As I mentioned, it also works for when a is an irrational interval from b. (p and q are still whole since they are members of the integers k) But this so far is still just an observation, not a proof. So why don't you work with me to help find this proof instead of going over old ground?

I'd be happy to, but I'm not going to get into this notion that
Fourier Analysis is "incomplete." It's a completely linear transform
and is as "complete" as the rest of mathematics. I don't think you
understand what the concept of a "virtual fundamental" really is, or
else you wouldn't be so adamant that is "wrong" (it most certainly is
not and is not incompatible with anything you're doing here).

The difference is that most theories so far have been working in the
frequency domain, and you are working in the time domain here. That
being said, everything you're doing would have a comparable frequency
domain analysis as well.

> >
> > > Therefore I disagree with the view that FA is complete. Yet it doesn't contradict FA either. While the theory of Almost periodic functions comes close (and will probably play some part yet), I've never come across any scientist who has even touched on this particular theory, let alone solved it. IOW its probably the humble beginnings of a new *scientific* theory. Have I ever said anything different?
> >
> > I don't understand what you mean by this. What particular theory?
> > You'll note that if let the two intervals mentioned run on long
> > enough, that they do end up diverging. How close is "close enough" for them to be considered equivalent?
>

> What two intervals? I'm not simply 'comparing' an 81/64 with a 5/4 for eg but saying directly that the former wave *already* contains slightly detuned cycles of the latter. For this example, (a + b)/(p + q) = 145/9 = 16.111..., 64/(64.111..) = 4 - (4/145) and 81/(64.111..) = 5 + (4/145). Since the ear/brain does not perceive or need to perceive every cycle of a wave, then the theory is that sub-harmonic matching does seem to have some physical basis in wave theory.

Of course it does. However, an actual sinusoid at the subharmonic
frequency where the periods sync up will not exist. And so the nerve
endings that correspond to that frequency in the cochlea will not get
stimulated. However, your brain will perform a very similar
mathematical analysis to what you're doing here (finding repeating
maxima) and find its existence that way, and "create" the sound for
you to hear it. Hence the term, "virtual pitch."

-Mike

> -Rick
> >
> >
>
>

🔗rick <rick_ballan@...>

Hi Mike,

Yeah I do understand that Fourier Transforms deal with aperiodic functions as the increment freq -> zero. When I did a FA in Mathematica on the wave in question (and others) it gave the usual infinite F series harmonics 1,2,...with small amplitudes except at a and b, as expected. What's going on here is that, given that the peaks occur at t = (2k + 1)/2(a + b), there are significant peaks at certain times that are characteristic of each interval. The first is always at k = 0. The next occurs at k = K and is characteristic of each pair (a, b) (which I'll get to). The approx GCD = (a + b)/K. Now, the wave has period T = K/(a + b) *only at these times*, that is t = [1/(a + b)] + [NK/(a + b)], N = 0,1,2,...The wave doesn't have this period for any in between times. Therefore it is neither strictly periodic or aperiodic. (Or is it 'almost periodic'?) So to test this I tried isolating one period, repeated it and put it to a FT. But all it gave was an infinite FS with large contributions from a and b.
>
> Again, given that (a, b) are coprime, sin(2pi*at) + sin(2pi*bt) =
> 2sin(pi(a + b)t)cos(pi(a - b)t), the maxima of this wave satisfy
> > 2sin(pi(a + b)t)cos(pi(a - b)t) = (+ or -) 2cos(pi(a - b)t) giving
> > sin (pi(a + b)t) = (+ or -)1.
>
> I don't get this jump here, can you explain? Why would the maxima
> satisfy that equation?

Because the envelope is (+or-)2cos(pi(a - b)t). The original wave and this meet only at extrema and finding the times when this occurs requires solving the simultaneous equation given above.
>
> > Solving for time gives t = (2k + 1)/2(a + b) where k = 0,1,2... The first largest maxima occurs at K = 0 and t = 1/2(a + b). The second occurs at k = (p + q), where p and q are the corresponding JI interval, and t = 2(p + q)/2(a + b). The difference is T = (p + q)/(a + b) and ~ GCD = 1/T. Since (p + q) is whole then whole numbered multiples of it will also belong to the set k. IOW this period will continue throughout the wave starting from the initial time.
>
> I don't have MATLAB on this computer to test that but I'll assume
> you're right. So your goal is to find the maxima of these functions
> and then show that they are "nearly" the maxima of some simpler
> function, right?

Something like that. Yesterday I said that I couldn't find a proof. But today something occurred to me. The integer K in ~ GCD = (a + b)/K is *given*. For 81/64 for eg it is K = 9. And from here we can now deduce our 'simpler' interval p/q as follows. Taking the ratio (interval) between our original freq's and this gives aK/(a + b) and bK/(a + b). These can be put in the 'quotient-remainder' form
aK/(a + b) = p + R/(a + b),
bK/(a + b) = q - R/(a + b)
where R = aq - pb. For a/b = 81/64,(a + b) = 145, p = 5, q = 4 and R = 4.
Adding the two values above we get K = (p + q), here 5 + 4 = 9. Thus, our approx GCD becomes (a + b)/(p + q). Finally the original intervals a and b can be regarded as slightly detuned 5th and 4th harmonics, respectively, of this 'fundamental'. More precisely, p*(a + b)/(p + q) = a - R and q*(a + b)/(p + q) = b + R. Interestingly, this holds for irrational intervals as well. For eg, if we take the tempered major third a = 64*2^(1/3) = 80.63494...and b = 64 then everything just said holds.

>
> > This frequency isn't revealed by a Fourier Analysis. It's maxima are not all equal, nor are the values of the function in between. But this is not to say that the wave itself is not Fourier analysable.
>
> WHAT frequency? The frequency of the sin (pi(a + b)t) term? No, it
> won't appear as an impulse in the Fourier transform of the wave, but
> its existence can be clearly deduced by finding the two spikes on the
> transformed curve, getting the distance between them, and dividing by
> two, as you have done mathematically here.
>
> > Now the remaining problem is to find out WHY k = (p + q) in a rigorous mathematical sense. Even though I've tested many examples, including totally random ones, and they all surprisingly give the expected result, I'm still at a loss to find *the* mathematical proof for it. As I mentioned, it also works for when a is an irrational interval from b. (p and q are still whole since they are members of the integers k) But this so far is still just an observation, not a proof. So why don't you work with me to help find this proof instead of going over old ground?
>
> I'd be happy to, but I'm not going to get into this notion that
> Fourier Analysis is "incomplete." It's a completely linear transform
> and is as "complete" as the rest of mathematics. I don't think you
> understand what the concept of a "virtual fundamental" really is, or
> else you wouldn't be so adamant that is "wrong" (it most certainly is
> not and is not incompatible with anything you're doing here).
>
> The difference is that most theories so far have been working in the
> frequency domain, and you are working in the time domain here. That
> being said, everything you're doing would have a comparable frequency
> domain analysis as well.

I hope what I said above has made this clearer. Unless I'm mistaken, it's not a 'frequency' in the strict Fourier sense and therefore won't be revealed within the Freq domain. Long before I ever heard of virtual pitch I was convinced that GCD, which is actually the frequency of the wave, explained tonality. But I was stumped at how we could go from say 80/64 with GCD 16 to 81/64 with GCD 1. This I think solves that problem. Now I can look back on my previous GCD work and generalise it. Observe for eg how GCD's can create their own Fourier Series where they are analysed into GCD's and not sine waves, in a manner of speaking. 6 and 9 have GCD = 3 and are the 2 and 3 harmonics. Taking the harmonic series of each gives 6, 12, 18, 24...and 9, 18, 27, 36...But these have GCD's 3, 6, 9, 12... which are themselves harmonics of the GCD; GCD of (12, 18) = 6 = 2 x 3 etc... IOW we could consider this a series where our 'fundamental' or 'element' is say SUM [0->N] [sin(2piN6t) + sin(2piN9t)] rather than a single sine wave. From this point of view, a single sine wave might be considered as just one special case (of course we could further analyse to sine waves but that's not my point).

