Yahoo Tuning Groups Ultimate Backup tuning Sethares equation

--- In tuning@yahoogroups.com, "victorcerullo" <moog@l...> wrote:
> I am wondering if it is possible to use a linear transform of
> Plomp-Levelt-Sethares's equation in order to build a dissonance
> metric to be used in conjunction with a SMACOF algorithm for a
> 2-D or 3-D representation of a scale in the euclidean space.
> Anyone working on that currently?
>
> Kind Regards,
> Victor Cerullo
> /16tone

I'd like very very much to help you, but what is SMACOF? If it's
anything like multidimensional scaling, I've already done this,
though I tend to use harmonic entropy instead of Plomp-Levelt-
Sethares's equation. But probably I'm missing something, as I don't
see why you'd need a linear transform.

Also, if you're going to get mathematical, it might be best to reply
to the tuning-math list; some are annoyed when the content here gets
overly mathematical.

Excited,
Paul

🔗Aaron K. Johnson <akjmicro@comcast.net>

🔗jjensen142000 <jjensen14@hotmail.com>

Hi Victor

I don't know if it will be of any use to you, but I
programmed up the Plomp-Levelt-Sethares's equation last year,
and the Java applet and source code are available at:

http://home.austin.rr.com/jmjensen/DissonanceCurve.html

--Jeff

🔗victorcerullo <moog@libero.it>

Hi Jeff,
thanks a lot for the link to your very informative page and related
Java applet. I am wondering if we can modify that equation in
order to build a metric where:

d(x,z) <= d(x,y) + d(y,z)

so that we can use the SMACOF algorithm to draw a scale in the
euclidean space.

Kind Regards,
Victor

> Hi Victor
>
> I don't know if it will be of any use to you, but I
> programmed up the Plomp-Levelt-Sethares's equation last
year,
> and the Java applet and source code are available at:
>
> http://home.austin.rr.com/jmjensen/DissonanceCurve.html
>
> --Jeff

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@yahoogroups.com, "victorcerullo" <moog@l...> wrote:
> > I am wondering if it is possible to use a linear transform of
> > Plomp-Levelt-Sethares's equation in order to build a dissonance
> > metric to be used in conjunction with a SMACOF algorithm for a
> > 2-D or 3-D representation of a scale in the euclidean space.
> > Anyone working on that currently?
> >
> > Kind Regards,
> > Victor Cerullo
> > /16tone
>
> Hi Victor
>
> I don't know if it will be of any use to you, but I
> programmed up the Plomp-Levelt-Sethares's equation last year,
> and the Java applet and source code are available at:
>
> http://home.austin.rr.com/jmjensen/DissonanceCurve.html
>
> --Jeff

Jeff, I'm afraid this site shows a confusion between *pitch-ratios*
and *interval-ratios* in the context of scale construction. The
diatonic scale (especially in tempered form, such as meantone) is
*maximally consonant* from many points of view, but only when
intervals are properly considered as intervals -- and there are 7*6/2
= 21 intervals to consider in a 7-tone scale.

This site also doesn't mention the scaling problem with Sethares's
equation, though that's a separate issue.

Bye for now,
Paul

🔗Paul Erlich <paul@stretch-music.com>

Victor,

Sorry to intervene, but I wonder if you saw my response? (Maybe you
answered on tuning-math, I'll check there . . .) I've drawn many
scales in Euclidean space using multidimensional scaling on a
dissonance function applied to the interval matrix of the scale. I
posted some of the results to this list back when it was on the Mills
server, so they should be in Robert Walker's archives. With
multidimensional scaling, any monotonic transform of the dissonance
function, I think, will lead to a meaningful configuration of the
scale in Euclidean space. So one doesn't need to worry about the
condition for the metric being satisfied -- the dimension reduction
implies some fitting anyway, and of course Euclidean space will end
up satisfying the metric anyway. Let me know if I'm misunderstanding
what you're trying to do, if you'd like to see some results, or if I
should just back off . . .

