Geometry of Numbers

I'm looking up pure mathematics again. I tried to find a good
algorithm for integer bases of a null space (which is to say finding
unison vectors). It turns out that the LLL algorithm can return such
along with the reduced basis for the lattice itself. But I don't
understand how.

Anyway, I also found a chapter about Minkowski's work on quadratic forms:

http://www.springerlink.com/content/xg7198587164p330/?p=50746fcb2def4c2a84c0c19d015a1a21&pi=8

I can read that because my employer has access to articles on this site.

This is all covered by the geometry of numbers. The premise should be
familiar to us: that geometric models can be used to solve arithmetic
problems. We already know that good equal temperaments are analogous
to short vectors in a lattice. The "parametric quadratic badness" I
covered in different places -- most simply
http://x31eq.com/lattice.pdf -- is a positive definite quadratic form.
As such, finding good equal temperaments is an example of something
called "the minimum of a positive (definite quadratic) form". It's
something Minkowski and others looked at. And also strongly related
to something called "reduction theory".

Search Google Books for the minimum of a quadratic form and you'll
hopefully find a book titled "Mathematics of the 19th Century" showing
a formula for the upper bound of the minimum. If we can work out how
to apply it, this will tell us a badness threshold below which we're
bound to find something. It can be explained in geometric terms as
related to the volume of a lattice cell. You know that a ball around
the origin must include at least one point if it's bigger than a full
cell's volume.

There may be other useful results but I'm still finding it difficult
work. I haven't found anything relating to higher rank temperaments.
In geometric terms, these (classes) are the planes (for rank 2) that
pass through lattice points and get close to the origin.
Alternatively, you apply a rectangular matrix to the quadratic form
instead of a vector and take the determinant. Although I've shown a
strong relationship between the badness of equal temperaments and that
of higher rank temperaments that involve them I haven't found
independent verification. And I don't know how many equal
temperaments you need to look at to find the best 10 distinct
temperaments of a higher rank. From our point of view the numbers are
small enough that the equal temperament case on its own isn't that
interesting.

I'm sure there are interesting things in 19th Century work on linear
least squares problems as well. I still haven't found a statement of
my formula for the optimal RMS error which is bound to have come up
somewhere. Introductory texts are only interested in the solutions
and active research is way beyond this level.

Graham

Hi Graham,

Don't know if this helps, but there are two chapters on
lattice basis reduction here:
http://www.cs.berkeley.edu/~karp/greatalgo/
The one that's available covers LLL.

-Carl

At 05:02 AM 12/2/2008, you wrote:
>I'm looking up pure mathematics again. I tried to find a good
>algorithm for integer bases of a null space (which is to say finding
>unison vectors). It turns out that the LLL algorithm can return such
>along with the reduced basis for the lattice itself. But I don't
>understand how.
>
>Anyway, I also found a chapter about Minkowski's work on quadratic forms:
>
>http://www.springerlink.com/content/xg7198587164p330/?p=50746fcb2def4c
>2a84c0c19d015a1a21&pi=8
>
>I can read that because my employer has access to articles on this site.
>
>This is all covered by the geometry of numbers. The premise should be
>familiar to us: that geometric models can be used to solve arithmetic
>problems. We already know that good equal temperaments are analogous
>to short vectors in a lattice. The "parametric quadratic badness" I
>covered in different places -- most simply
>http://x31eq.com/lattice.pdf -- is a positive definite quadratic form.
> As such, finding good equal temperaments is an example of something
>called "the minimum of a positive (definite quadratic) form". It's
>something Minkowski and others looked at. And also strongly related
>to something called "reduction theory".
>
>Search Google Books for the minimum of a quadratic form and you'll
>hopefully find a book titled "Mathematics of the 19th Century" showing
>a formula for the upper bound of the minimum. If we can work out how
>to apply it, this will tell us a badness threshold below which we're
>bound to find something. It can be explained in geometric terms as
>related to the volume of a lattice cell. You know that a ball around
>the origin must include at least one point if it's bigger than a full
>cell's volume.
>
>There may be other useful results but I'm still finding it difficult
>work. I haven't found anything relating to higher rank temperaments.
>In geometric terms, these (classes) are the planes (for rank 2) that
>pass through lattice points and get close to the origin.
>Alternatively, you apply a rectangular matrix to the quadratic form
>instead of a vector and take the determinant. Although I've shown a
>strong relationship between the badness of equal temperaments and that
>of higher rank temperaments that involve them I haven't found
>independent verification. And I don't know how many equal
>temperaments you need to look at to find the best 10 distinct
>temperaments of a higher rank. From our point of view the numbers are
>small enough that the equal temperament case on its own isn't that
>interesting.
>
>I'm sure there are interesting things in 19th Century work on linear
>least squares problems as well. I still haven't found a statement of
>my formula for the optimal RMS error which is bound to have come up
>somewhere. Introductory texts are only interested in the solutions
>and active research is way beyond this level.
>
>
> Graham
>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
Hi Carl,