If I've got virtual pitch all wrong then I apologise in advance for what I'm about to say. Please feel free to correct me. But I simply find it hard to believe that the ear or brain has some special innate capacity to just invent fundamentals that are not there in the wave somehow. It seems illogical and unnatural to me. Or if it sometimes does imagine stuff then I still figure that the brain/mind must be responding to *something*. I prefer to believe that we can sometimes simply hear the frequency or near frequency directly, that it is as real as the sine waves that go into making it, that single sine waves are just one special case (the greater set being the GCD's or approx ones) and that musical tonality has this basis in reality. And fully aware of the difficulties inherent in all aesthetics, the murky ground between perception and expectation, sensation and thought, I just can't help the feeling that psychoacoustics interpreted the facts to "match" the theory, that they had sine waves in the mind prior to experiment and followed the logical trail. As I said earlier this year, can you or anyone actually *prove* that we don't hear the GCD or approx GCD, that we're not responding to objective stimuli? To prove that its in the mind would require destroying the wave in the process. Sure, Terhardt says that we cannot have virtual pitch without at least one spectral component. But I find this distinction itself weighted towards sine waves, "spectral" = real = sine waves. At any rate, I suspect that the general 'scientific' belief that art is subjective destroyed any motive or willpower to look for alternate explanations.

-Rick
>
> > >
> > > > Therefore I disagree with the view that FA is complete. Yet it doesn't contradict FA either. While the theory of Almost periodic functions comes close (and will probably play some part yet), I've never come across any scientist who has even touched on this particular theory, let alone solved it. IOW its probably the humble beginnings of a new *scientific* theory. Have I ever said anything different?
> > >
> > > I don't understand what you mean by this. What particular theory?
> > > You'll note that if let the two intervals mentioned run on long
> > > enough, that they do end up diverging. How close is "close enough" for them to be considered equivalent?
> >
>
> > What two intervals? I'm not simply 'comparing' an 81/64 with a 5/4 for eg but saying directly that the former wave *already* contains slightly detuned cycles of the latter. For this example, (a + b)/(p + q) = 145/9 = 16.111..., 64/(64.111..) = 4 - (4/145) and 81/(64.111..) = 5 + (4/145). Since the ear/brain does not perceive or need to perceive every cycle of a wave, then the theory is that sub-harmonic matching does seem to have some physical basis in wave theory.
>
> Of course it does. However, an actual sinusoid at the subharmonic
> frequency where the periods sync up will not exist. And so the nerve
> endings that correspond to that frequency in the cochlea will not get
> stimulated. However, your brain will perform a very similar
> mathematical analysis to what you're doing here (finding repeating
> maxima) and find its existence that way, and "create" the sound for
> you to hear it. Hence the term, "virtual pitch."
>
> -Mike
>
> > -Rick
> > >
> > >
> >
> >
>

🔗rick <rick_ballan@...>

PS: I forgot to say that for a = 64*(2^(1/3) = 80.6349... K = 9 stays the same as for 81/64. In fact all major thirds will hold this value.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > > I don't know what you mean by "the scientists". I disagree with the particular psychoacoustic explanation, not science per se. And because these frequencies are not strictly periodic then they will not be revealed by Fourier Analysis either.
> > >
> > > Well, that's not true.
> >
> > Yes it is true. Stop guessing Mike.
>
> Fourier Analysis isn't limited to periodic functions and it never has
> been. A Fourier "Series" is limited to periodic functions. The
> classical Fourier Transform works on aperiodic functions from t = -Inf
> to Inf and the transform is a continuous function.
>
> Again, given that (a, b) are coprime, sin(2pi*at) + sin(2pi*bt) =
> 2sin(pi(a + b)t)cos(pi(a - b)t), the maxima of this wave satisfy
> > 2sin(pi(a + b)t)cos(pi(a - b)t) = (+ or -) 2cos(pi(a - b)t) giving
> > sin (pi(a + b)t) = (+ or -)1.
>
> I don't get this jump here, can you explain? Why would the maxima
> satisfy that equation?
>
> > Solving for time gives t = (2k + 1)/2(a + b) where k = 0,1,2... The first largest maxima occurs at K = 0 and t = 1/2(a + b). The second occurs at k = (p + q), where p and q are the corresponding JI interval, and t = 2(p + q)/2(a + b). The difference is T = (p + q)/(a + b) and ~ GCD = 1/T. Since (p + q) is whole then whole numbered multiples of it will also belong to the set k. IOW this period will continue throughout the wave starting from the initial time.
>
> I don't have MATLAB on this computer to test that but I'll assume
> you're right. So your goal is to find the maxima of these functions
> and then show that they are "nearly" the maxima of some simpler
> function, right?
>
> > This frequency isn't revealed by a Fourier Analysis. It's maxima are not all equal, nor are the values of the function in between. But this is not to say that the wave itself is not Fourier analysable.
>
> WHAT frequency? The frequency of the sin (pi(a + b)t) term? No, it
> won't appear as an impulse in the Fourier transform of the wave, but
> its existence can be clearly deduced by finding the two spikes on the
> transformed curve, getting the distance between them, and dividing by
> two, as you have done mathematically here.
>
> > Now the remaining problem is to find out WHY k = (p + q) in a rigorous mathematical sense. Even though I've tested many examples, including totally random ones, and they all surprisingly give the expected result, I'm still at a loss to find *the* mathematical proof for it. As I mentioned, it also works for when a is an irrational interval from b. (p and q are still whole since they are members of the integers k) But this so far is still just an observation, not a proof. So why don't you work with me to help find this proof instead of going over old ground?
>
> I'd be happy to, but I'm not going to get into this notion that
> Fourier Analysis is "incomplete." It's a completely linear transform
> and is as "complete" as the rest of mathematics. I don't think you
> understand what the concept of a "virtual fundamental" really is, or
> else you wouldn't be so adamant that is "wrong" (it most certainly is
> not and is not incompatible with anything you're doing here).
>
> The difference is that most theories so far have been working in the
> frequency domain, and you are working in the time domain here. That
> being said, everything you're doing would have a comparable frequency
> domain analysis as well.
>
> > >
> > > > Therefore I disagree with the view that FA is complete. Yet it doesn't contradict FA either. While the theory of Almost periodic functions comes close (and will probably play some part yet), I've never come across any scientist who has even touched on this particular theory, let alone solved it. IOW its probably the humble beginnings of a new *scientific* theory. Have I ever said anything different?
> > >
> > > I don't understand what you mean by this. What particular theory?
> > > You'll note that if let the two intervals mentioned run on long
> > > enough, that they do end up diverging. How close is "close enough" for them to be considered equivalent?
> >
>
> > What two intervals? I'm not simply 'comparing' an 81/64 with a 5/4 for eg but saying directly that the former wave *already* contains slightly detuned cycles of the latter. For this example, (a + b)/(p + q) = 145/9 = 16.111..., 64/(64.111..) = 4 - (4/145) and 81/(64.111..) = 5 + (4/145). Since the ear/brain does not perceive or need to perceive every cycle of a wave, then the theory is that sub-harmonic matching does seem to have some physical basis in wave theory.
>
> Of course it does. However, an actual sinusoid at the subharmonic
> frequency where the periods sync up will not exist. And so the nerve
> endings that correspond to that frequency in the cochlea will not get
> stimulated. However, your brain will perform a very similar
> mathematical analysis to what you're doing here (finding repeating
> maxima) and find its existence that way, and "create" the sound for
> you to hear it. Hence the term, "virtual pitch."
>
> -Mike
>
> > -Rick
> > >
> > >
> >
> >
>

🔗Mike Battaglia <battaglia01@...>

Rick, I will read the whole thing later and respond to it, but as for
now, about what you said here:

> I hope what I said above has made this clearer. Unless I'm mistaken, it's not a 'frequency' in the strict Fourier sense and therefore won't be revealed within the Freq domain. Long before I ever heard of virtual pitch I was convinced that GCD, which is actually the frequency of the wave, explained tonality. But I was stumped at how we could go from say 80/64 with GCD 16 to 81/64 with GCD 1. This I think solves that problem. Now I can look back on my previous GCD work and generalise it. Observe for eg how GCD's can create their own Fourier Series where they are analysed into GCD's and not sine waves, in a manner of speaking. 6 and 9 have GCD = 3 and are the 2 and 3 harmonics. Taking the harmonic series of each gives 6, 12, 18, 24...and 9, 18, 27, 36...But these have GCD's 3, 6, 9, 12... which are themselves harmonics of the GCD; GCD of (12, 18) = 6 = 2 x 3 etc... IOW we could consider this a series where our 'fundamental' or 'element' is say SUM [0->N] [sin(2piN6t) + sin(2piN9t)] rather than a single sine wave. From this point of view, a single sine wave might be considered as just one special case (of course we could further analyse to sine waves but that's not my point).

