Continued Fractions are Trig ID's within Waves

Hi everyone,

I finally found the answer. Take the ratio a/b where a > b and a/b is either coprime or irrational. If [1/1, p/q,....a/b] represents the convergents of the continued fraction for a/b then the following identity holds:

f(t) = sin(2piat) + sin(2pibt) =

sin(2pi (p((a + b)/(p + q))t + ((aq - pb)/(p + q))t) +
sin(2pi (q((a + b)/(p + q))t - ((aq - pb)/(p + q))t).

Since p and q are integers, then the original wave can be seen as the composite of two 'simpler' waves, the pth and qth harmonics of fundamental GCD = (a + b)/(p + q), with an equal and opposite time-dependent phase shift in each component, R = (aq - pb)/(p + q). (Observe that the term (aq - pb) equals the determinant of each convergent).

The extrema (peaks) of each (p, q) follows what seems to be a new type of envelope (amplitude modulation):

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

For the first term p/q = 1/1, this reduces to +/- 2cos(2pi(a - b)t) which suggests that this standard trigonometric identity now belongs to a larger class of identities. In fact, if there are N convergents, then their will be N - 1 such identities. Since irrationals have an infinite number of such convergents then there are an infinite number of identities. Also, for this first term the GCD becomes

p((a + b)/(p + q) = q((a + b)/(p + q) = (a + b)/2

which is just the average frequency.

Example, a/b = 81/64 gives convergents [1/1, 4/3, 5/4, 19/15, 81/64] which are the p/q's. This implies that the complex wave 'presents itself' as a very badly tuned unison, a badly tuned fourth, a not so bad JI major third, a pretty good 19/15 and a perfect Pythagorean Maj third, or something like that. The GCD's are, respectively, 145/2 (average), 145/7, 145/9, 145/34 and 145/145 = 1 where it's seen that they are decreasing. Again, the concept of GCD = 1 now seems to be a special case. Of course the intervals are unison 1(145/7):1(145/7), fourth 3(145/7):4(145/7), and so on. Notice also that the determinants aq - pb in the remainder frequencies are becoming increasingly smaller, equalling zero when p/q = 81/64.

This is the maths. Thinking of this interval as a possible unison or fourth seems anti-intuitive but apparently the waves don't make this distinction. For music, it might be that some cut-off point exists in the remainder or something?? In any case, this at least gives this whole area of number theory some basis in the physics of waves.

Cheers

Rick

Rick,

I'm trying to follow this. How would the fraction 13/8 work in this formula IE what would be the steps to find its convergents? I'd be quite interested to know what the convergent answers are, because to my ear 13/8 appears as smack in between two much stronger areas.

On Sat, May 22, 2010 at 1:08 AM, rick <rick_ballan@...> wrote:
>
> Hi everyone,
>
> I finally found the answer. Take the ratio a/b where a > b and a/b is either coprime or irrational. If [1/1, p/q,....a/b] represents the convergents of the continued fraction for a/b then the following identity holds:iuj899
>
> f(t) = sin(2piat) + sin(2pibt) =
>
> sin(2pi (p((a + b)/(p + q))t + ((aq - pb)/(p + q))t) +
> sin(2pi (q((a + b)/(p + q))t - ((aq - pb)/(p + q))t).
>
> Since p and q are integers, then the original wave can be seen as the composite of two 'simpler' waves, the pth and qth harmonics of fundamental GCD = (a + b)/(p + q), with an equal and opposite time-dependent phase shift in each component, R = (aq - pb)/(p + q). (Observe that the term (aq - pb) equals the determinant of each convergent).

In all seriousness, rick, I can't follow with this anymore. At least
take the time to format all of this so that it's readable, as I said
in my last post.

And unless I'm completely lost, note that putting a "time-dependent
phase shift" into a wave = changing the frequency of the wave.

-Mke

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> ... At least take the time to format all of this so that it's readable, as I said in my last post.

FWIW, I find the formulas readable enough ...

> And unless I'm completely lost, note that putting a "time-dependent
> phase shift" into a wave = changing the frequency of the wave.

... but I'm glad you said that, I was wondering if I was missing something.

Rick, can you clarify that you now use convergents as the way to get suitable p/q values? If so, what does that mean for the previous method of finding K (=p+q) and thence p and q? And what separates the reasonable values of p/q from the unreasonable? (e.g. 19/15, 5/4 vs 4/3, 1/1 in the case of a/b=81/64)

Steve M.

Mike, if you prefer:

Take the ratio a/b where a > b and a/b is either coprime or irrational. If [1/1, p/q,....a/b] represents the convergents of the continued fraction for a/b then the following identity holds:

f(t) = sin(a*t) + sin(b*t) =

sin(p*((a + b)/(p + q))t + ((aq - pb)/(p + q))t) +
sin(q*((a + b)/(p + q))t - ((aq - pb)/(p + q))t).

But really Mike, can't you just pull out a pen and paper? It's not as if I can help not having the "pi" symbol. I've found *an entire class of previously unidentified trigonometric identities*, of which the usual 2sin(A + B)cos(A - B) is just the first instance, and you can't even do it this service? And yes, putting a "time-dependent phase shift" into a wave = changing the frequency of the wave. But if you were to actually go through the steps you'll see that while these values are equal to a and b, respectively, the remainders can be separated into amplitude modulations. Hint, try it with exponentials.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, May 22, 2010 at 1:08 AM, rick <rick_ballan@...> wrote:
> >
> > Hi everyone,
> >
> > I finally found the answer. Take the ratio a/b where a > b and a/b is either coprime or irrational. If [1/1, p/q,....a/b] represents the convergents of the continued fraction for a/b then the following identity holds:
> >
> > f(t) = sin(2piat) + sin(2pibt) =
> >
> > sin(2pi (p((a + b)/(p + q))t + ((aq - pb)/(p + q))t) +
> > sin(2pi (q((a + b)/(p + q))t - ((aq - pb)/(p + q))t).
> >
> > Since p and q are integers, then the original wave can be seen as the composite of two 'simpler' waves, the pth and qth harmonics of fundamental GCD = (a + b)/(p + q), with an equal and opposite time-dependent phase shift in each component, R = (aq - pb)/(p + q). (Observe that the term (aq - pb) equals the determinant of each convergent).
>
> In all seriousness, rick, I can't follow with this anymore. At least
> take the time to format all of this so that it's readable, as I said
> in my last post.
>
> And unless I'm completely lost, note that putting a "time-dependent
> phase shift" into a wave = changing the frequency of the wave.
>
> -Mke
>

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

> But really Mike, can't you just pull out a pen and paper? It's not as if I can help not having the "pi" symbol. I've found *an entire class of previously unidentified trigonometric identities*, of which the usual 2sin(A + B)cos(A - B) is just the first instance, and you can't even do it this service?

Eh? You've not found any new trig identities, you are *using* trig identities. In particular, you've been using sin(A+B) + sin(A-B) = 2 sin(A)cos(B)

Mike, I don't know if you got the message but I also posted a file showing the envelopes for 81/64. There you'll see more clearly what I'm saying.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, May 22, 2010 at 1:08 AM, rick <rick_ballan@...> wrote:
> >
> > Hi everyone,
> >
> > I finally found the answer. Take the ratio a/b where a > b and a/b is either coprime or irrational. If [1/1, p/q,....a/b] represents the convergents of the continued fraction for a/b then the following identity holds:iuj899
> >
> > f(t) = sin(2piat) + sin(2pibt) =
> >
> > sin(2pi (p((a + b)/(p + q))t + ((aq - pb)/(p + q))t) +
> > sin(2pi (q((a + b)/(p + q))t - ((aq - pb)/(p + q))t).
> >
> > Since p and q are integers, then the original wave can be seen as the composite of two 'simpler' waves, the pth and qth harmonics of fundamental GCD = (a + b)/(p + q), with an equal and opposite time-dependent phase shift in each component, R = (aq - pb)/(p + q). (Observe that the term (aq - pb) equals the determinant of each convergent).
>
> In all seriousness, rick, I can't follow with this anymore. At least
> take the time to format all of this so that it's readable, as I said
> in my last post.
>
> And unless I'm completely lost, note that putting a "time-dependent
> phase shift" into a wave = changing the frequency of the wave.
>
> -Mke
>

On Sun, May 23, 2010 at 5:40 AM, rick <rick_ballan@...> wrote:
>
> Mike, if you prefer:
>
> Take the ratio a/b where a > b and a/b is either coprime or irrational. If [1/1, p/q,....a/b] represents the convergents of the continued fraction for a/b then the following identity holds:
>
> f(t) = sin(a*t) + sin(b*t) =
>
> sin(p*((a + b)/(p + q))t + ((aq - pb)/(p + q))t) +
> sin(q*((a + b)/(p + q))t - ((aq - pb)/(p + q))t).
>
> But really Mike, can't you just pull out a pen and paper? It's not as if I can help not having the "pi" symbol. I've found *an entire class of previously unidentified trigonometric identities*, of which the usual 2sin(A + B)cos(A - B) is just the first instance, and you can't even do it this service?

Of course I can pull out a pen and paper. The point is that this has
to be near the 50th message on this subject so far, and every one
comes with an onslaught of equations that are barely readable. I threw
out a few suggestions on how to format it just so I can wrap my head
around what you're talking about. Using S(x) and C(x) for sin and
cosine (that was Gene's), getting rid of pi terms, etc. The picture
you sent of the envelope helped a bit.

Anyway, let's go back to this:

> f(t) = sin(a*t) + sin(b*t) =
>
> sin(p*((a + b)/(p + q))t + ((aq - pb)/(p + q))t) +
> sin(q*((a + b)/(p + q))t - ((aq - pb)/(p + q))t).

This is tautological and will work for ANY p and q, not just the
convergents of a/b. Work the algebra out: the first term simplifies to
sin(at) and the second term simplifies to sin(bt). That is, p*((a +
b)/(p + q))t + ((aq - pb)/(p + q))t really is just at, and q*((a +
b)/(p + q))t - ((aq - pb)/(p + q))t is really just bt.

p*((a + b)/(p + q))t + ((aq - pb)/(p + q))t <-- multiply through the p
((ap + pb)/(p + q))t + ((aq - pb)/(p + q))t <-- same denominator, add
numerators together
((ap + pb + aq - pb)/(p+q))t <-- get rid of the +/- pb terms
((ap+aq)/(p+q))t <-- factor out a
a(p+q)/(p+q)t
at

The same thing works out for the second term as well.

And then here:

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

I'm not seeing how you've derived that. The original equation you've
written above isn't really sin(u+v) + sin(u-v), since the "u" terms
are different in both terms.

Nonetheless, I would expect that this is more or less tautological,
and wouldn't be limited to p/q as convergents of a/b.

> And yes, putting a "time-dependent phase shift" into a wave = changing the frequency of the wave. But if you were to actually go through the steps you'll see that while these values are equal to a and b, respectively, the remainders can be separated into amplitude modulations. Hint, try it with exponentials.

What values are equal to a and b, respectively?

-Mike

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
>
> > But really Mike, can't you just pull out a pen and paper? It's not as if I can help not having the "pi" symbol. I've found *an entire class of previously unidentified trigonometric identities*, of which the usual 2sin(A + B)cos(A - B) is just the first instance, and you can't even do it this service?
>
> Eh? You've not found any new trig identities, you are *using* trig identities. In particular, you've been using sin(A+B) + sin(A-B) = 2 sin(A)cos(B)
>
Which is now the special case for p/q = 1/1.

Ok Mike, I've read what you say below. Of course I'm aware that the 'form' of the conclusion is tautological and the p, q in a = pg + R and b = qg + R will cancel out. However, this in itself is a necessary but insufficient condition. Not all other values of p and q are in the wave. Remember that I *deduced* them by finding t = (2k + 1)/2(a + b), solved for k to get k = (p + q). The period is the difference between this k and k = 0 giving T = (p + q)/(a + b).

Now, it might be that other values of p and q also occur in the wave besides the convergents. The mediants 62/49, 42/34 come to mind for the 81/64 (especially since (aq - pq) is Bezout's Identity). I'll test this when I get the time. However, this still does not mean that we can choose any p, q at random. As I said a few days ago, even though I 'know' it's there I still can't find the key to a direct proof. I've probably already done it without realising. If you think of one, please let me know. In the meantime, this doesn't stop me from working out some consequences. More often than not I find that proofs often just 'pop in' on their own time.

