Re: On Harmonic Entropy

>Our _brain_ determines what pitch we'll hear when we listen to
>a sound. It does so by trying to match the frequencies in the
>sound's spectrum (timbre) with a harmonic series. The pitch we
>hear is high or low depending on whether the frequency of the
>fundamental of the best-fit harmonic series is high or low.

This raises a point that hasn't been discussed much: where
exactly do(es) the mechanism(s) responsible for tonalness sit?
On top of both the place and periodicity mechanisms? Or are
they/it only part of the periodicity mechanism? That is: should
harmonic entropy only apply to frequencies up to about 4K?
A typical complex tone may have components both above and
below the 4K threshold...

In general, the applications of harmonic entropy we've seen
so far have been fairly abstracted... only fundamentals of
complex tones of dyads are taken as inputs, for example.
Probably a pretty good approximation of what would happen if
we assume some generic spectra for our tones and input an
entire FFT of the dyad, but...

It would be cool to see if harmonic entropy can also tell us
something about how we hear timbre... great stuff for
listening tests.

-Carl

>This raises a point that hasn't been discussed much: where
>exactly do(es) the mechanism(s) responsible for tonalness sit?
>On top of both the place and periodicity mechanisms?

We don't know much about periodicity mechanisms, but we do know that virtual
pitch is a _central_ process, since experiments which distribute the
"potential harmonics" between the two ears give the same results as if one
or both ears received the full stimulus. Place mechanisms are definitely
ear-based, so virtual pitch comes after place mechanisms.

>Or are
>they/it only part of the periodicity mechanism?

May be.

>That is: should
>harmonic entropy only apply to frequencies up to about 4K?

You mean fundamentals up to 4K?

>A typical complex tone may have components both above and
>below the 4K threshold...

Yes, and virtual pitch works very well for high "potential harmonics"
detected through the place mechanism. That doesn't mean there isn't a
periodicity mechanism behind all this, as long as the _fundamental_ is below
4K.

>It would be cool to see if harmonic entropy can also tell us
>something about how we hear timbre... great stuff for
>listening tests.

What do you have in mind?

> We don't know much about periodicity mechanisms, but we do know
> that virtual pitch is a _central_ process, since experiments which
> distribute the "potential harmonics" between the two ears give the
> same results as if one or both ears received the full stimulus.
> Place mechanisms are definitely ear-based, so virtual pitch comes
> after place mechanisms.

Right...

> Or are they/it only part of the periodicity mechanism?
>
> May be.

Fantastic how cutting-edge the whole subject is.

>That is: should harmonic entropy only apply to frequencies up to
>about 4K?
>
> You mean fundamentals up to 4K?

I meant any frequency. I didn't realize (or remember) that
virtual pitch works on frequency data from the place mechanism.

I guess the main point I was unclear on is how little h.e.
may actually have to do with the periodicity mechanism. It's
tempting to create "length of period" arguments, especially in
light of n*d, but they may not have anything to do with how
h.e. really works.

>>A typical complex tone may have components both above and
>>below the 4K threshold...
>
> Yes, and virtual pitch works very well for high "potential
> harmonics" detected through the place mechanism. That doesn't mean
> there isn't a periodicity mechanism behind all this, as long as
> the _fundamental_ is below 4K.

Not clear what the fundamental has to do with the 4K threshold.

>>It would be cool to see if harmonic entropy can also tell us
>>something about how we hear timbre... great stuff for
>>listening tests.
>
> What do you have in mind?

A listening test like the one that we did in the early days of
this list, except with timbres instead of chords.

-Carl

>Not clear what the fundamental has to do with the 4K threshold.

I thought you were claiming that periodicity only works up to 4KHz. If this
is so, it's the (possibly missing) _fundamental_, not the partials, that
must be below 4KHz.

Do you still have questions outstanding?

>>Not clear what the fundamental has to do with the 4K threshold.
>
>I thought you were claiming that periodicity only works up to 4KHz.