The idea is that the brain creates a "virtual pitch" that corresponds
to something close to the GCD of the wave coming in. But since the
auditory system doesn't have until infinity to see what the real GCD
of two sinusoids is, it just takes a small chunk of time instead. This
chunk of time or "window size" I believe would most likely be
frequency dependent, which explains the recent trend in using wavelets
instead of the STFT in analyzing sound.

But I don't think this is really going to explain all of tonality. For
example, this would break down for a 10:12:15 chord. The root of a
10:12:15 chord is almost always heard as 5, not 1. This is because
each dyad of the chord has its own "GCD" and the one produced by 10:15
is going to be stronger than the rest of them, with the 12:15 one
coming in at a close second.

It should be noted that you are using the term "GCD" here in the same
way that everyone else is using the term "virtual fundamental," but
for some reason you swear you're talking about something else.

> I prefer to believe that we can sometimes simply hear the frequency or near frequency directly, that it is as real as the sine waves that go into making it, that single sine waves are just one special case (the greater set being the GCD's or approx ones) and that musical tonality has this basis in reality.

The reason that it's called virtual is, again, that two pressure waves
at frequencies 400 Hz and 500 Hz simply will not make the region of
the cochlea corresponding to 100 Hz resonate. Your brain simply
recreates it based off of the periodicity information in the signal.
And since the people who discovered this can call it whatever they
want, they went with "virtual pitch" instead. Or "phantom
fundamental," or "missing fundamental," or "virtual fundamental," or
any one of those.

-Mike

🔗genewardsmith <genewardsmith@...>

🔗cameron <misterbobro@...>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗rick <rick_ballan@...>

Ok Mike, thanks. I suppose as a musician I can tend to go on the defensive a bit concerning the term "virtual" since it implies "fake". But of course this is not sense in which it was meant. As for the 10:12:15 chord, let me sleep on this and get back (Autumn (fall?) has arrived here and me and my dog seem to have come down with something. Strange, he keeps sneezing. Whoever heard of that?).

Cheers mate

-Rick

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Rick, I will read the whole thing later and respond to it, but as for
> now, about what you said here:
>
> > I hope what I said above has made this clearer. Unless I'm mistaken, it's not a 'frequency' in the strict Fourier sense and therefore won't be revealed within the Freq domain. Long before I ever heard of virtual pitch I was convinced that GCD, which is actually the frequency of the wave, explained tonality. But I was stumped at how we could go from say 80/64 with GCD 16 to 81/64 with GCD 1. This I think solves that problem. Now I can look back on my previous GCD work and generalise it. Observe for eg how GCD's can create their own Fourier Series where they are analysed into GCD's and not sine waves, in a manner of speaking. 6 and 9 have GCD = 3 and are the 2 and 3 harmonics. Taking the harmonic series of each gives 6, 12, 18, 24...and 9, 18, 27, 36...But these have GCD's 3, 6, 9, 12... which are themselves harmonics of the GCD; GCD of (12, 18) = 6 = 2 x 3 etc... IOW we could consider this a series where our 'fundamental' or 'element' is say SUM [0->N] [sin(2piN6t) + sin(2piN9t)] rather than a single sine wave. From this point of view, a single sine wave might be considered as just one special case (of course we could further analyse to sine waves but that's not my point).
>
> The idea is that the brain creates a "virtual pitch" that corresponds
> to something close to the GCD of the wave coming in. But since the
> auditory system doesn't have until infinity to see what the real GCD
> of two sinusoids is, it just takes a small chunk of time instead. This
> chunk of time or "window size" I believe would most likely be
> frequency dependent, which explains the recent trend in using wavelets
> instead of the STFT in analyzing sound.
>
> But I don't think this is really going to explain all of tonality. For
> example, this would break down for a 10:12:15 chord. The root of a
> 10:12:15 chord is almost always heard as 5, not 1. This is because
> each dyad of the chord has its own "GCD" and the one produced by 10:15
> is going to be stronger than the rest of them, with the 12:15 one
> coming in at a close second.
>
> It should be noted that you are using the term "GCD" here in the same
> way that everyone else is using the term "virtual fundamental," but
> for some reason you swear you're talking about something else.
>
> > I prefer to believe that we can sometimes simply hear the frequency or near frequency directly, that it is as real as the sine waves that go into making it, that single sine waves are just one special case (the greater set being the GCD's or approx ones) and that musical tonality has this basis in reality.
>
> The reason that it's called virtual is, again, that two pressure waves
> at frequencies 400 Hz and 500 Hz simply will not make the region of
> the cochlea corresponding to 100 Hz resonate. Your brain simply
> recreates it based off of the periodicity information in the signal.
> And since the people who discovered this can call it whatever they
> want, they went with "virtual pitch" instead. Or "phantom
> fundamental," or "missing fundamental," or "virtual fundamental," or
> any one of those.
>
> -Mike
>

🔗rick <rick_ballan@...>

Hi Mike,

I couldn't relax thinking of the problem you posed. I've posted a pdf called 10;12;15. You mentioned that we hear something close to 5. I don't know why yet but there are definite peaks at the times 0.01, 0.21, 0.41, 0.61 etc...T = 0.2 = 1/5. IOW this frequency appears to be a product of the wave.

I was also looking at quasi/almost periodic functions in the Wiki article you suggested to Cam. I suspect that what I'm looking at might have to do with quasiperiodic signals (I know, not functions) because it just looks right. Study study!

-Rick

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

Because you never hear "just" the chord coming in. You hear an unbroken wash
of sound and the brain separates it out and finds different periodicities in
it. In the case of 10:12:15, there are 4 periodicities, and they are, from
strongest to weakest:

10:15 (as 3:2)
12:15 (as 5:4)
10:12 (as 6:5)
10:12:15 (as itself)

-Mike

On Thu, Apr 22, 2010 at 8:02 AM, rick <rick_ballan@...> wrote:

>
>
> Hi Mike,
>
> I couldn't relax thinking of the problem you posed. I've posted a pdf
> called 10;12;15. You mentioned that we hear something close to 5. I don't
> know why yet but there are definite peaks at the times 0.01, 0.21, 0.41,
> 0.61 etc...T = 0.2 = 1/5. IOW this frequency appears to be a product of the
> wave.
>
> I was also looking at quasi/almost periodic functions in the Wiki article
> you suggested to Cam. I suspect that what I'm looking at might have to do
> with quasiperiodic signals (I know, not functions) because it just looks
> right. Study study!
>
>
> -Rick
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Mike Battaglia
> <battaglia01@...> wrote:
> >
> > Rick, I will read the whole thing later and respond to it, but as for
> > now, about what you said here:
> >
> > > I hope what I said above has made this clearer. Unless I'm mistaken,
> it's not a 'frequency' in the strict Fourier sense and therefore won't be
> revealed within the Freq domain. Long before I ever heard of virtual pitch I
> was convinced that GCD, which is actually the frequency of the wave,
> explained tonality. But I was stumped at how we could go from say 80/64 with
> GCD 16 to 81/64 with GCD 1. This I think solves that problem. Now I can look
> back on my previous GCD work and generalise it. Observe for eg how GCD's can
> create their own Fourier Series where they are analysed into GCD's and not
> sine waves, in a manner of speaking. 6 and 9 have GCD = 3 and are the 2 and
> 3 harmonics. Taking the harmonic series of each gives 6, 12, 18, 24...and 9,
> 18, 27, 36...But these have GCD's 3, 6, 9, 12... which are themselves
> harmonics of the GCD; GCD of (12, 18) = 6 = 2 x 3 etc... IOW we could
> consider this a series where our 'fundamental' or 'element' is say SUM
> [0->N] [sin(2piN6t) + sin(2piN9t)] rather than a single sine wave. From this
> point of view, a single sine wave might be considered as just one special
> case (of course we could further analyse to sine waves but that's not my
> point).
> >
> > The idea is that the brain creates a "virtual pitch" that corresponds
> > to something close to the GCD of the wave coming in. But since the
> > auditory system doesn't have until infinity to see what the real GCD
> > of two sinusoids is, it just takes a small chunk of time instead. This
> > chunk of time or "window size" I believe would most likely be
> > frequency dependent, which explains the recent trend in using wavelets
> > instead of the STFT in analyzing sound.
> >
> > But I don't think this is really going to explain all of tonality. For
> > example, this would break down for a 10:12:15 chord. The root of a
> > 10:12:15 chord is almost always heard as 5, not 1. This is because
> > each dyad of the chord has its own "GCD" and the one produced by 10:15
> > is going to be stronger than the rest of them, with the 12:15 one
> > coming in at a close second.
> >
> > It should be noted that you are using the term "GCD" here in the same
> > way that everyone else is using the term "virtual fundamental," but
> > for some reason you swear you're talking about something else.
> >
> > > I prefer to believe that we can sometimes simply hear the frequency or
> near frequency directly, that it is as real as the sine waves that go into
> making it, that single sine waves are just one special case (the greater set
> being the GCD's or approx ones) and that musical tonality has this basis in
> reality.
> >
> > The reason that it's called virtual is, again, that two pressure waves
> > at frequencies 400 Hz and 500 Hz simply will not make the region of
> > the cochlea corresponding to 100 Hz resonate. Your brain simply
> > recreates it based off of the periodicity information in the signal.
> > And since the people who discovered this can call it whatever they
> > want, they went with "virtual pitch" instead. Or "phantom
> > fundamental," or "missing fundamental," or "virtual fundamental," or
> > any one of those.
> >
> > -Mike
> >
>
>
>

🔗rick <rick_ballan@...>

Hi Marcel,

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> Hi Rick,
>
> Uhm, the greatest common divisor for 5/4 is 1, not 16. Just like it's 1.
> >
> > The GCD of 81/64 is 16 is what I said. 81/64 = (5 x 16)/(4 x 16) = 5/4.
> > Therefore 16 is to (81, 64) what 1 is to (5, 4).
> >
> Aah ok, I didn't follow your logic before.
> Ok but that said, 80/64 (I caugt your error) has 16 repetitions in it, 81/64
> has 1 repetition.

I'm not exactly sure what you mean by "16 repetitions". The GCD 16 divides into 64 four times and into 80 five times. That is, 64/16 = 4 and 80/16 = 5. Therefore there are 4 repetitions of 16 to make up 64 and 5 to give 80. The wave oscillates at the GCD frequency 16 and by doing so places 64 and 80 in the role of the 4th and 5th harmonics of this fundamental.

> And you're seeing 81/64 as an almost 5/4, as 81/64 has almost 16 "near repetitions" in it's wave?

Close. I'm seeing that 81/64 as an "almost 5/4" has almost 4 repetitions of the approx GCD = 16.111...to make up 64 and almost 5 repetitions to give 81. Since I found that this frequency too is 'in the wave' then it sets up 64 and 81 as slightly detuned 4th and 5th harmonics. Or what comes to the same thing, 16.111...as a slightly detuned 16 fundamental.

This goes for any interval close enough to 5/4, rational and irrational?

Apparently, and not just 5/4. If we begin with any large numbered pair b:a, where a and b have some distance between them (not 81 and 82 for eg), then they appear to map back onto some smaller "JI" interval q:p. I suppose that the criteria is that a/b ~ = p/q. But I don't know how to put it into proper mathematical terms yet. And yes, 'a' can be irrational. For eg, if instead of 81 we take 64*(2^(1/3) = 80.6349... so that a/b = (2^(1/3) then it still maps back onto a 5/4. The reason for this has to do with the fact that t = (2k + 1)/2(a + b), k = 0,1,2,...and k is an integer.

> And you have a formula for when an interval for instance
> is somehwere between 6/5 and 5/4, to determine whether that interval has a stronger resemblance to 5/4 or to 6/5? So in effect whether it is more of a mistuned 6/5 or mistuned 5/4?

Yes, this was an unforeseen consequence. With ~ GCD = (a + b)/(p + q), then p*(a + b)/(p + q) = a - R and q*(a + b)/(p + q) = b + R where R = (aq - pb)/(p + q). Obviously the smaller R gives the greater approximation. But this only applies when there might be some doubt as to what (p, q) are. In most cases they will simple be given.
>
> Marcel
>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

>
> I'm not exactly sure what you mean by "16 repetitions".

(actually that should have been 15 repetitions, making 16 5/4 wavelengths)
Well, in 80/64 and 81/64 the common number is 64.
It means that in the time that the lower wave does 64 sine wave cycles, the
higher wave does 80 or 81.
The resulting combination wave of the low wave 64 and the high wave 80 has a
wavelength itself before it repeats.
The wavelength is the ratio of the 2 waves divided by the GCD. So the
wavelength for 80/64 is 5/4, and the wavelength of 5/4 will be 16 times in
80/64.
For 81/64 the GCD is 1, so 81/64 is the smallest combination wavelength that
repeats itself.
So 81/64 has 16 5/4 waves in it's wave, at thesame length as 81/64.

> The GCD 16 divides into 64 four times and into 80 five times. That is,
> 64/16 = 4 and 80/16 = 5. Therefore there are 4 repetitions of 16 to make up
> 64 and 5 to give 80. The wave oscillates at the GCD frequency 16 and by
> doing so places 64 and 80 in the role of the 4th and 5th harmonics of this
> fundamental.
>
> > And you're seeing 81/64 as an almost 5/4, as 81/64 has almost 16 "near
> repetitions" in it's wave?
>
> Close. I'm seeing that 81/64 as an "almost 5/4" has almost 4 repetitions of
> the approx GCD = 16.111...to make up 64 and almost 5 repetitions to give 81.
>

Huh I don't get it, either I'm again not following your logic, or your math
is wrong here.
5/4 has 16 cycles in the time 81/64 does 1.

> Since I found that this frequency too is 'in the wave' then it sets up 64
> and 81 as slightly detuned 4th and 5th harmonics. Or what comes to the same
> thing, 16.111...as a slightly detuned 16 fundamental.
>
> This goes for any interval close enough to 5/4, rational and irrational?
>
> Apparently, and not just 5/4. If we begin with any large numbered pair b:a,
> where a and b have some distance between them (not 81 and 82 for eg), then
> they appear to map back onto some smaller "JI" interval q:p. I suppose that
> the criteria is that a/b ~ = p/q. But I don't know how to put it into proper
> mathematical terms yet. And yes, 'a' can be irrational. For eg, if instead
> of 81 we take 64*(2^(1/3) = 80.6349... so that a/b = (2^(1/3) then it still
> maps back onto a 5/4. The reason for this has to do with the fact that t =
> (2k + 1)/2(a + b), k = 0,1,2,...and k is an integer.
>
> > And you have a formula for when an interval for instance
> > is somehwere between 6/5 and 5/4, to determine whether that interval has
> a stronger resemblance to 5/4 or to 6/5? So in effect whether it is more of
> a mistuned 6/5 or mistuned 5/4?
>
> Yes, this was an unforeseen consequence. With ~ GCD = (a + b)/(p + q), then
> p*(a + b)/(p + q) = a - R and q*(a + b)/(p + q) = b + R where R = (aq -
> pb)/(p + q). Obviously the smaller R gives the greater approximation. But
> this only applies when there might be some doubt as to what (p, q) are. In
> most cases they will simple be given.