I don't usually top-post but you did:) So I am going to read this
lecture and try to get back in the loop here. I need to ask a really
naive question, though, to you or Graham, to ask what the connection
is between these kinds of lattices and group-theoretical kinds
of lattices, such as the Leech Lattice, and whether or not those
lattices have any value in a tuning-math framework, (what the heck,
why not E8 while we are at it?)

> Hi Graham,
>
> Don't know if this helps, but there are two chapters on
> lattice basis reduction here:
> http://www.cs.berkeley.edu/~karp/greatalgo/
> The one that's available covers LLL.
>
> -Carl
>
> At 05:02 AM 12/2/2008, you wrote:
> >I'm looking up pure mathematics again. I tried to find a good
> >algorithm for integer bases of a null space (which is to say
finding
> >unison vectors). It turns out that the LLL algorithm can return
such
> >along with the reduced basis for the lattice itself. But I don't
> >understand how.
> >
> >Anyway, I also found a chapter about Minkowski's work on quadratic
forms:
> >
> >http://www.springerlink.com/content/xg7198587164p330/?
p=50746fcb2def4c
> >2a84c0c19d015a1a21&pi=8
> >
> >I can read that because my employer has access to articles on this
site.
> >
> >This is all covered by the geometry of numbers. The premise
should be
> >familiar to us: that geometric models can be used to solve
arithmetic
> >problems. We already know that good equal temperaments are
analogous
> >to short vectors in a lattice. The "parametric quadratic badness"
I
> >covered in different places -- most simply
> >http://x31eq.com/lattice.pdf -- is a positive definite quadratic
form.
> > As such, finding good equal temperaments is an example of
something
> >called "the minimum of a positive (definite quadratic) form". It's
> >something Minkowski and others looked at. And also strongly
related
> >to something called "reduction theory".
> >
> >Search Google Books for the minimum of a quadratic form and you'll
> >hopefully find a book titled "Mathematics of the 19th Century"
showing
> >a formula for the upper bound of the minimum. If we can work out
how
> >to apply it, this will tell us a badness threshold below which
we're
> >bound to find something. It can be explained in geometric terms as
> >related to the volume of a lattice cell. You know that a ball
around
> >the origin must include at least one point if it's bigger than a
full
> >cell's volume.
> >
> >There may be other useful results but I'm still finding it
difficult
> >work. I haven't found anything relating to higher rank
temperaments.
> >In geometric terms, these (classes) are the planes (for rank 2)
that
> >pass through lattice points and get close to the origin.
> >Alternatively, you apply a rectangular matrix to the quadratic form
> >instead of a vector and take the determinant. Although I've shown
a
> >strong relationship between the badness of equal temperaments and
that
> >of higher rank temperaments that involve them I haven't found
> >independent verification. And I don't know how many equal
> >temperaments you need to look at to find the best 10 distinct
> >temperaments of a higher rank. From our point of view the numbers
are
> >small enough that the equal temperament case on its own isn't that
> >interesting.
> >
> >I'm sure there are interesting things in 19th Century work on
linear
> >least squares problems as well. I still haven't found a statement
of
> >my formula for the optimal RMS error which is bound to have come up
> >somewhere. Introductory texts are only interested in the solutions
> >and active research is way beyond this level.
> >
> >
> > Graham
> >
>

2008/12/4 Carl Lumma <carl@lumma.org>:
> Hi Graham,
>
> Don't know if this helps, but there are two chapters on
> lattice basis reduction here:
> http://www.cs.berkeley.edu/~karp/greatalgo/
> The one that's available covers LLL.