I was. My understanding based on past threads (way past...) was
that neurons can fire up to about 1000 hz., and gang together (in
groups of four?) to follow stimuli up to about 4000 hz. The
critical band is very wide at low frequencies, but the two
mechanisms have an octave or two of significant overlap.

>If this is so, it's the (possibly missing) _fundamental_, not the
>partials, that must be below 4KHz.

Are you saying that virtual fundamentals are actually fed back
through the periodicity mechanism (the virtual fundamental is
actually "played" on neurons)?

.------------. .-------.
|periodicity |--+--| place |
'------------' | '-------'
^ |
| v
| .------------------.
o--| virtual pitch |
'------------------'

> Do you still have questions outstanding?

Not other than this, thanks.

-Carl

>>If this is so, it's the (possibly missing) _fundamental_, not the
>>partials, that must be below 4KHz.

>Are you saying that virtual fundamentals are actually fed back
>through the periodicity mechanism (the virtual fundamental is
>actually "played" on neurons)?

Maybe, but that's not what I'm saying. I don't even want to assume that
virtual pitch is a distinct phenomenon from periodicity pitch. What I'm
saying is that low-frequency periodicities will be detected in a signal,
even if all the energy at low frequencies is filtered out.

>>>If this is so, it's the (possibly missing) _fundamental_, not the
>>>partials, that must be below 4KHz.
>
>>Are you saying that virtual fundamentals are actually fed back
>>through the periodicity mechanism (the virtual fundamental is
>>actually "played" on neurons)?
>
> Maybe, but that's not what I'm saying.

Is that to say you know of nobody actually measuring pulses
on an EEG at a frequency where there was no acoustic energy?

> I don't even want to assume that virtual pitch is a distinct
> phenomenon from periodicity pitch.

Um. I thought those two terms were simply synonymous; what
difference between them don't you want to assume?

> What I'm saying is that low-frequency periodicities will be
> detected in a signal, even if all the energy at low frequencies
> is filtered out.

Detected by the listener; a phenomenon called virtual fundamental,
virtual pitch, or periodicity pitch.

Until this thread, I thought:

.-------------. .-------.
| periodicity | | place |
'-------------' '-------'
| |
| |
.---------------. .-----------.
| tonalness/ | | roughness |
| virtual pitch | '-----------'
'---------------' /
\ /
\ /
\ /
--------------
| concordance/ |
| discordance |
--------------

When really:

.------------. .-------.
|periodicity | | place |
'------------' '-------'
^ \ / \
? \ / \
? .----------------. .-----------.
? | tonalness/ | | roughness |
?--| virtual pitch | '-----------'
'----------------' /
\ /
\ /
\ /
.--------------.
| concordance/ |
| discordance |
'--------------'

Right?

-Carl

>>>Are you saying that virtual fundamentals are actually fed back
>>>through the periodicity mechanism (the virtual fundamental is
>>>actually "played" on neurons)?
>
>> Maybe, but that's not what I'm saying.

>Is that to say you know of nobody actually measuring pulses
>on an EEG at a frequency where there was no acoustic energy?

I wouldn't be surprised, but no, I don't know of a specific case of this.

>> I don't even want to assume that virtual pitch is a distinct
>> phenomenon from periodicity pitch.

>Um. I thought those two terms were simply synonymous; what
>difference between them don't you want to assume?

Well you're putting them in different boxes in your diagram . . . if you
thought that they were synonymous, why did you do that?

>> Maybe, but that's not what I'm saying.
>
> Is that to say you know of nobody actually measuring pulses
> on an EEG at a frequency where there was no acoustic energy?
////
> .------------. .-------.
> |periodicity | | place |
> '------------' '-------'
> ^ \ / \
> ? \ / \
> ? .----------------. .-----------.
> ? | tonalness/ | | roughness |
> ?--| virtual pitch | '-----------'
> '----------------' /
> \ /
> \ /
> .--------------.
> | concordance/ |
> | discordance |
> '--------------'

So if the ?'s are deleted, I still don't understand why
a virtual fundamental couldn't be rendered above 4K
(since the virtual pitch mechanism is getting frequencies
high enough to create a v.f. above 4K from the place
mechanism).