Ok that's nice!

But what are the other implications of your math, what's your goal?
I still don't understand this.

Marcel

🔗rick <rick_ballan@...>

Yeah that's what I figured too. Being the stronger interval the fifth overrides the fact that the 12 is not one of its upper harmonics. But the brain still 'reasons' it that way, or something like that.

-Rick
>
> On Thu, Apr 22, 2010 at 8:02 AM, rick <rick_ballan@...> wrote:
>
> >
> >
> > Hi Mike,
> >
> > I couldn't relax thinking of the problem you posed. I've posted a pdf
> > called 10;12;15. You mentioned that we hear something close to 5. I don't
> > know why yet but there are definite peaks at the times 0.01, 0.21, 0.41,
> > 0.61 etc...T = 0.2 = 1/5. IOW this frequency appears to be a product of the
> > wave.
> >
> > I was also looking at quasi/almost periodic functions in the Wiki article
> > you suggested to Cam. I suspect that what I'm looking at might have to do
> > with quasiperiodic signals (I know, not functions) because it just looks
> > right. Study study!
> >
> >
> > -Rick
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Mike Battaglia
> > <battaglia01@> wrote:
> > >
> > > Rick, I will read the whole thing later and respond to it, but as for
> > > now, about what you said here:
> > >
> > > > I hope what I said above has made this clearer. Unless I'm mistaken,
> > it's not a 'frequency' in the strict Fourier sense and therefore won't be
> > revealed within the Freq domain. Long before I ever heard of virtual pitch I
> > was convinced that GCD, which is actually the frequency of the wave,
> > explained tonality. But I was stumped at how we could go from say 80/64 with
> > GCD 16 to 81/64 with GCD 1. This I think solves that problem. Now I can look
> > back on my previous GCD work and generalise it. Observe for eg how GCD's can
> > create their own Fourier Series where they are analysed into GCD's and not
> > sine waves, in a manner of speaking. 6 and 9 have GCD = 3 and are the 2 and
> > 3 harmonics. Taking the harmonic series of each gives 6, 12, 18, 24...and 9,
> > 18, 27, 36...But these have GCD's 3, 6, 9, 12... which are themselves
> > harmonics of the GCD; GCD of (12, 18) = 6 = 2 x 3 etc... IOW we could
> > consider this a series where our 'fundamental' or 'element' is say SUM
> > [0->N] [sin(2piN6t) + sin(2piN9t)] rather than a single sine wave. From this
> > point of view, a single sine wave might be considered as just one special
> > case (of course we could further analyse to sine waves but that's not my
> > point).
> > >
> > > The idea is that the brain creates a "virtual pitch" that corresponds
> > > to something close to the GCD of the wave coming in. But since the
> > > auditory system doesn't have until infinity to see what the real GCD
> > > of two sinusoids is, it just takes a small chunk of time instead. This
> > > chunk of time or "window size" I believe would most likely be
> > > frequency dependent, which explains the recent trend in using wavelets
> > > instead of the STFT in analyzing sound.
> > >
> > > But I don't think this is really going to explain all of tonality. For
> > > example, this would break down for a 10:12:15 chord. The root of a
> > > 10:12:15 chord is almost always heard as 5, not 1. This is because
> > > each dyad of the chord has its own "GCD" and the one produced by 10:15
> > > is going to be stronger than the rest of them, with the 12:15 one
> > > coming in at a close second.
> > >
> > > It should be noted that you are using the term "GCD" here in the same
> > > way that everyone else is using the term "virtual fundamental," but
> > > for some reason you swear you're talking about something else.
> > >
> > > > I prefer to believe that we can sometimes simply hear the frequency or
> > near frequency directly, that it is as real as the sine waves that go into
> > making it, that single sine waves are just one special case (the greater set
> > being the GCD's or approx ones) and that musical tonality has this basis in
> > reality.
> > >
> > > The reason that it's called virtual is, again, that two pressure waves
> > > at frequencies 400 Hz and 500 Hz simply will not make the region of
> > > the cochlea corresponding to 100 Hz resonate. Your brain simply
> > > recreates it based off of the periodicity information in the signal.
> > > And since the people who discovered this can call it whatever they
> > > want, they went with "virtual pitch" instead. Or "phantom
> > > fundamental," or "missing fundamental," or "virtual fundamental," or
> > > any one of those.
> > >
> > > -Mike
> > >
> >
> >
> >
>

🔗rick <rick_ballan@...>

🔗genewardsmith <genewardsmith@...>

🔗rick <rick_ballan@...>

Another possibility might be that we hear a slightly mistuned 16:19:24 i.e (8/5) x 10:12:15 = 16: 19.2: 24, and 5 becomes 8? Just a thought.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Because you never hear "just" the chord coming in. You hear an unbroken wash
> of sound and the brain separates it out and finds different periodicities in
> it. In the case of 10:12:15, there are 4 periodicities, and they are, from
> strongest to weakest:
>
> 10:15 (as 3:2)
> 12:15 (as 5:4)
> 10:12 (as 6:5)
> 10:12:15 (as itself)
>
> -Mike
>
>
> On Thu, Apr 22, 2010 at 8:02 AM, rick <rick_ballan@...> wrote:
>
> >
> >
> > Hi Mike,
> >
> > I couldn't relax thinking of the problem you posed. I've posted a pdf
> > called 10;12;15. You mentioned that we hear something close to 5. I don't
> > know why yet but there are definite peaks at the times 0.01, 0.21, 0.41,
> > 0.61 etc...T = 0.2 = 1/5. IOW this frequency appears to be a product of the
> > wave.
> >
> > I was also looking at quasi/almost periodic functions in the Wiki article
> > you suggested to Cam. I suspect that what I'm looking at might have to do
> > with quasiperiodic signals (I know, not functions) because it just looks
> > right. Study study!
> >
> >
> > -Rick
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Mike Battaglia
> > <battaglia01@> wrote:
> > >
> > > Rick, I will read the whole thing later and respond to it, but as for
> > > now, about what you said here:
> > >
> > > > I hope what I said above has made this clearer. Unless I'm mistaken,
> > it's not a 'frequency' in the strict Fourier sense and therefore won't be
> > revealed within the Freq domain. Long before I ever heard of virtual pitch I
> > was convinced that GCD, which is actually the frequency of the wave,
> > explained tonality. But I was stumped at how we could go from say 80/64 with
> > GCD 16 to 81/64 with GCD 1. This I think solves that problem. Now I can look
> > back on my previous GCD work and generalise it. Observe for eg how GCD's can
> > create their own Fourier Series where they are analysed into GCD's and not
> > sine waves, in a manner of speaking. 6 and 9 have GCD = 3 and are the 2 and
> > 3 harmonics. Taking the harmonic series of each gives 6, 12, 18, 24...and 9,
> > 18, 27, 36...But these have GCD's 3, 6, 9, 12... which are themselves
> > harmonics of the GCD; GCD of (12, 18) = 6 = 2 x 3 etc... IOW we could
> > consider this a series where our 'fundamental' or 'element' is say SUM
> > [0->N] [sin(2piN6t) + sin(2piN9t)] rather than a single sine wave. From this
> > point of view, a single sine wave might be considered as just one special
> > case (of course we could further analyse to sine waves but that's not my
> > point).
> > >
> > > The idea is that the brain creates a "virtual pitch" that corresponds
> > > to something close to the GCD of the wave coming in. But since the
> > > auditory system doesn't have until infinity to see what the real GCD
> > > of two sinusoids is, it just takes a small chunk of time instead. This
> > > chunk of time or "window size" I believe would most likely be
> > > frequency dependent, which explains the recent trend in using wavelets
> > > instead of the STFT in analyzing sound.
> > >
> > > But I don't think this is really going to explain all of tonality. For
> > > example, this would break down for a 10:12:15 chord. The root of a
> > > 10:12:15 chord is almost always heard as 5, not 1. This is because
> > > each dyad of the chord has its own "GCD" and the one produced by 10:15
> > > is going to be stronger than the rest of them, with the 12:15 one
> > > coming in at a close second.
> > >
> > > It should be noted that you are using the term "GCD" here in the same
> > > way that everyone else is using the term "virtual fundamental," but
> > > for some reason you swear you're talking about something else.
> > >
> > > > I prefer to believe that we can sometimes simply hear the frequency or
> > near frequency directly, that it is as real as the sine waves that go into
> > making it, that single sine waves are just one special case (the greater set
> > being the GCD's or approx ones) and that musical tonality has this basis in
> > reality.
> > >
> > > The reason that it's called virtual is, again, that two pressure waves
> > > at frequencies 400 Hz and 500 Hz simply will not make the region of
> > > the cochlea corresponding to 100 Hz resonate. Your brain simply
> > > recreates it based off of the periodicity information in the signal.
> > > And since the people who discovered this can call it whatever they
> > > want, they went with "virtual pitch" instead. Or "phantom
> > > fundamental," or "missing fundamental," or "virtual fundamental," or
> > > any one of those.
> > >
> > > -Mike
> > >
> >
> >
> >
>