-Paul

--- In tuning@yahoogroups.com, "victorcerullo" <moog@l...> wrote:
> Hi Jeff,
> thanks a lot for the link to your very informative page and related
> Java applet. I am wondering if we can modify that equation in
> order to build a metric where:
>
> d(x,z) <= d(x,y) + d(y,z)
>
> so that we can use the SMACOF algorithm to draw a scale in the
> euclidean space.
>
> Kind Regards,
> Victor
>
> > Hi Victor
> >
> > I don't know if it will be of any use to you, but I
> > programmed up the Plomp-Levelt-Sethares's equation last
> year,
> > and the Java applet and source code are available at:
> >
> > http://home.austin.rr.com/jmjensen/DissonanceCurve.html
> >
> > --Jeff

🔗victorcerullo <moog@libero.it>

Paul,

thanks a lot for your two messages, and sorry for the late reply.
SMACOF is actually a multidimensional scaling algorithm. I am
trying to use Sethares's equation in conjunction with
multidimensional scaling, with unsatisfying results at the
moment.

>With multidimensional scaling, any monotonic transform
>of the dissonance function, I think, will lead to a meaningful
>configuration of the scale in Euclidean space.

That's exactly what I am looking for: a meaningful monotonic
transform of that equation suitable for feeding a
multidimensional scaling algorithm. I will check Robert Walker's
archives.

>So one doesn't need to worry about the condition for the metric
>being satisfied -- the dimension reduction implies some fitting
>anyway, and of course Euclidean space will end up satisfying
>the metric anyway. Let me know if I'm misunderstanding what
>you're trying to do, if you'd like to see some results, or if I
>should just back off . . .

No misunderstanding at all: you are definitely on target.

Regards,
Victor

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "victorcerullo" <moog@l...> wrote:
> Paul,
>
> thanks a lot for your two messages, and sorry for the late reply.
> SMACOF is actually a multidimensional scaling algorithm. I am
> trying to use Sethares's equation in conjunction with
> multidimensional scaling, with unsatisfying results at the
> moment.

OK, just to be sure we're on the same page, here's what i've done.
Let's take the example of the diatonic scale. There are 7*6/2 = 21
intervals to consider. So you plug in the dissonances of the 21
intervals in the appropriate places in the 7-dimensional distance
matrix, and you use MDS to scale that down to 2 or 3 dimensions.
These, and similar results for some 10- and 12-note scales (and up to
4 dimensions), are what i posted. They show the structure of the
scales most wonderfully. I could easily do more.

Here's the catch. "Intervals" above should really be "interval
classes" since the scale itself normally repeats over many octaves.
You don't want to consider such a huge matrix of intervals, so you
use an octave-equivalent dissonance measure to start with. If you
wish to use a curve of the Plomp-Levelt-Sethares family, the
simplest "cheat" would be to simply average the curve from 0 to 1
octave with the "mirror" curve going from 1 octave to 0. This would
build octave-equivalence into your dissonance measure, as required.
However, I've avoided Sethares's curves because they have a scaling
problem, and I've used octave-equivalent harmonic entropy for this
purpose instead.

> >So one doesn't need to worry about the condition for the metric
> >being satisfied -- the dimension reduction implies some fitting
> >anyway, and of course Euclidean space will end up satisfying
> >the metric anyway. Let me know if I'm misunderstanding what
> >you're trying to do, if you'd like to see some results, or if I
> >should just back off . . .
>
> No misunderstanding at all: you are definitely on target.

The above should have answered your other question as well, I hope.

🔗jjensen142000 <jjensen14@hotmail.com>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> > http://home.austin.rr.com/jmjensen/DissonanceCurve.html
> >
> > --Jeff
>
> Jeff, I'm afraid this site shows a confusion between *pitch-ratios*
> and *interval-ratios* in the context of scale construction. The
> diatonic scale (especially in tempered form, such as meantone) is
> *maximally consonant* from many points of view, but only when
> intervals are properly considered as intervals -- and there are
7*6/2
> = 21 intervals to consider in a 7-tone scale.
>
> This site also doesn't mention the scaling problem with Sethares's
> equation, though that's a separate issue.
>
> Bye for now,
> Paul

Hi, Paul. Nice to talk to you again.

Could you elaborate on what you mean with regards to pitch vs interval
ratios? It seems to me what I did was only to identify which
pitches where the most consonant with middle C, namely F=4/3 and
G=3/2 and then complete the scale by building major chords on F & G.
I seem to remember talking about summing over all pairwise intervals,
but that doesn't seem to really be necessary, although it might
be interesting.

I say not necessary because I'm thinking that it is not having
many maximally consonant intervals that causes this scale to support
a lot of music that I like (although I bet it is a big factor),
but I think the central point is the enabling of the G7 --> C
cadence.