Rick

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, May 23, 2010 at 5:40 AM, rick <rick_ballan@...> wrote:
> >
> > Mike, if you prefer:
> >
> > Take the ratio a/b where a > b and a/b is either coprime or irrational. If [1/1, p/q,....a/b] represents the convergents of the continued fraction for a/b then the following identity holds:
> >
> > f(t) = sin(a*t) + sin(b*t) =
> >
> > sin(p*((a + b)/(p + q))t + ((aq - pb)/(p + q))t) +
> > sin(q*((a + b)/(p + q))t - ((aq - pb)/(p + q))t).
> >
> > But really Mike, can't you just pull out a pen and paper? It's not as if I can help not having the "pi" symbol. I've found *an entire class of previously unidentified trigonometric identities*, of which the usual 2sin(A + B)cos(A - B) is just the first instance, and you can't even do it this service?
>
> Of course I can pull out a pen and paper. The point is that this has
> to be near the 50th message on this subject so far, and every one
> comes with an onslaught of equations that are barely readable. I threw
> out a few suggestions on how to format it just so I can wrap my head
> around what you're talking about. Using S(x) and C(x) for sin and
> cosine (that was Gene's), getting rid of pi terms, etc. The picture
> you sent of the envelope helped a bit.
>
> Anyway, let's go back to this:
>
> > f(t) = sin(a*t) + sin(b*t) =
> >
> > sin(p*((a + b)/(p + q))t + ((aq - pb)/(p + q))t) +
> > sin(q*((a + b)/(p + q))t - ((aq - pb)/(p + q))t).
>
> This is tautological and will work for ANY p and q, not just the
> convergents of a/b. Work the algebra out: the first term simplifies to
> sin(at) and the second term simplifies to sin(bt). That is, p*((a +
> b)/(p + q))t + ((aq - pb)/(p + q))t really is just at, and q*((a +
> b)/(p + q))t - ((aq - pb)/(p + q))t is really just bt.
>
> p*((a + b)/(p + q))t + ((aq - pb)/(p + q))t <-- multiply through the p
> ((ap + pb)/(p + q))t + ((aq - pb)/(p + q))t <-- same denominator, add
> numerators together
> ((ap + pb + aq - pb)/(p+q))t <-- get rid of the +/- pb terms
> ((ap+aq)/(p+q))t <-- factor out a
> a(p+q)/(p+q)t
> at
>
> The same thing works out for the second term as well.
>
> And then here:
>
> +/- 2cos(2pi((aq - pb)/(p + q)]t +/- pi((p - q)(2k + 1)/(2(p + q))).
>
> I'm not seeing how you've derived that. The original equation you've
> written above isn't really sin(u+v) + sin(u-v), since the "u" terms
> are different in both terms.
>
> Nonetheless, I would expect that this is more or less tautological,
> and wouldn't be limited to p/q as convergents of a/b.
>
> > And yes, putting a "time-dependent phase shift" into a wave = changing the frequency of the wave. But if you were to actually go through the steps you'll see that while these values are equal to a and b, respectively, the remainders can be separated into amplitude modulations. Hint, try it with exponentials.
>
> What values are equal to a and b, respectively?
>
> -Mike
>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > ... At least take the time to format all of this so that it's readable, as I said in my last post.
>
> FWIW, I find the formulas readable enough ...

Thanks Steve,

So far I've been accused of either being completely confusing or completely obvious (not to mention any names like, for example, MIKE...that was a lol).

>
> > And unless I'm completely lost, note that putting a "time-dependent
> > phase shift" into a wave = changing the frequency of the wave.
>
> ... but I'm glad you said that, I was wondering if I was missing something.
>
> Rick, can you clarify that you now use convergents as the way to get suitable p/q values? If so, what does that mean for the previous method of finding K (=p+q) and thence p and q? And what separates the reasonable values of p/q from the unreasonable? (e.g. 19/15, 5/4 vs 4/3, 1/1 in the case of a/b=81/64)
>
> Steve M.
>
Well I can confirm that the convergents are all there. Since p and q are whole then (p + q) is whole and therefore satisfy t = (2k + 1)/2(a + b) with k = (p + q). It's also easy to prove that whole-numbered multiples of k also satisfy this condition i.e. that they are periodic. Convergents also appear to have all the 'clues' I was looking for like the appearance of the determinant aq - pb. Their extrema do follow the curve of frequency R = (aq - pb)/(p + q). This I suspect is the key to any immanent proof (because only a select few p, q's do this). But whether or not this is the complete list I'm not exactly sure yet. The convergents might still be a sub-class of a greater Stern-Brocot tree which for the 81/64 would include 62/49, 43/34 etc...but then these won't exist for irrational a/b?? This should all become clear when I understand more the connection between this branch of number theory and wave motion.

As for what separates the reasonable from unreasonable intervals, there is still the problem that the *largest* or near largest peak occurs for the 5/4 which seems to separate it out. This has proved to be the most difficult thing to pin-down. I've been looking at suprema's instead of just extrema's but no luck yet. It might give the closest GCD to (a + b)/(a - b) or something? You know more than anyone what I'm talking about. (Did you check out the file I posted? You might be able to see something I missed).

And thanks for all your help so far Steve. You're influence is in there.

Cheers

Rick

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
> The convergents might still be a sub-class of a greater Stern-Brocot tree which for the 81/64 would include 62/49, 43/34 etc...but then these won't exist for irrational a/b??

Convergents are a subclass of the semiconvergents, which do exist for irrational x. For example, the semiconvergents to 81/64 go 1, 4/3, 5/4, 9/7, 14/11, 19/15, 81/64.

This might be something. For a/b = 81/64 the convergents for (a + b)/(a - b) gives {8, 9, 17/2, 145/17}. Taking (p + q)/(p - q) gives {2, 7, 9, 17/2, 145/17}. The two 'converge' after 9 which is the p/q = 5/4. Or another eg, a/b = 51/32 gives {1, 2, 3/2, 8/5, 51/32}, sum/diff gives {4, 9/2, 13/3, 35/8, 83/19}. Again, the 13/3 = (5 + 8)/(8 - 5) is the one we want.

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > > ... At least take the time to format all of this so that it's readable, as I said in my last post.
> >
> > FWIW, I find the formulas readable enough ...
>
> Thanks Steve,
>
> So far I've been accused of either being completely confusing or completely obvious (not to mention any names like, for example, MIKE...that was a lol).
>
> >
> > > And unless I'm completely lost, note that putting a "time-dependent
> > > phase shift" into a wave = changing the frequency of the wave.
> >
> > ... but I'm glad you said that, I was wondering if I was missing something.
> >
> > Rick, can you clarify that you now use convergents as the way to get suitable p/q values? If so, what does that mean for the previous method of finding K (=p+q) and thence p and q? And what separates the reasonable values of p/q from the unreasonable? (e.g. 19/15, 5/4 vs 4/3, 1/1 in the case of a/b=81/64)
> >
> > Steve M.
> >
> Well I can confirm that the convergents are all there. Since p and q are whole then (p + q) is whole and therefore satisfy t = (2k + 1)/2(a + b) with k = (p + q). It's also easy to prove that whole-numbered multiples of k also satisfy this condition i.e. that they are periodic. Convergents also appear to have all the 'clues' I was looking for like the appearance of the determinant aq - pb. Their extrema do follow the curve of frequency R = (aq - pb)/(p + q). This I suspect is the key to any immanent proof (because only a select few p, q's do this). But whether or not this is the complete list I'm not exactly sure yet. The convergents might still be a sub-class of a greater Stern-Brocot tree which for the 81/64 would include 62/49, 43/34 etc...but then these won't exist for irrational a/b?? This should all become clear when I understand more the connection between this branch of number theory and wave motion.
>
> As for what separates the reasonable from unreasonable intervals, there is still the problem that the *largest* or near largest peak occurs for the 5/4 which seems to separate it out. This has proved to be the most difficult thing to pin-down. I've been looking at suprema's instead of just extrema's but no luck yet. It might give the closest GCD to (a + b)/(a - b) or something? You know more than anyone what I'm talking about. (Did you check out the file I posted? You might be able to see something I missed).
>
> And thanks for all your help so far Steve. You're influence is in there.
>
> Cheers
>
> Rick
>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > The convergents might still be a sub-class of a greater Stern-Brocot tree which for the 81/64 would include 62/49, 43/34 etc...but then these won't exist for irrational a/b??
>
> Convergents are a subclass of the semiconvergents, which do exist for irrational x. For example, the semiconvergents to 81/64 go 1, 4/3, 5/4, 9/7, 14/11, 19/15, 81/64.
>
Ah I see, thanks Gene.

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> For a/b = 81/64 the convergents for (a + b)/(a - b)
> gives {8, 9, 17/2, 145/17}.
> Taking (p + q)/(p - q) gives {2, 7, 9, 17/2, 145/17}.

Hi Rick,
here attend the Pythagorean-Triple: 17^2 + 144^2 = 145^2

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html
"17, 144, 145 primitive m=9, m=8 "

and also the other 3 possible solutions with hypothenouse=145
"
Triples with hypotenuse=145:
1) 87, 116, 145 =29× 3, 4, 5 P=348 A=5046 r=29 (no m,n)
2) 17, 144, 145 primitive P=306 A=1224 r=8 m=9 n=8
3) 24, 143, 145 primitive P=312 A=1716 r=11 m=12 n=1
4) 100, 105, 145 =5× 20, 21, 29 P=350 A=5250 r=30 (no m,n)
"
and finally
http://www.research.att.com/~njas/sequences/A084648
Hypotenuses for which there exist 4 distinct integer triangles:
65, 85, 130, 145,....."

bye
A.S.

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
> For a/b = 81/64 the convergents for (a + b)/(a - b)
> gives {8, 9, 17/2, 145/17}.

Hi Rick,
attend, here the special 'Pythagorean-Triple' 17^2 + 144^2 = 145^2

http://www.research.att.com/~njas/sequences/A084648
"
Hypotenuses for which there exist 4 distinct integer triangles:
65, 85, 130, 145,.....
"

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html
"17, 144, 145 primitive r=8, m=9, n=8 "

and also note the 3 others possibilities for hyp=145 apart from 2):

Triples with hypotenuse=145:
1) 87, 116, 145 =29× 3, 4, 5 P=348 A=5046 r=29 (no m,n)
2) 17, 144, 145 primitive P=306 A=1224 r=8 m=9 n=8
3) 24, 143, 145 primitive P=312 A=1716 r=11 m=12 n=1
4) 100, 105, 145 =5× 20, 21, 29 P=350 A=5250 r=30 (no m,n)

bye
A.S.

Thanks A.S., yeah there does some to be some connection. I'll definitely look into this.

Rick

--- In tuning@yahoogroups.com, "a_sparschuh" <a_sparschuh@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > For a/b = 81/64 the convergents for (a + b)/(a - b)
> > gives {8, 9, 17/2, 145/17}.
> > Taking (p + q)/(p - q) gives {2, 7, 9, 17/2, 145/17}.
>
> Hi Rick,
> here attend the Pythagorean-Triple: 17^2 + 144^2 = 145^2
>
> http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html
> "17, 144, 145 primitive m=9, m=8 "
>
> and also the other 3 possible solutions with hypothenouse=145
> "
> Triples with hypotenuse=145:
> 1) 87, 116, 145 =29× 3, 4, 5 P=348 A=5046 r=29 (no m,n)
> 2) 17, 144, 145 primitive P=306 A=1224 r=8 m=9 n=8
> 3) 24, 143, 145 primitive P=312 A=1716 r=11 m=12 n=1
> 4) 100, 105, 145 =5× 20, 21, 29 P=350 A=5250 r=30 (no m,n)
> "
> and finally
> http://www.research.att.com/~njas/sequences/A084648
> Hypotenuses for which there exist 4 distinct integer triangles:
> 65, 85, 130, 145,....."
>
> bye
> A.S.
>

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
> Thanks Steve,
> So far I've been accused of either being completely confusing or completely obvious ...

Rick,
Don't thank me too much; yes I find the formulas readable, but sometimes I am confused by the words :-) Also remember I said that I was not convinced by the way you derived the K=p+q by looking for peaks in the combined waveform (although the resulting formula looks reasonable).

It is perhaps useful to reiterate what I think the aim is. Given a large-number frequency ratio a/b, or even letting a and b be irrational, find an expression for the "virtual fundamental" (VF) for a dyad with those frequencies. Sometimes we "know" the answer, e.g. 500/400 has VF 100 (same as the GCD). Other times it is not so easy e.g. 501/399 (the GCD is 3, surely the VF is not 3?). In an offlist exchange (which I hope he will not mind me discussing) Carl said it is approximately 399/4, or 501/5. Yes, but can't we find a single answer? Ideally of course it would be *the* value that most closely captures what most listeners actually perceive; failing that, it should at least be a value that "makes sense", e.g. varies continuously with a and b, is between 399/4 and 501/5, etc. (I have mentioned another criterion in an earlier post, but I'll save that for later.)

The approach is to first find a smaller-number ratio p/q which approximates a/b as a ratio and thence derive approximate frequencies a' and b' near a and b, with a'/b' = p/q and thence the VF (GCD of p and q). (Or, derive these things in a different order.)

BTW any statement of the results should say that the formula (with "aq-bp" and all that) is an identity, and that we are restricting ourselves to looking at certain values of p,q and explain why. (The way you usually state it seems to imply that the identity holds for just those values of p, q.) Having said that, that formula is not an essential part of the results, in my version of the problem stated above.

I am pretty certain that the best way to find p/q is to use a convergent or semi-convergent; it only remains to decide the balance between accuracy and small-numberedness in choosing p/q. Other methods such as "rounding (a+b)/(a-b)" will probably turn out to be equivalent anyway (but perhaps I was the only one considering that). Re the method of finding p/q in the graph of the combined waveform - I find this unconvincing. I assume that the 64.pdf file is showing that each of the several envelopes representing different convergent frequencies does indeed touch the original combined waveform at certain peaks of the latter (but it misses many others!). And I assume that this is what you mean by "the convergents are all there"? But I don't believe that this is signficant - they wouldn't be heard, surely. Particularly as the peaks are not even "high peaks" as they were in your original derivation of K. The most convincing envelope is the first one - but to me this is just the difference tone. (And furthermore the frequency of the last two is so low that we can hardly see anything happening in the brief window of time shown.)

> ... Well I can confirm that the convergents are all there. ...
> Rick

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> .... yeah there does some to be some connection....
> >....attend the Pythagorean-Triple: 17^2 + 144^2 = 145^2 .....

Hi Rick,
hence that Pythagorean triangle contains an angle phi
with an superpartcular ratio for the cosine

cos(phi) = 144/145

sin(phi) = 17/145

tan(phi) = 17/144

asin(17/145) = ~0.11751164543145...radiant

or about

(180/pi)*asin(17/145) = ~6.73292132685961°...degree out of 360°

Formula in:
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html

Note 75.23 Pythagorean triples Chris Evans Mathematical Gazette 75 (1991), page 317.