🔗Mike Battaglia <battaglia01@...>

I think that you hear all of those things in a single minor chord. The map
doesn't have to be self-consistent. Take the 12-tet chord C-E-A-D-G, for
example. You are going to hear all of those fourths as approximate 4/3's,
and then you're going to hear the C-E as an approximate 5/4, and then you'll
hear the C-G at the end as an approximate 3/1. Note that this doesn't
actually work out if you do the math.

-Mike

On Sat, Apr 24, 2010 at 1:20 AM, rick <rick_ballan@...> wrote:

>
>
> Another possibility might be that we hear a slightly mistuned 16:19:24 i.e
> (8/5) x 10:12:15 = 16: 19.2: 24, and 5 becomes 8? Just a thought.
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Mike Battaglia
> <battaglia01@...> wrote:
> >
> > Because you never hear "just" the chord coming in. You hear an unbroken
> wash
> > of sound and the brain separates it out and finds different periodicities
> in
> > it. In the case of 10:12:15, there are 4 periodicities, and they are,
> from
> > strongest to weakest:
> >
> > 10:15 (as 3:2)
> > 12:15 (as 5:4)
> > 10:12 (as 6:5)
> > 10:12:15 (as itself)
> >
> > -Mike
> >
> >
> > On Thu, Apr 22, 2010 at 8:02 AM, rick <rick_ballan@...> wrote:
> >
> > >
> > >
> > > Hi Mike,
> > >
> > > I couldn't relax thinking of the problem you posed. I've posted a pdf
> > > called 10;12;15. You mentioned that we hear something close to 5. I
> don't
> > > know why yet but there are definite peaks at the times 0.01, 0.21,
> 0.41,
> > > 0.61 etc...T = 0.2 = 1/5. IOW this frequency appears to be a product of
> the
> > > wave.
> > >
> > > I was also looking at quasi/almost periodic functions in the Wiki
> article
> > > you suggested to Cam. I suspect that what I'm looking at might have to
> do
> > > with quasiperiodic signals (I know, not functions) because it just
> looks
> > > right. Study study!
> > >
> > >
> > > -Rick
> > >
> > > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com> <tuning%
> 40yahoogroups.com>, Mike Battaglia
>
> > > <battaglia01@> wrote:
> > > >
> > > > Rick, I will read the whole thing later and respond to it, but as for
> > > > now, about what you said here:
> > > >
> > > > > I hope what I said above has made this clearer. Unless I'm
> mistaken,
> > > it's not a 'frequency' in the strict Fourier sense and therefore won't
> be
> > > revealed within the Freq domain. Long before I ever heard of virtual
> pitch I
> > > was convinced that GCD, which is actually the frequency of the wave,
> > > explained tonality. But I was stumped at how we could go from say 80/64
> with
> > > GCD 16 to 81/64 with GCD 1. This I think solves that problem. Now I can
> look
> > > back on my previous GCD work and generalise it. Observe for eg how
> GCD's can
> > > create their own Fourier Series where they are analysed into GCD's and
> not
> > > sine waves, in a manner of speaking. 6 and 9 have GCD = 3 and are the 2
> and
> > > 3 harmonics. Taking the harmonic series of each gives 6, 12, 18,
> 24...and 9,
> > > 18, 27, 36...But these have GCD's 3, 6, 9, 12... which are themselves
> > > harmonics of the GCD; GCD of (12, 18) = 6 = 2 x 3 etc... IOW we could
> > > consider this a series where our 'fundamental' or 'element' is say SUM
> > > [0->N] [sin(2piN6t) + sin(2piN9t)] rather than a single sine wave. From
> this
> > > point of view, a single sine wave might be considered as just one
> special
> > > case (of course we could further analyse to sine waves but that's not
> my
> > > point).
> > > >
> > > > The idea is that the brain creates a "virtual pitch" that corresponds
> > > > to something close to the GCD of the wave coming in. But since the
> > > > auditory system doesn't have until infinity to see what the real GCD
> > > > of two sinusoids is, it just takes a small chunk of time instead.
> This
> > > > chunk of time or "window size" I believe would most likely be
> > > > frequency dependent, which explains the recent trend in using
> wavelets
> > > > instead of the STFT in analyzing sound.
> > > >
> > > > But I don't think this is really going to explain all of tonality.
> For
> > > > example, this would break down for a 10:12:15 chord. The root of a
> > > > 10:12:15 chord is almost always heard as 5, not 1. This is because
> > > > each dyad of the chord has its own "GCD" and the one produced by
> 10:15
> > > > is going to be stronger than the rest of them, with the 12:15 one
> > > > coming in at a close second.
> > > >
> > > > It should be noted that you are using the term "GCD" here in the same
> > > > way that everyone else is using the term "virtual fundamental," but
> > > > for some reason you swear you're talking about something else.
> > > >
> > > > > I prefer to believe that we can sometimes simply hear the frequency
> or
> > > near frequency directly, that it is as real as the sine waves that go
> into
> > > making it, that single sine waves are just one special case (the
> greater set
> > > being the GCD's or approx ones) and that musical tonality has this
> basis in
> > > reality.
> > > >
> > > > The reason that it's called virtual is, again, that two pressure
> waves
> > > > at frequencies 400 Hz and 500 Hz simply will not make the region of
> > > > the cochlea corresponding to 100 Hz resonate. Your brain simply
> > > > recreates it based off of the periodicity information in the signal.
> > > > And since the people who discovered this can call it whatever they
> > > > want, they went with "virtual pitch" instead. Or "phantom
> > > > fundamental," or "missing fundamental," or "virtual fundamental," or
> > > > any one of those.
> > > >
> > > > -Mike
> > > >
> > >
> > >
> > >
> >
>
>
>

🔗rick <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I think that you hear all of those things in a single minor chord. The map
> doesn't have to be self-consistent. Take the 12-tet chord C-E-A-D-G, for
> example. You are going to hear all of those fourths as approximate 4/3's,
> and then you're going to hear the C-E as an approximate 5/4, and then you'll
> hear the C-G at the end as an approximate 3/1. Note that this doesn't
> actually work out if you do the math.
>
> -Mike
>
> When you say that this doesn't work out when you do the math, I assume you mean that if we take 5/4 x (4/3)^3 = 80/27 = 2.9629... then it doesn't equal 3/1? However, applying my method, with a = 80, b = 27 the Bezout's equations ax + by = 1 gives x = -1, y = 3 and p/q = 3/1.

