harmonic entropy continuing

Paul Erlich, TD 646

Hi Paul....

In answer to your question concerning the Harmonic Entropy article:

> There is a very strong propensity for the ear to try to fit what it >hears into one or a small number of harmonic series, and the >fundamentals of these series, even if not physically present, are either >heard outright, or provide a more subtle sense of overall pitch known to >musicians as the "root". As a component of consonance, the ease with >which the ear/brain system can resolve the fundamental is known as >"tonalness." I have proposed a concept called "relative harmonic >entropy" to model this component of dissonance.

This part of the article is quite clear to me and VERY interesting...
After just re-reading Helmholz... it seems like he is mostly concerned
with "roughness" or "beating" causing dissonance... but doesn't really
get involved with the way the brain tries to match intervals with the
"nearest" lower ratios as determining "tonalness..."

>For inharmonic timbres, Sethares can find a whole new set of intervals
>that have low roughness (= are consonant), but they don't have high >tonalness, because they don't imply a fundamental.

This is exceptionally interesting... one could maybe base a "whole new
music" on these materials...

>For triads or higher-ads - the roughness of Otonal chords is the same as
>that of Utonal chords because they have the same intervals, but the >tonalness of Otonal chords is much greater, because they imply a much >simpler set of harmonics over a fundamental.

I find this quite peculiar... but now that I've re-read Helmholz, I'm
going to go back and re-read Partch... and may have more to say about
this...

>The harmonic entropy is based on the concept that the critical band
>represents a certain >degree of uncertainty in the perception of pitch

Personally, for what it's worth, I like the phrasing of this sentence...
even though the "particulars" of the specifically-defined "critical
band"
might be questionable -- since it brings into play the idea of a
continuous spectrum of pitches...

>i.e., 301/200 will be heard as a mistuned 3/2.

Well, yup. But what's more interesting, of course, is when things get
really complicated...

>Mediants define the conceptual boundaries of the ratios and since >simpler ratios are farther away from more complex ones and complex ones >cluster together, there are more possible actual intervals (rational or >otherwise) that can be conceived as simpler ones than for more complex >ones.

Wazzat?? Here's where I start to get lost (it had to happen). Is this
because of the bell curve??

>The simpler-integer ratios take up a lot of room, defined as the >interval between the mediant below and the mediant above, in interval >space, and so are associated with. large "slices" of the
>probability distribution, while the more complex ratios are more crowded
>and therefore are associated with smaller "slices."

This is obviously associated with the previous paragraph, and since I
didn't understand THAT one, the probability that I would understand THIS
one is, indeed, low...

>When the true interval is near a simple-integer ratio, there will be one
>large probability and many much smaller ones. When the true interval is
>far from any simple-integer ratios, many more complex ratios will all >have roughly equal probabilities. The entropy function will come out >quite small in the former case, and quite large in the latter case.

Using "fuzzy logic," I'm kind of understanding this... It is related to
the whole idea of the complex ratios being "clustered together" which
you, hopefully, will explain a bit more in layman's terms. However,
accepting that fact, the above paragraph certainly makes sense...
although, admittedly I don't fully understand the math behind it...

>The function is x*log(x).

>In the case of 700 cents, 3/2 will have far more probabilty than any >other ratio, and the harmonic entropy is nearly minimal.

>In the case of 300 cents, 6/5 will have the largest probability in most
>cases, but 7/6, 13/11, and 19/16 will all have non-negligible amounts of >probability, so the harmonic entropy is moderate.

>In the case of 100 cents, 15/14, 16/15, 17/16, 18/17, 19/18, 20/19, and
>1/1 will all have significant probability, and the harmonic entropy is >nearly maximal.

Well, this certainly came at the right time... since these examples are
very clear, and make a lot of sense...

>Also, there is some evidence that the ear-brain system adjusts its
>sampling rate according to >the rate of harmonic change in the music.
>(Monz)

This is fascinating, and I'd like to know more about it...

>In other words, re: Farey series of order infinity - even if we could
>conceive of intervals as accurately as we want, we would still >substitute simpler ratios for more complex ones, because
>of our perceptual limitations (= the bell curve doesn't change, no >matter how high the order), >and the fact that simpler ratios always >take up more room than more complex ones, no matter >how high the order >in the Farey series

This is related to the other paragraphs... I guess I'm really not
getting the "more room" concept... and it seems rather significant...
Does it have something to do with the fact that lower ratios are
actually LARGER intervals??? Or is this wrong. ...

Regarding the Partch discussion... it seems as though the Utonal chords
and Otonal chords should really sound quite different... Is this so? I
guess I really need to hear some examples for me to fully understand
this...

Could someone possibly construct such synthesized examples?? Otherwise,
I guess Dean Drummond would have to play something.

Thanks for all the help!

Joe
___________ ______ ___ ___ _
Joseph Pehrson

>Regarding the Partch discussion... it seems as though the Utonal chords
>and Otonal chords should really sound quite different... Is this so? I
>guess I really need to hear some examples for me to fully understand
>this...

Of course, the major and minor triads in 5-limit western music are otonal
and utonal chords. An interesting question is: what _are_ "major" and
"minor"?

Can we define a "major" consonance as one with low roughness and high
tonalness, and a "minor" consonance as one with low roughness and low
tonalness? Is the sound of minor an un-expectedly weak sense of root?
If so, Partch's otonal = major, utonal = minor conjecture would be
justified up to the 9- or 11-limit (and maybe higher for unsaturated
chords), when the utonal chords become dissonant due to combination
tone effects and/or very low tonalness. My experiments with chord
progressions in extended JI sound, to my ear, in support of this idea.