It's a good reference, but it doesn't mention null spaces, or exactly
what kinds of lattices the algorithm works for.

After that it talks about integer programming which my be relevant.

Graham

2008/12/4 Paul H <phjelmstad@msn.com>:
> Hi Carl,
>
> I don't usually top-post but you did:) So I am going to read this
> lecture and try to get back in the loop here. I need to ask a really
> naive question, though, to you or Graham, to ask what the connection
> is between these kinds of lattices and group-theoretical kinds
> of lattices, such as the Leech Lattice, and whether or not those
> lattices have any value in a tuning-math framework, (what the heck,
> why not E8 while we are at it?)

What's a Leech Lattice?-)

These are the "lattices" as defined in group theory, although you can
also see them as linear algebra with integers. A group with a
metric/norm is the simplest definition. The norm could give you the
pitch difference of an interval or its complexity in ratio space.
That complexity relates to the kind of error you want to measure.

This "val space" as Gene called it, where the lattice points are ET
mappings or vals, is the dual of ratio space. But, as a continuous
space, each point is a rank 1 temperament class with a tuning, AKA
"tuned temperament" AKA "equal temperament". The simplest metric is
the usual Euclidean distance which gives the mean squared error. You
add weighting by using a rectangular instead of square lattice. A
tuned temperament's error is the distance from its point in this space
(probably not on the lattice) to a point (never on the lattice for
harmonic timbres) representing just intonation: the JI point.

The full temperament class, including octave stretches, is a line
drawn through the origin and the defining val point. In linear
algebra terminology, the "span" of the val. Calling this a
"temperament" may raise hackles because most (an infinite proportion)
of the points along the line are useless temperaments. The complexity
of a temperament class is the distance from the val to the origin.
The optimal error is the smallest distance from the temperament class
line to the JI point. This also happens to be the sine of the angle
between the temperament class line and the span of the JI point (the
JI line).

A rank r temperament class is defined by r equal temperament mappings
(vals). The set of possible tunings is the span of those vals. The
complexity is the distance of the line (plane, etc) joining the vals
from the origin. The optimal error is the distance from the plane
(hyperplane) spanning the vals to the JI point or the angle between
the (hyper)plane and the JI line.

The scalar badness I defined is the distance from a val to the JI
line. For a Euclidean metric, it's equal to the error times the
complexity. You can model this geometrically by projecting the val
parallel to the JI line onto a (hyper)plane that includes the origin
and is perpendicular to the JI line. Then the badness is the distance
to the origin. The parametric badness is a generalization of this,
giving an arbitrary balance between error and complexity. It's
defined by a quadratic form.

I got the geometric model for higher rank temperaments in badness
space wrong before. I'm not sure what it comes out as, but the span
of a set of vals still defines a temperament class, and the distance
to the origin is the badness.

The trouble with LLL reduction is that it I believe it doesn't work
with arbitrary quadratic forms. Only with a "Euclidean metric" which
seems to be the Euclidean distance on a rectangular lattice. For
these badness spaces, the metric can still be Euclidean, but the
lattice gets skewed, or something. A lattice basis reduction
algorithm for quadratic norms would be nice as we could feed in the
prime intervals and get out the optimal equal temperament mapping.

LLL reduction can be used for finding the simplest set of ET mappings
or unison vectors for a given temperament class. However, it doesn't
remove torsion. I'm wondering if a Gram-Schmidt orthogonalization can
be guaranteed to reveal torsion...

Graham

Sorry, the complexity of a temperament class is the (hyper)volume of
the (hyper)parallelapiped with the vals and origins as the corners.
Not the distance to the origin. Similarly for the parametric scalar
badness.