-Rick

> On Sat, Apr 24, 2010 at 1:20 AM, rick <rick_ballan@...> wrote:
>
> >
> >
> > Another possibility might be that we hear a slightly mistuned 16:19:24 i.e
> > (8/5) x 10:12:15 = 16: 19.2: 24, and 5 becomes 8? Just a thought.
> >
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Mike Battaglia
> > <battaglia01@> wrote:
> > >
> > > Because you never hear "just" the chord coming in. You hear an unbroken
> > wash
> > > of sound and the brain separates it out and finds different periodicities
> > in
> > > it. In the case of 10:12:15, there are 4 periodicities, and they are,
> > from
> > > strongest to weakest:
> > >
> > > 10:15 (as 3:2)
> > > 12:15 (as 5:4)
> > > 10:12 (as 6:5)
> > > 10:12:15 (as itself)
> > >
> > > -Mike
> > >
> > >
> > > On Thu, Apr 22, 2010 at 8:02 AM, rick <rick_ballan@> wrote:
> > >
> > > >
> > > >
> > > > Hi Mike,
> > > >
> > > > I couldn't relax thinking of the problem you posed. I've posted a pdf
> > > > called 10;12;15. You mentioned that we hear something close to 5. I
> > don't
> > > > know why yet but there are definite peaks at the times 0.01, 0.21,
> > 0.41,
> > > > 0.61 etc...T = 0.2 = 1/5. IOW this frequency appears to be a product of
> > the
> > > > wave.
> > > >
> > > > I was also looking at quasi/almost periodic functions in the Wiki
> > article
> > > > you suggested to Cam. I suspect that what I'm looking at might have to
> > do
> > > > with quasiperiodic signals (I know, not functions) because it just
> > looks
> > > > right. Study study!
> > > >
> > > >
> > > > -Rick
> > > >
> > > > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com> <tuning%
> > 40yahoogroups.com>, Mike Battaglia
> >
> > > > <battaglia01@> wrote:
> > > > >
> > > > > Rick, I will read the whole thing later and respond to it, but as for
> > > > > now, about what you said here:
> > > > >
> > > > > > I hope what I said above has made this clearer. Unless I'm
> > mistaken,
> > > > it's not a 'frequency' in the strict Fourier sense and therefore won't
> > be
> > > > revealed within the Freq domain. Long before I ever heard of virtual
> > pitch I
> > > > was convinced that GCD, which is actually the frequency of the wave,
> > > > explained tonality. But I was stumped at how we could go from say 80/64
> > with
> > > > GCD 16 to 81/64 with GCD 1. This I think solves that problem. Now I can
> > look
> > > > back on my previous GCD work and generalise it. Observe for eg how
> > GCD's can
> > > > create their own Fourier Series where they are analysed into GCD's and
> > not
> > > > sine waves, in a manner of speaking. 6 and 9 have GCD = 3 and are the 2
> > and
> > > > 3 harmonics. Taking the harmonic series of each gives 6, 12, 18,
> > 24...and 9,
> > > > 18, 27, 36...But these have GCD's 3, 6, 9, 12... which are themselves
> > > > harmonics of the GCD; GCD of (12, 18) = 6 = 2 x 3 etc... IOW we could
> > > > consider this a series where our 'fundamental' or 'element' is say SUM
> > > > [0->N] [sin(2piN6t) + sin(2piN9t)] rather than a single sine wave. From
> > this
> > > > point of view, a single sine wave might be considered as just one
> > special
> > > > case (of course we could further analyse to sine waves but that's not
> > my
> > > > point).
> > > > >
> > > > > The idea is that the brain creates a "virtual pitch" that corresponds
> > > > > to something close to the GCD of the wave coming in. But since the
> > > > > auditory system doesn't have until infinity to see what the real GCD
> > > > > of two sinusoids is, it just takes a small chunk of time instead.
> > This
> > > > > chunk of time or "window size" I believe would most likely be
> > > > > frequency dependent, which explains the recent trend in using
> > wavelets
> > > > > instead of the STFT in analyzing sound.
> > > > >
> > > > > But I don't think this is really going to explain all of tonality.
> > For
> > > > > example, this would break down for a 10:12:15 chord. The root of a
> > > > > 10:12:15 chord is almost always heard as 5, not 1. This is because
> > > > > each dyad of the chord has its own "GCD" and the one produced by
> > 10:15
> > > > > is going to be stronger than the rest of them, with the 12:15 one
> > > > > coming in at a close second.
> > > > >
> > > > > It should be noted that you are using the term "GCD" here in the same
> > > > > way that everyone else is using the term "virtual fundamental," but
> > > > > for some reason you swear you're talking about something else.
> > > > >
> > > > > > I prefer to believe that we can sometimes simply hear the frequency
> > or
> > > > near frequency directly, that it is as real as the sine waves that go
> > into
> > > > making it, that single sine waves are just one special case (the
> > greater set
> > > > being the GCD's or approx ones) and that musical tonality has this
> > basis in
> > > > reality.
> > > > >
> > > > > The reason that it's called virtual is, again, that two pressure
> > waves
> > > > > at frequencies 400 Hz and 500 Hz simply will not make the region of
> > > > > the cochlea corresponding to 100 Hz resonate. Your brain simply
> > > > > recreates it based off of the periodicity information in the signal.
> > > > > And since the people who discovered this can call it whatever they
> > > > > want, they went with "virtual pitch" instead. Or "phantom
> > > > > fundamental," or "missing fundamental," or "virtual fundamental," or
> > > > > any one of those.
> > > > >
> > > > > -Mike
> > > > >
> > > >
> > > >
> > > >
> > >
> >
> >
> >
>

🔗rick <rick_ballan@...>

Just one question Mike,

You said "The reason that it's called virtual is, again, that two pressure waves at frequencies 400 Hz and 500 Hz simply will not make the region of the cochlea corresponding to 100 Hz resonate. Your brain simply recreates it based off of the periodicity information in the signal."

I just want to ask, does a sine wave of 100Hz make this cochlea region resonate?

-Rick

🔗martinsj013 <martinsj@...>

🔗rick <rick_ballan@...>

Hi Steve,

Sorry I missed this post. I'll get to your other questions in a sec. Concerning doubts as to the derivation, this is not the problem. There's no doubt here. You can check it yourself again if you like (I'll drop the 2pi as Mike suggested. I'll also present it from the beginning because people are coming in half way and misunderstanding).

1. f(t) = sin[at] + sin[bt] = 2sin[(a + b)t/2]cos[(a - b)t/2].
2. Envelope is +/- 2cos[(a - b)t/2].
3. Extrema occurs when f(t) equals envelope:
2sin[(a + b)t/2]cos[(a - b)t/2] = +/- 2cos[(a - b)t/2] or
sin[(a + b)t/2] = +/- 1.
4. Solving for t gives t = (2k + 1)/2(a + b), k = 0,1,2...

First maxima occurs at t = 1/2(a + b). Therefore the period between this and the other extrema is given by
5. T = (2k + 1)/2(a + b) - 1/2(a + b) = k/(a + b).

Everything said so far is mathematically correct. It's finding the 'target' cycle, mathematically, that presents the problem i.e. essentially finding 'k'.

You know how I guessed at them. By comparing the waves with their 'JI counterparts' (say, ratio p/q) and noting where the maxima's approximately coincide, the values *always* gave k = p + q. 32/27 gives k = 11 = 5 + 6, 81/64 gives k = 9 = 4 + 5 etc...Now, once k is known it is easy to deduce p and q.

