Yahoo Tuning Groups Ultimate Backup tuning-math Finding closest JI interval

The problem here lies with the fact that there IS no "smallest"
numbered JI interval for a more complex one, from a purely
mathematical standpoint. Different organisms have
differently-structured auditory systems and will hence perceive these
sorts of things a bit differently.

For example, at least according to a guest lecturer at UM last year,
the critical bandwidth of a cat is much wider than that of a human
being. The net result of this would be that cats will have much better
time resolution and much worse pitch resolution. So to a cat, for
example, a whole step might be perceived as is a quarter tone for us,
and a half step might be perceived as ultra-rapid beating. And who
knows how any sort of periodicity detection in these animals might
differ from how we see it.

So you're not going to find an irrefutable mathematical proof that an
interval of high complexity "is really" an interval at an arbitrary
point of lower complexity. Keep in mind that we're looking at sound
waves, and that there are tons and tons of perspectives to take and
ways to interpret the data. The human auditory system is just one way
to do it. With that in mind, a restatement of your goal should be to
model, mathematically, the specific approach that the human auditory
system takes - as best at possible.

-Mike

PS: from a quick google search, it seems that some more recent
research about animal auditory processing suggests that the critical
bandwidth for various non-human animals might have been overestimated
a bit. Without belaboring the point too much, let's just say that if
the critical bandwidth WERE wider than in humans, it would have
drastic effects on our perception of sound.

On Tue, May 18, 2010 at 10:47 AM, rick <rick_ballan@yahoo.com.au> wrote:
>
> I'm (still) looking for a straightforward formula or method to deduce the smallest numbered 'JI' interval for any given large numbered coprime or incommensurate interval. For example, 51/32 is close to 51.2/32 = 8/5 or 81/64 ~ 80/64 = 5/4 etc...Continued fractions don't give the correct result and while it seems like an intuitively obvious problem - there will always be some smallest interval- I'm at a loss proving this.
>
> Thanks
>
> Rick

🔗Mike Battaglia <battaglia01@gmail.com>

🔗Carl Lumma <carl@lumma.org>

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, May 19, 2010 at 2:59 AM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
>
> >For example, at least according to a guest lecturer at UM last year,
> >the critical bandwidth of a cat is much wider than that of a human
> >being. The net result of this would be that cats will have much better
> >time resolution and much worse pitch resolution.
>
> ? The critical band isn't the limit of frequency resolution,
> even for pure tones.
>
> -Carl

Not directly as in it determining the "just noticeable difference" or
something like that, but in a certain way, it would have an effect.

To take it to extremes, imagine if the entire cochlea responded as one
huge, singular, broadband auditory filter that passed everything. So
the whole thing would effectively make up a filterbank of one filter.
Thus there'd be no frequency resolution at all, really, since every
frequency would stimulate every part of the cochlea. Some processing
further downstream might yield some very limited frequency data from
this anyway (and it might be this where the actual "JND" value comes
from). Although I doubt it. But maybe.

Now let's take it to the opposite extreme - let's say that there were
an infinite amount of hairs or nerve endings or whatever they are in
the cochlea, and they all correspond to "a single" frequency. So an
infinite amount of filters in a filterbank corresponding to a single
sinusoidal frequency. This would "sort of" equivocate to a classical
Fourier transform, and we'd have no time resolution at all, because
the hairs would never stop resonating. A single impulse would set all
of them oscillating until the end of time.

So although the "JND" for pitch is much smaller than the usually cited
critical bandwidth, to make the width of the auditory filter much
broader would certainly have some very significant effects on our
perception of pitch - in this case by altering the time/frequency
resolution of the entire audible spectrum. If the bands got wide
enough, an interval like 33/32 (or 16/15), for example, wouldn't even
be perceived as two notes anymore, but one note with a lot of very
fast beating.

Or, in other words, if Rick were a cat, he might be looking for a way
for 1601/1500 to simplify to 1/1, not 16/15. It's all somewhat
arbitrary.

Although I am admittedly very lacking in knowledge of the -neural-
side of all of this - so perhaps there is something I'm overlooking.
Just another one of my half-assed theories really, but it makes sense
from a signal processing standpoint.

-Mike

🔗Carl Lumma <carl@lumma.org>

Mike wrote:

>To take it to extremes, imagine if the entire cochlea responded as one
>huge, singular, broadband auditory filter that passed everything. So
>the whole thing would effectively make up a filterbank of one filter.
>Thus there'd be no frequency resolution at all, really, since every
>frequency would stimulate every part of the cochlea. Some processing
>further downstream might yield some very limited frequency data from
>this anyway (and it might be this where the actual "JND" value comes
>from). Although I doubt it. But maybe.

