Re: [tuning-math] Digest Number 2323 - Maybe map it in 3D as a spiral around a cone

Maybe the most interesting and informative way to approach the topology would be to position the "notes" on a spiral around a "deformed" cylinder.

I am not sure what the deformation of the cylinder should be. Maybe a cone, or a cone with "curves", or maybe even a torus with curves.

It may be found that there is a particular proportion of diameter, to height, which can resolve the paradoxes if mapped onto a particular curved surface of a "distorted" cylinder.

My attempts at this resulted in a spiral around a cylinder, and the only "mapped" component of it was in the positions around the cylinder.

The height to diameter value was just arbitrary, as was the choice of a cylinder instead of a cone or curved surface; nevertheless this page shows what I did, and could serve as a good starting point for further visualisations.

see this page for QT video and to download Mac interactive application.

http://www.lucytune.com/new_to_lt/recipe.html

Maybe some very smart mathematician/programmer can develop something enlightening in vrml or similar;-)

On 9 Dec 2008, at 17:22, tuning-math@yahoogroups.com wrote:

> tuning-math
> Messages In This Digest (2 Messages)
> 1a.
> Re: Geometry of Numbers From: Paul H
> 1b.
> Re: Geometry of Numbers From: Graham Breed
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> 1a.
> Re: Geometry of Numbers
> Posted by: "Paul H" phjelmstad@msn.com paulhjelmstad
> Mon Dec 8, 2008 2:26 pm (PST)
>
> --- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
> >
> > 2008/12/5 Paul H <phjelmstad@...>:
> > <snip>
> > > Of course going to continuous Lie groups is a lot harder and
> > > involves derivatives and stuff and of course going from
> > > finite to infinite groups and lattices is also more subtle
> > >
> > > (Is it the same thing? Is a continuous group really like
> > > an infinite group? I guess so...)
> >
> > I'm missing (at least) one of your messages in this thread, so I'm
> not
> > sure of the context.
> >
> > Anyway, the lattices I started out talking about, with reference to
> > the geometry of numbers, are free abelian groups. Nothing else
> > special about them as groups that I know of.
> >
> > Octave-equivalent equal temperaments are cyclic groups. The fashion
> > now is to do the calculations with octave-specific vectors because
> > removing the octaves causes more problems than it solves. If
> there's
> > a theory of octave-equivalent regular temperaments it might be
> > interesting and Gene's done some work towards it.
> >
> > Maybe the odd-limit lattices have more interesting group structure.
> > Probably that's what you've been talking about. There's a problem
> in
> > the 9-limit in that 9:8 and 3:2, which are 9:1 and 3:1
> > octave-equivalently, are both primary consonances. You can solve
> this
> > with "wormholes" but I don't know if the result is a valid lattice
> > norm. For a Euclidean metric to work the space would have to be
> > curved, so that 1:1, 3:2, and 9:8 don't lie on a straight line.
> That
> > may disqualify the resulting algebra from being a lattice.
> >
> > There are also 2-D octave-equivalent lattices that I use to
> visualize
> > harmony, but without lattice distance having a special meaning. See
> >
> > See http://x31eq.com/lattice.htm#alt7limit
> >
> > and on down the page. I don't think they're interesting from the
> > point of view of group theory because they end up as 2-D square
> > lattices. Maybe there's a clever metric you can apply to one of
> them
> > that gives sensible complexity results.
> >
> >
> >
> > Graham
>
> Interesting. So, is a continuous group, really just an infinite group?
> I am trying to make the leap from discrete groups to Lie groups etc.
>
> Is there any relationship at all between like the Leech Lattice
> and the lattices discussed on this newsgruoup?
>
> Thanks
>
> PGH
>
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> Messages in this topic (24)
> 1b.
> Re: Geometry of Numbers
> Posted by: "Graham Breed" gbreed@gmail.com x31eq
> Mon Dec 8, 2008 5:13 pm (PST)
>
> 2008/12/9 Paul H <phjelmstad@msn.com>:
>
> > Interesting. So, is a continuous group, really just an infinite > group?
> > I am trying to make the leap from discrete groups to Lie groups etc.
>
> I checked Wikipedia, and it says that Lie groups are differentiable
> manifolds. I know about them because they're used in physics. The
> tuning space with a Euclidean metric (giving e.g. TOP-RMS error) is a
> differentiable manifold. It's Euclidean space after all. The same
> vectors with the metric for TOP-max error (dual to a taxicab metric)
> don't give a differentiable manifold. However I assume it is a
> continuous group (I don't know the terminology).
>
> This ties in with my complaint about taxicab metrics (and their duals)
> not giving angles. But then I only know about differentiable
> manifolds and there may be other ways of getting geometries to work.
> Measuring complexity for higher rank temperaments in TOP-max space
> means finding the minimax equivalent of an angle. The highest
> absolute coefficient in the weighted wedgie looks good but I don't
> have any theory to say why.
>
> A discrete group can be infinite, can't it? Like the integers under > addition?
>
> > Is there any relationship at all between like the Leech Lattice
> > and the lattices discussed on this newsgruoup?
>
> I don't know anything about that.
>
> Graham
>
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> Messages in this topic (24)
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Charles Lucy
lucy@lucytune.com

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