-Carl

I don't think the periodicity mechanism necessarily gets its frequencies
from the place mechanism anyway!

I find it very unlikely that frequency of sound is encoded anywhere in the
brain as frequency of neural firing except below about 40 Hz. Is there
evidence?
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

When a noise signal is added to a few-ms delayed version of itself, one
hears a pitch corresponding to the inverse of the delay time, even though
there is no energy peak at that frequency.

-----Original Message-----
From: David C Keenan [mailto:D.KEENAN@...]
Sent: Monday, August 13, 2001 8:56 PM
To: harmonic_entropy@yahoogroups.com
Subject: Re: [harmonic_entropy] Re: On Harmonic Entropy

I find it very unlikely that frequency of sound is encoded anywhere in the
brain as frequency of neural firing except below about 40 Hz. Is there
evidence?
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

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>>>I don't even want to assume that virtual pitch is a distinct
>>>phenomenon from periodicity pitch.
>>
>>Um. I thought those two terms were simply synonymous; what
>>difference between them don't you want to assume?
>
>Well you're putting them in different boxes in your diagram . . .
>if you thought that they were synonymous, why did you do that?

I've been distinguishing between "periodicity pitch" (which I
take to be the a sound people claim to hear in certain
circumstances, and which is created by something shown in my
diagrams as "tonalness/virtual pitch") and "the periodicity
mechanism" (which is a way of counting zero crossings to
determine frequency, and appears in my diagrams as "periodicity").

>I don't think the periodicity mechanism necessarily gets its
>frequencies from the place mechanism anyway!

They aren't connected that way in my diagram. But the "virtual
pitch" mechanism, which I take to be a higher-level process
that results in tonalness/harmonic entropy, does take information
from the place mechanism, according to you (least, that's what
I thought you said):

>Yes, and virtual pitch works very well for high "potential
>harmonics" detected through the place mechanism.

And is connected that way in my 'current' diagrams, but not in
the one reflecting my pre-thread understanding.

-Carl

>I've been distinguishing between "periodicity pitch" (which I
>take to be the a sound people claim to hear in certain
>circumstances, and which is created by something shown in my
>diagrams as "tonalness/virtual pitch") and "the periodicity
>mechanism" (which is a way of counting zero crossings to
>determine frequency, and appears in my diagrams as "periodicity").

We may or may not have such a periodicity mechanism. If we do, as it seems
we do, it seems likely that virual pitch is a result of it, though as far as
I know, the issue isn't settled.

>They aren't connected that way in my diagram. But the "virtual
>pitch" mechanism, which I take to be a higher-level process
>that results in tonalness/harmonic entropy, does take information
>from the place mechanism, according to you (least, that's what
>I thought you said):

>>Yes, and virtual pitch works very well for high "potential
>>harmonics" detected through the place mechanism.

>And is connected that way in my 'current' diagrams, but not in
>the one reflecting my pre-thread understanding.

Perhaps I should have phrased that better:

"virtual pitch works very well for high "potential
harmonics", ones high enough to only be detectable by themselves through the
place mechanism."

or

"Virtual pitch works very well for high "potential harmonics", which can be
directly heard out if attention is drawn to them, which almost certainly
means calling attention to the data from the place mechanism".

> We may or may not have such a periodicity mechanism. If we do, as
> it seems we do, it seems likely that virual pitch is a result of
> it, though as far as I know, the issue isn't settled.

Wow.

> Perhaps I should have phrased that better:
>
> "virtual pitch works very well for high "potential
> harmonics", ones high enough to only be detectable
> by themselves through the place mechanism."
>
> or
>
> "Virtual pitch works very well for high "potential
> harmonics", which can be directly heard out if
> attention is drawn to them, which almost certainly
> means calling attention to the data from the place
> mechanism".

Tricky topic. Can we say that harmonic entropy
calculations are defined on all of the partials
coming in, and that our usual way of inputting
only the fundamentals is an approximation of this?

-Carl

>Tricky topic. Can we say that harmonic entropy
>calculations are defined on all of the partials
>coming in, and that our usual way of inputting
>only the fundamentals is an approximation of this?