Or is the sound of minor simply somthing we take from our exposure to
the diatonic scale? Is it just a scalar effect that we carry over to our
experience of bare harmonies? Or perhaps the major/minor duality of
music has no substantial basis at all -- maybe we create it in our music
by features of composition, done by custom now, by total accident at
first...

-Carl

>Clearly these words "major" and "minor" are used to mean several different
>things, and I think trying to find a single underlying meaning for these
>words is an endeavor which is too oriented on linguistic accident and
>ultimately misses the point.

I see that I wasn't too clear. It's obvious that these words can mean
different things at different times, a combination of things at a given
time, etc. I was referring specifically to the (admittedly subjective)
sound of otonal vs. utonal chords. Do you feel that the difference in
the sound between otonal and utonal 5-limit chords is similar to that
between otonal and utonal 7-limit chords? If not, then so be it. But
if so, isn't it possible that there is a teleological explanation?

>I think Harry Partch's definition of "tonality" is the source of a lot
>of trouble. He speaks as if tonality was nothing more than one chord,
>the tonic chord.

Of course tonality can be more than this, but my post was about the
experience of listening to bare chords only, so Partch's observations
are relavent.

>As you know, I (perhaps indoctrinated by contemporary theory :)) believe
>that tonality is founded, instead, on not only the tonicity of the tonic
>chord but also the resolving tendencies of the notes in the scale. While
>Partch might lead you to believe that the major-minor duality in Western
>music is nothing more than the Otonal-Utonal duality in the particular
>case of the 5-limit, the fact of the matter is that we had not two but
>six or seven equally viable modes in Western music, and it was only
>through the increasing use of dissonance in the Baroque period that the
>tritone achieved its pivotal role as the characteristic dissonance of the
>diatonic scale, and the resolving tendencies of the tritone reduced the
>number of viable modes to two: major and minor. The fact that this
>duality, by conicidence, appears parallel to the otonal-utonal duality,
>and that this parallelism is linguistically supported, may have misled
>Partch, and may be misleading you now.

Paul, re-read the first sentence of the last paragraph of my post.
Remember that I was discussing only the experience of listening to bare
chords. It seems you are answering that sentence affirmatively.

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

>Do you feel that the difference
in
> the sound between otonal and utonal 5-limit chords is similar to
that
> between otonal and utonal 7-limit chords?

I think otonal/utonal pairs sound more and more different the higher
the limit (in the 3-limit they're the same), though the intervallic
constitution can be considered a "motivic" similarity if so projected
by the music (say, as a melodic inversion).

>If not, then so be it.
But
> if so, isn't it possible that there is a teleological explanation?

What does teleological mean?

> Paul, re-read the first sentence of the last paragraph of my post.
> Remember that I was discussing only the experience of listening to
bare
> chords. It seems you are answering that sentence affirmatively.
>

Yes, I think so. I think that the consonance of utonal chords
deteriorates rapidly with increasing limit, and so in the 5-limit,
the
utonal triad emerges as one of the most consonant triads in the
diatonic scale (though still threatened by otonal interpretations
like
16:19:24 and even 6:7:9), while beyond the 9-limit, utonal chords
just
sound "wrong" to me. I like Kyle Gann's use of a tuning which
provides
9-limit utonalities and 11-limit otonalities; he's breaking Partch's
symmetry, no doubt influenced by the lack of psychoacoustical
symmetry
between otonal and utonal.
>
> -Carl

>I think otonal/utonal pairs sound more and more different the higher
>the limit (in the 3-limit they're the same),

I agree. But what I was asking -- is there a similarity between the
differences? Does the 1/(4:5:6:7) sound like a "minor" version of the
4:5:6:7? Using "minor" here simply to describe the sound difference
between 5-limit chords (wether that difference comes from scale theory,
psychoacoustics, or both).

BTW- major and minor 3-limit chords are not the same. Perfect fourth
vs. perfect fifth?!

Teleological means purposeful, or having causes in the laws of nature.
I first read the word in Kierkegaard, "is there a teleological
abridgement of the dialectic?..." (the answer is no).

-Carl

What I'm driving at here is... does minor sound sad? Major happy? Do
the chords alone sound that way, or only in context of the diatonic scale?
Tritone explains why two modes were selected during the developement of
tonal music, but not why they sound different. That difference must come
from the psychoacoustic quality of the tonic triad...

-Carl

I can't seem to remember if this web
page has already been discussed...

http://www.sohl.com/sohl/mt/maptone.html

-C.

Another thing -- by my proposed definition of minor, all ASS's would be
minor.

-Carl

Carl wrote:

>What I'm driving at here is... does minor sound sad? Major happy? Do
>the chords alone sound that way, or only in context of the diatonic scale?

Well, I've never believed it was anything more than conditioning. I don't know how many (Western) cultures share the tradition of a bugler (trumpeter?) playing "taps" at a funeral, but can someone explain to me how the simple phrasing of a major triad cause so many people to weep? Context seems to be the key...

Best all,
Jon

`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`
Real Life: Orchestral Percussionist
Web Life: "Corporeal Meadows" - about Harry Partch
http://www.corporeal.com/

>>I can't seem to remember if this web
>>page has already been discussed...
>>http://www.sohl.com/sohl/mt/maptone.html
>
>Yup, I think I mentioned that it approaches the problem of triadic
>consonance from what one might call the Helmholtz/Plomp/Sethares
>angle, and so fails to differentiate between otonal and utonal . . .

Was anybody aware that his proggies play the sonorities on the graph?
Since he's already done the bulk of the work, perhaps he could be
persuaded to differentiate between o- and utonal, to add voronoi cells,
etc.

-Carl