Graham

>Hi Carl,
>
>I don't usually top-post but you did:) So I am going to read this
>lecture and try to get back in the loop here. I need to ask a really
>naive question, though, to you or Graham, to ask what the connection
>is between these kinds of lattices and group-theoretical kinds
>of lattices, such as the Leech Lattice, and whether or not those
>lattices have any value in a tuning-math framework, (what the heck,
>why not E8 while we are at it?)

Hi Paul! Around here the space usually dictates the lattice. We've
worked with tonespace, chordspace (the dual to chordspace), tuningspace,
valspace (which may or may not be tuningspace depending who you ask),
and more. Tonespace is the best-traveled, but it's really only been
tapped up to rank 3. It depends which higher-dimensional lattices are
the correct analogs of the rank 2 and 3 Dn lattices we're been using.
My guess is that E8 has too many symmetries. The idea of a tonespace
is that all consonant dyads and only consonant dyads have distance 1.

-Carl

>This "val space" as Gene called it, where the lattice points are ET
>mappings or vals, is the dual of ratio space.

I think you're calling tonespace "ratio space" here. I think I got
the bit about its dual wrong. What I called chordspace is I think
just a different lattice in tonespace. The lattice may be the
geometric dual of the tone lattice. See:
http://lumma.org/tuning/gws/sevlat.htm

-Carl

2008/12/4 Carl Lumma <carl@lumma.org>:
>>This "val space" as Gene called it, where the lattice points are ET
>>mappings or vals, is the dual of ratio space.
>
> I think you're calling tonespace "ratio space" here. I think I got
> the bit about its dual wrong. What I called chordspace is I think
> just a different lattice in tonespace. The lattice may be the
> geometric dual of the tone lattice. See:
> http://lumma.org/tuning/gws/sevlat.htm

"Ratio space" is what i thought it was called when I originally joined
the tuning list. A space containing ratios. For harmonic timbres
that'd be like a tone-space, but without each consonance needing a
distance of 1, as you said in the other message. As such it
generalizes very easily to higher dimensions.

I don't know what your chord-space is.

The idea of dual spaces is well established. It means that unison
vectors are the null space of a temperament class defined by mappings
and a set of mappings (vals) is the null space of a set of unison
vectors. Also, the wedge products derived either way are complements
of each other.

I think this kind of duality is called "Poincaré duality". That
should mean any formula in one space can be converted to work in the
other. However, I still don't know how to get TOP-RMS error and
complexity working via linear algebra as a function of more than one
unison vector. That's one of the niggling details I'd like to clear
up. It can be done by finding some mappings and applying the
tuning-space formulae, but that's cheating.

There are simple formulae involving wedge products so it can be done
that way. You can derive them from the geometric model.

Graham

Graham wrote;
>"Ratio space" is what i thought it was called when I originally joined
>the tuning list. A space containing ratios.

I'm sure it's been called many things, but tonespace is the most
widely used -- McLaren, Monzo, Erlich, and others.

>For harmonic timbres that'd be like a tone-space, but without each
>consonance needing a distance of 1,

Huh?

>I don't know what your chord-space is.

It's Gene's and I gave a link.

-Carl

2008/12/4 Carl Lumma <carl@lumma.org>:
> Graham wrote;
>>"Ratio space" is what i thought it was called when I originally joined
>>the tuning list. A space containing ratios.
>
> I'm sure it's been called many things, but tonespace is the most
> widely used -- McLaren, Monzo, Erlich, and others.

I see it's in Monz's dictionary, anyway, looking like ratio space.

>>For harmonic timbres that'd be like a tone-space, but without each
>>consonance needing a distance of 1,
>
> Huh?

From earlier in this thread, "The idea of a tonespace
is that all consonant dyads and only consonant dyads have distance 1."
A ratio space doesn't have this constraint. However, a ratio space
is only properly defined for harmonic timbres because otherwise the
"JI" intervals won't be ratios.

>>I don't know what your chord-space is.
>
> It's Gene's and I gave a link.

I must have missed it then. Is it something I'm likely to be interested in?