Yes. But it should be an extremely good approximation if all of the tones
have harmonic partials, and similar spectra. Think about it. The only effect
harmonic partials should have is to decrease s relative to no harmonic
partials.

I wrote:
>>I find it very unlikely that frequency of sound is encoded anywhere in the
>>brain as frequency of neural firing except below about 40 Hz. Is there
>>evidence?

Paul erlich replied:
>When a noise signal is added to a few-ms delayed version of itself, one
>hears a pitch corresponding to the inverse of the delay time, even though
>there is no energy peak at that frequency.

So? I don't see that as evidence of frequency encoded as frequency (or time
encoded as time) in the brain. Just because you can imagine no other way,
doesn't make it so.

Surely someone has tortured some poor animal to find this out?

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

Sorry Dave -- I read your question too quickly. My understanding is that
higher frequencies _are_ encoded as frequency of neural firing, only most of
the firings are actually skipped -- so evidently some periodicity-detection
must be going on behind the scenes. That's what I remember reading.

-----Original Message-----
From: David C Keenan [mailto:D.KEENAN@...]
Sent: Monday, August 13, 2001 10:37 PM
To: harmonic_entropy@yahoogroups.com
Subject: RE: [harmonic_entropy] Re: On Harmonic Entropy

I wrote:
>>I find it very unlikely that frequency of sound is encoded anywhere in the
>>brain as frequency of neural firing except below about 40 Hz. Is there
>>evidence?

Paul erlich replied:
>When a noise signal is added to a few-ms delayed version of itself, one
>hears a pitch corresponding to the inverse of the delay time, even though
>there is no energy peak at that frequency.

So? I don't see that as evidence of frequency encoded as frequency (or time
encoded as time) in the brain. Just because you can imagine no other way,
doesn't make it so.

Surely someone has tortured some poor animal to find this out?

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

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> Yes. But it should be an extremely good approximation if all of
> the tones have harmonic partials, and similar spectra. Think about
> it.

Oh, I totally agree...

>>Probably a pretty good approximation of what would happen if
>>we assume some generic spectra for our tones and input an
>>entire FFT of the dyad

> The only effect harmonic partials should have is to decrease s
> relative to no harmonic partials.

Really? I don't follow that.

-Carl

It would decrease s because the harmonics give you more independent sources
of information about what the fundamental frequency is. There would be fewer
ways that an incorrect fundamental could fool you into thinking it was the
correct one.

-----Original Message-----
From: carl@... [mailto:carl@...]
Sent: Monday, August 13, 2001 10:45 PM
To: harmonic_entropy@yahoogroups.com
Subject: [harmonic_entropy] Re: On Harmonic Entropy

> Yes. But it should be an extremely good approximation if all of
> the tones have harmonic partials, and similar spectra. Think about
> it.

Oh, I totally agree...

>>Probably a pretty good approximation of what would happen if
>>we assume some generic spectra for our tones and input an
>>entire FFT of the dyad

> The only effect harmonic partials should have is to decrease s
> relative to no harmonic partials.

Really? I don't follow that.

-Carl

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>It would decrease s because the harmonics give you more independent
>sources of information about what the fundamental frequency is.
>There would be fewer ways that an incorrect fundamental could fool
>you into thinking it was the correct one.

Ooo! right. If our brains can really use multiple points to
reduce s, it seems s could be eliminated through a much higher
JI limit than our listening seems to indicate. Perhaps our
brains can't reconnoiter high harmonic relationships for other
reasons besides s...

-Carl

>Ooo! right. If our brains can really use multiple points to
>reduce s, it seems s could be eliminated through a much higher
>JI limit than our listening seems to indicate.

What exactly do you mean by "s could be eliminated through a much higher JI
limit" and "our listening seems to indicate"?

>I find it very unlikely that frequency of sound is encoded anywhere
>in the brain as frequency of neural firing except below about 40 Hz.
>Is there evidence?