For pure tones, the JND comes from the fact that basilar membrane
displacement is not uniform within a critical band, and the fact
that hair cell bundles count wave periods (at least up to ~ 4KHz).
For complex tones, it additionally comes from the fact that multiple
signals are being combined.

>So although the "JND" for pitch is much smaller than the usually cited
>critical bandwidth, to make the width of the auditory filter much
>broader would certainly have some very significant effects on our
>perception of pitch - in this case by altering the time/frequency
>resolution of the entire audible spectrum.

If the displacement behavior is scaled appropriately, there would
be no effect on the melodic JND.

>If the bands got wide
>enough, an interval like 33/32 (or 16/15), for example, wouldn't even
>be perceived as two notes anymore, but one note with a lot of very
>fast beating.

Are you talking about harmonic intervals again?

-Carl

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, May 19, 2010 at 3:32 AM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
>
> >To take it to extremes, imagine if the entire cochlea responded as one
> >huge, singular, broadband auditory filter that passed everything. So
> >the whole thing would effectively make up a filterbank of one filter.
> >Thus there'd be no frequency resolution at all, really, since every
> >frequency would stimulate every part of the cochlea. Some processing
> >further downstream might yield some very limited frequency data from
> >this anyway (and it might be this where the actual "JND" value comes
> >from). Although I doubt it. But maybe.
>
> For pure tones, the JND comes from the fact that basilar membrane
> displacement is not uniform within a critical band, and the fact
> that hair cell bundles count wave periods (at least up to ~ 4KHz).
> For complex tones, it additionally comes from the fact that multiple
> signals are being combined.

What do you mean by counting wave periods, exactly...? Can you
reference me to some more reading on this?

Also, when you say that the basilar displacement differs within the
critical band, you mean that the hair cell corresponding to the
frequency of the note coming in will displace the basilar membrane
more than the surrounding hair cells, right? So that the hair at the
"peak" frequency at the "center" of the critical band will vibrate
more than the surrounding frequencies.

In fact, to make this even more clear, I drew a nice little mspaint
picture here:

http://tech.dir.groups.yahoo.com/group/tuning-math/files/jnd.gif

So the two filters have the same "bandwidth," as defined by that they
have the same cutoff point, but the one on the left will yield a more
coarse "JND" because of its more uniform amplitude. Right?

> >So although the "JND" for pitch is much smaller than the usually cited
> >critical bandwidth, to make the width of the auditory filter much
> >broader would certainly have some very significant effects on our
> >perception of pitch - in this case by altering the time/frequency
> >resolution of the entire audible spectrum.
>
> If the displacement behavior is scaled appropriately, there would
> be no effect on the melodic JND.

If I understand you correctly above, then for the melodic JND, for a
single note, I guess it wouldn't. For more than one note, it will
certainly make a difference. But I concede your point.

> >If the bands got wide
> >enough, an interval like 33/32 (or 16/15), for example, wouldn't even
> >be perceived as two notes anymore, but one note with a lot of very
> >fast beating.
>
> Are you talking about harmonic intervals again?
>
> -Carl

Er, what do you mean? As opposed to what?

-Mike

🔗Carl Lumma <carl@lumma.org>

Mike wrote:

>What do you mean by counting wave periods, exactly...? Can you
>reference me to some more reading on this?

I guess start here:
http://en.wikipedia.org/wiki/Temporal_theory

>Also, when you say that the basilar displacement differs within the
>critical band, you mean that the hair cell corresponding to the
>frequency of the note coming in will displace the basilar membrane
>more than the surrounding hair cells, right? So that the hair at the
>"peak" frequency at the "center" of the critical band will vibrate
>more than the surrounding frequencies.
>
>In fact, to make this even more clear, I drew a nice little mspaint
>picture here:
>http://tech.dir.groups.yahoo.com/group/tuning-math/files/jnd.gif

Uh, yes. The displacement isn't perfectly symmetrical... the
peak is biased toward the low frequencies -- or high (I can never
remember which).

>> Are you talking about harmonic intervals again?
>
>Er, what do you mean? As opposed to what?

Single tones.

-Carl

🔗Mike Battaglia <battaglia01@gmail.com>

Another thing, check out this auditory clip here:

http://www.sfu.ca/sonic-studio/handbook/Sound/Law_Of_Uncertainty.aiff

Those are short bursts of what sounds like a 1000 Hz sine wave, going
from 20ms down to 2ms. Note how the "pitchedness" of it drops as you
get closer to 2ms, until it starts to just sound like an unpitched
impulse or noise or something.