The only reference I can find is from Paul Erlich:

/tuning-math/message/13853

Graham

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >Hi Carl,
> >
> >I don't usually top-post but you did:) So I am going to read this
> >lecture and try to get back in the loop here. I need to ask a
really
> >naive question, though, to you or Graham, to ask what the
connection
> >is between these kinds of lattices and group-theoretical kinds
> >of lattices, such as the Leech Lattice, and whether or not those
> >lattices have any value in a tuning-math framework, (what the heck,
> >why not E8 while we are at it?)
>
> Hi Paul! Around here the space usually dictates the lattice. We've
> worked with tonespace, chordspace (the dual to chordspace),
tuningspace,
> valspace (which may or may not be tuningspace depending who you
ask),
> and more. Tonespace is the best-traveled, but it's really only been
> tapped up to rank 3. It depends which higher-dimensional lattices
are
> the correct analogs of the rank 2 and 3 Dn lattices we're been
using.
> My guess is that E8 has too many symmetries. The idea of a
tonespace
> is that all consonant dyads and only consonant dyads have distance
1.
>
> -Carl

Cool. Is there any chance this work might tie into a TOE?

Is Dn Lattice related to Dn Lie Groups, Dynkin diagrams, and so forth?

... i mean, just the integration of music theory, mathematics,
physics theology and cosmology:)

Graham wrote:
>From earlier in this thread, "The idea of a tonespace is that all
>consonant dyads and only consonant dyads have distance 1." A ratio
>space doesn't have this constraint.

Show me.

>However, a ratio space
>is only properly defined for harmonic timbres because otherwise the
>"JI" intervals won't be ratios.

You say stuff like this in your pdfs too, but from my perspective
it's way out in left field. Simple rationals are approximately the
most consonant intervals for any pitched timbre, first off. Second
off, there's little consensus in the spectral consonance camp on how
to calculate the minima of spectral consonance for arbitrary timbres,
and I'm unaware of any experimental data showing that any of the
competing methods, including Sethares', actually works. Finally,
assuming that there was such a method and it produced significant
deviations from JI, it could presumably be incorporated into JI as
corrections to f0. And failing that, it could be incorporated by
extracting the appropriate irrational 'prime' intervals from the
timbre and building a tonesspace with those. In conclusion, I think
the issue is a non-starter, as far as mentioning it in passing in
theory documents or as one-sentence disclaimers in discussions.

>>>I don't know what your chord-space is.
>>
>> It's Gene's and I gave a link.
>
>I must have missed it then. Is it something I'm likely to be interested in?

It was even in the bit you quoted in your first reply:
http://lumma.org/tuning/gws/sevlat.htm

-Carl

Paul wrote:
>Cool. Is there any chance this work might tie into a TOE?

Physics TOE? I doubt it. Though I am hoping some of Lisi's extra
particles show up when the LHC finally starts running.

>Is Dn Lattice related to Dn Lie Groups,

Probably.

>Dynkin diagrams,

I don't know.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Paul wrote:
> >Cool. Is there any chance this work might tie into a TOE?
>
> Physics TOE? I doubt it. Though I am hoping some of Lisi's extra
> particles show up when the LHC finally starts running.
>
> >Is Dn Lattice related to Dn Lie Groups,
>
> Probably.
>
> >Dynkin diagrams,
>
> I don't know.
>
> -Carl

Okay let me research this. What then is your "Dn" Lattice?

Thx

- Paul
>

> Okay let me research this. What then is your "Dn" Lattice?

D3 lattice:
http://www.research.att.com/~njas/lattices/D3.html

According to this page, the E8 lattice is the union of
two D8 lattices:
http://www.valdostamuseum.org/hamsmith/E8.html

Here's Gene's theory page with much info:
http://lumma.org/tuning/gws/theory.htm

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > Okay let me research this. What then is your "Dn" Lattice?
>
> D3 lattice:
> http://www.research.att.com/~njas/lattices/D3.html
>
> According to this page, the E8 lattice is the union of
> two D8 lattices:
> http://www.valdostamuseum.org/hamsmith/E8.html
>
> Here's Gene's theory page with much info:
> http://lumma.org/tuning/gws/theory.htm
>
> -Carl

Great. So we are dealing with continuous groups.

There is some correspondence between some of the sporadic groups
and some of the Lie groups. I am into M12 and M24 especially.

