back to harmonic entropy vs. noble mediants

Interesting how the list goes dead, and then a topic comes along that
gets 100 posts a day!

Anyway, I don't have heat or lightto shed on this subject, but a
question, and maybe it can spark new investigations:

Harmonic Entropy defenders--Why doesn't the noble mediant algorithm
produce results that precisely agree with HE? Is it justthe limit of
human hearing and the overtone series?

-AKJ.

i asked this same question

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Aaron Krister Johnson wrote:
>
> Interesting how the list goes dead, and then a topic comes along that
> gets 100 posts a day!
>
> Anyway, I don't have heat or lightto shed on this subject, but a
> question, and maybe it can spark new investigations:
>
> Harmonic Entropy defenders--Why doesn't the noble mediant algorithm
> produce results that precisely agree with HE? Is it justthe limit of
> human hearing and the overtone series?
>
> -AKJ.
>
>

here is a link BTW to Helmholtz chart on page 192
http://anaphoria.com/helmholtz192.gif
it should be scanned high enough to read

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Aaron Krister Johnson wrote:
>
> Interesting how the list goes dead, and then a topic comes along that
> gets 100 posts a day!
>
> Anyway, I don't have heat or lightto shed on this subject, but a
> question, and maybe it can spark new investigations:
>
> Harmonic Entropy defenders--Why doesn't the noble mediant algorithm
> produce results that precisely agree with HE? Is it justthe limit of
> human hearing and the overtone series?
>
> -AKJ.
>
>

> Aaron Krister Johnson wrote:
> > Anyway, I don't have heat or lightto shed on this subject, but a
> > question, and maybe it can spark new investigations:
> >
> > Harmonic Entropy defenders--Why doesn't the noble mediant algorithm
> > produce results that precisely agree with HE? Is it justthe limit of
> > human hearing and the overtone series?

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> i asked this same question

Sorry I missed it Kraig. Hi Aaron.

It's a fine question, and one doesn't have to be a defender of either,
to find it interesting, or to explain it, since it is a question about
the relation of two mathematical objects and not about the relation of
either of them to the real world.

I wish I knew the answer. The form of the question that I have been
giving much thought to is:

Since many HE local maxima are within a few cents of simple noble
intervals, why do simple nobles utterly fail to predict the HE maxima
nearest to the most consonant intervals?

Aaron, you suggest it may relate to the limit of human hearing and the
overtone series.

The former seems more likely to me. Both HE and noble mediants take
the overtone series into account in some sense, and both give more
weight to simpler ratios. But HE takes into account human pitch
discrimination, while noble mediants are far too simple for that.

However, it seems as though noble mediants might be more likely to
agree with HE for very _poor_ pitch discrimination rather than very
good discrimination.

Thanks Kraig, for the Helmholtz graph, and for emailing me the pages
that describe its derivation. It uses the same kind of calculations as
Sethares' "sensory dissonance" (SD) except that Sethares has the
benefit of Plomp and Levelt's measurements of the actual shape of the
critical band function, where Helmholtz just used the simplest
plausible one he could think of.

Harmonic entropy begins from an entirely different basis. This is most
evident in the fact that HE does not take into account the actual
partials of the tones, but assumes "typical harmonic timbres", and so
it does not stretch with stretched partials for example. With
inharmonic partials HE and SD disagree with each other but I
understand the idea is that both inform the actual experience.

As I mentioned earlier, HE is blind to difference-tone effects, but SD
may accomodate them if they are added as additional partials before
running the model.

I just noticed that in an earlier message I addressed my remarks to a
mysterious "Karl". That was intended to be "Kraig" and appears to be
some kind of freudian slip in which you were merged with Carl. My
apologies to both of you.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:

> mysterious "Karl". That was intended to be "Kraig" and appears to be
> some kind of freudian slip in which you were merged with Carl. My
> apologies to both of you.