As far as scaling in Sethare's equation, are you referring to
the fact that the dissonance "bump function" of 2 pure tones
changes shape depending on the absolute pitch of the tones?

Anyway, I am working on Parncutt's "Harmony: A PsychoAcoustical
Approach" these days and I *may* have some more applets rolling
out, like calculating virtual pitch, etc...

--Jeff

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> > > http://home.austin.rr.com/jmjensen/DissonanceCurve.html
> > >
> > > --Jeff
> >
> > Jeff, I'm afraid this site shows a confusion between *pitch-
ratios*
> > and *interval-ratios* in the context of scale construction. The
> > diatonic scale (especially in tempered form, such as meantone) is
> > *maximally consonant* from many points of view, but only when
> > intervals are properly considered as intervals -- and there are
> 7*6/2
> > = 21 intervals to consider in a 7-tone scale.
> >
> > This site also doesn't mention the scaling problem with
Sethares's
> > equation, though that's a separate issue.
> >
> > Bye for now,
> > Paul
>
> Hi, Paul. Nice to talk to you again.
>
> Could you elaborate on what you mean with regards to pitch vs
interval
> ratios? It seems to me what I did was only to identify which
> pitches where the most consonant with middle C, namely F=4/3 and
> G=3/2 and then complete the scale by building major chords on F & G.

Well, that's another issue -- constructing the diatonic scale -- that
I don't want to get into in this thread (though we can start another
one . . .)

> I seem to remember talking about summing over all pairwise
intervals,
> but that doesn't seem to really be necessary, although it might
> be interesting.
>
> I say not necessary because I'm thinking that it is not having
> many maximally consonant intervals that causes this scale to support
> a lot of music that I like (although I bet it is a big factor),
> but I think the central point is the enabling of the G7 --> C
> cadence.

Yes, the dissonant interval that enables this is B-F, but I don't see
any tritones mentioned on your site. Maybe I'm somehow reading your
site wrong, but it appears that you make at least some confused
statements:

"How do we justify the G chord being part of the scale? G is
excellent with C alone, but B and D are terrible, they do not even
appear as local minima; 3 octaves up at 2112Hz, we get a deep but
wide minima near D and a bit higher than B-flat ( but never B!).
Perhaps this very weakness can be construed as a strength, though.
The notes of the G chord serve two functions:

G, B and D fill in large interval gaps in (C-E-F-A-C)
B has the leading tone property to C ( cf. [Helmholtz 1877 p.285 ]) "

When B and D are sounded harmonically in Western major-scale music,
they are *not* sounded against C. So their Plomp-Levelt-Sethares
dissonance with C is completely irrelevant, since Plomp-Levelt-
Sethares dissonance concerns the behavior of *simultaneously sounding
tones*, not the behavior of tones vs. a keynote, or anything else.

It seemed to me your page was treating the notes of the major scale
as if they were to be sounded against C and against C only, never
against one another. This may be what happens in some Eastern and
isolated Western styles, but not in the 'mainstream' of Western music
from the Renaissance through Romantic periods.

> As far as scaling in Sethare's equation, are you referring to
> the fact that the dissonance "bump function" of 2 pure tones
> changes shape depending on the absolute pitch of the tones?

No, but that's another point where harmonic entropy ends up being
more convenient.

🔗victorcerullo <moog@libero.it>

Paul,

> So you plug in the dissonances of the 21
> intervals in the appropriate places in the
> 7-dimensional distance matrix, and you use
> MDS to scale that down to 2 or 3 dimensions.

May I ask you what monotonic transform of the dissonance
function did you use in the case of Sethares's equation?

> If you wish to use a curve of the
> Plomp-Levelt-Sethares family, the
> simplest "cheat" would be to simply average
> the curve from 0 to 1 octave with the "mirror"
> curve going from 1 octave to 0. This would
> build octave-equivalence into your dissonance
> measure, as required. However, I've avoided
> Sethares's curves because they have a scaling
> problem, and I've used octave-equivalent harmonic
> entropy for this purpose instead.

I think one point in favour of the Plomp-Levelt approach is that it
could be used with an MDS algorithm to represent a scale in the
euclidean space even in some cases where other methods
don't make sense (i.e. the Euler-Barlow prime numbers
approach in the case of a 12-tet scale). I still have to consider the
harmonic entropy method, though.

