Re: [tuning] Digest Number 6106--Cook and Fujisawa Triadic consonance

I found the site from which one can download the Seeing Harmony software.

http://www.res.kutc.kansai-u.ac.jp/~cook/harmony.html

--John

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:
>
> I found the site from which one can download the Seeing Harmony
> software.
>
> http://www.res.kutc.kansai-u.ac.jp/~cook/harmony.html
>
> --John

Thanks John. I wasn't awore of Cook/Fujisawa. Here are some
thoughts:

The Psychophysics of Harmony Perception: Harmony is a
Three-Tone Phenomenon
http://www.res.kutc.kansai-u.ac.jp/~cook/EMR(2006).PDF

The title was certainly promising, but I have serious problems
with the contents.

Table 1 is doing nothing for morale on page 3 of 21! They
conveniently don't underline their own erroneous predictions.

The hypothesis that triads whose interior intervals are the
same size are more tense could easily work out, since the
harmonic series contains consecutive intervals of shrinking
size, and the relative shrinkage is most pronounced low in
the series.

http://www.res.kutc.kansai-u.ac.jp/~cook/AmSci.pdf

Again, we're off to a good start:

"We believe ... that the different emotional responses to minor
and major have a biological basis."

I agree. But then...

"If interval dissonance and triad tension were the only factors
determining the sonority of triads, we should expect that all
of the major and minor chords would sound rather similar."

To solve this problem, they simply declare that if the bottom
interior interval is the smaller one, you get minor 'sadness',
and if it's the larger, major 'happiness'. That's not any kind
of explanation of the asymmetry they (correctly) take pains to
point out.

In extended JI, we know this asymmetry between major and minor
continues, and indeed gets rapidly more severe. But even in
the 5-limit, minor chords are more tense and not as consonant
as major chords. By drawing conclusions from two models that
are symmetrical -- the usual P&L sensory dissonance, and their
"tension" based on "intervallic equidistance" -- they can
hardly expect to explain this.

The triad graphs are of course headed in the right direction,
and they certainly produce familiar patterns, but I think it's
a superficial similarity to the triad plots produced by Erlich
and others. In fact it's evident they are very nearly mirror
symmetric about the diagonal emanating from the origin.

In general, it's all too easy to number juggle and create
something that isn't totally falsified by observations of
12-ET constructs. Since the authors don't seem to be
considering the full continuum of pitch, they miss out on
a lot that could potentially falsify their models and are
therefore much more likely to go astray.

Nevertheless, at least they're trying! And their citations
of experiments showing the major/minor asymmetry in children
and so on are good gunpowder.

-Carl

Carl Lumma wrote:

> The Psychophysics of Harmony Perception: Harmony is a
> Three-Tone Phenomenon
> http://www.res.kutc.kansai-u.ac.jp/~cook/EMR(2006).PDF
> > The title was certainly promising, but I have serious problems
> with the contents.
> > Table 1 is doing nothing for morale on page 3 of 21! They
> conveniently don't underline their own erroneous predictions.

It's not good, at any rate, that Sethares gets an underlining for agreeing with them. They aren't listing the part of Fig. 1 for non-musicians, where the chord types aren't separated. They might agree more with it.

> The hypothesis that triads whose interior intervals are the
> same size are more tense could easily work out, since the
> harmonic series contains consecutive intervals of shrinking
> size, and the relative shrinkage is most pronounced low in
> the series.

It's supposed to come from cognitive science. I can see that there would be more confusion with chords that have equal intervals. I've tried listening to magic chords from this point of view and it does make some sense. The augmented triads sound most tense when they're played as two consecutive major thirds, for example.

It's interesting, at least, that they have a formula that agrees well with musical common sense, and takes account of the timbre, but doesn't invoke a rational spelling. It means it works for chords that depend on temperament, like the augmented triads in 12-equal.

> http://www.res.kutc.kansai-u.ac.jp/~cook/AmSci.pdf

> To solve this problem, they simply declare that if the bottom
> interior interval is the smaller one, you get minor 'sadness',
> and if it's the larger, major 'happiness'. That's not any kind
> of explanation of the asymmetry they (correctly) take pains to
> point out.

That's in the other paper as well. But still no rationale for why it works. "Major and minor have different moods. Let's assign different moods to major and minor. Oh look, with this model major and minor have different moods!"

> In extended JI, we know this asymmetry between major and minor
> continues, and indeed gets rapidly more severe. But even in
> the 5-limit, minor chords are more tense and not as consonant
> as major chords. By drawing conclusions from two models that
> are symmetrical -- the usual P&L sensory dissonance, and their
> "tension" based on "intervallic equidistance" -- they can
> hardly expect to explain this.

We assign the asymmetry to otonal/utonal which they don't consider. In the other EMR paper there are "Fujisawa, Konaka & Cook, 2006" results that do a better job of distinguishing major and minor. I don't know how they do it.