I had assumed that an increase in the critical bandwidth would
basically make the transition from pitch to noise happen more quickly.
So while a 10ms might be on the bare threshold of "pitchedness" for
us, I had assumed it would be well within the realm of "impulseness"
for a cat. And if the "place theory" were basically all that were
going on, it would.

But alas, this "temporal" model of hearing basically destroys that
hypothesis. A victim of outdated research am I. So I stand corrected.

-Mike

On Wed, May 19, 2010 at 4:22 AM, Carl Lumma <carl@lumma.org> wrote:
>
>
>
> Mike wrote:
>
> >What do you mean by counting wave periods, exactly...? Can you
> >reference me to some more reading on this?
>
> I guess start here:
> http://en.wikipedia.org/wiki/Temporal_theory
>
> >Also, when you say that the basilar displacement differs within the
> >critical band, you mean that the hair cell corresponding to the
> >frequency of the note coming in will displace the basilar membrane
> >more than the surrounding hair cells, right? So that the hair at the
> >"peak" frequency at the "center" of the critical band will vibrate
> >more than the surrounding frequencies.
> >
> >In fact, to make this even more clear, I drew a nice little mspaint
> >picture here:
> >http://tech.dir.groups.yahoo.com/group/tuning-math/files/jnd.gif
>
> Uh, yes. The displacement isn't perfectly symmetrical... the
> peak is biased toward the low frequencies -- or high (I can never
> remember which).
>
> >> Are you talking about harmonic intervals again?
> >
> >Er, what do you mean? As opposed to what?
>
> Single tones.
>
> -Carl

🔗Mike Battaglia <battaglia01@gmail.com>

After a bit of reading it seems like the "temporal theory" is
basically a more precise description of what we usually just call the
"periodicity mechanism," detailing how it actually works.

So, awesome. I've been looking for this for a while. Hopefully this
will connect some dots into where my model is failing. Perhaps if I
can get a detailed model of just HOW these nerves tend to fire, it
might yield a better archetypical model for a "harmonic filter" than
my damped comb filter model.

-Mike

On Wed, May 19, 2010 at 4:42 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> Another thing, check out this auditory clip here:
>
> http://www.sfu.ca/sonic-studio/handbook/Sound/Law_Of_Uncertainty.aiff
>
> Those are short bursts of what sounds like a 1000 Hz sine wave, going
> from 20ms down to 2ms. Note how the "pitchedness" of it drops as you
> get closer to 2ms, until it starts to just sound like an unpitched
> impulse or noise or something.
>
> I had assumed that an increase in the critical bandwidth would
> basically make the transition from pitch to noise happen more quickly.
> So while a 10ms might be on the bare threshold of "pitchedness" for
> us, I had assumed it would be well within the realm of "impulseness"
> for a cat. And if the "place theory" were basically all that were
> going on, it would.
>
> But alas, this "temporal" model of hearing basically destroys that
> hypothesis. A victim of outdated research am I. So I stand corrected.
>
> -Mike

🔗rick <rick_ballan@yahoo.com.au>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗rick <rick_ballan@yahoo.com.au>

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > I'm (still) looking for a straightforward formula or method to deduce the smallest numbered 'JI' interval for any given large numbered coprime or incommensurate interval. For example, 51/32 is close to 51.2/32 = 8/5 or 81/64 ~ 80/64 = 5/4 etc...Continued fractions don't give the correct result and while it seems like an intuitively obvious problem - there will always be some smallest interval- I'm at a loss proving this.
>
> Why do you say continued fractions aren't working? The convergents to 51/32 go 1, 2, 3/2, 8/5, 51/32 and the convergents to 81/64 go 1, 4/3, 5/4, 19/15, 81/64 and that's what you said you wanted.
>
Yes you're absolutely right Gene. Its been so many years that I completely forgot about the method for figuring out the convergents (and was too sure of myself to look it up). Also, the remainder r = aq - pb obviously relates to the determinant formula. The question now is to find out why the wave is 'selecting' the 5/4 over the greater or lesser ones.

Thanks

🔗rick <rick_ballan@yahoo.com.au>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> At 07:47 AM 5/18/2010, you wrote:
> >I'm (still) looking for a straightforward formula or method to deduce
> >the smallest numbered 'JI' interval for any given large numbered
> >coprime or incommensurate interval. For example, 51/32 is close to
> >51.2/32 = 8/5 or 81/64 ~ 80/64 = 5/4 etc...Continued fractions don't
> >give the correct result and while it seems like an intuitively obvious
> >problem - there will always be some smallest interval- I'm at a loss
> >proving this.
> >
> >Thanks
> >
> >Rick
>
> You need to consider the convergents and semiconvergents.
> Then you will always get the simplest ratio within a given
> factor of the starting ratio. Many programming languages
> come with such a function built-in.
>
> -Carl
>
Yes thanks Carl. As I said to Gene I've been so into the nitty gritty that I forgot about the obvious. Do you happen to know the Mathematica function?