Regards,
Victor

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "victorcerullo" <moog@l...> wrote:
> Paul,
>
> > So you plug in the dissonances of the 21
> > intervals in the appropriate places in the
> > 7-dimensional distance matrix, and you use
> > MDS to scale that down to 2 or 3 dimensions.
>
> May I ask you what monotonic transform of the dissonance
> function did you use in the case of Sethares's equation?

I didn't even try Sethares's equation, because of the scaling
problem, and the large number of parameters (such the amplitudes of
all the tones' partials). However, it *should* work fine, especially
if "octave-equivalencized" as below, even without any transform
applied, since the MDS result will automatically satisfy the triangle
inequality even if the original data doesn't.

> > If you wish to use a curve of the
> > Plomp-Levelt-Sethares family, the
> > simplest "cheat" would be to simply average
> > the curve from 0 to 1 octave with the "mirror"
> > curve going from 1 octave to 0. This would
> > build octave-equivalence into your dissonance
> > measure, as required.

Did you try it?

> I think one point in favour of the Plomp-Levelt approach is that it
> could be used with an MDS algorithm to represent a scale in the
> euclidean space even in some cases where other methods
> don't make sense (i.e. the Euler-Barlow prime numbers
> approach in the case of a 12-tet scale).

I would never use the Euler or Barlow approaches under any
circumstances, 12-equal or otherwise. Among other severe problems,
which affect even the simplest JI constructs, they are discontinuous
(indeed, *everywhere* discontinuous, no?), as a function of interval
size, so clearly they fail miserably to capture any psychophysical
reality. Tenney's harmonic distance is far preferable, anyhow, within
this class of 'rational-only' (and everywhere discontinuous) distance
measures. Harmonic entropy roughly agrees with Tenney for the
simplest ratios, but connects these points with smooth curves instead
of infinite numbers of spikes and jumps.

> I still have to consider the
> harmonic entropy method, though.

Clearly, this approach gives terrific results at least for 12-equal,
for diatonic scales, and for decatonic scales.

🔗Paul Erlich <paul@stretch-music.com>

Hi Victor,

I've uploaded two plots of 3D multidimensional scaling results (using
a particular octave-equivalent harmonic entropy curve for distance --
the only transformation i needed was to subtract the minimum; i.e., a
constant shift so that 0 cents represents 0 distance) here:

/tuning/files/Erlich/

The diatonic scale, if you can visualize the 3D graph well enough (i
tried to make it clear but didn't do a great job), has its single
chain of fifths, F-C-G-D-A-E-B, twisted into a helix.

The 12-equal scale shows up as a perfect circle of fifths when
projected onto the lower plane (which is in fact the 2D
multidimensional scaling solution), while the third dimension tries
to bend the circle. 3D is particularly bad for 12-equal, in fact
technically should not be used for that case at all, since the fourth
eigenvalue of the {configuration matrix times its transpose} is equal
to the third. One must go to 4D to reap the benefits of the 'bending'
of the circle of fifths.

Best,
Paul

🔗victorcerullo <moog@libero.it>

Very interesting diagrams, Paul. Thanks. I will look for more
information about harmonic entropy in the next few days.

Kind Regards,
Victor

> Hi Victor,
>
> I've uploaded two plots of 3D multidimensional scaling results
(using
> a particular octave-equivalent harmonic entropy curve for
distance --
> the only transformation i needed was to subtract the minimum;
i.e., a
> constant shift so that 0 cents represents 0 distance) here:
>
> /tuning/files/Erlich/
>
> The diatonic scale, if you can visualize the 3D graph well
enough (i
> tried to make it clear but didn't do a great job), has its single
> chain of fifths, F-C-G-D-A-E-B, twisted into a helix.
>
> The 12-equal scale shows up as a perfect circle of fifths when
> projected onto the lower plane (which is in fact the 2D
> multidimensional scaling solution), while the third dimension
tries
> to bend the circle. 3D is particularly bad for 12-equal, in fact
> technically should not be used for that case at all, since the
fourth
> eigenvalue of the {configuration matrix times its transpose} is
equal
> to the third. One must go to 4D to reap the benefits of the
'bending'
> of the circle of fifths.
>
> Best,
> Paul