In a sense M12 and stuff with Outer(Aut(S6)-> M12 construction
and discrete set theory is about only tempered systems

but I guess your geometric stuff is the continuous version of
these groups (and corresponding lattices)

As most ppl can see I am kind of stuck in discrete groups
and also 12-tET, but that is where most music theory lies
in the Western world anyway

I would love to see how unimodular lattices and such relate
to these lattices so I guess I will look at Genes page and
also refresh on D3 especially

I know that SPLAG talks about the same theory that is discussed
here with the Neimeier stuff and cubic octahedrons and the
hexanies and octads etc I think that is right on page 3 of SPLAG
(Sphere Packings Lattices and Groups)

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...)

PGH

2008/12/5 Carl Lumma <carl@lumma.org>:
> Graham wrote:
> >From earlier in this thread, "The idea of a tonespace is that all
>>consonant dyads and only consonant dyads have distance 1." A ratio
>>space doesn't have this constraint.
>
> Show me.

I don't know how I'm supposed to show the absence of a constraint on a
concept without a formal definition. Tone-space, at least, is defined
and described here:

http://tonalsoft.com/enc/t/tone-space.aspx

Nothing about distances being 1 for consonances. And the second
diagram seems to show 6:5 further from the origin than 3:2 and 5:4.

This discussion started with the space of Tenney-weighted lattices.
Whatever you call such a space (with either Euclidean or taxi-cab
metric) it manifestly doesn't give all consonant dyads a distance of
1.

>>However, a ratio space
>>is only properly defined for harmonic timbres because otherwise the
>>"JI" intervals won't be ratios.
>
> You say stuff like this in your pdfs too, but from my perspective
> it's way out in left field. Simple rationals are approximately the
> most consonant intervals for any pitched timbre, first off. Second
> off, there's little consensus in the spectral consonance camp on how
> to calculate the minima of spectral consonance for arbitrary timbres,
> and I'm unaware of any experimental data showing that any of the
> competing methods, including Sethares', actually works. Finally,
> assuming that there was such a method and it produced significant
> deviations from JI, it could presumably be incorporated into JI as
> corrections to f0. And failing that, it could be incorporated by
> extracting the appropriate irrational 'prime' intervals from the
> timbre and building a tonesspace with those. In conclusion, I think
> the issue is a non-starter, as far as mentioning it in passing in
> theory documents or as one-sentence disclaimers in discussions.

You're welcome to your opinion. It won't stop me covering them.

You've got a lot of arguments squashed into one paragraph there, so
I'll have to paraphrase them.

- Simple ratios are always most consonant

Maybe, but this is something you're stating without evidence. And it
depends on "pitched timbres" which you defined on the tuning list as
having harmonic or near-harmonic partials. With that definition it's
a tautology.

- No consensus on calculating

Sure, and not much work done in the field either. Any method based on
the critical bandwidth will give coincident partials as roughness
minima, though. That's all I need to optimize the temperament.

- Unaware of experimental data

The first edition of "Tuning, Timbre, Spectrum, Scale" gave figures
for the tuning and timbre of gamelans. The second edition has more
details on Thai classical music, which I've now read as PDFs. That
gives a very good match to 7-equal for mixed timbres including those
of in inharmonic instrument. (By the usual definition, a pitched
instrument.)

Here's a link in case you can read it as well:

http://www.springerlink.com/content/ww1922243836p375/?p=86551bf884f4412885032d2d674a244d&pi=14

- Corrections to f0

You can map anything to the prime numbers, and you can also not
bother. It doesn't make much sense to talk about ratios if they
aren't even vaguely the right size.

- Irrational "prime" intervals

Yes, that's exactly what I do.

- Mention in passing

My standard test code includes full calculations for the partials of
ideal tubulongs. I've posted the results for a gamelan timbre on one
of the mailing lists. I helped Tony Salinas find a tuning for his
conical bells. I can work out other examples if you'd like.

>>>>I don't know what your chord-space is.
>>>
>>> It's Gene's and I gave a link.
>>
>>I must have missed it then. Is it something I'm likely to be interested in?
>
> It was even in the bit you quoted in your first reply:
> http://lumma.org/tuning/gws/sevlat.htm

If you're going to link to a page that has no mention of the term it's
defining, it isn't enough to say "I gave a link". You have to
actually say what that link has to do what you were talking about. So
the lattice in there original to Gene is the BCC one with Euclidean
norm. Is that "chord space"?