Rick

🔗rick <rick_ballan@yahoo.com.au>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> The problem here lies with the fact that there IS no "smallest"
> numbered JI interval for a more complex one, from a purely
> mathematical standpoint. Different organisms have
> differently-structured auditory systems and will hence perceive these
> sorts of things a bit differently.
>
> For example, at least according to a guest lecturer at UM last year,
> the critical bandwidth of a cat is much wider than that of a human
> being. The net result of this would be that cats will have much better
> time resolution and much worse pitch resolution. So to a cat, for
> example, a whole step might be perceived as is a quarter tone for us,
> and a half step might be perceived as ultra-rapid beating. And who
> knows how any sort of periodicity detection in these animals might
> differ from how we see it.
>
> So you're not going to find an irrefutable mathematical proof that an
> interval of high complexity "is really" an interval at an arbitrary
> point of lower complexity. Keep in mind that we're looking at sound
> waves, and that there are tons and tons of perspectives to take and
> ways to interpret the data. The human auditory system is just one way
> to do it. With that in mind, a restatement of your goal should be to
> model, mathematically, the specific approach that the human auditory
> system takes - as best at possible.
>
> -Mike
>
> PS: from a quick google search, it seems that some more recent
> research about animal auditory processing suggests that the critical
> bandwidth for various non-human animals might have been overestimated
> a bit. Without belaboring the point too much, let's just say that if
> the critical bandwidth WERE wider than in humans, it would have
> drastic effects on our perception of sound.

As always what you say here is interesting Mike (and a little bit funny). In fact, Lou Reed and his wife Laurie Anderson are out here now doing a musical concert exclusively for dogs. I'm not joking.

You're forgetting that this "approx GCD" and the ratio p/q represents the solution to the problem, and wasn't chosen or designed by me. This means that, in the context of this problem at least, there DOES appear to be a smallest pair p/q for any given complex pair a/b. As I said, the solution to 'input' values 81/64 IS 145/9 = (81 + 64)/(5 + 4), 32/27 IS 59/11 and so on. The question now is, why? Besides, something interesting is going on here. Run with it for a sec Mike. If we take

f(t) = sin(2piat) + sin(2pibt) =
sin(2pi (p*(a + b)/(p + q)t + (aq -pb)t/2(p + q))) +
sin(2pi (q*(a + b)/(p + q)t - (aq -pb)t/2(p + q))),

we have a JI wave, interval p/q, GCD = (a + b)/(p + q), which becomes progressively more removed from its 'pure' state with increasing time t. In fact, we could say that ideally it *is* a JI interval at t = 0. At any rate, the wave seems to 'present itself' as one. Now, looking at these waves over long time periods, it is seen that the extrema of these 'JI waves' follow a set of overlapping envelopes that are different from the usual (1/2(a - b) or secondary combo tones. I calculated these to be

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

This seems to suggest that this JI wave is constantly 'refreshing' itself. But where do these envelopes come from? Why do they seem to involve these p, q numbers? Perhaps this might be answered if we take the original wave in exponential form? Something like

f(t) = exp[i2piat] + exp[i2pibt] =
exp[i2pi Rt)]*exp[i2pi pGt] +
exp[-i2pi Rt]*exp[i2pi qGt],

R = ((aq - pb)/(p + q)), G = (a + b)/(p + q). I haven't figured this out fully yet but I'm sure there's something going on here. You wouldn't study the hearing of cats when learning Fourier analysis so I don't see how it relates.

Rick

> On Tue, May 18, 2010 at 10:47 AM, rick <rick_ballan@...> wrote:
> >
> > I'm (still) looking for a straightforward formula or method to deduce the smallest numbered 'JI' interval for any given large numbered coprime or incommensurate interval. For example, 51/32 is close to 51.2/32 = 8/5 or 81/64 ~ 80/64 = 5/4 etc...Continued fractions don't give the correct result and while it seems like an intuitively obvious problem - there will always be some smallest interval- I'm at a loss proving this.
> >
> > Thanks
> >
> > Rick
>