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48590.html#48647

> Hi Victor,
>
> I've uploaded two plots of 3D multidimensional scaling results
(using
> a particular octave-equivalent harmonic entropy curve for distance -
-
> the only transformation i needed was to subtract the minimum; i.e.,
a
> constant shift so that 0 cents represents 0 distance) here:
>
> /tuning/files/Erlich/
>
> The diatonic scale, if you can visualize the 3D graph well enough
(i
> tried to make it clear but didn't do a great job), has its single
> chain of fifths, F-C-G-D-A-E-B, twisted into a helix.
>
> The 12-equal scale shows up as a perfect circle of fifths when
> projected onto the lower plane (which is in fact the 2D
> multidimensional scaling solution), while the third dimension tries
> to bend the circle. 3D is particularly bad for 12-equal, in fact
> technically should not be used for that case at all, since the
fourth
> eigenvalue of the {configuration matrix times its transpose} is
equal
> to the third. One must go to 4D to reap the benefits of
the 'bending'
> of the circle of fifths.
>
> Best,
> Paul

***These are really beautiful graphics, but I'm not understanding
them... any suggestions??? Thanks!

Joseph

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_48590.html#48647
>
> > Hi Victor,
> >
> > I've uploaded two plots of 3D multidimensional scaling results
> (using
> > a particular octave-equivalent harmonic entropy curve for
distance -
> -
> > the only transformation i needed was to subtract the minimum;
i.e.,
> a
> > constant shift so that 0 cents represents 0 distance) here:
> >
> > /tuning/files/Erlich/
> >
> > The diatonic scale, if you can visualize the 3D graph well enough
> (i
> > tried to make it clear but didn't do a great job), has its single
> > chain of fifths, F-C-G-D-A-E-B, twisted into a helix.
> >
> > The 12-equal scale shows up as a perfect circle of fifths when
> > projected onto the lower plane (which is in fact the 2D
> > multidimensional scaling solution), while the third dimension
tries
> > to bend the circle. 3D is particularly bad for 12-equal, in fact
> > technically should not be used for that case at all, since the
> fourth
> > eigenvalue of the {configuration matrix times its transpose} is
> equal
> > to the third. One must go to 4D to reap the benefits of
> the 'bending'
> > of the circle of fifths.
> >
> > Best,
> > Paul
>
>
> ***These are really beautiful graphics, but I'm not understanding
> them... any suggestions??? Thanks!
>
> Joseph

I'm afraid they're not the best, but let's start with the 12-equal
one -- it's sort of more possible to see what's going on there. First
of all, can you see how each note has 'projections' onto each of the
three walls (or two walls and floor), helping suggest the note's
position in three-dimensional space? If you can visualize the notes
(the BIG letters) this way, we're most of the way there.

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48590.html#48812

> I'm afraid they're not the best, but let's start with the 12-equal
> one -- it's sort of more possible to see what's going on there.
First
> of all, can you see how each note has 'projections' onto each of
the
> three walls (or two walls and floor), helping suggest the note's
> position in three-dimensional space? If you can visualize the notes
> (the BIG letters) this way, we're most of the way there.

***Well, that much I can see, but the question is why the note is
being projected in three-dimensional space in the first place... I
notice that on the "floor" a *circle* is produced... Looks purdy,
though, rather like the pipes on the Pompidou Centre...

http://www.cnac-gp.fr/Pompidou/Accueil.nsf/tunnel?OpenForm

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_48590.html#48812
>
> > I'm afraid they're not the best, but let's start with the 12-
equal
> > one -- it's sort of more possible to see what's going on there.
> First
> > of all, can you see how each note has 'projections' onto each of
> the
> > three walls (or two walls and floor), helping suggest the note's
> > position in three-dimensional space? If you can visualize the
notes
> > (the BIG letters) this way, we're most of the way there.
>
>
> ***Well, that much I can see, but the question is why the note is
> being projected in three-dimensional space in the first place... I
> notice that on the "floor" a *circle* is produced... Looks purdy,
> though, rather like the pipes on the Pompidou Centre...
>
> http://www.cnac-gp.fr/Pompidou/Accueil.nsf/tunnel?OpenForm
>
> :)
>
> JP

OK, great. Now the idea of multidimensional scaling is to take a set
of objects (in this case, pitch-classes in a scale),
some 'dissimilarity' function which you've calculated for each *pair*
of objects, and find a configuration for the objects in space where
the distance between each pair of objects is as close as possible a
match to the original dissimilarity you put in. In other words, the
configuration of the objects should immediately *visually* suggest to
the viewer much of the information that would come from studying the
n*(n-1)/2 'dissimilarities'. In our case, the latter represent
diadic 'dissonances', and I've used octave-equivalent harmonic
entropy to quantify the dissonance between pitch-classes here.