> The triad graphs are of course headed in the right direction,
> and they certainly produce familiar patterns, but I think it's
> a superficial similarity to the triad plots produced by Erlich
> and others. In fact it's evident they are very nearly mirror
> symmetric about the diagonal emanating from the origin.

All triad plots are going to look similar. They are getting better results than dyadic measures -- distinguishing different inversions (with root position first), and making major and minor better than augmented and "suspended". For a simple model that's pretty good. None of the other "theoretical" rankings they cite does as well. And average otonal numbers wouldn't either: 3:4:5 would always be simpler than 4:5:6 and diminished chords as 5:6:7 would beat 5-limit minors (even as 6:7:9).

> In general, it's all too easy to number juggle and create
> something that isn't totally falsified by observations of
> 12-ET constructs. Since the authors don't seem to be
> considering the full continuum of pitch, they miss out on
> a lot that could potentially falsify their models and are
> therefore much more likely to go astray.

Their plots have the full continuum, and some of them seem to suggest JI tunings. In one of the papers (maybe the one you didn't mention) they say they're looking at the implications for alternative scales as well -- which in that case would lead to quartertones. The fact that it can be falsified is good. I don't think the tension part has been.

> Nevertheless, at least they're trying! And their citations
> of experiments showing the major/minor asymmetry in children
> and so on are good gunpowder.

It's more evidence that standard music theory was getting it right all along.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> > The hypothesis that triads whose interior intervals are the
> > same size are more tense could easily work out, since the
> > harmonic series contains consecutive intervals of shrinking
> > size, and the relative shrinkage is most pronounced low in
> > the series.
>
> It's supposed to come from cognitive science.

Cognitive psychology? Howso?

> > The triad graphs are of course headed in the right direction,
> > and they certainly produce familiar patterns, but I think it's
> > a superficial similarity to the triad plots produced by Erlich
> > and others. In fact it's evident they are very nearly mirror
> > symmetric about the diagonal emanating from the origin.
>
> All triad plots are going to look similar. They are getting
> better results than dyadic measures -- distinguishing
> different inversions (with root position first), and making
> major and minor better than augmented and "suspended". For
> a simple model that's pretty good.

I wouldn't call it simple, as it considers all triples of
partials, apples their arbitrary major-minor sine wave thing,
sums and then gloms that together (somehow) with sensory
dissonance, and IIRC they mention an arbitrary .6 normalizing
factor or something.

> None of the other
> "theoretical" rankings they cite does as well. And average
> otonal numbers wouldn't either: 3:4:5 would always be
> simpler than 4:5:6 and diminished chords as 5:6:7 would beat
> 5-limit minors (even as 6:7:9).

This is why I've suggested that harmonic entropy be applied
via a greedy algorithm, where n-ads of partials are removed
from a sound, such that it would be likely that 3:2 is
pulled before 6:7:9 from 6:7:9, which is I think what tends
to happen with this chord, unless the listener is primed with
4:6:7:9.

> > In general, it's all too easy to number juggle and create
> > something that isn't totally falsified by observations of
> > 12-ET constructs. Since the authors don't seem to be
> > considering the full continuum of pitch, they miss out on
> > a lot that could potentially falsify their models and are
> > therefore much more likely to go astray.
>
> Their plots have the full continuum,

Yes, but they only test 12-ET cases.

-Carl

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>>> The hypothesis that triads whose interior intervals are the
>>> same size are more tense could easily work out, since the
>>> harmonic series contains consecutive intervals of shrinking
>>> size, and the relative shrinkage is most pronounced low in
>>> the series.
>> It's supposed to come from cognitive science.
> > Cognitive psychology? Howso?

Sorry, Gestalt psychology is what they say. You'll have to get the original paper for the logic behind it. It makes sense, though, that any algorithm to assign a root has to start with identifying an asymmetry. If a chord's completely symmetric there's nothing you can do. Well, you can find the lowest note, but we're capable of assigning virtual pitches. Fortunately the harmonic series has unequally spaced pitches that lead to a single fundamental.

>>> The triad graphs are of course headed in the right direction,
>>> and they certainly produce familiar patterns, but I think it's
>>> a superficial similarity to the triad plots produced by Erlich
>>> and others. In fact it's evident they are very nearly mirror
>>> symmetric about the diagonal emanating from the origin.
>> All triad plots are going to look similar. They are getting >> better results than dyadic measures -- distinguishing >> different inversions (with root position first), and making >> major and minor better than augmented and "suspended". For >> a simple model that's pretty good.
> > I wouldn't call it simple, as it considers all triples of
> partials, apples their arbitrary major-minor sine wave thing,
> sums and then gloms that together (somehow) with sensory > dissonance, and IIRC they mention an arbitrary .6 normalizing
> factor or something.

They assume that symmetric triples lead to tension. Beyond that it's as simple as possible. They have to consider the full timbre so they sum (yes, sum, not "glom together somehow") the triples. There has to be a function to determine the asymmetry so they chose a gaussian.