I think you need to work harder. That was a slip of higher orders of
magnitude than you give credit.

Cheers,
Jon

Thanks!
is there somewhere where one can see the Plomp and Levelt's measurements
or better possibly explain their difference to HH.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Dave Keenan wrote:
>
> >
>
> Thanks Kraig, for the Helmholtz graph, and for emailing me the pages
> that describe its derivation. It uses the same kind of calculations as
> Sethares' "sensory dissonance" (SD) except that Sethares has the
> benefit of Plomp and Levelt's measurements of the actual shape of the
> critical band function, where Helmholtz just used the simplest
> plausible one he could think of.
>
>
>
> -- Dave Keenan
>
>

i know!
the abuse we all have to take ........for the betterment of art :)
but at least no one mentioned that that also makes us the 'K.K.K.'

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Jon Szanto wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, "Dave > Keenan" <d.keenan@...> wrote:
>
> > mysterious "Karl". That was intended to be "Kraig" and appears to be
> > some kind of freudian slip in which you were merged with Carl. My
> > apologies to both of you.
>
> I think you need to work harder. That was a slip of higher orders of
> magnitude than you give credit.
>
> Cheers,
> Jon
>
>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Thanks!
> is there somewhere where one can see the Plomp and Levelt's measurements
> or better possibly explain their difference to HH.

Probably in Sethares' book "Tuning Timbre Spectrum Scale". But there
may be enough information for you in Dave Benson's wonderfully
downloadable PDF version of his "Music a Mathematical Offering"
http://www.maths.abdn.ac.uk/~bensondj/html/maths-music.html
http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf
(10 MB). See PDF page 155.

-- Dave Keenan

Alright, since the conversation's moved here, I'll repost this, as I'd
really like to get some feedback on these ideas.

I think this was originally in response to one of Dave's posts, but I
don't remember.

I was just writing in response to Margo's thread that when I come to
perceive the relationship between two justly tuned intervals as being
whole unto itself, I get a sense of "depth" to the interval. 3/2, and
stacks of 3/2's, are much more shallow than 5/4, which sounds like
it's going "in" more or "back" or "deep" or something. I
synaesthetically perceive it to be the aural equivalent of the visual
process by which the brain puts parallax imagery between two eyes
together to form one three dimensional image. The distance by which
the two images diverge correlates with the brain's propensity and
ability to merge the two images into a cohesive, three dimensional
whole. Similarly, the harmonic "distance" by which the two tones
diverge correlates with the brain's propensity to merge the two tones
together into a cohesive interval. I use the term distance here to
mean in some kind of multi-dimensional JI space, and not distance in
cents or in pitch.

So if I perceive my ability to place increasingly higher-limit
harmonic intervals as an increase in awareness of "depth," and as I
perceive my ability to determine finer and finer pitch variations as
an increase of resolution (as in a monitor), could it be that this
sensation of depth is the actual experience of the periodicity
mechanism working?

So a few questions I have from this observation:

1) Are there any models of harmonic entropy that incorporate a free
variable that describes the ability of the periodicity mechanism to
operate, as there is with the ability of the place mechanism? Is that
what the "N" parameter is for?

and 2) has any experimentation ever been done on with having the sigma
value vary across the pitch spectrum? I'm thinking along the lines of
having sigma "peaks" at 100, 200, 300 ... up to 1200 cents to simulate
the harmonic entropy of an average listener who has been mainly
exposed to 12-tet... Of course, some other intervals would likely be
thrown in as well to make it really authentic, like perhaps some
smaller peaks for the just intervals, like one at 7/4 and one at 350
cents. I can already tell you that my pitch discrimination at around
350 cents is much stronger than my pitch discrimination at 550/600
cents, for example -- I sometimes get 550 and 600 cent intervals
confused, especially if they're at the top of a chord (although
considerably less so recently), but I rarely ever actually confuse 350
for 400 cents.