For the each note in 12-equal, it's pretty clear that the two notes
least dissonant with it are the fifth above and the fifth below.
Hence it's not surprising that the two-dimensional multidimensional
scaling solution arranges the 12 notes into a perfect circle of
fifths, where the notes least distant from each note lie a fifth
below and a fifth above. Are you with me so far?

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48590.html#48869

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> >
> > /tuning/topicId_48590.html#48812
> >
> > > I'm afraid they're not the best, but let's start with the 12-
> equal
> > > one -- it's sort of more possible to see what's going on there.
> > First
> > > of all, can you see how each note has 'projections' onto each
of
> > the
> > > three walls (or two walls and floor), helping suggest the
note's
> > > position in three-dimensional space? If you can visualize the
> notes
> > > (the BIG letters) this way, we're most of the way there.
> >
> >
> > ***Well, that much I can see, but the question is why the note is
> > being projected in three-dimensional space in the first place...
I
> > notice that on the "floor" a *circle* is produced... Looks
purdy,
> > though, rather like the pipes on the Pompidou Centre...
> >
> > http://www.cnac-gp.fr/Pompidou/Accueil.nsf/tunnel?OpenForm
> >
> > :)
> >
> > JP
>
> OK, great. Now the idea of multidimensional scaling is to take a
set
> of objects (in this case, pitch-classes in a scale),
> some 'dissimilarity' function which you've calculated for each
*pair*
> of objects, and find a configuration for the objects in space where
> the distance between each pair of objects is as close as possible a
> match to the original dissimilarity you put in. In other words, the
> configuration of the objects should immediately *visually* suggest
to
> the viewer much of the information that would come from studying
the
> n*(n-1)/2 'dissimilarities'. In our case, the latter represent
> diadic 'dissonances', and I've used octave-equivalent harmonic
> entropy to quantify the dissonance between pitch-classes here.
>
> For the each note in 12-equal, it's pretty clear that the two notes
> least dissonant with it are the fifth above and the fifth below.
> Hence it's not surprising that the two-dimensional multidimensional
> scaling solution arranges the 12 notes into a perfect circle of
> fifths, where the notes least distant from each note lie a fifth
> below and a fifth above. Are you with me so far?

***I guess I can kinda see how, if the dissonance level is low in
both dimensions, the fifths would be close to one another, but I'm
not quite sure why they're marching around in a circle like this...

🔗Kurt Bigler <kkb@breathsense.com>

on 12/2/03 8:57 PM, Joseph Pehrson <jpehrson@rcn.com> wrote:

> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_48590.html#48869
>
>> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
> wrote:
>>> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>>>
>>> /tuning/topicId_48590.html#48812
>>>
>>>> I'm afraid they're not the best, but let's start with the 12-
>> equal
>>>> one -- it's sort of more possible to see what's going on there.
>>> First
>>>> of all, can you see how each note has 'projections' onto each
> of
>>> the
>>>> three walls (or two walls and floor), helping suggest the
> note's
>>>> position in three-dimensional space? If you can visualize the
>> notes
>>>> (the BIG letters) this way, we're most of the way there.
>>>
>>>
>>> ***Well, that much I can see, but the question is why the note is
>>> being projected in three-dimensional space in the first place...
> I
>>> notice that on the "floor" a *circle* is produced... Looks
> purdy,
>>> though, rather like the pipes on the Pompidou Centre...
>>>
>>> http://www.cnac-gp.fr/Pompidou/Accueil.nsf/tunnel?OpenForm
>>>
>>> :)
>>>
>>> JP
>>
>> OK, great. Now the idea of multidimensional scaling is to take a
> set
>> of objects (in this case, pitch-classes in a scale),
>> some 'dissimilarity' function which you've calculated for each
> *pair*
>> of objects, and find a configuration for the objects in space where
>> the distance between each pair of objects is as close as possible a
>> match to the original dissimilarity you put in. In other words, the
>> configuration of the objects should immediately *visually* suggest
> to
>> the viewer much of the information that would come from studying
> the
>> n*(n-1)/2 'dissimilarities'. In our case, the latter represent
>> diadic 'dissonances', and I've used octave-equivalent harmonic
>> entropy to quantify the dissonance between pitch-classes here.
>>
>> For the each note in 12-equal, it's pretty clear that the two notes
>> least dissonant with it are the fifth above and the fifth below.
>> Hence it's not surprising that the two-dimensional multidimensional
>> scaling solution arranges the 12 notes into a perfect circle of
>> fifths, where the notes least distant from each note lie a fifth
>> below and a fifth above. Are you with me so far?
>
>
> ***I guess I can kinda see how, if the dissonance level is low in
> both dimensions, the fifths would be close to one another, but I'm
> not quite sure why they're marching around in a circle like this...
>
> JP

Because that shape happenned to make the distances you see between the
points match the dissimilarity rating. You want it to look like you can see
dissimilarity by how far apart they are. Or maybe it is easier to say that
you want to be able to see similarity by how close things are. You want to
be able to read similarity right off the picture in an obvious way.