>> None of the other >> "theoretical" rankings they cite does as well. And average >> otonal numbers wouldn't either: 3:4:5 would always be >> simpler than 4:5:6 and diminished chords as 5:6:7 would beat >> 5-limit minors (even as 6:7:9).
> > This is why I've suggested that harmonic entropy be applied
> via a greedy algorithm, where n-ads of partials are removed
> from a sound, such that it would be likely that 3:2 is
> pulled before 6:7:9 from 6:7:9, which is I think what tends
> to happen with this chord, unless the listener is primed with
> 4:6:7:9.

Do you have that working? I've tried subharmonic matching before, which makes some sense. Somebody gave me some code to implement Terhardt's virtual pitch algorithm but I didn't look at it.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >>> The hypothesis that triads whose interior intervals are the
> >>> same size are more tense could easily work out, since the
> >>> harmonic series contains consecutive intervals of shrinking
> >>> size, and the relative shrinkage is most pronounced low in
> >>> the series.
> >> It's supposed to come from cognitive science.
> > Cognitive psychology? Howso?
>
> Sorry, Gestalt psychology is what they say. You'll have to
> get the original paper for the logic behind it. It makes
> sense, though, that any algorithm to assign a root has to
> start with identifying an asymmetry. If a chord's
> completely symmetric there's nothing you can do. Well, you
> can find the lowest note, but we're capable of assigning
> virtual pitches. Fortunately the harmonic series has
> unequally spaced pitches that lead to a single fundamental.

That's my point.

> >> All triad plots are going to look similar. They are getting
> >> better results than dyadic measures -- distinguishing
> >> different inversions (with root position first), and making
> >> major and minor better than augmented and "suspended". For
> >> a simple model that's pretty good.
> >
> > I wouldn't call it simple, as it considers all triples of
> > partials, apples their arbitrary major-minor sine wave thing,
> > sums and then gloms that together (somehow) with sensory
> > dissonance, and IIRC they mention an arbitrary .6 normalizing
> > factor or something.
>
> They assume that symmetric triples lead to tension. Beyond
> that it's as simple as possible. They have to consider the
> full timbre so they sum (yes, sum, not "glom together
> somehow") the triples.

I said sum the triples. They glom that result together with
sensory dissonance (somehow).

> > This is why I've suggested that harmonic entropy be applied
> > via a greedy algorithm, where n-ads of partials are removed
> > from a sound, such that it would be likely that 3:2 is
> > pulled before 6:7:9 from 6:7:9, which is I think what tends
> > to happen with this chord, unless the listener is primed with
> > 4:6:7:9.
>
> Do you have that working?

Not even started. But I'm willing to bet software like
Melodyne DNA and Visual Vox Polyphonic use a similar
technique.

> I've tried subharmonic matching before, which makes some
> sense.

How does it make sense? From an evolutionary biology
standpoint, harmonics make sense, not subharmonics.

> Somebody gave me some code to implement Terhardt's virtual
> pitch algorithm but I didn't look at it.

I read that paper years ago and it was interesting.
Should re-read it. Terhardt does smack of a lot of
'just so' explanations, however. That said, he probably
did more to advance psychoacoustics than anyone else in
recent times.

-Carl

Carl Lumma wrote:

> I said sum the triples. They glom that result together with
> sensory dissonance (somehow).

Tension is the sum of the triples. Instability is a weighted sum of the tension and sensory dissonance.

>> I've tried subharmonic matching before, which makes some
>> sense.
> > How does it make sense? From an evolutionary biology
> standpoint, harmonics make sense, not subharmonics.

Subharmonic matching gives you the fundamental of a harmonic series. The fundamental being the pitch that's a common subharmonic of all partials. It's explained here:

http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/virtualp.html

What you can do is take a signal, lower it by each harmonic interval, add the results together, and take the highest frequency in the result as the fundamental. There are lots of free parameters: the weight of each interval and the tolerance you allow in choosing the best frequency. You also have to treat octaves as special because that's how we hear.

>> Somebody gave me some code to implement Terhardt's virtual
>> pitch algorithm but I didn't look at it.
> > I read that paper years ago and it was interesting.
> Should re-read it. Terhardt does smack of a lot of
> 'just so' explanations, however. That said, he probably
> did more to advance psychoacoustics than anyone else in
> recent times.

He's still being cited as the authority on virtual pitch. I don't think there are enough data to back up the details of his algorithm though.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > I said sum the triples. They glom that result together with
> > sensory dissonance (somehow).
>
> Tension is the sum of the triples. Instability is a
> weighted sum of the tension and sensory dissonance.

Right. And what's the weighting? Arbitrary IIRC.

> >> I've tried subharmonic matching before, which makes some
> >> sense.
> >
> > How does it make sense? From an evolutionary biology
> > standpoint, harmonics make sense, not subharmonics.
>
> Subharmonic matching gives you the fundamental of a harmonic
> series. The fundamental being the pitch that's a common
> subharmonic of all partials. It's explained here:
>
> http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/
> virtualp.html

Ok, was just thrown by the term. This is an excellent
article by the way, and I agree with just about everything
said here.

-Carl