Dave,

I went back and found the post I remembered...

x~

Date: Fri, 8 May 1998 19:07:45 -0700 (PDT)
From: Jesse Bagshaw Gay <jgay@...>
To: tuning@...
Subject: ear architecture
Message-ID: <Pine.OSF.3.96.980508185734.1296A-100000@...>

re: several things about the ear from several recent posts...

one of my texts(Sensation and Perception, by Coren et al) says:

1. Individual neurons can fire up to 1000 time/sec.
The volley theory says that groups of neurons cooperate in response
to higher frequencies( for example, to represent 2000 Hz, 2 neurons
could fire at 1000 Hz each, 180 degrees out of phase with each other)
Such cooperation has been shown up to about 4000 Hz.

2. In general, there are two ways in which sounds are differentiated
by the ear.

a. Place theory: The basilar membrane is shaped like a very long
triangle. Each region resonates at a different frequency.
Consequently, the place on the basilar membrane that gets most
excited in response to any incoming sound wave provides information
about the frequency.

b. Frequency theory: Individual neurons fire at the rate of the
incoming sound.(Or groups, as per the volley theory.)

The place theory does not hold below about 500 Hz, and the frequency
theory does not hold above about 4000Hz. Between 500 and 4000 hz,
both mechanisms play a role.

3. Some kind of higher processing occurs which enables the brain to
infer a fundamental from a set of sufficiently defined harmonics.

~x

-Carl

>>Ooo! right. If our brains can really use multiple points to
>>reduce s, it seems s could be eliminated through a much higher
>>JI limit than our listening seems to indicate.
>
>What exactly do you mean by "s could be eliminated through a much
>higher JI limit" and "our listening seems to indicate"?

Well, if s is as small as 1% for single sine tones, and our brain
assumes everything to be in 19-limit JI, and if each note in a
dyad has 19 partials, and the dyad really is in 19-limit JI, then
how much uncertainty can be left about the dyad? Our collective
listening over the past few years on these lists seems to say
that for raw dyads, only the "under-two" intervals have a life of
their own beyond the 9-limit or so.

-Carl

>Well, if s is as small as 1% for single sine tones,

Near 3000 Hz. Move much away from that, and s goes to 3%, 5%, and higher.

>and our brain
>assumes everything to be in 19-limit JI,

Now where is that coming from? Have you seen the harmonic entropy curves for
s=1%? 0.6%?

>and if each note in a
>dyad has 19 partials, and the dyad really is in 19-limit JI, then
>how much uncertainty can be left about the dyad? Our collective
>listening over the past few years on these lists seems to say
>that for raw dyads, only the "under-two" intervals have a life of
>their own beyond the 9-limit or so.

What does "under-two" mean?

>> and our brain assumes everything to be in 19-limit JI,
>
> Now where is that coming from? Have you seen the harmonic entropy
> curves for s=1%? 0.6%?

I've seen the curves... so just say that the brain assumes
everything to be in JI. That's a central idea of h.e.

>>and if each note in a dyad has 19 partials, and the dyad really
>>is in 19-limit JI, then how much uncertainty can be left about
>>the dyad? Our collective listening over the past few years on
>>these lists seems to say that for raw dyads, only the "under-two"
>>intervals have a life of their own beyond the 9-limit or so.
>
> What does "under-two" mean?

Oops - Partchian under. Ratios with a base-2 denominator.

-Carl

Paul wrote:
>When a noise signal is added to a few-ms delayed version of itself, one
>hears a pitch corresponding to the inverse of the delay time, even though
>there is no energy peak at that frequency.

I think it's wrong to assume there's no energy peak then. The
autocorrelation
of white noise is a pulse; the autocorrelation of white noise plus a
delayed
version of that noise will be two pulses. Since the power spectral density
is the Fourier transform of the autocorrelation, that won't be a horizontal
straight line anymore. I suspect there will be a peak at the inverse of the
delay time, but haven't got the formula ready.

Manuel

>>> and our brain assumes everything to be in 19-limit JI,
>
>> Now where is that coming from? Have you seen the harmonic entropy
>> curves for s=1%? 0.6%?