The actual arrangement is just a "trick" - it is sort of arbitrary - the
computer just needs to find a shape that happens to create the right
distance between the individual objects. A strange analogy: suppose you
decide you want to be near your friends and you want to design a
neighborhood arrangement that accomplishes that everyone is near their
friends and farther from the people they don't like. The arrangement is
secondary - the shape doesn't matter. It is the distances that are
important.

-Kurt

🔗Paul Erlich <paul@stretch-music.com>

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

/tuning/topicId_48590.html#48976

> Because that shape happenned to make the distances you see between
the
> points match the dissimilarity rating. You want it to look like
you can see
> dissimilarity by how far apart they are. Or maybe it is easier to
say that
> you want to be able to see similarity by how close things are. You
want to
> be able to read similarity right off the picture in an obvious way.
>
> The actual arrangement is just a "trick" - it is sort of arbitrary -
the
> computer just needs to find a shape that happens to create the right
> distance between the individual objects. A strange analogy:
suppose you
> decide you want to be near your friends and you want to design a
> neighborhood arrangement that accomplishes that everyone is near
their
> friends and farther from the people they don't like. The
arrangement is
> secondary - the shape doesn't matter. It is the distances that are
> important.
>
> -Kurt

***Thanks, Kurt, for the help. What confuses me, though, is the fact
that it looks like the notes have both an x and a y axis so they
would have to be in a very particular place rather than
an "arbitrary" one... I must not be "getting" the underlying setup of
it.

Thanks!

Joseph

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> /tuning/topicId_48590.html#48976
>
> > Because that shape happenned to make the distances you see
between
> the
> > points match the dissimilarity rating. You want it to look like
> you can see
> > dissimilarity by how far apart they are. Or maybe it is easier
to
> say that
> > you want to be able to see similarity by how close things are.
You
> want to
> > be able to read similarity right off the picture in an obvious
way.
> >
> > The actual arrangement is just a "trick" - it is sort of
arbitrary -
> the
> > computer just needs to find a shape that happens to create the
right
> > distance between the individual objects. A strange analogy:
> suppose you
> > decide you want to be near your friends and you want to design a
> > neighborhood arrangement that accomplishes that everyone is near
> their
> > friends and farther from the people they don't like. The
> arrangement is
> > secondary - the shape doesn't matter. It is the distances that
are
> > important.
> >
> > -Kurt
>
>
> ***Thanks, Kurt, for the help. What confuses me, though, is the
fact
> that it looks like the notes have both an x and a y axis so they
> would have to be in a very particular place rather than
> an "arbitrary" one... I must not be "getting" the underlying setup
of
> it.

The notes are all in a particular place in 3D; they each have a
particular position along the x, y, *and* z axes. The solution is not
arbitrary in that it "best" matches the assumed 'dissimilarities'
under some specific mathematical assumptions about what is "best" --
you may search the internet for "classical multidimensional scaling"
for more details.

🔗Kurt Bigler <kkb@breathsense.com>

on 12/8/03 7:55 AM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
>> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>>
>> /tuning/topicId_48590.html#48976
>>
>>> Because that shape happenned to make the distances you see
> between
>> the
>>> points match the dissimilarity rating. You want it to look like
>> you can see
>>> dissimilarity by how far apart they are. Or maybe it is easier
> to
>> say that
>>> you want to be able to see similarity by how close things are.
> You
>> want to
>>> be able to read similarity right off the picture in an obvious
> way.
>>>
>>> The actual arrangement is just a "trick" - it is sort of
> arbitrary -
>> the
>>> computer just needs to find a shape that happens to create the
> right
>>> distance between the individual objects. A strange analogy:
>> suppose you
>>> decide you want to be near your friends and you want to design a
>>> neighborhood arrangement that accomplishes that everyone is near
>> their
>>> friends and farther from the people they don't like. The
>> arrangement is
>>> secondary - the shape doesn't matter. It is the distances that
> are
>>> important.
>>>
>>> -Kurt
>>
>>
>> ***Thanks, Kurt, for the help. What confuses me, though, is the
> fact
>> that it looks like the notes have both an x and a y axis so they
>> would have to be in a very particular place rather than
>> an "arbitrary" one... I must not be "getting" the underlying setup
> of
>> it.
>
> The notes are all in a particular place in 3D; they each have a
> particular position along the x, y, *and* z axes. The solution is not
> arbitrary in that it "best" matches the assumed 'dissimilarities'
> under some specific mathematical assumptions about what is "best" --
> you may search the internet for "classical multidimensional scaling"
> for more details.