🔗Carl Lumma <carl@lumma.org>

🔗rick <rick_ballan@yahoo.com.au>

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > Just curious, is the use of convergents here derived from a particular
> > psychoacoustic or perceptual construct? Or is it more of a neat
> > mathematical trick that happens to work? If memory serves right it's
> > related to the Stern-Brocot tree, yes?
>
> Yes, and to the Euclidean algorithm and Farey sequences and a whole lotta stuff. But really, it's the *first* thing about this stuff to learn.
>
> If you want to amuse yourself with ideas for mathematical widgets possibly code, you could look at my article on MOS on the Xenharmonic Wiki. Lots of continued fractions and Farey sequences in there for the fans of such things.
>
Gene, I did do allot of work a few months back on Bezout's ID, Farey series, the Stern-Brocot tree and even the determinants of matrices, but I realised that (a, b) wasn't restricted to rationals. What I didn't know was their connection to continued fractions which I stupidly overlooked and which I now see is all there. However, in answer to Mike's statement above, the whole point of what I'm driving at is that what might have previously been seen as mere mathematical tricks, or derived via purely perceptual experiments, now finds some *physical* basis in wave theory. Given a 64:81 wave we can now rewrite it as a 4:5 wave with modulated amplitudes. Therefore, if the ear/brain can extract a virtual pitch of GCD = 16 from the 4:5 (i.e. 64:80) wave and perceive it as the tonic then it can also extract the ~ GCD = 16.111...from the 64:81. As for the absence of beating being a criterion for musical 'pleasantness', I use beating harmonies all the time in my compositions. Besides, it's obvious that questions of differences are a consequence of ratios and division.

Just to bring you up to speed, what Mike was referring to in an earlier post about me "being on a crusade to change the term 'virtual pitch'" was that I've never understood why when we hear the pitch of a GCD or near GCD wave it is called 'virtual' while sine waves are called 'spectral'. Surely the ear/brain must still be referring to *something* in the outside world? It seems far more realistic to me to call all pitch virtual and all frequency spectral and *then* show that a gap can exist between the two. I mean I've taught my fair share of tone-deaf people who can't hear the correct pitch even from a pitch fork. Is this 'virtual pitch'? Should we base a theory of music on such things? Mike has also pointed out that there's no resonance at the virtual pitch, hence the name. But what if the definition of resonance itself has been unconsciously biased to detect only sine waves in order to fit what the theoretical part of our brains are saying? If the hairs in the cochlea are not responding at the sine wave location for the GCD then perhaps this initial assumption is flawed? And then we have the whole problem of aesthetics. At any rate, psychoacoustics will not think twice about citing mathematical proofs when it happens to correspond with theory. Then why shouldn't it be a two way process?

-Rick

🔗a_sparschuh <a_sparschuh@yahoo.com>

--- In tuning-math@yahoogroups.com, "rick" <rick_ballan@...> wrote:
> For example, 51/32
> is close to 51.2/32 = 8/5 or 81/64 ~ 80/64 = 5/4 etc...

Hi Rick,

simply try out
http://superspace.epfl.ch/approximator/
with yours fraction as input:

51 / 32 = 1.59375

delivers yours both ratios as desired, when choosing

@ 0.1 tolerance: 3/2 with +0.09375 error
@ 0.01 tolerance: 8/5 with -0.00625 error

For more generally information and other useful algoritms consult:

http://reference.wolfram.com/mathematica/guide/ContinuedFractionsAndRationalApproximations.html

http://reference.wolfram.com/mathematica/ref/Rationalize.html

Also attend there the "Egyptian"-fraction-expansion calculation in

http://www.wolframalpha.com/input/?i=rationalize+[1.59375]

Hope that helps.

bye
A.S.

🔗rick <rick_ballan@yahoo.com.au>

--- In tuning-math@yahoogroups.com, "a_sparschuh" <a_sparschuh@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > For example, 51/32
> > is close to 51.2/32 = 8/5 or 81/64 ~ 80/64 = 5/4 etc...
>
> Hi Rick,
>
> simply try out
> http://superspace.epfl.ch/approximator/
> with yours fraction as input:
>
> 51 / 32 = 1.59375
>
> delivers yours both ratios as desired, when choosing
>
> @ 0.1 tolerance: 3/2 with +0.09375 error
> @ 0.01 tolerance: 8/5 with -0.00625 error
>
> For more generally information and other useful algoritms consult:
>
> http://reference.wolfram.com/mathematica/guide/ContinuedFractionsAndRationalApproximations.html
>
> http://reference.wolfram.com/mathematica/ref/Rationalize.html
>
>
> Also attend there the "Egyptian"-fraction-expansion calculation in
>
> http://www.wolframalpha.com/input/?i=rationalize+[1.59375]
>
> Hope that helps.
>
> bye
> A.S.
>
Thanks A.S. very handy

Rick