It might be interesting to generate a sigma curve and multiply THAT
into the equation rather than keeping sigma constant. You could plot a
variety of the most common scales and intervals you've heard and do
some kind of regression analysis to get a sigma curve, then see if the
result holds any resemblance to how you first heard microtonal music
when you were previously only exposed to 12tet or whatever your
background is. Might provide a decent model of how accordant music
from other cultures will be to us and how accordant western music will
be to those in other cultures. So we might be able to compute an HE
"map" for a specific person this way.

It might also be interesting to see if there is an optimal level of
harmonic entropy that makes intervals easiest to learn as well, as
opposed to ones that are already learned and ones that might just
sound like noise for quite a while until they are placed.

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> Alright, since the conversation's moved here, I'll repost this, as I'd
> really like to get some feedback on these ideas.
>
> I think this was originally in response to one of Dave's posts, but I
> don't remember.
>
> I was just writing in response to Margo's thread that when I come to
> perceive the relationship between two justly tuned intervals as being
> whole unto itself, I get a sense of "depth" to the interval. 3/2, and
> stacks of 3/2's, are much more shallow than 5/4, which sounds like
> it's going "in" more or "back" or "deep" or something. I
> synaesthetically perceive it to be the aural equivalent of the visual
> process by which the brain puts parallax imagery between two eyes
> together to form one three dimensional image. The distance by which
> the two images diverge correlates with the brain's propensity and
> ability to merge the two images into a cohesive, three dimensional
> whole. Similarly, the harmonic "distance" by which the two tones
> diverge correlates with the brain's propensity to merge the two tones
> together into a cohesive interval. I use the term distance here to
> mean in some kind of multi-dimensional JI space, and not distance in
> cents or in pitch.
>
> So if I perceive my ability to place increasingly higher-limit
> harmonic intervals as an increase in awareness of "depth," and as I
> perceive my ability to determine finer and finer pitch variations as
> an increase of resolution (as in a monitor), could it be that this
> sensation of depth is the actual experience of the periodicity
> mechanism working?

It's possible. But that's about all you could say. I don't see any
evidence one way or the other.

I had to look it up and remind myself that periodicity refers to time
domain type neural signals and place refers to frequency domain (roughly).

> So a few questions I have from this observation:
>
> 1) Are there any models of harmonic entropy that incorporate a free
> variable that describes the ability of the periodicity mechanism to
> operate, as there is with the ability of the place mechanism? Is that
> what the "N" parameter is for?

There has sometimes been an assumption that because the
Helmholtz/Plomp/Levelt/Sethares Sensory Dissonance (SD) is purely
derived from the place mechanism, depending as it does on the shape of
its filter response curve (the critical band curve), that therefore
harmonic entropy (HE) must relate purely to the periodicity mechanism.
But Paul doesn't actually exclude the possibility that both place and
periodicity mechanisms may feed into the neural net that we imagine
harmonic entropy (HE) might be modelling. This may be particularly
relevant to higher frequencies where the periodicity mechanism can't
keep up.

The s (sigma) parameter is what describes the accuracy with which
frequency can be measured by the combination of mechanisms. I
understand it is simply the standard error expressed as a percentage
of the frequency.

> and 2) has any experimentation ever been done on with having the sigma
> value vary across the pitch spectrum?

I don't think so, because it has always been given as a function only
of the width of the interval without absolute frequencies. But varying
sigma across the spectrum sounds like a perfectly valid thing to do.

> I'm thinking along the lines of
> having sigma "peaks" at 100, 200, 300 ... up to 1200 cents to simulate
> the harmonic entropy of an average listener who has been mainly
> exposed to 12-tet...

Interesting idea. I had in mind just making sigma higher towards the
ends of the spectrum.