Besides the shape, I meant to say that the position and rotation of the
result was also arbitrary. This page seems to confirm that:

http://rweb.stat.umn.edu/R/library/mva/html/cmdscale.html

since there are no inputs that could be used to establish a position or
rotation of the results. I seem to remember from my grad school days that
people did MDS and *afterward* looked for meanings in the resulting axes,
possibly rotating if necessary.

The shape is of course *less* arbitrary, but only in the sense that it is an
optimized solution. There is no argument for any a priori existence of what
the resulting dimensions refer to, and in fact even the number of dimensions
is in question. As I recall people try MDS with a different number of
desired dimensions in a result and the result that gives a "good" error in
the least number of dimensions is usually considered most valuable. Thus if
the number of dimensions is not really known, how can the shape be
considered to be anything but arbitrary. Of course there will be cases
where the choice of dimensionality (3D in this case) is somewhat less than
arbitrary, and the shape will be intuitively meaningful, such as Carl's
response already indicated.

It sounds like you are saying something very different, and my apologies if
both my memories and the reference I just found are somehow misleading me.

-Kurt

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> > The notes are all in a particular place in 3D; they each have a
> > particular position along the x, y, *and* z axes. The solution is
not
> > arbitrary in that it "best" matches the assumed 'dissimilarities'
> > under some specific mathematical assumptions about what
is "best" --
> > you may search the internet for "classical multidimensional
scaling"
> > for more details.
>
> Besides the shape, I meant to say that the position and rotation of
the
> result was also arbitrary.

Yes, of course the position and rotation are arbitrary -- obviously
these don't affect the distances.

> The shape is of course *less* arbitrary, but only in the sense that
it is an
> optimized solution. There is no argument for any a priori
existence of what
> the resulting dimensions refer to, and in fact even the number of
dimensions
> is in question.

Right, but *given* the number of dimensions (three in this case), the
solution that you get is not arbitrary (except for rotation and
position, of course) -- it best fits the criteria of the procedure.

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48590.html#49311

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >
> > /tuning/topicId_48590.html#48976
> >
> > > Because that shape happenned to make the distances you see
> between
> > the
> > > points match the dissimilarity rating. You want it to look
like
> > you can see
> > > dissimilarity by how far apart they are. Or maybe it is easier
> to
> > say that
> > > you want to be able to see similarity by how close things are.
> You
> > want to
> > > be able to read similarity right off the picture in an obvious
> way.
> > >
> > > The actual arrangement is just a "trick" - it is sort of
> arbitrary -
> > the
> > > computer just needs to find a shape that happens to create the
> right
> > > distance between the individual objects. A strange analogy:
> > suppose you
> > > decide you want to be near your friends and you want to design a
> > > neighborhood arrangement that accomplishes that everyone is
near
> > their
> > > friends and farther from the people they don't like. The
> > arrangement is
> > > secondary - the shape doesn't matter. It is the distances that
> are
> > > important.
> > >
> > > -Kurt
> >
> >
> > ***Thanks, Kurt, for the help. What confuses me, though, is the
> fact
> > that it looks like the notes have both an x and a y axis so they
> > would have to be in a very particular place rather than
> > an "arbitrary" one... I must not be "getting" the underlying
setup
> of
> > it.
>
> The notes are all in a particular place in 3D; they each have a
> particular position along the x, y, *and* z axes. The solution is
not
> arbitrary in that it "best" matches the assumed 'dissimilarities'
> under some specific mathematical assumptions about what is "best" --

> you may search the internet for "classical multidimensional
scaling"
> for more details.

***Well, I just thought it looks pretty cool and seems pretty deep...
so I'll leave it at that... :)