[Fwd: SETI bioastro: Composer Reveals Musical Chords' Hidden Geometry]

-------- Original Message --------
Subject: SETI bioastro: Composer Reveals Musical Chords' Hidden Geometry
Date: Fri, 07 Jul 2006 16:19:02 -0400
From: "LARRY KLAES" <ljk4@msn.com>
To: bioastro@setileague.org

Composer Reveals Musical Chords' Hidden Geometry

Princeton NJ (SPX) Jul 07, 2006

Composers often speak of fitting chords and melodies together, as though
sounds were physical objects with geometric shape -- and now a Princeton
University musician has shown that advanced geometry actually does offer
a
tool for understanding musical structure.

http://www.terradaily.com/reports/Composer_Reveals_Musical_Chords_Hidden_Geometry_999.html

See also: http://music.princeton.edu/~dmitri/ChordGeometries.html

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> Composers often speak of fitting chords and melodies together, as though > sounds were physical objects with geometric shape -- and now a Princeton > University musician has shown that advanced geometry actually does offer
> a > tool for understanding musical structure.
> > http://www.terradaily.com/reports/Composer_Reveals_Musical_Chords_Hidden_Geometry_999.html
> > See also: http://music.princeton.edu/~dmitri/ChordGeometries.html

I read the first part of the paper, and it looks promising. He makes big claims which may turn out to be overblown, but at least he's testing it against music. The basic idea is that chords should divide the octave equally but not quite equally.

It all assumes Z12, although there's a bit where he mentions higher dimensions. I don't know what he meant by it. Still, if anybody can understand it, there's an opportunity to generalize it to 7+12 and explain a few more things.

Graham

>I read the first part of the paper, and it looks promising. He makes
>big claims which may turn out to be overblown, but at least he's testing
>it against music. The basic idea is that chords should divide the
>octave equally but not quite equally.
>
>It all assumes Z12, although there's a bit where he mentions higher
>dimensions. I don't know what he meant by it. Still, if anybody can
>understand it, there's an opportunity to generalize it to 7+12 and
>explain a few more things.

The visualizations themselves are completely underwhelming in
my opinion.

-Carl

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

> It all assumes Z12, although there's a bit where he mentions higher
> dimensions. I don't know what he meant by it. Still, if anybody can
> understand it, there's an opportunity to generalize it to 7+12 and
> explain a few more things.

I don't see why you think it assumes C12.

My feeling is that there's too much machinery in there; I'd get rid of
the multisets to start with. If he's got to use orbifolds, then use
n-tuples, and sort them to get the class representative. But couldn't
we make life much easier by not using octave equivalence, sorting
according to size, and then using these for your orbifolds? Another
approach would be to get rid of the orbifolds if you could, and simply
regard n-note chords as points in Rn, with the distance the usual
metric, and use this for voice leading, and then try to deal with
octave equivalence and permuations of the chord from there.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>It all assumes Z12, although there's a bit where he mentions higher >>dimensions. I don't know what he meant by it. Still, if anybody can >>understand it, there's an opportunity to generalize it to 7+12 and >>explain a few more things.
> > I don't see why you think it assumes C12.

I don't know what C12 is. It's actually 12Z he says, not Z12. But 12 notes anyway, and he says it a lot.

> My feeling is that there's too much machinery in there; I'd get rid of
> the multisets to start with. If he's got to use orbifolds, then use
> n-tuples, and sort them to get the class representative. But couldn't
> we make life much easier by not using octave equivalence, sorting
> according to size, and then using these for your orbifolds? Another
> approach would be to get rid of the orbifolds if you could, and simply
> regard n-note chords as points in Rn, with the distance the usual
> metric, and use this for voice leading, and then try to deal with
> octave equivalence and permuations of the chord from there.

Um, yeah, what he said...

I thought he needed octave equivalence because the music uses octave equivalence. You have to apply the octave equivalence before you sort according to size, and I though the whole point was that the music works like this when it doesn't have to. But if you think you can simplify it, go for it, as there's obviously an audience for this kind of thing.

Graham

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

> > I don't see why you think it assumes C12.
>
> I don't know what C12 is. It's actually 12Z he says, not Z12. But 12
> notes anyway, and he says it a lot.

C12 is the cyclic group of order 12; topologists use Z12 for that but
number theorists use that for something else. But 12Z is something
diffierent; it is the set of integers which are multiples of 12. R/12Z
is the circle group; it is the group of pitch classes in this case,
and it's not a discrete group, but continuous.

> I thought he needed octave equivalence because the music uses octave
> equivalence. You have to apply the octave equivalence before you sort
> according to size, and I though the whole point was that the music
works
> like this when it doesn't have to.

Well, but he's trying to model voice leading, and it isn't clear to me
that works best using octave equivalence. B to C, up a semitone, is
obviously not much like B a seventh down to C.