>I've seen the curves... so just say that the brain assumes
>everything to be in JI. That's a central idea of h.e.

The brain tries to fit everything into JI . . . that's different.

>>>and if each note in a dyad has 19 partials, and the dyad really
>>>is in 19-limit JI, then how much uncertainty can be left about
>>>the dyad?

Plenty -- that's exactly what the harmonic entropy calculation tells you. I
think s=0.6% is generous for most cases, even for timbres rich in harmonic
partials, because typically only a couple of partials from each note will
fall within the "fovea" (I think that's the name for that optimal region
near 3000Hz).

Questions outstanding... yes, I forgot a very important
one. Assuming that generalized Tenney complexity passes
the validation excercise for triads, how do we calculate
harmonic entropy for irrational triads without mediants?

-Carl

The same way we did for dyads! We didn't need the mediants there . . .

-----Original Message-----
From: carl@... [mailto:carl@...]
Sent: Thursday, August 16, 2001 7:30 PM
To: harmonic_entropy@yahoogroups.com
Subject: [harmonic_entropy] Re: On Harmonic Entropy

Questions outstanding... yes, I forgot a very important
one. Assuming that generalized Tenney complexity passes
the validation excercise for triads, how do we calculate
harmonic entropy for irrational triads without mediants?

-Carl

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>The same way we did for dyads! We didn't need the mediants
>there . . .

Okay. The best source I can find on the 2-widths approximation
is:

>I verified that the weighted sum of these remains essentially
>constant as one moves one's Gaussian window along the interval
>axis -- and renormalize these to define one's probability
>function, and get essentially the same results as using the
>mediant-defined partition. In the triadic case, we don't know
>the partition sizes but if we can verify that the weighted sum
>of the inverse geometric means remains constant as one moves
>one's Gaussian window around the triadic plane, we can probably
>live with that approximate solution and never have to worry about
>the "real" one.

Now, I understand what you're doing, but I should note that I
don't know what weighted sums are, or renormalization. I
assume these have to do with the fact that n*d widths may
overlap or not cover the log-freq. line in places.

Nor do I know exactly how the 2-widths approximation was
used to calculate h.e... We still draw a bell curve, and
then around each ratio under the curve we center a region
proportional to 1/sqrt(n*d), and then get p's as with
mediants, with the addition of something to get the p's to
sum to 1?

-Carl

By "weighted sum" I just mean the sum of the "widths" (1/sqrt(n*d)) when
each is multiplied by the height of the point n/d on the relevant bell curve
-- in order to approximate the integral under the bell curve. Since the
total probability should be 1 no matter where you put the bell curve, it's a
sign that no significant distortion is taking place if the "weighted sum" is
essentially constant. The "renormalization" just refers to scaling each of
the products of "width" times "height" by a constant so that they sum to 1,
and can be intepreted as probabilities (p) in the formula for entropy
(sum(p*log(p))).

Got it?

-----Original Message-----
From: carl@... [mailto:carl@...]
Sent: Thursday, August 16, 2001 8:55 PM
To: harmonic_entropy@yahoogroups.com
Subject: [harmonic_entropy] Re: On Harmonic Entropy

>The same way we did for dyads! We didn't need the mediants
>there . . .

Okay. The best source I can find on the 2-widths approximation
is:

>I verified that the weighted sum of these remains essentially
>constant as one moves one's Gaussian window along the interval
>axis -- and renormalize these to define one's probability
>function, and get essentially the same results as using the
>mediant-defined partition. In the triadic case, we don't know
>the partition sizes but if we can verify that the weighted sum
>of the inverse geometric means remains constant as one moves
>one's Gaussian window around the triadic plane, we can probably
>live with that approximate solution and never have to worry about
>the "real" one.

Now, I understand what you're doing, but I should note that I
don't know what weighted sums are, or renormalization. I
assume these have to do with the fact that n*d widths may
overlap or not cover the log-freq. line in places.

Nor do I know exactly how the 2-widths approximation was
used to calculate h.e... We still draw a bell curve, and
then around each ratio under the curve we center a region
proportional to 1/sqrt(n*d), and then get p's as with
mediants, with the addition of something to get the p's to
sum to 1?