> Of course, some other intervals would likely be
> thrown in as well to make it really authentic, like perhaps some
> smaller peaks for the just intervals, like one at 7/4 and one at 350
> cents. I can already tell you that my pitch discrimination at around
> 350 cents is much stronger than my pitch discrimination at 550/600
> cents, for example -- I sometimes get 550 and 600 cent intervals
> confused, especially if they're at the top of a chord (although
> considerably less so recently), but I rarely ever actually confuse 350
> for 400 cents.
>
> It might be interesting to generate a sigma curve and multiply THAT
> into the equation rather than keeping sigma constant. You could plot a
> variety of the most common scales and intervals you've heard and do
> some kind of regression analysis to get a sigma curve, then see if the
> result holds any resemblance to how you first heard microtonal music
> when you were previously only exposed to 12tet or whatever your
> background is. Might provide a decent model of how accordant music
> from other cultures will be to us and how accordant western music will
> be to those in other cultures. So we might be able to compute an HE
> "map" for a specific person this way.

Definitely an interesting idea. Have you tried contacting Paul Erlich
by email? Try replying to this message and choosing Paul Erlich in the
"To:" menu instead of the whole list.
/tuning/topicId_50781.html#50953

> It might also be interesting to see if there is an optimal level of
> harmonic entropy that makes intervals easiest to learn as well, as
> opposed to ones that are already learned and ones that might just
> sound like noise for quite a while until they are placed.

Maybe so.

-- Dave Keenan

Quite a good book it appears thanks for the reference!

If Plomp is correct in asserting that the 6/5 is more consonant than the 5/4,
why would so many object to it ( preferring the 19/16) especially and consistently on this list for so many years i wonder?
both these 5 limit thirds deviate from 12 et intervals pretty equally so i don't see how cultural conditioning might play in.

Outside of this , it is easy to see the Sterns and Stern/Brocot pattern- [possibly could be called the Stern(s)/Brocot consonance tree]

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Dave Keenan wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > Thanks!
> > is there somewhere where one can see the Plomp and Levelt's measurements
> > or better possibly explain their difference to HH.
>
> Probably in Sethares' book "Tuning Timbre Spectrum Scale". But there
> may be enough information for you in Dave Benson's wonderfully
> downloadable PDF version of his "Music a Mathematical Offering"
> http://www.maths.abdn.ac.uk/~bensondj/html/maths-music.html > <http://www.maths.abdn.ac.uk/%7Ebensondj/html/maths-music.html>
> http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf > <http://www.maths.abdn.ac.uk/%7Ebensondj/html/music.pdf>
> (10 MB). See PDF page 155.
>
> -- Dave Keenan
>
>

I assume you're referring to the graph on PDF page 158 of
http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf

I suspect the prediction of a slightly more consonant 5:6 than 4:5 may
be an artifact of their specific choice of timbre for that graph --
namely one that has no harmonics beyond the sixth.

-- Dave Keenan

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Quite a good book it appears thanks for the reference!
>
> If Plomp is correct in asserting that the 6/5 is more consonant than
the
> 5/4,
> why would so many object to it ( preferring the 19/16) especially and
> consistently on this list for so many years i wonder?
> both these 5 limit thirds deviate from 12 et intervals pretty equally
> so i don't see how cultural conditioning might play in.
>
> Outside of this , it is easy to see the Sterns and Stern/Brocot
> pattern- [possibly could be called the Stern(s)/Brocot consonance tree]
>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Thanks!
> is there somewhere where one can see the Plomp and Levelt's
> measurements or better possibly explain their difference to HH.

http://dare.ubn.kun.nl/bitstream/2066/15403/1/5544.pdf

Helmholtz just said that the maximum dissonance was when
beats were 33 Hz. Plompt and Levelt actually had people
assess intervals and averaged the results.

-Carl

> > I can already tell you that my pitch discrimination at around
> > 350 cents is much stronger than my pitch discrimination at 550/600
> > cents, for example -- I sometimes get 550 and 600 cent intervals
> > confused,

Mike- first off, sigma is about sine tones in an experimental
setting, not normal timbres. Secondly, it's more along the
lines of a JND-type concept than what intervals you confuse.