Can we find a formula for these triples?
You will have noticed that the smallest sides are the odd numbers 3, 5, 7, 9,... So the smallest sides are of the form 2i+1.
The other sides, as a series are 4, 12, 24, 40, 60... Can we find a formula here?
We notice they are all multiples of 4: 4×1, 4×3, 4×6, 4×10, 4×15,.. . The series of multiples: 1, 3, 6, 10, 15,... are the Triangle Numbers with a formula i(i+1)/2.
So our second sides are 4 times each of these, or, simply, just 2i(i+1).
The third side is just one more than the second side: 2i(i+1)+1, so our formula is as follows:
shortest side = 2i+1; longest side = 2i(i+1); hypotenuse = 1+2i(i+1)
Check now that the sum of the squares of the two sides is the same as the square of the hypotenuse (Pythagoras's Theorem).

[Use-Fixed-Width-Font]

i a:2i+1 b:2i(i+1) h=b+1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1 3 2×1×2=4 5
2 5 2×2×3=12 13
3 7 2×3×4=24 25
4 9 2×4×5=40 41
5 11 2×5×6=60 61
6 13 2×6×7=84 85
7 15 2×7×8=112 113
8 17 2x8x9=144 145

especially here attend last case
i=8 with yours concluding 17^2 + 144^2 = 145^2
for understanding the prior convergents.

bye
A.S.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> I am pretty certain that the best way to find p/q is to use a
> convergent or semi-convergent; it only remains to decide the
> balance between accuracy and small-numberedness in choosing p/q.

One has to work the other way - from the rationals to the target.
This is because some rationals approximate simpler rationals.
So you need to determine which field of attraction you're in,
and the simplest way to ensure you've gone out far enough from
the target is to cut up the entire band (octave, say) in advance
and precompute them. This is what harmonic entropy does.

-Carl

Hi Steve,

Well I will thank you, Mike, Carl, Gene and everyone else simply for considering the problem whether you agree with the idea or not. I have found all your criticisms very helpful. Likewise, I hope that my previous doubts about VP are taken in the same vain.

However, I can't see how your 501/399 example creates doubt about what I'm proposing. Firstly, the ~ GCD would be (501 + 399)/(5 + 4) = 100 which *is* between 399/4 and 501/5. Of course this *is* the time between the first two largest maxima. Secondly, this example is in fact much closer to a 'pure' 5/4 than the previous 81/64 because the envelope frequency is 2cos[2pit + pi/18] i.e. (aq - pb)/(p + q) = 9/9. This means that *the maxima are decreasing at a rate that is very small compared to the component frequencies*. Therefore, the maxima occur at almost precisely the same times as a 5/4 wave with GCD = 100 while the curve shape too is almost the same for long time periods.

As for all the other stuff about 5/4 being only one of the convergents and so on, well that's just me putting ideas/results out there. I still 'know' that for an 81/64 that the 5/4 is the most reasonable, for a 32/27 it is a 6/5 etc... I'm just trying to say it in a more mathematical way. It might be something as simple as the fact that 81/64 is closer to 80/64 = 5/4 than any other? Or it might be that (p + q)/(p - q) is closest to (a + b)/(a - b). I don't know yet. But I have tested it in many many graphic examples, some chosen at random, and it always checks out. Certainly enough 'probable cause' to look into it in more depth. At any rate, we shouldn't let my own lack of mathematical skills get in the way.

Rick

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > Thanks Steve,
> > So far I've been accused of either being completely confusing or completely obvious ...
>
> Rick,
> Don't thank me too much; yes I find the formulas readable, but sometimes I am confused by the words :-) Also remember I said that I was not convinced by the way you derived the K=p+q by looking for peaks in the combined waveform (although the resulting formula looks reasonable).
>
> It is perhaps useful to reiterate what I think the aim is. Given a large-number frequency ratio a/b, or even letting a and b be irrational, find an expression for the "virtual fundamental" (VF) for a dyad with those frequencies. Sometimes we "know" the answer, e.g. 500/400 has VF 100 (same as the GCD). Other times it is not so easy e.g. 501/399 (the GCD is 3, surely the VF is not 3?). In an offlist exchange (which I hope he will not mind me discussing) Carl said it is approximately 399/4, or 501/5. Yes, but can't we find a single answer? Ideally of course it would be *the* value that most closely captures what most listeners actually perceive; failing that, it should at least be a value that "makes sense", e.g. varies continuously with a and b, is between 399/4 and 501/5, etc. (I have mentioned another criterion in an earlier post, but I'll save that for later.)
>
> The approach is to first find a smaller-number ratio p/q which approximates a/b as a ratio and thence derive approximate frequencies a' and b' near a and b, with a'/b' = p/q and thence the VF (GCD of p and q). (Or, derive these things in a different order.)
>
> BTW any statement of the results should say that the formula (with "aq-bp" and all that) is an identity, and that we are restricting ourselves to looking at certain values of p,q and explain why. (The way you usually state it seems to imply that the identity holds for just those values of p, q.) Having said that, that formula is not an essential part of the results, in my version of the problem stated above.
>
> I am pretty certain that the best way to find p/q is to use a convergent or semi-convergent; it only remains to decide the balance between accuracy and small-numberedness in choosing p/q. Other methods such as "rounding (a+b)/(a-b)" will probably turn out to be equivalent anyway (but perhaps I was the only one considering that). Re the method of finding p/q in the graph of the combined waveform - I find this unconvincing. I assume that the 64.pdf file is showing that each of the several envelopes representing different convergent frequencies does indeed touch the original combined waveform at certain peaks of the latter (but it misses many others!). And I assume that this is what you mean by "the convergents are all there"? But I don't believe that this is signficant - they wouldn't be heard, surely. Particularly as the peaks are not even "high peaks" as they were in your original derivation of K. The most convincing envelope is the first one - but to me this is just the difference tone. (And furthermore the frequency of the last two is so low that we can hardly see anything happening in the brief window of time shown.)
>
> > ... Well I can confirm that the convergents are all there. ...
> > Rick
>

Hi A.S.,

So we have (m + n)^2 + 2mn = m^2 + n^2. With m = 8, n = 9 we get

17^2 + 144 = 145.

Of course n^2/m^2 = 81/64. If we form the convergents of their sum and difference we seem to get [m, n,...]. I'm not sure where to go from here. Is this just an observation or did you have something more specific in mind?

Rick

--- In tuning@yahoogroups.com, "a_sparschuh" <a_sparschuh@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > For a/b = 81/64 the convergents for (a + b)/(a - b)
> > gives {8, 9, 17/2, 145/17}.
> > Taking (p + q)/(p - q) gives {2, 7, 9, 17/2, 145/17}.
>
> Hi Rick,
> here attend the Pythagorean-Triple: 17^2 + 144^2 = 145^2
>
> http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html
> "17, 144, 145 primitive m=9, m=8 "
>
> and also the other 3 possible solutions with hypothenouse=145
> "
> Triples with hypotenuse=145:
> 1) 87, 116, 145 =29× 3, 4, 5 P=348 A=5046 r=29 (no m,n)
> 2) 17, 144, 145 primitive P=306 A=1224 r=8 m=9 n=8
> 3) 24, 143, 145 primitive P=312 A=1716 r=11 m=12 n=1
> 4) 100, 105, 145 =5× 20, 21, 29 P=350 A=5250 r=30 (no m,n)
> "
> and finally
> http://www.research.att.com/~njas/sequences/A084648
> Hypotenuses for which there exist 4 distinct integer triangles:
> 65, 85, 130, 145,....."
>
> bye
> A.S.
>

On Thu, May 27, 2010 at 8:16 AM, rick <rick_ballan@...> wrote:
>
> Hi Steve,
>
> Well I will thank you, Mike, Carl, Gene and everyone else simply for considering the problem whether you agree with the idea or not. I have found all your criticisms very helpful. Likewise, I hope that my previous doubts about VP are taken in the same vain.

...What doubts do you have? I thought at this point you just weren't a
fan of the terminology used.

-Mike

On Mon, May 24, 2010 at 6:07 AM, rick <rick_ballan@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> >
> > Eh? You've not found any new trig identities, you are *using* trig identities. In particular, you've been using sin(A+B) + sin(A-B) = 2 sin(A)cos(B)
> >
> Which is now the special case for p/q = 1/1.

OK, unless I'm missing something... ANY p/q will work. You have
basically taken sin(A+B) + sin(A-B) = 2sin(A)cos(B) and added a few
extra terms in such a way that they cancel out and yield the original
identity. The original identity isn't a "special case" for p/q = 1/1,
but rather the original identity is a "special case" for p/q =
ANYTHING, since no matter what you set p and q to the extraneous terms
cancel out to yield the original identity.

Perhaps there's some nuance that I'm missing...

-Mike

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
> However, I can't see how your 501/399 example creates doubt about what I'm proposing. Firstly, the ~ GCD would be (501 + 399)/(5 + 4) = 100 which *is* between 399/4 and 501/5.

You misunderstand me - I was merely re-stating what we are trying to do. And I think your formula does do it. But I don't buy your proof of it, and I don't know how to choose the "right" value of p/q.

> As for all the other stuff about 5/4 being only one of the convergents and so on, well that's just me putting ideas/results out there. I still 'know' that for an 81/64 that the 5/4 is the most reasonable, for a 32/27 it is a 6/5 etc...

I think you agree we need to do better than just "know" it. Have you seen Carl's recent suggestion?

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> I think you agree we need to do better than just "know" it. Have you seen Carl's recent suggestion?

Something like that needs to be done; you aren't going to get 5/4 rather than 19/15 from 81/64 using only math, since it's not about only math, it's also and primarily about hearing.

I have been following this conversation and have a question - so this is
(finally) proof that irrational ratios do indeed have a particular
frequency?

That was probably the first controversial issue that came into my inbox when
I joined this list.

Chris

On Thu, May 27, 2010 at 4:20 PM, martinsj013 <martinsj@...> wrote:

>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "rick"
> <rick_ballan@...> wrote:
> > However, I can't see how your 501/399 example creates doubt about what
> I'm proposing. Firstly, the ~ GCD would be (501 + 399)/(5 + 4) = 100 which
> *is* between 399/4 and 501/5.
>
> You misunderstand me - I was merely re-stating what we are trying to do.
> And I think your formula does do it. But I don't buy your proof of it, and I
> don't know how to choose the "right" value of p/q.
>
> > As for all the other stuff about 5/4 being only one of the convergents
> and so on, well that's just me putting ideas/results out there. I still
> 'know' that for an 81/64 that the 5/4 is the most reasonable, for a 32/27 it
> is a 6/5 etc...
>
> I think you agree we need to do better than just "know" it. Have you seen
> Carl's recent suggestion?
>
> Steve M.
>
>
>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
>
> > I think you agree we need to do better than just "know" it. Have you seen Carl's recent suggestion?
>
> Something like that needs to be done; you aren't going to get
> 5/4 rather than 19/15 from 81/64 using only math, since it's not
> about only math, it's also and primarily about hearing.

Of course for dyads it has been done. -Carl

Steve> I am pretty certain that the best way to find p/q is to use a
> > convergent or semi-convergent; it only remains to decide the
> > balance between accuracy and small-numberedness in choosing p/q.

Carl> One has to work the other way - from the rationals to the target.
> This is because some rationals approximate simpler rationals.
> So you need to determine which field of attraction you're in,
> and the simplest way to ensure you've gone out far enough from
> the target is to cut up the entire band (octave, say) in advance
> and precompute them. This is what harmonic entropy does.

Hi Carl,

(re my proposal - it sounds like you think I was proposing to work "backwards" through the convergents, getting to simpler fractions, but further from the target; I actually meant to work "forwards" i.e. getting more complicated fractions, but closer to the target. Either way, I don't have a scoring system that tells me when to stop.)

More importantly, then:

I assume you are suggesting using H.E. itself, rather than just *a* function that cuts up the octave in a similar way. Either way, it seems a great idea worthy of exploration.

Some details remain for Rick to sort out :-)

Is H.E. entirely appropriate? it is not a measure of consonance but of recognizability(?) - is that right for finding VF?
I am assuming that by "fields of attraction" you mean the areas that surround local minima of the H.E. curve?
Should we always go all the way to the minimum of the field of attraction or can we stop part way down the curve?
Aren't there only 13 local minima within the octave - this seems rather a restricted choice to "force" to?
In general the minimum is not precisely at a rational number but only close to it - should we use the rational or the precise minimum?

Steve M.

Steve, you said "Re the method of finding p/q in the graph of the combined waveform - I find this unconvincing. I assume that the 64.pdf file is showing that each of the several envelopes representing different convergent frequencies does indeed touch the original combined waveform at certain peaks of the latter (but it misses many others!)."

But you've not exactly understood what the graphs represent. The envelopes in the graphs are exact. First of all, the general equation is:

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

k = 0,1,2,...I've only represented the case of k = 0 which starts at t = 0. But they 'refresh' themselves at every value of k so they're all over the place. Secondly, there is no need to 'assume' that the envelopes touch the waveform. They do. They represent the solution to the extrema of each curve. If we take the 5/4 for eg then we can think of this as a pure 5/4 wave with an amplitude modulation of frequency (aq - pb)/(p + q) = 4/9. Which brings me to the third point that it doesn't "miss many other" peaks. The peaks that are not 'touched' by this envelope k = 0 are touched by others, the tail end of a previous one etc...Or they 'belong' to one of the other envelopes.

"The most convincing envelope is the first one - but to me this is just the difference tone. (And furthermore the frequency of the last two is so low that we can hardly see anything happening in the brief window of time shown.)"

If by "the first one" you mean that p/q = 1/1 then yes, it is the difference tone. If you substitute these values in all of the equations you'll see that they reduce to the standard trig ID 2sin((A + B)/2)cos((A - B)/2) suggesting that this now is just one instance of a more general formula. And yes, the last two are so slow that they wouldn't be heard (which is probably why they are not so relevant for tuning and which will likely be part of any 'selection' process). Nevertheless, they're there, mathematically speaking. They are necessary for completeness. However, the p/q = 1/1 is *not* close to the predicted VP. For the 81/64, only the 5/4 is.