-Carl

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> multiplied by the height of the point n/d on the relevant bell
> curve -- in order to approximate the integral under the bell curve.

I was so busy thinking about the (n*d)-ness of the widths, I
didn't remember to connect that they would be the bases of the
rectangles, and so "height" was just throwing me off, if a view
into my incoherent thought process means anything. I _do_ remember
you showing me this approximation on my first trip to Boston!
(I take it cubes can be used to approximate intergrals under an
umbrella.)

> Since the total probability should be 1 no matter where you put
> the bell curve, it's a sign that no significant distortion is
> taking place if the "weighted sum" is essentially constant.

Right. That's the one thing I had from the beginning.

> The "renormalization" just refers to scaling each of the products
> of "width" times "height" by a constant so that they sum to 1, and
> can be intepreted as probabilities (p) in the formula for entropy
> (sum(p*log(p))).
>
> Got it?

I could show a friend.

Thank you.

-Carl

--- In harmonic_entropy@y..., carl@l... wrote:

> > Got it?
>
> I could show a friend.
>
> Thank you.

Cool.

Now for the triadic case, a triangular plot of
harmonic series triads replaces the line of harmonic
series dyads (i.e., Farey series or Tenney series).
See the "GEOMETRY OF TRIANGULAR
PLOTS" posts archived in this forum, and the
associated graphics in the Files folder.

> Now for the triadic case, a triangular plot of
> harmonic series triads replaces the line of harmonic
> series dyads (i.e., Farey series or Tenney series).
> See the "GEOMETRY OF TRIANGULAR
> PLOTS" posts archived in this forum, and the
> associated graphics in the Files folder.

Looks like message #93 (and corresponding correction
in message #102). Ah, the wonders of the 30-60-90
triangle. Great work! I know it's elementary stuff,
but it always impresses me anyway.

So:

() Why use a triangular plot? Sure, it's nice having
the "third" interval of the triad built-in, but is there
another reason?

() How does the triangular plot affect the integral
approximation? Not at all? If we view the 1/cbrt(abc)
value as representing a circular area centered on the
triad. . .

() The Chalmers-style pics in the files section look
hexagonal (?), and have a shocking symmetry!

() I admit that I didn't follow the application of
Chalmers' 'triangle' to the i1/i2/i3 coordinate plane...
Is there a point? Why not just make your plots on
the plane?

>There's one more subtlety I want to bring up . . . later . . .

What was that?

-Carl

--- In harmonic_entropy@y..., carl@l... wrote:
> > Now for the triadic case, a triangular plot of
> > harmonic series triads replaces the line of harmonic
> > series dyads (i.e., Farey series or Tenney series).
> > See the "GEOMETRY OF TRIANGULAR
> > PLOTS" posts archived in this forum, and the
> > associated graphics in the Files folder.
>
> Looks like message #93 (and corresponding correction
> in message #102). Ah, the wonders of the 30-60-90
> triangle. Great work! I know it's elementary stuff,
> but it always impresses me anyway.
>
> So:
>
> () Why use a triangular plot? Sure, it's nice having
> the "third" interval of the triad built-in, but is there
> another reason?

All three intervals in the triad are treated symmetrically. There's no other Euclidean way to do
that. Perhaps there are other messages in the archives on this subject that you need to read?
>
> () How does the triangular plot affect the integral
> approximation? Not at all? If we view the 1/cbrt(abc)
> value as representing a circular area centered on the
> triad. . .

It can't represent a circular area -- finite circles don't fit together without gaps. It represents some
weird polygonal shape, or perhaps none at all -- as long as the validation exercise works out,
we're in good shape.

The bell curve, though, _must_ be circularly symmetrical in this triangular lattice, not in a square
(a:b vs. b:c or whatever) lattice, since the uncertainty s must apply equally to all three intervals.
>
> () The Chalmers-style pics in the files section look
> hexagonal (?), and have a shocking symmetry!