> > You could plot a
> > variety of the most common scales and intervals you've heard

Sigma has nothing to do with interval recognition. It's
simply the error of the hearing apparatus. It will vary a
bit from person to person, depending on age for example.

-Carl

On Wed, Jun 25, 2008 at 12:58 PM, Carl Lumma <carl@...> wrote:
>> > I can already tell you that my pitch discrimination at around
>> > 350 cents is much stronger than my pitch discrimination at 550/600
>> > cents, for example -- I sometimes get 550 and 600 cent intervals
>> > confused,
>
> Mike- first off, sigma is about sine tones in an experimental
> setting, not normal timbres. Secondly, it's more along the
> lines of a JND-type concept than what intervals you confuse.

Even with sine tones, I find that I can recognize 350 cents vs 400
cents easier than 550 vs 600 cents. Although nowadays it might be
different, since I've been listening to a lot more 11 limit music.

>> > You could plot a
>> > variety of the most common scales and intervals you've heard
>
> Sigma has nothing to do with interval recognition. It's
> simply the error of the hearing apparatus. It will vary a
> bit from person to person, depending on age for example.
>
> -Carl

I'm saying that my sense of pitch is more finely tuned at certain
intervals than others. I think that it might vary not only from person
to person and from one age group to another, but within an individual
person across the pitch spectrum.

-Mike

Mike wrote:
> I'm saying that my sense of pitch is more finely tuned at certain
> intervals than others.

How do you know?

-Carl

It's just an observation of mine. I can distinguish quarter tones
pretty easily at any interval (except between 550 and 600), but I can
distinguish third tones and even smaller around a major third, and
most likely nowhere near that around a minor seventh or a major
seventh.

On Wed, Jun 25, 2008 at 1:38 PM, Carl Lumma <carl@...> wrote:
> Mike wrote:
>> I'm saying that my sense of pitch is more finely tuned at certain
>> intervals than others.
>
> How do you know?
>
> -Carl
>
>

I wrote:
> > > You could plot a
> > > variety of the most common scales and intervals you've heard
>
> Sigma has nothing to do with interval recognition.

Although interestingly it was measured via an interval
recognition task! (The data was heavily cooked.) A full
investigation of this isn't in the cards for me right
now. If you're interested, contact me offlist and I
can send you some papers.

-Carl

At this point , personally, while this appears that this is as far as research has been done, that i can not even begin to put much into any theory or consonance and dissonance based on just dyads. It does not hold up even in the face of my own feeble and limited attempts and demonstrations. If someone wishes to make a sequence of all the say up to 4 note chords possible with the harmonics up to 11 ( with no spaces bigger that an octave) from predicted consonance to dissonance in order, then we can compare.

Helmholtz i think is being greatly underrated here. His investigations into the con/dis factors of chords are NOT based on his graph of dyads, and this is a critical point. Like the great scientist he was, he approached from a different perspective as foreseeing the very problems inherent at looking at it merely that way. He himself does not believe that the properties of dyads are of much use, otherwise he would approach the problem that way. Whether or not difference tones play a factor, it does support his investigations into the inversions of chords. His examples hold in the limited way they can be applied. At least he is recognizing the breath of the problem. We have only began to explore the continuum and would say we are still in a dark age in understanding even these simplest of things.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > Thanks!
> > is there somewhere where one can see the Plomp and Levelt's
> > measurements or better possibly explain their difference to HH.
>
> http://dare.ubn.kun.nl/bitstream/2066/15403/1/5544.pdf > <http://dare.ubn.kun.nl/bitstream/2066/15403/1/5544.pdf>
>
> Helmholtz just said that the maximum dissonance was when
> beats were 33 Hz. Plompt and Levelt actually had people
> assess intervals and averaged the results.
>
> -Carl
>
>