But getting back to basics, the debate was whether or not VP's are there 'in the wave'. The fact that these are often *not* close to either the difference frequency or the GCD led theorists to believe that the brain was finding sub-harmonic matches, was somehow compensating for the "missing fundamental". Now Mike has given me many examples to test this and, as in the case of Carl's recent example 501/399, *each and every time* the formula ~ GCD = (a + b)/(p + q) has predicted a value close to the VP. And a graph of each example has shown significant peaks early on at the expected values. Further, just to prove a point, Mike has set many problems which are plain useless for musical harmony. Are these "unconvincing"?

But I thought that certain things went without saying: If the original freq's are close to upper harmonics of some ~ GCD then their difference tone will be close to this 'fundamental' i.e. if p/q has GCD = 1 then p - q = 1. Therefore, the two approaches are not mutually exclusive. For the 501/399 and VP around 100, then 501 - 399 = 102 *is* close to ~ GCD = 100. Therefore I don't know what you mean by ""The most convincing envelope is the first one - but to me this is just the difference tone".

Rick

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > Thanks Steve,
> > So far I've been accused of either being completely confusing or completely obvious ...
>
> Rick,
> Don't thank me too much; yes I find the formulas readable, but sometimes I am confused by the words :-) Also remember I said that I was not convinced by the way you derived the K=p+q by looking for peaks in the combined waveform (although the resulting formula looks reasonable).
>
> It is perhaps useful to reiterate what I think the aim is. Given a large-number frequency ratio a/b, or even letting a and b be irrational, find an expression for the "virtual fundamental" (VF) for a dyad with those frequencies. Sometimes we "know" the answer, e.g. 500/400 has VF 100 (same as the GCD). Other times it is not so easy e.g. 501/399 (the GCD is 3, surely the VF is not 3?). In an offlist exchange (which I hope he will not mind me discussing) Carl said it is approximately 399/4, or 501/5. Yes, but can't we find a single answer? Ideally of course it would be *the* value that most closely captures what most listeners actually perceive; failing that, it should at least be a value that "makes sense", e.g. varies continuously with a and b, is between 399/4 and 501/5, etc. (I have mentioned another criterion in an earlier post, but I'll save that for later.)
>
> The approach is to first find a smaller-number ratio p/q which approximates a/b as a ratio and thence derive approximate frequencies a' and b' near a and b, with a'/b' = p/q and thence the VF (GCD of p and q). (Or, derive these things in a different order.)
>
> BTW any statement of the results should say that the formula (with "aq-bp" and all that) is an identity, and that we are restricting ourselves to looking at certain values of p,q and explain why. (The way you usually state it seems to imply that the identity holds for just those values of p, q.) Having said that, that formula is not an essential part of the results, in my version of the problem stated above.
>
> I am pretty certain that the best way to find p/q is to use a convergent or semi-convergent; it only remains to decide the balance between accuracy and small-numberedness in choosing p/q. Other methods such as "rounding (a+b)/(a-b)" will probably turn out to be equivalent anyway (but perhaps I was the only one considering that). Re the method of finding p/q in the graph of the combined waveform - I find this unconvincing. I assume that the 64.pdf file is showing that each of the several envelopes representing different convergent frequencies does indeed touch the original combined waveform at certain peaks of the latter (but it misses many others!). And I assume that this is what you mean by "the convergents are all there"? But I don't believe that this is signficant - they wouldn't be heard, surely. Particularly as the peaks are not even "high peaks" as they were in your original derivation of K. The most convincing envelope is the first one - but to me this is just the difference tone. (And furthermore the frequency of the last two is so low that we can hardly see anything happening in the brief window of time shown.)
>
> > ... Well I can confirm that the convergents are all there. ...
> > Rick
>

> But getting back to basics, the debate was whether or not VP's are there 'in the wave'. The fact that these are often *not* close to either the difference frequency or the GCD led theorists to believe that the brain was finding sub-harmonic matches, was somehow compensating for the "missing fundamental".

They're obviously close to the GCD for dyads. When chords get
involved, things get tricky.

> Now Mike has given me many examples to test this and, as in the case of Carl's recent example 501/399, *each and every time* the formula ~ GCD = (a + b)/(p + q) has predicted a value close to the VP. And a graph of each example has shown significant peaks early on at the expected values. Further, just to prove a point, Mike has set many problems which are plain useless for musical harmony. Are these "unconvincing"

I've set many problems which are plain useless for musical harmony?
Like what? Like the fact that EVERYTHING you're trying to do is
dependent on the setup of the auditory system? Like the fact that
there's no mathematical "proof" that 81/64 "is really 5/4," because
for a different organism it might not be? Like the fact that all of
this ONLY applies to dyads, and that adding additional notes will bias
how you perceive the original ones? Like the fact that your equations
in the OP are tautological and have nothing to do with convergents at
all? Like the fact that your new "class of trigonometric identities"
is actually just the original identity and reduces to it for all p/q?

What of that is irrelevant?

> But I thought that certain things went without saying: If the original freq's are close to upper harmonics of some ~ GCD then their difference tone will be close to this 'fundamental' i.e. if p/q has GCD = 1 then p - q = 1.

No. The difference tone for 5/3 is 2, but it will be perceived as
having a root of 1. The difference tone for 8/5 is 3, but it will be
perceived as having a root of 1. etc. The difference tone for 6/5 is
1, and will be perceived as having a root of 1. Unless I treat it like
a faux 19/16 and create a huge overtonal chord where 19/16 is
sharpened 18 whole cents to be 6/5. Then it's going to not be
perceived the same way anymore.

Seems like this is pretty relevant to me. But by all means, carry on.
What am I but just a member of the tuning establishment, trying to
maintain the status quo?

-Mike

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> (re my proposal - it sounds like you think I was proposing to
> work "backwards" through the convergents, getting to simpler
> fractions, but further from the target; I actually meant to
> work "forwards" i.e. getting more complicated fractions, but
> closer to the target. Either way, I don't have a scoring
> system that tells me when to stop.)

Yes, I do the same thing, using mediants. I didn't realize
you were making a suggestion. Maybe you can state the problem
you're trying to solve again for my benefit.

One nice thing about H.E. is that it converges rapidly as more
complex rationals are considered. The width of the uncertainty
Gaussian around the target interval is the only parameter.

> Is H.E. entirely appropriate? it is not a measure of consonance
> but of recognizability(?) - is that right for finding VF?

It's a measure of consonance, keeping in mind that highly
recognizable ratios are highly consonant (by definition).

> I am assuming that by "fields of attraction" you mean the
> areas that surround local minima of the H.E. curve?

They can be interpreted that way.

> Aren't there only 13 local minima within the octave - this
> seems rather a restricted choice to "force" to?

It depends on size of the parameter mentioned above, but yes,
some fans of extended JI will be disappointed. That said, I
haven't seen a lot of evidence that more complex intervals can
be tuned by ear, etc.

> In general the minimum is not precisely at a rational number
> but only close to it - should we use the rational or the
> precise minimum?

Paul only calculated the entropy in cent increments, so none
of the minima can be rational except the octave. Nevertheless,
some of the minima are pushed one way or the other by their
neighbors.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, May 24, 2010 at 6:07 AM, rick <rick_ballan@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > >
> > > Eh? You've not found any new trig identities, you are *using* trig identities. In particular, you've been using sin(A+B) + sin(A-B) = 2 sin(A)cos(B)
> > >
> > Which is now the special case for p/q = 1/1.
>
> OK, unless I'm missing something... ANY p/q will work. You have
> basically taken sin(A+B) + sin(A-B) = 2sin(A)cos(B) and added a few
> extra terms in such a way that they cancel out and yield the original
> identity. The original identity isn't a "special case" for p/q = 1/1,
> but rather the original identity is a "special case" for p/q =
> ANYTHING, since no matter what you set p and q to the extraneous terms
> cancel out to yield the original identity.
>
> Perhaps there's some nuance that I'm missing...
>
> -Mike
>
Sure, I started with the usual trig ID to find the equation for the extrema which was t = (2k + 1)/2(a + b). I then compared the 'pure' JI wave to find the closest peaks and found that they always come out with T = K/(a + b) where K is *given*. For major thirds it is always K = 9, minor thirds K = 11 etc. *Why* this is is the million dollar question. My statements about Bezout, convergents etc...are all trying to answer this one (seemingly simple) question.

However, assuming that K is just given so we can see what happens (something might come out?), we can easily deduce p and q by dividing a and b by this frequency and taking it in quotient + remainder form.
aK/(a + b) = p + (aq - pb)/(a + b),
bK/(a + b) = q - (aq - pb)/(a + b),
Adding gives K = (p + q). Multiplying the (a + b)/K back out gives us the identities.

Now, the fact that these identities work for any p and q is not the point. By p and q I mean those numbers that were deduced by knowing K. For major thirds it is 9 = 5 + 4, minor's 11 = 6 + 5. And graphing these values has revealed that they do look like pure JI waves with a modulated amplitude of (aq - pb)/(a + b). But I've also since learnt that all of the convergents seem to be there, but not just any p, q chosen at random. The case K = 2 = 1 + 1 is the usual trig ID. Again, until I know why this is, what the selective process is etc..., the theory is not finished. But once I do then I can see in advance that the usual trig ID will become a special case. (Contrary to what's often said, tautologies are often used in proofs; "this equals that, therefore...". As I said, it is necessary but insufficient).

Rick

Rick

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > However, I can't see how your 501/399 example creates doubt about what I'm proposing. Firstly, the ~ GCD would be (501 + 399)/(5 + 4) = 100 which *is* between 399/4 and 501/5.
>
> You misunderstand me - I was merely re-stating what we are trying to do. And I think your formula does do it. But I don't buy your proof of it, and I don't know how to choose the "right" value of p/q.

Oh, sorry Steve. I thought for a horrible minute that you somehow completely missed the point (which I did think was strange). In that case forget what I've written before reading this message. For the record I haven't claimed to have a proof. Mike said that "I seemed to have it covered" and I said thanks.

> > As for all the other stuff about 5/4 being only one of the convergents and so on, well that's just me putting ideas/results out there. I still 'know' that for an 81/64 that the 5/4 is the most reasonable, for a 32/27 it is a 6/5 etc...
>
> I think you agree we need to do better than just "know" it. Have you seen Carl's recent suggestion?
>
> Steve M.
>
Absolutely. Just knowing it is hardly satisfactory. Yeah I read Carl's suggestion and have been thinking about it. I'll re-read it and get back.

Cheers

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
>
> > I think you agree we need to do better than just "know" it. Have you seen Carl's recent suggestion?
>
> Something like that needs to be done; you aren't going to get 5/4 rather than 19/15 from 81/64 using only math, since it's not about only math, it's also and primarily about hearing.
>
That's precisely the issue at stake. Is what we call 'virtual pitch' created in the ear/brain, is it in the wave or somehow both? I could be wrong but I'm not convinced that there isn't something in the wave that makes us distinguish a 5/4 from a 19/15. First of all it's ~ GCD = 16.111..is closer to the audible range than the ~ GCD = 4.2647...Second, the corresponding period of the 5/4 is much shorter than the 19/15 and so appears first and most often. Finally, just looking at it it appears much 'simpler', its extrema occurring earlier and more consistently.

However, even brushing aside where it comes from, doesn't the VP for this wave lie around 16 and not 4?

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I have been following this conversation and have a question - so this is
> (finally) proof that irrational ratios do indeed have a particular
> frequency?

I think so, yes. Or rather, if 'frequency = periodic function' then they correspond to an almost periodic function with 'almost frequency = ~ GCD.
>
> That was probably the first controversial issue that came into my inbox when
> I joined this list.
>
> Chris
>
> On Thu, May 27, 2010 at 4:20 PM, martinsj013 <martinsj@...> wrote:
>
> >
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "rick"
> > <rick_ballan@> wrote:
> > > However, I can't see how your 501/399 example creates doubt about what
> > I'm proposing. Firstly, the ~ GCD would be (501 + 399)/(5 + 4) = 100 which
> > *is* between 399/4 and 501/5.
> >
> > You misunderstand me - I was merely re-stating what we are trying to do.
> > And I think your formula does do it. But I don't buy your proof of it, and I
> > don't know how to choose the "right" value of p/q.
> >
> > > As for all the other stuff about 5/4 being only one of the convergents
> > and so on, well that's just me putting ideas/results out there. I still
> > 'know' that for an 81/64 that the 5/4 is the most reasonable, for a 32/27 it
> > is a 6/5 etc...
> >
> > I think you agree we need to do better than just "know" it. Have you seen
> > Carl's recent suggestion?
> >
> > Steve M.
> >
> >
> >
>

I didn't mean to insult you Mike. I was just saying that the examples you've given me so far helped lead me to this ~ GCD and was pointing out that sometimes we can go out of our way simply to prove a point. As for me saying that difference = GCD, I should have said consecutive harmonics (der). And while I'm aware that I've only been dealing with dyads, I have pointed out that the complexity of these greatly increases when we start to compound ~ GCD's or add more detuned upper harmonics of them.
Eg given f and g as two ~ GCD's, do these produce a third (f + g)/(p + q) etc...?