They're hexagonal for the same reason that the Farey or Tenney series in the 1-d case is a full
line, and not just a ray extending in one direction from the origin. In that case, we allow the interval
to be positive or negative. In the 2D case, we allow _any_ of the three intervals to be positive
or negative, causing _six_ copies of the basic triangular plot to occur, forming a hexagon.
>
> () I admit that I didn't follow the application of
> Chalmers' 'triangle' to the i1/i2/i3 coordinate plane...
> Is there a point? Why not just make your plots on
> the plane?

See above -- let me know if you still have any questions.
>
> >There's one more subtlety I want to bring up . . . later . . .
>
> What was that?

I forgot. What was the context in which I wrote that?

>> () How does the triangular plot affect the integral
>> approximation? Not at all? If we view the 1/cbrt(abc)
>> value as representing a circular area centered on the
>> triad. . .
>
>/.../ It represents some weird polygonal shape,
>or perhaps none at all -- as long as the validation exercise
>works out, we're in good shape.

Right.

> The bell curve, though, _must_ be circularly symmetrical in this
> triangular lattice, not in a square (a:b vs. b:c or whatever)
> lattice, since the uncertainty s must apply equally to all three
> intervals.

Okay. That's the key point I was missing.

>> () The Chalmers-style pics in the files section look
>> hexagonal (?), and have a shocking symmetry!
>
> They're hexagonal for the same reason that the Farey or Tenney
> series in the 1-d case is a full line, and not just a ray extending
> in one direction from the origin. In that case, we allow the
> interval to be positive or negative.

Hmm. Why do we allow that?

> In the 2D case, we allow _any_ of the three intervals to be
> positive or negative, causing _six_ copies of the basic triangular
> plot to occur, forming a hexagon.

So they are exact copies then...

> > >There's one more subtlety I want to bring up . . . later . . .
> >
> > What was that?
>
> I forgot. What was the context in which I wrote that?

At the end of message #93, which I gather was a response to
Jon Wild (I think the thread started over on TBL), after showing
how to get (x,y) from (x',y') and how to turn i1,i2,i3 into
a Chalmers-style triangle plot.

-Carl

--- In harmonic_entropy@y..., carl@l... wrote:

> >
> > They're hexagonal for the same reason that the Farey or Tenney
> > series in the 1-d case is a full line, and not just a ray extending
> > in one direction from the origin. In that case, we allow the
> > interval to be positive or negative.
>
> Hmm. Why do we allow that?

So that the bell curve won't hit a wall for small
intervals (those near 1/1). There will be other
(negative) intervals on the other side of the origin.
Negative intervals just mean that the perceptual
error makes you wrong about which note is higher
and which is lower.
>
> > In the 2D case, we allow _any_ of the three intervals to be
> > positive or negative, causing _six_ copies of the basic triangular
> > plot to occur, forming a hexagon.
>
> So they are exact copies then...

Yes -- each is a mirror-image of its neighbors.

--- In harmonic_entropy@y..., carl@l... wrote:

>
> > > >There's one more subtlety I want to bring up . . . later . . .
> > >
> > > What was that?
> >
> > I forgot. What was the context in which I wrote that?
>
> At the end of message #93,

I posted the subtlety in message #97.

Aha!

>>> They're hexagonal for the same reason that the Farey or Tenney
>>> series in the 1-d case is a full line, and not just a ray
>>> extending in one direction from the origin. In that case, we
>>> allow the interval to be positive or negative.
>>
>> Hmm. Why do we allow that?
>
> So that the bell curve won't hit a wall for small
> intervals (those near 1/1). There will be other
> (negative) intervals on the other side of the origin.
> Negative intervals just mean that the perceptual
> error makes you wrong about which note is higher
> and which is lower.
>
>>> In the 2D case, we allow _any_ of the three intervals to be
>>> positive or negative, causing _six_ copies of the basic
>>> triangular plot to occur, forming a hexagon.
>>
>> So they are exact copies then...
>
> Yes -- each is a mirror-image of its neighbors.

Thanks!

> I posted the subtlety in message #97.