6. Assuming that (a + b)/k is our 'fundamental' then dividing the original frequencies by this gives

ak/(a + b) = p + r/(a + b), bk/(a + b) = q - r/(a + b) where r = aq - pb. [Technically speaking, p is the 'floor' of ak/(a + b) and q is the 'ceiling' of bk/(a + b). I've since learned that if we go through successive values of k = 1,2,3,...then not all the convergents are in there. Only the 'target' one's are, which means I'm getting somewhere].

7. Multiplying (a + b)/k and adding gives k = (p + q) and ~ GCD = (a + b)/(p + q).

Therefore this result is correct. It's finding the conditions to arrive at 'k' (or 'k's) that's the problem. Interestingly your product formula below does put us in close to the expected values as well. Why this is I don't know either.

But about your expectation that p/q = 6/5 should point to the same fundamental as its inverse p/q = 5/3, well they do approximately. Once we have accepted that there's a 'give' around the intervals and noting that ~ GCD doesn't mean real GCD then this is to be expected. 6 x (59/11) = 32.1818..., dividing 2 gives 16.0909... or 3 x (43/8) = 16.125 and x 2 gives 32.25. And of course 59/11 = 5.3636 ~= 43/8 = 5.375.

You might want to run through the floor/ceiling of some examples and maybe see something I've missed. I have noticed that for 81/64 all the even k give p = q and give remainders equal to the average frequency 17/2, which might be something.

Anyway, sorry for another long maths email.

Rick

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >...As mentioned, the two largest maxima in the wave sin (2pi a*t) + sin (2pi b*t) is T = (p + q)/(a + b). Since 1/T approximates a GCD we take p/T = a - R and q/T = b + R where R = (aq - pb)/(p + q). The (aq - pb) seems to relate to Diophantine equations etc...
> >
>
> Rick,
> This formula gives plausible answers, i.e. R is quite small provided you make a sensible choice for p/q. As I said "offline", though, I was not convinced by your derivation of it. I have spotted a consistency problem with it that may need addressing, for example:
>
> a/b=32/27, p/q=6/5: virtual fundamental = 59/11 = 5+4/11; but
> a/b=27/16, p/q=5/3: VF = 43/8 = 5+3/8.
> I would have expected the two VF's to be the same.
>
> I propose for consideration the formula VF = SQRT(a*b)/SQRT(p*q), which I cannot prove (any more than I can disprove yours) but which also gives plausible answers. The two VF's in my example would be equal, with this formula.
>
> Both formulae imply that a slight increase in "a" (not leading to a change in p/q) gives a slight increase in the VF. The same is true for "b" - not surprising if you think about it. There must be experimental research that confirms or denies this but I cannot recall seeing it.
>
> Steve M.
>

🔗rick <rick_ballan@...>

🔗martinsj013 <martinsj@...>

🔗Mike Battaglia <battaglia01@...>

On Mon, May 31, 2010 at 5:11 PM, martinsj013 <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
> > ... Concerning doubts as to the derivation, this is not the problem.
>
> Perhaps I misled you by saying "derivation". I agree that your calculations are correct for finding: (this is a mouthful) the time between the first two "high" points where the combined wave touches the envelope. My problem is that I don't think this implies a frequency that "is there" and "refreshes".
>
> Mike B and now Carl L have pointed out also that experiments have shown that VF perception depends on absolute frequency.
>
> Steve M.

I feel like I'm beating a dead horse, so this is the last thing I'm
going to say on this before I dip out.

Yes, VF perception depends on absolute frequency. It isn't
specifically this that is my problem here - harmonic entropy, for
example, doesn't take this into account either. And there are lots of
little holes in HE as well. But it's a great idea nonetheless.

My problem is in the approach being taken here in the first place.
Perhaps Rick will discover some clever mathematical thing that turns
out being close to something that the auditory system is doing. That
would be neat.

But it should be kept in mind that the goal is to model the auditory
system. But this isn't what we're seeing. Instead, we're seeing
attempts to prove that phantom fundamental pitches aren't "virtual,"
that the theorem that every signal can be represented as a sum of sine
waves is a long-standing scientific conspiracy, that all of the
research done in the last century on psychoacoustics is hopelessly
wrong and that the answer lies in pure mathematics (how could this
possibly be true?), and a few rash statements that a brand new set of
trigonometric identities that have eluded the minds of centuries of
the world's finest mathemeticians have just been discovered.

So I ask Rick: what is the goal here? To learn how music works, or to
become "famous" in an extremely obscure niche of music theory? If you
really want it so bad, we could label your approx GCD method the
"Ballan Transform" when it's done, but I don't share your agenda of
overthrowing the "Fourier establishment" as it were.

-Mike

🔗rick <rick_ballan@...>

Hi Steve,

Sorry I didn't put the equation onto the envelope graphs but being a programming language they don't come out too well. The equation is

+/-2cos(2pi[(aq - pb)/(p + q)]t +/- pi[(p - q)(2k + 1)/2(p + q)]].

k = 0,1,2,...The examples I've given are *only for the first case* where the last term (phase angle) has k = 0. But they 'refresh' for all k from + to - infinity. So the 5/4 has the two largest peaks from the time t = 0 and dies down, rises up, dies again in an oscillation of freq (aq - pb)/(p + q). But in its low ebb, another one is at its highest level and so on. Seen in isolation it does seem to meet the condition of an almost periodic function with a modulated amplitude of freq (aq - pb)/(p + q).

I'm not sure what absolute frequency is (The article is missing bits) so I can't comment yet.

Rick

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, May 31, 2010 at 5:11 PM, martinsj013 <martinsj@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > > ... Concerning doubts as to the derivation, this is not the problem.
> >
> > Perhaps I misled you by saying "derivation". I agree that your calculations are correct for finding: (this is a mouthful) the time between the first two "high" points where the combined wave touches the envelope. My problem is that I don't think this implies a frequency that "is there" and "refreshes".
> >
> > Mike B and now Carl L have pointed out also that experiments have shown that VF perception depends on absolute frequency.
> >
> > Steve M.
>
> I feel like I'm beating a dead horse, so this is the last thing I'm
> going to say on this before I dip out.
>
> Yes, VF perception depends on absolute frequency. It isn't
> specifically this that is my problem here - harmonic entropy, for
> example, doesn't take this into account either. And there are lots of
> little holes in HE as well. But it's a great idea nonetheless.
>
> My problem is in the approach being taken here in the first place.
> Perhaps Rick will discover some clever mathematical thing that turns
> out being close to something that the auditory system is doing. That
> would be neat.
>
> But it should be kept in mind that the goal is to model the auditory
> system. But this isn't what we're seeing. Instead, we're seeing
> attempts to prove that phantom fundamental pitches aren't "virtual,"
> that the theorem that every signal can be represented as a sum of sine
> waves is a long-standing scientific conspiracy, that all of the
> research done in the last century on psychoacoustics is hopelessly
> wrong and that the answer lies in pure mathematics (how could this
> possibly be true?), and a few rash statements that a brand new set of
> trigonometric identities that have eluded the minds of centuries of
> the world's finest mathemeticians have just been discovered.
>
> So I ask Rick: what is the goal here? To learn how music works, or to
> become "famous" in an extremely obscure niche of music theory? If you
> really want it so bad, we could label your approx GCD method the
> "Ballan Transform" when it's done, but I don't share your agenda of
> overthrowing the "Fourier establishment" as it were.
>
> -Mike
>

🔗rick <rick_ballan@...>

Mike, I think that Fourier Analysis is one of the most elegant and beautiful theories in all of mathematics. Sure I would love to find something that's even half that good. And I never ever said that it didn't apply.

Concerning psychoacoustics, I was exploring this stuff long before I even knew it existed. I'm not trying to debunk anything. If I accidentally challenge some old cherished theories then so be it. It's not really the end of the world.