Rick

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > But getting back to basics, the debate was whether or not VP's are there 'in the wave'. The fact that these are often *not* close to either the difference frequency or the GCD led theorists to believe that the brain was finding sub-harmonic matches, was somehow compensating for the "missing fundamental".
>
> They're obviously close to the GCD for dyads. When chords get
> involved, things get tricky.
>
> > Now Mike has given me many examples to test this and, as in the case of Carl's recent example 501/399, *each and every time* the formula ~ GCD = (a + b)/(p + q) has predicted a value close to the VP. And a graph of each example has shown significant peaks early on at the expected values. Further, just to prove a point, Mike has set many problems which are plain useless for musical harmony. Are these "unconvincing"
>
> I've set many problems which are plain useless for musical harmony?
> Like what? Like the fact that EVERYTHING you're trying to do is
> dependent on the setup of the auditory system? Like the fact that
> there's no mathematical "proof" that 81/64 "is really 5/4," because
> for a different organism it might not be? Like the fact that all of
> this ONLY applies to dyads, and that adding additional notes will bias
> how you perceive the original ones? Like the fact that your equations
> in the OP are tautological and have nothing to do with convergents at
> all? Like the fact that your new "class of trigonometric identities"
> is actually just the original identity and reduces to it for all p/q?
>
> What of that is irrelevant?
>
> > But I thought that certain things went without saying: If the original freq's are close to upper harmonics of some ~ GCD then their difference tone will be close to this 'fundamental' i.e. if p/q has GCD = 1 then p - q = 1.
>
> No. The difference tone for 5/3 is 2, but it will be perceived as
> having a root of 1. The difference tone for 8/5 is 3, but it will be
> perceived as having a root of 1. etc. The difference tone for 6/5 is
> 1, and will be perceived as having a root of 1. Unless I treat it like
> a faux 19/16 and create a huge overtonal chord where 19/16 is
> sharpened 18 whole cents to be 6/5. Then it's going to not be
> perceived the same way anymore.
>
> Seems like this is pretty relevant to me. But by all means, carry on.
> What am I but just a member of the tuning establishment, trying to
> maintain the status quo?
>
> -Mike
>

The bottom line Mike is that non convergents simply don't work. I've posted a file called eg of non convergent showing an attempt at calling 81/64 an 8/5.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, May 24, 2010 at 6:07 AM, rick <rick_ballan@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > >
> > > Eh? You've not found any new trig identities, you are *using* trig identities. In particular, you've been using sin(A+B) + sin(A-B) = 2 sin(A)cos(B)
> > >
> > Which is now the special case for p/q = 1/1.
>
> OK, unless I'm missing something... ANY p/q will work. You have
> basically taken sin(A+B) + sin(A-B) = 2sin(A)cos(B) and added a few
> extra terms in such a way that they cancel out and yield the original
> identity. The original identity isn't a "special case" for p/q = 1/1,
> but rather the original identity is a "special case" for p/q =
> ANYTHING, since no matter what you set p and q to the extraneous terms
> cancel out to yield the original identity.
>
> Perhaps there's some nuance that I'm missing...
>
> -Mike
>

However, there is one thing here that shows what I was talking about: "Like the fact that there's no mathematical "proof" that 81/64 "is really 5/4," because for a different organism it might not be?"

Well what would Fourier analysis be to a cat? Yet psychoacoustics use it all the time. Did Fourier himself consider the auditory system of a cat when he was working on his coefficients? Or does the fact that my dog doesn't appreciate Bach mean that the principle of superposition couldn't possibly be true? Would these "different organisms" appreciate the importance of mathematical proofs (or my lack thereof)? Well which is it Mike? Nor did I ever say that "81/64 "is really 5/4"", but am simply pointing out that this branch of number theory now finds a basis in wave theory. You also know as well as I that the 16.111 predicts the VP, which can be been confirmed by experience, and that we have excluded the difference tone 1/1 and the 19/15, which gives a VP that is too far removed from the initial freq's (at around 4). Once again, would you doubt that these two intervals are similar if they were used to back, say, harmonic entropy?

For the record, if it ends up that there is nothing mathematical to distinguish the 5/4 from the other convergents then this does not invalidate the convergents themselves or their 'presence' in the waves. It just means that there are now other considerations to take into account, perhaps human considerations as in HE. I just want to be sure first. But if this ends up being the case, then I'll concentrate my efforts on proving that all the convergents are there and this will demarcate the theory. We could then say "the convergents are physically in the wave but the selection process is determined by the human auditory system" etc...It really doesn't bother me one way or the other. In fact it just occurred to me that this is probably what I should do.

Rick

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > But getting back to basics, the debate was whether or not VP's are there 'in the wave'. The fact that these are often *not* close to either the difference frequency or the GCD led theorists to believe that the brain was finding sub-harmonic matches, was somehow compensating for the "missing fundamental".
>
> They're obviously close to the GCD for dyads. When chords get
> involved, things get tricky.
>
> > Now Mike has given me many examples to test this and, as in the case of Carl's recent example 501/399, *each and every time* the formula ~ GCD = (a + b)/(p + q) has predicted a value close to the VP. And a graph of each example has shown significant peaks early on at the expected values. Further, just to prove a point, Mike has set many problems which are plain useless for musical harmony. Are these "unconvincing"
>
> I've set many problems which are plain useless for musical harmony?
> Like what? Like the fact that EVERYTHING you're trying to do is
> dependent on the setup of the auditory system? Like the fact that
> there's no mathematical "proof" that 81/64 "is really 5/4," because
> for a different organism it might not be? Like the fact that all of
> this ONLY applies to dyads, and that adding additional notes will bias
> how you perceive the original ones? Like the fact that your equations
> in the OP are tautological and have nothing to do with convergents at
> all? Like the fact that your new "class of trigonometric identities"
> is actually just the original identity and reduces to it for all p/q?
>
> What of that is irrelevant?
>
> > But I thought that certain things went without saying: If the original freq's are close to upper harmonics of some ~ GCD then their difference tone will be close to this 'fundamental' i.e. if p/q has GCD = 1 then p - q = 1.
>
> No. The difference tone for 5/3 is 2, but it will be perceived as
> having a root of 1. The difference tone for 8/5 is 3, but it will be
> perceived as having a root of 1. etc. The difference tone for 6/5 is
> 1, and will be perceived as having a root of 1. Unless I treat it like
> a faux 19/16 and create a huge overtonal chord where 19/16 is
> sharpened 18 whole cents to be 6/5. Then it's going to not be
> perceived the same way anymore.
>
> Seems like this is pretty relevant to me. But by all means, carry on.
> What am I but just a member of the tuning establishment, trying to
> maintain the status quo?
>
> -Mike
>

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
>
> I didn't mean to insult you Mike. I was just saying that the examples you've given me so far helped lead me to this ~ GCD

If you are going to talk about an approximate GCD to the horror of number theorists everywhere would you at least define it?

On Fri, May 28, 2010 at 12:21 PM, rick <rick_ballan@...> wrote:
>
> However, there is one thing here that shows what I was talking about: "Like the fact that there's no mathematical "proof" that 81/64 "is really 5/4," because for a different organism it might not be?"
>
> Well what would Fourier analysis be to a cat? Yet psychoacoustics use it all the time. Did Fourier himself consider the auditory system of a cat when he was working on his coefficients? Or does the fact that my dog doesn't appreciate Bach mean that the principle of superposition couldn't possibly be true?

Fourier analysis isn't inherently tied to psychoacoustics, the realm
of percept, or of sound at all, really. It's a generic mathematical
transform. I don't understand what you mean by "the principle of
superposition couldn't possibly be true."

> Would these "different organisms" appreciate the importance of mathematical proofs (or my lack thereof)? Well which is it Mike? Nor did I ever say that "81/64 "is really 5/4"", but am simply pointing out that this branch of number theory now finds a basis in wave theory.

The point I'm making is that you are looking for "the way" to reduce
81/64 to 5/4 without taking the auditory system's setup into
consideration and using just pure mathematics. It also seems like you
want to do it without ever using Fourier analysis, although you've
been using it the whole time anyway. Your motive is to proclaim that
this problem is solved completely by "wave theory" or "number theory,"
and that the missing fundamental phenomenon isn't a "virtual" pitch
after all because it really exists in the wave. You will talk about
how you did this without using Fourier analysis, despite the fact that
referring to a complex waveform as "81/64" to begin with already
involves Fourier analysis.

And as I've pointed out a number of times, which "convergent" the
81/64 will reduce to is going to be dependent on a lot of things (like
musical context and what other notes are in the chord). I don't think
81/64 is going to be particularly limited to representing only its
convergents either. You are trying to take a frequency-domain
representation (81/64) and turn it into a mixed time-frequency
representation (5/4 with time-varying amplitude and phase shifts).
There are an infinite amount of ways to do this, and the human
auditory system represents one.

> You also know as well as I that the 16.111 predicts the VP, which can be been confirmed by experience, and that we have excluded the difference tone 1/1 and the 19/15, which gives a VP that is too far removed from the initial freq's (at around 4). Once again, would you doubt that these two intervals are similar if they were used to back, say, harmonic entropy?

Who doubts that 81/64 and 5/4 are perceptually similar? But we're in
the realm of percept here. And I'm telling you that they're only
perceptually similar because of how the auditory system is set up.
We've been over this constantly. You're trying, mathematically, to
come up with THE ONE mixed time-frequency system FOR ALL TIME!!! And
it isn't going to happen and is logically impossible.

-Mike

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> >
> > I didn't mean to insult you Mike. I was just saying that the examples you've given me so far helped lead me to this ~ GCD
>
> If you are going to talk about an approximate GCD to the horror of number theorists everywhere would you at least define it?
>
I've defined it many times. I would suggest you get your facts together before saying sarcastic comments.

On 29 May 2010 09:19, rick <rick_ballan@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>>
>> If you are going to talk about an approximate GCD to the horror of number theorists everywhere would you at least define it?
>>
> I've defined it many times. I would suggest you get your facts together before saying sarcastic comments.

Why is it Gene's responsibility to trawl through your messages
searching for a definition?

Graham

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, May 28, 2010 at 12:21 PM, rick <rick_ballan@...> wrote:
> >
> > However, there is one thing here that shows what I was talking about: "Like the fact that there's no mathematical "proof" that 81/64 "is really 5/4," because for a different organism it might not be?"
> >
> > Well what would Fourier analysis be to a cat? Yet psychoacoustics use it all the time. Did Fourier himself consider the auditory system of a cat when he was working on his coefficients? Or does the fact that my dog doesn't appreciate Bach mean that the principle of superposition couldn't possibly be true?
>
> Fourier analysis isn't inherently tied to psychoacoustics, the realm
> of percept, or of sound at all, really. It's a generic mathematical
> transform. I don't understand what you mean by "the principle of
> superposition couldn't possibly be true."

Well obviously I'm being sarcastic. Of course the POS is true but I don't need to consider the musical tastes of my dog to determine this.
>
> > Would these "different organisms" appreciate the importance of mathematical proofs (or my lack thereof)? Well which is it Mike? Nor did I ever say that "81/64 "is really 5/4"", but am simply pointing out that this branch of number theory now finds a basis in wave theory.
>
> The point I'm making is that you are looking for "the way" to reduce
> 81/64 to 5/4 without taking the auditory system's setup into
> consideration and using just pure mathematics. It also seems like you
> want to do it without ever using Fourier analysis, although you've
> been using it the whole time anyway. Your motive is to proclaim that
> this problem is solved completely by "wave theory" or "number theory,"
> and that the missing fundamental phenomenon isn't a "virtual" pitch
> after all because it really exists in the wave. You will talk about
> how you did this without using Fourier analysis, despite the fact that
> referring to a complex waveform as "81/64" to begin with already
> involves Fourier analysis.

I've said all along that the waves are Fourier analysable but they contain harmonies that are not revealed by FA. No I haven't been using FA.
>
> And as I've pointed out a number of times, which "convergent" the
> 81/64 will reduce to is going to be dependent on a lot of things (like
> musical context and what other notes are in the chord). I don't think
> 81/64 is going to be particularly limited to representing only its
> convergents either. You are trying to take a frequency-domain
> representation (81/64) and turn it into a mixed time-frequency
> representation (5/4 with time-varying amplitude and phase shifts).
> There are an infinite amount of ways to do this, and the human
> auditory system represents one.
>
> > You also know as well as I that the 16.111 predicts the VP, which can be been confirmed by experience, and that we have excluded the difference tone 1/1 and the 19/15, which gives a VP that is too far removed from the initial freq's (at around 4). Once again, would you doubt that these two intervals are similar if they were used to back, say, harmonic entropy?
>
> Who doubts that 81/64 and 5/4 are perceptually similar? But we're in
> the realm of percept here. And I'm telling you that they're only
> perceptually similar because of how the auditory system is set up.
> We've been over this constantly. You're trying, mathematically, to
> come up with THE ONE mixed time-frequency system FOR ALL TIME!!! And
> it isn't going to happen and is logically impossible.
>
"Who doubts that 81/64 and 5/4 are perceptually similar?". I was saying that they are not only 'perceptually' similar but their wave forms are as well.
>
"they're only perceptually similar because of how the auditory system is set up." Yet even in a simple dyad between sine waves both the GCD and ~ GCD show up as a feature of the principle of superposition. Since *all* waves are composed of such things then nothing has been oversimplified. On the contrary. If f and g are two ~ GCD's then we can take their ~ GCD's for triads, and so on ad infinitum. Given any time-frequency system, it is by definition composed of such sine waves. Therefore I'm not trying to come up with "the one time-frequency system for all time". I hate utopianism and you're putting words in my mouth.

I've also said many times that the Cartesian duality between "perception - reality" went out with the horse and cart.

On Sat, May 29, 2010 at 1:57 AM, rick <rick_ballan@...> wrote:
>
> Well obviously I'm being sarcastic. Of course the POS is true but I don't need to consider the musical tastes of my dog to determine this.

That strawman isn't even good.

> > > Would these "different organisms" appreciate the importance of mathematical proofs (or my lack thereof)? Well which is it Mike? Nor did I ever say that "81/64 "is really 5/4"", but am simply pointing out that this branch of number theory now finds a basis in wave theory.

> > You will talk about
> > how you did this without using Fourier analysis, despite the fact that
> > referring to a complex waveform as "81/64" to begin with already
> > involves Fourier analysis.
>
> I've said all along that the waves are Fourier analysable but they contain harmonies that are not revealed by FA. No I haven't been using FA.

Read again: "You will talk about how you did this without using
Fourier analysis, despite the fact that referring to a complex
waveform as "81/64" to begin with already involves Fourier analysis."

