Klein-bottle Tonnetze

I read this article by Peck

http://societymusictheory.org/mto/issues/mto.03.9.3/toc.9.3.html

and wonder now what temperaments map for Klein-bottles?

Gabor

--- In tuning@yahoogroups.com, "alternativetuning"

/tuning/topicId_46646.html#46646

<alternativetuning@y...> wrote:
> I read this article by Peck
>
> http://societymusictheory.org/mto/issues/mto.03.9.3/toc.9.3.html
>
> and wonder now what temperaments map for Klein-bottles?
>
> Gabor

***Does this make any sense at all?? I've known composers who
were "on the bottle," but never those consciously using "bottle
theory..."

J. Pehrson

> I read this article by Peck
>
> http://societymusictheory.org/mto/issues/mto.03.9.3/toc.9.3.html

I can't make heads or tales of this, or most of the other
stuff on MTO. But here's a review by Peck of the AMS
conference on music theory (must be the one Gene mentioned
a while back?)...

http://tinyurl.com/lpjh

...Jon Wild and Stephen Soderberg are or were on this list.
And Eytan Agmon has been 'round the way. The latter two
demonstrated profound aversion toward the materials discussed
here.

>[4] Jack Douthett (jdouthett@tvi.cc.nm.us), of TVI Community
>College, spoke next, about the concept of maximal evenness.

Ugh.

But this sounds interesting...

>[5] The first presentation session concluded with the work of
>two graduate students. The first was Panayotis Mavromatis
>(pm@theory.esm.rochester.edu) of the Eastman School of Music.
>Mavromatis's presentation, "Minimal Description Length: An
>Information-Theoretic Approach to Music Model Building,"
>provided an approach to oral-based musical traditions, for
>which no explicit documentation of rule systems exists or is
>known. These rule systems may involve such activities as
>composition, improvisation, or listening. The particular
>repertoire he considered is modern Greek church chant. Drawing
>on statistical techniques, he built a stochastic model of melody
>in this corpus. Specifically, he defined a Hidden Markov Model
>(HMM), using a variant of Stolcke and Omohundro's state merging
>algorithm, with Rissanen's Minimal Description Length (MDL) as
>the termination criterion. Among the questions raised after the
>talk was how this model might be used to analyze jazz
>improvisation.

//

>The final speaker in the first presentation session was Jonathan
>Wild (wild@fas.harvard.edu), a graduate student at Harvard
>University. His talk, "Tessellating the Chromatic," dealt with
>concepts related to combinatoriality as applied to pitch-space,
>rather than pitch-class space.

Whoa, stop the press! Pitch space? How trendy, how neo-neo.
My, pitch space and music. Dear me.

-Carl

>http://tinyurl.com/lpjh

Oh, and this sounds cool.

>[19] The next presentation was by John Rahn (jrahn@u.washington.edu)
>of the University of Washington. Rahn's talk was titled "Some Recent
>Developments in Mathematics Applied to Music Theory." Rahn presented
>summaries of two large and significant contributions to the field of
>mathematics and the arts. The first book he discussed was Guerino
>Mazzola's The Topos of Music (1999). Drawing on Mazzola's earlier
>work, this book puts forth a classification theory of musical objects.
>It contains further topologies for various musical domains, including
>melody, rhythm, and harmony, especially in the contexts of counterpoint
>and modulation. It also deals significantly with a theory of musical
>performance; using Lie algebra, it extends its concepts to object-
>oriented software environments. The second book Rahn discussed was
>Michael Leyton's A Generative Theory of Shape (2001). Dealing not
>only with music but with the arts in general, Leyton's book presents
>a theory of (complex) shape with regard to two properties of
>intelligence: transfer of structure and recoverability of the
>generative operations, with the ultimate goal of deriving
>understandability from complexity. To this end, Leyton defines a
>class of unfolding groups, which "unfold" complex shapes from their
>maximally collapsed versions.

NB Gene, they're having another one in 04, in Phoenix.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

I haven't had time to digest this yet, but it seems there is a trend
to move from abelian to nonabelian tone groups. How this can possibly
make sense beats me, but I also find this (note that SL2(Z) is
nonabelian)

7] The second presentation session opened with a talk by Thomas Noll
(noll@cs.tu-berlin.de), of the Technical University Berlin. His
presentation was titled "A Mathematical Model for Tone Apperception."
It applied mathematical concepts to apperception psychology, as a
means of modeling musical ambiguity. Drawing on ideas of Wilhelm
Wundt (such as Wundt's eye metaphor), Noll described apperception in
terms of inner vision, wherein ideas enter the scope of attention
either passively or actively. This process may be modeled via a
symplectic geometry. It has implications for Riemann's musical motion
metaphors, Lewin's GIS-model, Gollin's transformational approach to
enharmonicity, and Meeus' Neo-Rameauean approach to tonal
progressions. Specifically, Noll invoked the discrete subgroup of
integral symplectic matrices SL2(Z), wherein the upper triangle
matrix corresponds to the "passive fifth step" (e.g., I/I -> V/I),
while the lower triangle matrix corresponds to the "active fifth
shift" (e.g., I/I -> I/V).

I also note that people are now using hidden Markov models, which
I've long thought might be worth exploring. I also see Lie algebas
being used, which makes no sense to me at this point. Other people
are using combinatorics, wreath products, and the like which does
make sense. Someone is also using discrete Fourier transforms, and I
don't know why. I've tried to figure out how Pontryagin duality,
which in abstract harmonic analysis translates the category of
finitely generated abelian groups into compact groups, could possibly
be useful, and came up dry.