> "Who doubts that 81/64 and 5/4 are perceptually similar?". I was saying that they are not only 'perceptually' similar but their wave forms are as well.

I see. So their waveforms are, perceptually, visually similar.

> "they're only perceptually similar because of how the auditory system is set up." Yet even in a simple dyad between sine waves both the GCD and ~ GCD show up as a feature of the principle of superposition.

After reading this, I'm not convinced you know what the principle of
superposition is.

> Since *all* waves are composed of such things then nothing has been oversimplified.

Are composed of what things? Could you be any more vague?

> On the contrary. If f and g are two ~ GCD's then we can take their ~ GCD's for triads, and so on ad infinitum. Given any time-frequency system, it is by definition composed of such sine waves. Therefore I'm not trying to come up with "the one time-frequency system for all time". I hate utopianism and you're putting words in my mouth.

Fine, Rick, then here's your task. Take the interval 25/24, the
chromatic semitone. I want you to tell me what the approx GCD is for
this interval if the "24" is set to 110 Hz.

Then I want you to tell me what the approx GCD is for this interval if
the "24" is set to 1760 Hz, 4 octaves higher.

I hope you come up with a different answer for each one.

-Mike

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

> > If you are going to talk about an approximate GCD to the horror of number theorists everywhere would you at least define it?
> >
> I've defined it many times. I would suggest you get your facts together before saying sarcastic comments.
>

Rubbish. A definition goes like this: "Given two nonzero real numbers a and b, we may define the approximate GCD of a and b, denoted ~GCD(a, b), by means of the following procedure." Then you give a procedure which defines an exact answer, and which someone could write a computer program for. This you have not done. You have NOT given a definition.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 29 May 2010 09:19, rick <rick_ballan@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >>
> >> If you are going to talk about an approximate GCD to the horror of number theorists everywhere would you at least define it?
> >>
> > I've defined it many times. I would suggest you get your facts together before saying sarcastic comments.
>
> Why is it Gene's responsibility to trawl through your messages
> searching for a definition?
>
>
> Graham
>
What do these adjectives "horror" and "trawl" imply? Male bonding perhaps? I've defined it more than once directly to him. Here it is again, ~ GCD = (a + b)/(p + q) where p/q are the convergents of a/b.

On Sat, May 29, 2010 at 5:19 AM, rick <rick_ballan@...> wrote:
>
> What do these adjectives "horror" and "trawl" imply? Male bonding perhaps? I've defined it more than once directly to him. Here it is again, ~ GCD = (a + b)/(p + q) where p/q are the convergents of a/b.

LOL, yes, you've figured it out. Not only is there a conspiracy to
keep your ideas down (and preserve the Fourier status quo), but it's a
gay conspiracy on top of it. A gay conspiracy between Gene and Graham.
Clearly you have it all sorted out on your end.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, May 29, 2010 at 1:57 AM, rick <rick_ballan@...> wrote:
> >
> > Well obviously I'm being sarcastic. Of course the POS is true but I don't need to consider the musical tastes of my dog to determine this.
>
> That strawman isn't even good.
>
> > > > Would these "different organisms" appreciate the importance of mathematical proofs (or my lack thereof)? Well which is it Mike? Nor did I ever say that "81/64 "is really 5/4"", but am simply pointing out that this branch of number theory now finds a basis in wave theory.
>
> > > You will talk about
> > > how you did this without using Fourier analysis, despite the fact that
> > > referring to a complex waveform as "81/64" to begin with already
> > > involves Fourier analysis.
> >
> > I've said all along that the waves are Fourier analysable but they contain harmonies that are not revealed by FA. No I haven't been using FA.
>
> Read again: "You will talk about how you did this without using
> Fourier analysis, despite the fact that referring to a complex
> waveform as "81/64" to begin with already involves Fourier analysis."

Again, sinA + sinB can be thought of as the result of some FA but the deduction of these near periodic waves don't seem to be revealed by one.
>
> > "Who doubts that 81/64 and 5/4 are perceptually similar?". I was saying that they are not only 'perceptually' similar but their wave forms are as well.
>
> I see. So their waveforms are, perceptually, visually similar.

And isn't the entire set of mathematical 'proofs' also based on ocular-centric notions like "dimensionless number SPACE", "phase SPACE", <, >, = and so on? You want to have your cake and eat it.
>
> > "they're only perceptually similar because of how the auditory system is set up." Yet even in a simple dyad between sine waves both the GCD and ~ GCD show up as a feature of the principle of superposition.
>
> After reading this, I'm not convinced you know what the principle of
> superposition is.

Don't be ridiculous. You're deliberately trying to sabotage me in the view of the others because I've obviously hit a raw nerve. Stop showing off to Graham and Gene.
>
> > Since *all* waves are composed of such things then nothing has been oversimplified.
>
> Are composed of what things? Could you be any more vague?

These are cheap tricks Mike. You say I'm too detailed or too vague whenever its convenient.
>
> > On the contrary. If f and g are two ~ GCD's then we can take their ~ GCD's for triads, and so on ad infinitum. Given any time-frequency system, it is by definition composed of such sine waves. Therefore I'm not trying to come up with "the one time-frequency system for all time". I hate utopianism and you're putting words in my mouth.
>
> Fine, Rick, then here's your task. Take the interval 25/24, the
> chromatic semitone. I want you to tell me what the approx GCD is for
> this interval if the "24" is set to 110 Hz.

No, it doesn't apply to epimoric intervals as I said originally.
>
> Then I want you to tell me what the approx GCD is for this interval if
> the "24" is set to 1760 Hz, 4 octaves higher.
>
> I hope you come up with a different answer for each one.
>
> -Mike
>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > > If you are going to talk about an approximate GCD to the horror of number theorists everywhere would you at least define it?
> > >
> > I've defined it many times. I would suggest you get your facts together before saying sarcastic comments.
> >
>
> Rubbish. A definition goes like this: "Given two nonzero real numbers a and b, we may define the approximate GCD of a and b, denoted ~GCD(a, b), by means of the following procedure." Then you give a procedure which defines an exact answer, and which someone could write a computer program for. This you have not done. You have NOT given a definition.
>
You're expecting a finished product. Did I not ask you and everyone else capable on the list to help search for a possible proof, not dismiss it because of obstacles that have yet to be overcome? Yes I did. And leave out the adjectives like "horror" please. They are not helpful.

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

> What do these adjectives "horror" and "trawl" imply? Male bonding perhaps? I've defined it more than once directly to him. Here it is again, ~ GCD = (a + b)/(p + q) where p/q are the convergents of a/b.
>

If a/b is irrational, this gives an infinite number of different answers.

On Sat, May 29, 2010 at 5:34 AM, rick <rick_ballan@...> wrote:
>
> > Read again: "You will talk about how you did this without using
> > Fourier analysis, despite the fact that referring to a complex
> > waveform as "81/64" to begin with already involves Fourier analysis."
>
> Again, sinA + sinB can be thought of as the result of some FA but the deduction of these near periodic waves don't seem to be revealed by one.

What do you mean "the deduction of these near periodic waves?" I
already gave you a way to find pseudoperiodicities using the
transform.

> > After reading this, I'm not convinced you know what the principle of
> > superposition is.
>
> Don't be ridiculous. You're deliberately trying to sabotage me in the view of the others because I've obviously hit a raw nerve. Stop showing off to Graham and Gene.

LOL, sabotage? Yes, you have hit a raw nerve. The nerve you've hit is
that I've spent lots of time trying to help you out and explain things
to you, and your hopelessly transparent reaction was to label me as
part of some kind of conspiracy keeping you down. So that time was
wasted, I believe.

> > > Since *all* waves are composed of such things then nothing has been oversimplified.
> >
> > Are composed of what things? Could you be any more vague?
>
> These are cheap tricks Mike. You say I'm too detailed or too vague whenever its convenient.

LOL dude, in all honesty, I give up. If the purpose of your messages
isn't to faciliate communication, then what is it?

> > > On the contrary. If f and g are two ~ GCD's then we can take their ~ GCD's for triads, and so on ad infinitum. Given any time-frequency system, it is by definition composed of such sine waves. Therefore I'm not trying to come up with "the one time-frequency system for all time". I hate utopianism and you're putting words in my mouth.
> >
> > Fine, Rick, then here's your task. Take the interval 25/24, the
> > chromatic semitone. I want you to tell me what the approx GCD is for
> > this interval if the "24" is set to 110 Hz.
>
> No, it doesn't apply to epimoric intervals as I said originally.

Fine, do it to 2501/2400 then.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Yes, I do the same thing, using mediants. I didn't realize
> you were making a suggestion. Maybe you can state the problem
> you're trying to solve again for my benefit.

Yes of course. With the proviso that I am beginning to wonder if this is going anywhere. And, if this is a good idea,m I am sure I got it from someone in this forum. If it isn't, I'll take the blame :-)

Given (my) statement of Rick's problem ...
"Given a large-number frequency ratio a/b, or even letting a and b be irrational, find an expression for the "virtual fundamental" (VF) for a dyad with those frequencies. The approach is to first find a smaller-number ratio p/q which approximates a/b and then take the VF to be (a+b)/(p+q)."

... I wanted to choose the "best" p/q, and a single "best" VF. To do this, consider all the convergents (*) in order, score each of them ("somehow") and stop when you reach the best one. Easy!

((*) Or semi-convergents, or mediants - not sure I'm distinguishing correctly between them yet.)

Rick had ideas of finding "the best p/q" within the graph, and I tried to help, but I don't think we can. Or, if we can, it will turn out to be merely equivalent to using number theory anyway.

But I wish I'd had the idea of using H.E. (again, "somehow"). Thanks for your answers to my questions - all make good sense to me.

Steve M.

If you're so convinced Mike that this wave theoretical approach to the problem of VP is "logically impossible" then why don't you find an indirect proof to back your claims? Until then, I'll take what you say about perception as a matter of opinion - in MY opinion a philosophically naive one - and will ignore statements like "I am telling you that...!". And please stop treating me like an errant schoolboy who 'won't learn his lessons'. If you read some Karl Popper you'll see that scientific theories are never actually true but only ever unfalsified up to a certain point in time. Virtual theory is no exception.

Rick

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, May 28, 2010 at 12:21 PM, rick <rick_ballan@...> wrote:
> >
> > However, there is one thing here that shows what I was talking about: "Like the fact that there's no mathematical "proof" that 81/64 "is really 5/4," because for a different organism it might not be?"
> >
> > Well what would Fourier analysis be to a cat? Yet psychoacoustics use it all the time. Did Fourier himself consider the auditory system of a cat when he was working on his coefficients? Or does the fact that my dog doesn't appreciate Bach mean that the principle of superposition couldn't possibly be true?
>
> Fourier analysis isn't inherently tied to psychoacoustics, the realm
> of percept, or of sound at all, really. It's a generic mathematical
> transform. I don't understand what you mean by "the principle of
> superposition couldn't possibly be true."
>
> > Would these "different organisms" appreciate the importance of mathematical proofs (or my lack thereof)? Well which is it Mike? Nor did I ever say that "81/64 "is really 5/4"", but am simply pointing out that this branch of number theory now finds a basis in wave theory.
>
> The point I'm making is that you are looking for "the way" to reduce
> 81/64 to 5/4 without taking the auditory system's setup into
> consideration and using just pure mathematics. It also seems like you
> want to do it without ever using Fourier analysis, although you've
> been using it the whole time anyway. Your motive is to proclaim that
> this problem is solved completely by "wave theory" or "number theory,"
> and that the missing fundamental phenomenon isn't a "virtual" pitch
> after all because it really exists in the wave. You will talk about
> how you did this without using Fourier analysis, despite the fact that
> referring to a complex waveform as "81/64" to begin with already
> involves Fourier analysis.
>
> And as I've pointed out a number of times, which "convergent" the
> 81/64 will reduce to is going to be dependent on a lot of things (like
> musical context and what other notes are in the chord). I don't think
> 81/64 is going to be particularly limited to representing only its
> convergents either. You are trying to take a frequency-domain
> representation (81/64) and turn it into a mixed time-frequency
> representation (5/4 with time-varying amplitude and phase shifts).
> There are an infinite amount of ways to do this, and the human
> auditory system represents one.
>
> > You also know as well as I that the 16.111 predicts the VP, which can be been confirmed by experience, and that we have excluded the difference tone 1/1 and the 19/15, which gives a VP that is too far removed from the initial freq's (at around 4). Once again, would you doubt that these two intervals are similar if they were used to back, say, harmonic entropy?
>
> Who doubts that 81/64 and 5/4 are perceptually similar? But we're in
> the realm of percept here. And I'm telling you that they're only
> perceptually similar because of how the auditory system is set up.
> We've been over this constantly. You're trying, mathematically, to
> come up with THE ONE mixed time-frequency system FOR ALL TIME!!! And
> it isn't going to happen and is logically impossible.
>
> -Mike
>

On Sat, May 29, 2010 at 8:11 AM, rick <rick_ballan@...> wrote:
>
> If you're so convinced Mike that this wave theoretical approach to the problem of VP is "logically impossible" then why don't you find an indirect proof to back your claims?

First off, what exactly is "wave theory" and how does it differ from
Fourier analysis?

And it is logically impossible unless you start making some recourse
to how the auditory system is set up. What kind of proof exactly would
you like? Haven't I "proved" this already? One way I proved it was my
(I thought clever) example about cats and critical bands that you seem
to have read about 3 words of and then thrown out as irrelevant.
Another was the example I told you to work out in the last email.
2501/2400 when played at a low frequency is going to sound like 1/1
with amplitude modulation. When played at a high frequency it's going
to instead sound like a half step. Is there a mathematical reason for
that?

For proof where there is no single time-frequency representation of a
signal, look up the "Fourier Uncertainty Principle."

> Until then, I'll take what you say about perception as a matter of opinion - in MY opinion a philosophically naive one - and will ignore statements like "I am telling you that...!". And please stop treating me like an errant schoolboy who 'won't learn his lessons'. If you read some Karl Popper you'll see that scientific theories are never actually true but only ever unfalsified up to a certain point in time. Virtual theory is no exception.

LOL, and now what's "virtual theory?"

-Mike

Rick,
--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
> But you've not exactly understood what the graphs represent. The envelopes in the graphs are exact. First of all, the general equation is ...

OK, I think you're right there. Rather than respond to all your points, just a few from me:

* It would be helpful if the file included more text to explain what it is showing. e.g. "the general equation" for what - I'm afraid I've lost track of that.

* I saw that the graphs are exact, but didn't realize about k=0, t=0. But are you really saying that any of this is significant? Just because every peak is hit by one or another of the envelopes in the set, this means the frequencies "are there"? Can this be perceived by the listener?

* Similarly, I don't understand the "refresh" idea. If you find two peaks (e.g.) 1ms apart, and if this is "refreshed" every 1ms, I feel sure that would be straightforwardly perceived. But if instead it is refreshed every (e.g.) 1.1ms, then whatever is perceived I doubt will have period 1ms or 1.1ms but something else. Are you even able to say that the "refresh" occurs regularly?

* I have never said that the (a+b)/(p+q) formula is unconvincing, just the way you derived it. Noted that you don't claim to have a proof, indeed you are hoping someone here will help find one. The presence in the graph of "significant peaks early on at the expected values", the idea of "refresh", and the set of envelopes, aren't leading me towards a proof :-(. I wonder if Mike B has any thoughts on these.

* The difference tone may be a red herring, please ignore that comment.

Steve M.

On Sat, May 29, 2010 at 10:44 AM, martinsj013 <martinsj@...> wrote:
>
> Rick,
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
> > But you've not exactly understood what the graphs represent. The envelopes in the graphs are exact. First of all, the general equation is ...
>
> OK, I think you're right there. Rather than respond to all your points, just a few from me:
>
> * It would be helpful if the file included more text to explain what it is showing. e.g. "the general equation" for what - I'm afraid I've lost track of that.

^ This.

> * I saw that the graphs are exact, but didn't realize about k=0, t=0. But are you really saying that any of this is significant? Just because every peak is hit by one or another of the envelopes in the set, this means the frequencies "are there"? Can this be perceived by the listener?

^ This.

> * Similarly, I don't understand the "refresh" idea. If you find two peaks (e.g.) 1ms apart, and if this is "refreshed" every 1ms, I feel sure that would be straightforwardly perceived. But if instead it is refreshed every (e.g.) 1.1ms, then whatever is perceived I doubt will have period 1ms or 1.1ms but something else. Are you even able to say that the "refresh" occurs regularly?

That's my exact point. The "refresh" has to do with how the auditory
system is set up. I believe from reading Paul's HE the analogous
parameter is the "s" free parameter (if I'm doing the DSP math in my
head right), and he says the auditory system can change it dynamically
depending on the stimulus (I.E. speech will have a faster "refresh"
than a long droning chord or something).

> * I have never said that the (a+b)/(p+q) formula is unconvincing, just the way you derived it. Noted that you don't claim to have a proof, indeed you are hoping someone here will help find one. The presence in the graph of "significant peaks early on at the expected values", the idea of "refresh", and the set of envelopes, aren't leading me towards a proof :-(. I wonder if Mike B has any thoughts on these.

I think he may have found something interesting whereby the sinusoid
with the frequency equaling the GCD of the convergents intersect with
the wave at different regular points (how's that for a mouthful). Why
this is, I'm not sure. Probably has something to do with how the
convergents are derived. I really don't think it has any particular
psychoacoustic significance.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> 2501/2400 when played at a low frequency is going to sound like 1/1 with amplitude modulation. When played at a high frequency it's going to instead sound like a half step. Is there a mathematical reason for
> that?

It's a good point, the method does not take account of whether frequencies a, b and the VF are audible. On the contrary, to double a and b is to double the VF. I take it as read that it would be usable only when they are audible. It may be a usable empirical formula.

FWIW I think that (given a method of choosing unique p/q) the method would give:

for 2750/2640 :- 110
for 44000/42240 :- 1760
for 2501/2400 : 100.02

Would any of these be perceived?

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> "Given a large-number frequency ratio a/b, or even letting
> a and b be irrational, find an expression for the "virtual
> fundamental" (VF) for a dyad with those frequencies.

There are some problems with this goal. First and foremost,
I'm not aware of any good data on what humans hear, so it's
hard to know if the expression one finds is working.

Secondly, it is pretty clear that whatever VF one hears, it
is conditioned on preceding stimuli. Now, the same is true
for consonance and dissonance, but we know that there is some
raw sensory value that isn't conditioned on preceding stimuli,
that has broad effects on how the conditioning works.
The question is, is the same thing true of VF? My hunch is
yes, but we don't really know for sure.

>The approach is to first find a smaller-number ratio p/q which
>approximates a/b and then take the VF to be (a+b)/(p+q)."

Why would the VF obey this formula?

> ... I wanted to choose the "best" p/q, and a single "best" VF.
>To do this, consider all the convergents (*) in order, score
>each of them ("somehow") and stop when you reach the best one.
>Easy!

Indeed.

> ((*) Or semi-convergents, or mediants - not sure I'm
> distinguishing correctly between them yet.)

IIRC, if you drill down with mediants you get both convergents
and semiconvergents, and both are necessary to get the
right result. :)

> But I wish I'd had the idea of using H.E. (again, "somehow").
> Thanks for your answers to my questions - all make good sense
> to me.

The H.E. local minima look pretty sensible to me. At least,
I don't know of any data that falsifies them. If you think
about it, any ratio that can supply a VF should have a field
of attraction, and any ratio with a field of attraction should
cause a dip on the entropy curve.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, May 29, 2010 at 5:19 AM, rick <rick_ballan@...> wrote:
> >
> > What do these adjectives "horror" and "trawl" imply? Male bonding perhaps? I've defined it more than once directly to him. Here it is again, ~ GCD = (a + b)/(p + q) where p/q are the convergents of a/b.
>
> LOL, yes, you've figured it out. Not only is there a conspiracy to
> keep your ideas down (and preserve the Fourier status quo), but it's a
> gay conspiracy on top of it. A gay conspiracy between Gene and Graham.
> Clearly you have it all sorted out on your end.
>
> -Mike
>
I *knew* it! But in all seriousness, how can someone understand the nature of a mathematical problem without looking into it in some depth? Sometimes "trawling" is precisely what is required. I'm not presenting some 'final theory' here which is all nicely defined and laid out in order. Just as introductions are usually written *after* the main body of the work (Hegel say's they are really a form of conclusion), clear definitions and first principles usually are recognised at the *end* of an inquiry when the problem has already been solved. "Ah, *that's* what it's saying!*. They are then presented in reverse order for the benefit of the reader. And I DID ask everybody for some help.

What I take offence at is the assumption that the problem has already been solved either by some *other* approach, or in absentia, i.e. that there really *is* no problem. For example, I've been going through all the possible values for 81/64 and, in the context of *this* problem, it ends up that all the possible convergents up to 5/4 are not even in there, and so far it looks like there might be a problem with 19/15 as well. When I know why this is *then* I can define my terms precisely. "Given any coprime or irrational pair (a, b)...then there exists a smallest integer pair (p, q) such that..."

Rick

Here is another example about questioning mathematical assumptions. If we take a/b with a > b then this is (obviously) saying how many times the freq b goes into the freq a. But since inverse freq = period then this is also saying the number of times the period 1/a will go into the period 1/b. Now if we form the first continued fraction a/b = N + r/b, and take b/r, then this is saying how many times r goes into b or period 1/r goes into 1/b. What does this really *mean*? And if we continue the process, what do these mean? What 'effect' do these have on the waves, if any?

Bertrand Russell and Whitehead asked similar questions about "what is a number?". We feel we 'know' that 2 + 2 = 4, but what really is '2' and '4'? Their answer was that it is a 'class of classes'. The result gave rise to mathematical logic and set theory. Russell states that his aim is to work towards greater simplicity and away from complexity. Of course the danger with this type of inquiry is that it might appear as if the questioner doesn't know that 2 + 2 = 4! And *that* I feel is what certain people on this list are relying on. They are using virtual pitch as a default position to avoid facing basic assumptions in wave theory. They are trying to answer primitive problems by going in the opposite direction, using for eg advanced mathematics like statistical analysis. But my fear with this is that these unquestioned assumptions are then being compounded.

Whatever one's opinions may be, being good at maths is no excuse for bad manners. And harmony *is* a form of good manners.

Rick

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, May 29, 2010 at 5:19 AM, rick <rick_ballan@...> wrote:
> >
> > What do these adjectives "horror" and "trawl" imply? Male bonding perhaps? I've defined it more than once directly to him. Here it is again, ~ GCD = (a + b)/(p + q) where p/q are the convergents of a/b.
>
> LOL, yes, you've figured it out. Not only is there a conspiracy to
> keep your ideas down (and preserve the Fourier status quo), but it's a
> gay conspiracy on top of it. A gay conspiracy between Gene and Graham.
> Clearly you have it all sorted out on your end.
>
> -Mike
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
> > ... find an expression for the "virtual
> > fundamental" (VF) for a dyad with those frequencies.
>
> There are some problems with this goal.

That is no doubt true.

> ... what humans hear
> ... is ... isn't conditioned on preceding stimuli.
> The question is, is the same thing true of VF? My hunch is
> yes, but we don't really know for sure.

sounds like you don't disapprove of the goal?

> > ... take the VF to be (a+b)/(p+q).
>
> Why would the VF obey this formula?

I can only say it seems to give a reasonable answer (it is Rick's formula of course). Did you see my earlier post:
/tuning/topicId_87740.html#88299

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> I can only say it seems to give a reasonable answer (it is Rick's
> formula of course). Did you see my earlier post:
> /tuning/topicId_87740.html#88299
>
> Steve M.

I didn't. But if I extrapolate a bit from the results
reported here
http://lib.tkk.fi/Diss/2003/isbn9512263149/article4.pdf
I conclude that, if a & b are unreduced Hz, the VF would be

if |a - 400Hz| < |b - 400Hz|
then a/p
else b/q

How does that compare to Rick's formula in your opinion?

-Carl

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

> You're expecting a finished product. Did I not ask you and everyone else capable on the list to help search for a possible proof, not dismiss it because of obstacles that have yet to be overcome? Yes I did. And leave out the adjectives like "horror" please. They are not helpful.

I do know what number theorists are likely to think of the phrase, helpful or not. It's not a term designed to endear you to the mathematics community. But the main point is that since you've not given a definition, you can't state a conjecture using that definition. If you can't state your conjecture, you can't prove it either. Or disprove it, for that matter. I think that's helpful.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> OK, unless I'm missing something... ANY p/q will work. You have
> basically taken sin(A+B) + sin(A-B) = 2sin(A)cos(B) and added a few
> extra terms in such a way that they cancel out and yield the original
> identity. ...
> ... Perhaps there's some nuance that I'm missing...

I think Rick knows that it is an identity (but perhaps hasn't expressed it clearly); he chooses to restrict attention to certain p/q (e.g. convergents to a/b) because this makes aq-pb small; he then has a "family" of identities, each of which is a different (possibly meaningful) way to interpret the original a/b.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> ... I extrapolate a bit from the results reported here
> http://lib.tkk.fi/Diss/2003/isbn9512263149/article4.pdf
> I conclude that, if a & b are unreduced Hz, the VF would be
> if |a - 400Hz| < |b - 400Hz|
> then a/p
> else b/q
> How does that compare to Rick's formula in your opinion?

On the face of it, I like it less (but I've not read the article yet). It seems to take some account of the actual frequency (which Rick's doesn't, as Mike B pointed out, and mine doesn't either). But the effect seems small and unobvious.

Let a' = p*VF, b' = q*VF. Then Rick's formula places a' and b' arithmetically symmetrical wrt a and b. i.e. a-a' = b'-b. Mine places them geometrically i.e. a/a' = b'/b. The formula above makes either b'=b or a'=a depending on whether the A.M. of a and b is greater or less than 400Hz. But these are mathematical observations; not sure I've got any musical ones.

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> On the face of it, I like it less (but I've not read the article
> yet). It seems to take some account of the actual frequency (which
> Rick's doesn't, as Mike B pointed out, and mine doesn't either).
> But the effect seems small and unobvious.

It's based on a little thing known as experimental evidence.
The evidence is that human listeners take some account of
absolute frequency. That's well established also for phoneme
recognition. -C.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
>
> > You're expecting a finished product. Did I not ask you and everyone else capable on the list to help search for a possible proof, not dismiss it because of obstacles that have yet to be overcome? Yes I did. And leave out the adjectives like "horror" please. They are not helpful.
>
> I do know what number theorists are likely to think of the phrase, helpful or not. It's not a term designed to endear you to the mathematics community. But the main point is that since you've not given a definition, you can't state a conjecture using that definition. If you can't state your conjecture, you can't prove it either. Or disprove it, for that matter. I think that's helpful.
>
Gene, I know you're right. It's just that I'm still not sure how to define the initial conditions mathematically. And I'm sure that certain others on the list are sick of hearing it.

Basically, the conjecture is that if we detune one of the harmonics of a rational wave of interval p/q, then its period, defined as the inverse of the GCD between its component frequencies, is only slightly modified. If a/b is our new interval, then this slightly modified GCD is defined as ~ GCD = (a + b)/(p + q). The deduction of this I've stated from beginning to end in the post to Steve above. Musically, the implication is that the "missing fundamental" is not missing but remains in the wave as this ~ GCD.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> It's based on a little thing known as experimental evidence.
> The evidence is that human listeners take some account of
> absolute frequency. That's well established also for phoneme
> recognition. -C.
>
Accepted. (I said my comments were mathematical, not musical; and I probably should have added "not psychoacoustical" and/or several other adjectives.)