This posting probably made no sense. Sorry about that. :)

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> NB Gene, they're having another one in 04, in Phoenix.

Noted.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

I didn't find the attempt to show this was useful convincing.

on 8/30/03 10:19 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> I didn't find the attempt to show this was useful convincing.

I didn't find the assertion of unconvincing utility meaningful, in the
absense of a clear referent.

-Kurt

>> I didn't find the attempt to show this was useful convincing.
>
>I didn't find the assertion of unconvincing utility meaningful,
>in the absense of a clear referent.

I think Gene's referring to the article mentioned in the subject,
and linked to at the start of the thread...

http://societymusictheory.org/mto/issues/mto.03.9.3/mto.03.9.3.peck.pdf

-Carl

On Sunday 31 August 2003 00:10, Gene Ward Smith wrote:
>
> This posting probably made no sense. Sorry about that. :)

It makes sense, thanks for providing your thoughts

Carlos

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> 7] The second presentation session opened with a talk by Thomas
Noll
> (noll@c...), of the Technical University Berlin. His
> presentation was titled "A Mathematical Model for Tone
Apperception."
> It applied mathematical concepts to apperception psychology, as a
> means of modeling musical ambiguity. Drawing on ideas of Wilhelm
> Wundt (such as Wundt's eye metaphor), Noll described apperception
in
> terms of inner vision, wherein ideas enter the scope of attention
> either passively or actively. This process may be modeled via a
> symplectic geometry. It has implications for Riemann's musical
motion
> metaphors, Lewin's GIS-model, Gollin's transformational approach
to
> enharmonicity, and Meeus' Neo-Rameauean approach to tonal
> progressions. Specifically, Noll invoked the discrete subgroup of
> integral symplectic matrices SL2(Z), wherein the upper triangle
> matrix corresponds to the "passive fifth step" (e.g., I/I -> V/I),
> while the lower triangle matrix corresponds to the "active fifth
> shift" (e.g., I/I -> I/V).
>
> I also note that people are now using hidden Markov models, which
> I've long thought might be worth exploring. I also see Lie algebas
> being used, which makes no sense to me at this point.
<snip>
> This posting probably made no sense. Sorry about that. :)

Oh, it did. Actually, I have been reading stuff by Thomas Noll about
Lie algebas some time ago (it appears in The Topos of Music, too).
It seemed to make sense, but I am not through yet (Lie algebras are
hard stuff for me :-().
There was a Lie algebra structure defined for the 2-dimensional
Euler grid. The quotient with the commutator turned out to be
isomorphic to Z12. If I understand it correctly, this leads to a
description of enharmonicity as a kind of "uncertainty relation",
modeled by the commutator.
Questions worth raising would be: how canonical is this concrete Lie
algebra structure, are there others, and if yes, do they all have a
quotient isomorphic to Z12? Maybe this is already answered in the
paper, but I did not get this far...

Hans Straub

--- In tuning@yahoogroups.com, "alternativetuning"
<alternativetuning@y...> wrote:

> I read this article by Peck
>
> http://societymusictheory.org/mto/issues/mto.03.9.3/toc.9.3.html
>
> and wonder now what temperaments map for Klein-bottles?
>
> Gabor

hi gabor. klein bottles are non-orientable surfaces, which would mean
that, for example, a major triad and a minor triad could not be
distinguished from one another. the "dicot" or "neutral thirds" 5-
limit temperament, defined by 25:24 vanishing, so that the generator
must represent both the major third and minor third, has the property
that a major triad and a minor triad can't be distinguished from one
another. so the cylinder for this temperament can't be distinguished
from its mirror-image. now if one were to close this temperament, say
by rendering it in 7-equal or 10-equal, one would be connecting the
two ends of the cylinder, and since the handedness of the cylinder is
irrelevant, one could easily connect one end to the other from
the "inside", obtaining a klein bottle. of course, there's no reason
to do this rather than connecting the ends in the normal way to form
a torus, but it seems to me that if one really wanted a temperament
to map the 5-limit lattice to a klein bottle, 7-equal and 10-equal
would work.

gene, any ideas?

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> hi gabor. klein bottles are non-orientable surfaces, which would mean
> that, for example, a major triad and a minor triad could not be
> distinguished from one another.

I think this is mistaken; the idea seems to be to have more
distinctions, not fewer.

> gene, any ideas?

You seem to have a different take on this; the paper took a point of
view which required a Balzano setup. Anyway, I don't think the
geometry helps and may serve merely to confuse; if you look at the
group and "generalized interval space" he constructs, that really
tells the story.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> > hi gabor. klein bottles are non-orientable surfaces, which would
mean
> > that, for example, a major triad and a minor triad could not be
> > distinguished from one another.
>
> I think this is mistaken; the idea seems to be to have more
> distinctions, not fewer.

how is it mistaken, exactly? there is no distinction between mirror-
images on a klein bottle because one can be transformed into the
other through mere translation.

> > gene, any ideas?
>
> You seem to have a different take on this; the paper took a point of
> view which required a Balzano setup. Anyway, I don't think the
> geometry helps and may serve merely to confuse; if you look at the
> group and "generalized interval space" he constructs, that really
> tells the story.

the original question was about temperaments -- do you have a better
answer than the one i gave?