Thanks for the link to the article. Like others I've read, it talks of partials and virtual pitch, as opposed to a dyad and virtual fundamental. Are these situations not different? Any pointers you could give me would be welcome.

Steve M.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
> >
> > On the face of it, I like it less (but I've not read the article
> > yet). It seems to take some account of the actual frequency (which
> > Rick's doesn't, as Mike B pointed out, and mine doesn't either).
> > But the effect seems small and unobvious.
>
> It's based on a little thing known as experimental evidence.
> The evidence is that human listeners take some account of
> absolute frequency. That's well established also for phoneme
> recognition. -C.
>
Hi Carl,

For some reason I missed this conversation but it looks interesting. I've downloaded that article and look forward to reading it. I'm also interested to see how far the experimental findings agree/disagree with my definition i.e. hopefully, finding out how much psychoacoustics and wave theory agree/disagree.

Remember way back when when you tried to teach me about HE? Well I always intended to check this out more thoroughly when I'm clearer about what I'm doing at my end, perhaps by adding the remainder to the interval and plotting these against interval, or something like that. In the meantime, do you know of a way to do this?

Rick

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> Thanks for the link to the article. Like others I've read, it
> talks of partials and virtual pitch, as opposed to a dyad and
> virtual fundamental. Are these situations not different? Any
> pointers you could give me would be welcome.

Partials and dyads are different (the "extrapolation" I mentioned)
but virtual pitch and virtual fundamental aren't. -C.

Actually Steve, just a thought. After reading your and Carl's exchange it occurred to me that it might be easier to go the other way. It might also explain why your definition is close to mine.

Instead of starting with a/b, if we start with p/q as Carl suggested and slightly detune the p we might write something like (p*S + q)/(p + q), with S is a comma for instance. (Eg, p/q = 5/4 and S = 81/80, I forget what this is called. Then pS = 81/16 = 5 + (1/16) and the ~ GCD above becomes 145/144). The determinant aq - pb becomes pq(S - 1) and remainder R = pq(S - 1)/(p + q). (= 1/36).

Now it's completely slipped my mind where I was leading to about your definition. Hmmm.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > It's based on a little thing known as experimental evidence.
> > The evidence is that human listeners take some account of
> > absolute frequency. That's well established also for phoneme
> > recognition. -C.
> >
> Accepted. (I said my comments were mathematical, not musical; and I probably should have added "not psychoacoustical" and/or several other adjectives.)
>
> Thanks for the link to the article. Like others I've read, it talks of partials and virtual pitch, as opposed to a dyad and virtual fundamental. Are these situations not different? Any pointers you could give me would be welcome.
>
> Steve M.
>

Sorry I missed all these posts, busy weekend.

Lets clean the slate Mike and get back to my original intention. After rereading Erlich's site, I remembered that it was this quote that initially peaked my interest: "The pitch corresponding to the fundamental itself need not be physically present in the sound". Now I reasoned to myself, 'surely the ear must be responding to *something*, albeit badly'. As you know, I've always thought it illogical to define frequency = sine wave when the definition of GCD is the more general. IOW my conjecture was that when we detune harmonics, a 'residual' of the GCD remained in the wave. This point at least I think I have proved. Nor do I think it contradicts HE mathematically in any way, only perhaps in the possible explanation of the results.

As I just said to Carl and Steve, and following their suggestion, instead of starting with some 'large numbered' interval a/b and trying (badly) to deduce a small numbered p/q, begin with our small p/q and detune, say, the p. Reinterpreting the results I've got so-far, perhaps the easiest way to do this is to take ~ GCD = (p*S + q)/(p + q) where S is the ratio representing the extent of the detuning. For eg, if p/q = 5/4 then we might take S = 81/80, the "whatsit? comma".

Now, the conditions for choosing S seems tailor-made for HE. For eg, the determinant aq - pb now becomes pSq - pq = pq(S - 1). By following through the logic of the Farey Series, where the distances between mediants around the simple numbers are greater but cluster around the complex intervals, we should arrive at a similar model to Paul's. The only difference now would be that a wave corresponds to each dyad point; the further the distance from p/q, the further is the ~ GCD from the original. A good place to start might be to select S such that the determinant is 1?

Rick

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, May 29, 2010 at 10:44 AM, martinsj013 <martinsj@...> wrote:
> >
> > Rick,
> > --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > > But you've not exactly understood what the graphs represent. The envelopes in the graphs are exact. First of all, the general equation is ...
> >
> > OK, I think you're right there. Rather than respond to all your points, just a few from me:
> >
> > * It would be helpful if the file included more text to explain what it is showing. e.g. "the general equation" for what - I'm afraid I've lost track of that.
>
> ^ This.
>
> > * I saw that the graphs are exact, but didn't realize about k=0, t=0. But are you really saying that any of this is significant? Just because every peak is hit by one or another of the envelopes in the set, this means the frequencies "are there"? Can this be perceived by the listener?
>
> ^ This.
>
> > * Similarly, I don't understand the "refresh" idea. If you find two peaks (e.g.) 1ms apart, and if this is "refreshed" every 1ms, I feel sure that would be straightforwardly perceived. But if instead it is refreshed every (e.g.) 1.1ms, then whatever is perceived I doubt will have period 1ms or 1.1ms but something else. Are you even able to say that the "refresh" occurs regularly?
>
> That's my exact point. The "refresh" has to do with how the auditory
> system is set up. I believe from reading Paul's HE the analogous
> parameter is the "s" free parameter (if I'm doing the DSP math in my
> head right), and he says the auditory system can change it dynamically
> depending on the stimulus (I.E. speech will have a faster "refresh"
> than a long droning chord or something).
>
> > * I have never said that the (a+b)/(p+q) formula is unconvincing, just the way you derived it. Noted that you don't claim to have a proof, indeed you are hoping someone here will help find one. The presence in the graph of "significant peaks early on at the expected values", the idea of "refresh", and the set of envelopes, aren't leading me towards a proof :-(. I wonder if Mike B has any thoughts on these.
>
> I think he may have found something interesting whereby the sinusoid
> with the frequency equaling the GCD of the convergents intersect with
> the wave at different regular points (how's that for a mouthful). Why
> this is, I'm not sure. Probably has something to do with how the
> convergents are derived. I really don't think it has any particular
> psychoacoustic significance.
>
> -Mike
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Partials and dyads are different (the "extrapolation" I mentioned)
> but virtual pitch and virtual fundamental aren't. -C.
>
Carl, thanks. I've now read the article4.pdf and can see what your formula is doing. Just one question - did you mean 400Hz? The article talks of "dominant partials" at around 600Hz. Am I missing something or should your formula use 600Hz?

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > Partials and dyads are different (the "extrapolation" I mentioned)
> > but virtual pitch and virtual fundamental aren't. -C.
>
> Carl, thanks. I've now read the article4.pdf and can see what
> your formula is doing. Just one question - did you mean 400Hz?
> The article talks of "dominant partials" at around 600Hz. Am I
> missing something or should your formula use 600Hz?

400Hz seems more realistic to me. :) It doesn't much matter
what value is used, given the paucity of data. -Carl

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

>Reinterpreting the results I've got so-far, perhaps the easiest way to do this is to take ~ GCD = (p*S + q)/(p + q) where S is the ratio representing the extent of the detuning. For eg, if p/q = 5/4 then we might take S = 81/80, the "whatsit? comma".
>
> Now, the conditions for choosing S seems tailor-made for HE. For eg, the determinant aq - pb now becomes pSq - pq = pq(S - 1).

Haven't you more or less reinvented Tenney height here? The above equation says that ~GCD(p,q) = pq(S-1), which is Tenney(p/q)(S-1). Since S is arbitrary, the key thing this extracts is the Tenney height of p/q.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
>
> > What do these adjectives "horror" and "trawl" imply? Male bonding perhaps? I've defined it more than once directly to him. Here it is again, ~ GCD = (a + b)/(p + q) where p/q are the convergents of a/b.
> >
>
> If a/b is irrational, this gives an infinite number of different answers.
>
Sure, because there are an infinite number of convergents. So what is that one condition then that makes the '5/4' stand out above all others? I don't think this is a fools errand because the peaks of the wave *do* occur very close to a 5/4 with the aforementioned GCD. This is what made me think that it might be an almost periodic function with respect to the pure 5/4 and that the E < || [f(t)-g(t)] || condition might be the smallest, or something like that.

Hi Carl, my version of the article is missing all the figures and diagrams. Is this the same with yours and Steve's?

Just a note. If we take two pianos (or digitally modelled pianos) then the possibility of a *third* fundamental/near fundamental arises as a feature of superposition. This possibility led me to reason that something akin to the tonic could be present in the wave without an instrument being assigned to play that particular note.

Rick

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > > Partials and dyads are different (the "extrapolation" I mentioned)
> > > but virtual pitch and virtual fundamental aren't. -C.
> >
> > Carl, thanks. I've now read the article4.pdf and can see what
> > your formula is doing. Just one question - did you mean 400Hz?
> > The article talks of "dominant partials" at around 600Hz. Am I
> > missing something or should your formula use 600Hz?
>
> 400Hz seems more realistic to me. :) It doesn't much matter
> what value is used, given the paucity of data. -Carl
>

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> Hi Carl, my version of the article is missing all the figures
> and diagrams. Is this the same with yours and Steve's?

It is apparently a draft. -Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> 400Hz seems more realistic to me. :) It doesn't much matter
> what value is used, given the paucity of data. -Carl

Notwithstanding the paucity of data, and recognizing that I may be taking your formula more seriously than you intended (I am sure you'll tell me if that's the case):
consider the dyads 445.5Hz:352.0Hz and 447.525Hz:353.6Hz (both are 81:64); your formula predicts VFs 89.1 and 88.4 respectively, i.e. a lower VF for the higher pair. Now I am not going to say this is wrong, but it reminds me that maybe we should be thinking in terms of a probability distribution function of the expected responses, and that it may well not be unimodal.

S.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > 400Hz seems more realistic to me. :) It doesn't much matter
> > what value is used, given the paucity of data. -Carl
>
> Notwithstanding the paucity of data, and recognizing that I may
> be taking your formula more seriously than you intended (I am
> sure you'll tell me if that's the case):
> consider the dyads 445.5Hz:352.0Hz and 447.525Hz:353.6Hz (both
> are 81:64); your formula predicts VFs 89.1 and 88.4 respectively,
> i.e. a lower VF for the higher pair.

Sure, you'll get boundary effects around any hard cutoff.
I picked 400 Hz, by the way, as a kind of average of 600 Hz
(mentioned in the paper), 440 Hz (concert pitch), and
262 Hz (middle C). Probably a bit low even so.

> Now I am not going to say this is wrong, but it reminds me
> that maybe we should be thinking in terms of a probability
> distribution function of the expected responses, and that it
> may well not be unimodal.

Sure it would be a better way to go about it. My formula
was just a simple summary of the results reported.

-Carl

What was the formula Carl? I can't seem to find it anywhere.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "martinsj013" <martinsj@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > > 400Hz seems more realistic to me. :) It doesn't much matter
> > > what value is used, given the paucity of data. -Carl
> >
> > Notwithstanding the paucity of data, and recognizing that I may
> > be taking your formula more seriously than you intended (I am
> > sure you'll tell me if that's the case):
> > consider the dyads 445.5Hz:352.0Hz and 447.525Hz:353.6Hz (both
> > are 81:64); your formula predicts VFs 89.1 and 88.4 respectively,
> > i.e. a lower VF for the higher pair.
>
> Sure, you'll get boundary effects around any hard cutoff.
> I picked 400 Hz, by the way, as a kind of average of 600 Hz
> (mentioned in the paper), 440 Hz (concert pitch), and
> 262 Hz (middle C). Probably a bit low even so.
>
> > Now I am not going to say this is wrong, but it reminds me
> > that maybe we should be thinking in terms of a probability
> > distribution function of the expected responses, and that it
> > may well not be unimodal.
>
> Sure it would be a better way to go about it. My formula
> was just a simple summary of the results reported.
>
> -Carl
>

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
> What was the formula Carl? I can't seem to find it anywhere.
Rick,

"if |a - 400Hz| < |b - 400Hz|
then a/p
else b/q"

from /tuning/topicId_89455.html#89670

S.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > What was the formula Carl? I can't seem to find it anywhere.
> Rick,
>
> "if |a - 400Hz| < |b - 400Hz|
> then a/p
> else b/q"
>
> from /tuning/topicId_89455.html#89670
>
> S.
>
Ah, thanks Steve, interesting. I was just working on something very similar myself (and my bones tell me this is going to be the right method). Given ak/(a + b) = p + r/(a + b), bk/(a + b) = q -/+ r/(a + b), r = aq - pb, we arrive at p and q simply by *rounding off*. So for 81/64 we get
k = 1, (p , q) = (1, 0),
k = 2, (p , q) = (1, 1),
k = 3, (p , q) = (2, 1),
k = 4, (p , q) = (2, 2),
k = 5, (p , q) = (3, 2),
k = 6, (p , q) = (3, 3),
k = 7, (p , q) = (4, 3),
k = 8, (p , q) = (4, 4),
k = 9, (p , q) = (5, 4),...
(p , q) = (1, 1) gives (a + b)/(p + q) = (a + b)/2, modulated freq r/(p + q) = (a - b)/2 which is just the usual beat frequency. We also see that all the convergents are there which explains why they worked.

But there are other things going on which eliminate most from the list. Even numbers up to k = 20 give p = q and since k = p + q this gives (a + b)/(p + q) = (a + b)/k ( = even) and r/(p + q) = (a - b) which are just cycles of the usual beat frequency.

Once I clean this up I should then begin to see why the 5/4 stands out above all other options, 3/2, 4/3 etc...Perhaps 'r > some value' (similar to Carl's basic idea) or the mod period is too long compared with the originals, and so on? Getting there slowly but surely.

Rick