Graham

--- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> 2008/12/5 Carl Lumma <carl@...>:
> > Graham wrote:
> > >From earlier in this thread, "The idea of a tonespace is that
> > >all consonant dyads and only consonant dyads have distance 1."
> > >A ratio space doesn't have this constraint.
> >
> > Show me.
>
> I don't know how I'm supposed to show the absence of a constraint
> on a concept without a formal definition.

You're the one talking about ratio space. I've seldom heard
the term before. All the An lattices we've discussed for the
past 11 years make all consonances the same length, and have
no edges that are shorter, hence I say they have length 1.
Gene gives the "Hahn norm" which gives those lengths here:
http://lumma.org/tuning/gws/sevlat.htm

> Tone-space, at least, is defined and described here:
>
> http://tonalsoft.com/enc/t/tone-space.aspx
>
> Nothing about distances being 1 for consonances. And the second
> diagram seems to show 6:5 further from the origin than 3:2 and 5:4.

Those are Zn rectangular lattices, which have always been
deprecated around here for precisely that reason.

> This discussion started with the space of Tenney-weighted
> lattices. Whatever you call such a space (with either Euclidean
> or taxi-cab metric) it manifestly doesn't give all consonant
> dyads a distance of 1.

That's Tenney space. Is that what you mean by ratio space then?

> You've got a lot of arguments squashed into one paragraph there,
> so I'll have to paraphrase them.
>
> - Simple ratios are always most consonant
>
> Maybe, but this is something you're stating without evidence.
> And it depends on "pitched timbres" which you defined on the
> tuning list as having harmonic or near-harmonic partials. With
> that definition it's a tautology.

That's right, both phenomena have the same cause.

> - Unaware of experimental data
>
> The first edition of "Tuning, Timbre, Spectrum, Scale" gave figures
> for the tuning and timbre of gamelans.

As I pointed out on the tuning list, mapping the minima for a
timbre directly to scale tones, as Sethares so often does
throughout his book, is meaningless. I'm not the first to
criticize his gamelan analysis, either.

> The second edition has more
> details on Thai classical music, which I've now read as PDFs.
> That gives a very good match to 7-equal for mixed timbres
> including those of in inharmonic instrument. (By the usual
> definition, a pitched instrument.)

Weak evidence even if it's correct. There's no need for
musicology here. The best evidence would be obtained by
finding consonance minima for a few arbitrary timbres and
then having subjects rank the intervals.

> My standard test code includes full calculations for the partials
> of ideal tubulongs. I've posted the results for a gamelan timbre
> on one of the mailing lists. I helped Tony Salinas find a tuning
> for his conical bells. I can work out other examples if you'd
> like.

The experimental protocol mentioned above is easily conducted
via mailing list. Meanwhile, I'm unaware that either Indonesian
ensembles or Tony have any aesthetic reason to (having spoken
with him), or practical success in, minimizing beating.

> > It was even in the bit you quoted in your first reply:
> > http://lumma.org/tuning/gws/sevlat.htm
>
> If you're going to link to a page that has no mention of the
> term it's defining, it isn't enough to say "I gave a link".
> You have to actually say what that link has to do what you
> were talking about. So the lattice in there original to Gene
> is the BCC one with Euclidean norm. Is that "chord space"?

Yes.

-Carl

I wrote:
> > > http://lumma.org/tuning/gws/sevlat.htm
> >
> > If you're going to link to a page that has no mention of the
> > term it's defining, it isn't enough to say "I gave a link".
> > You have to actually say what that link has to do what you
> > were talking about. So the lattice in there original to Gene
> > is the BCC one with Euclidean norm. Is that "chord space"?
>
> Yes.

Er, no, sorry. It's just a cubic lattice. The BCC lattice
he talks about is (octave-equivalent) valspace.

-Carl

2008/12/5 Carl Lumma <carl@lumma.org>:
> I wrote:
>> > > http://lumma.org/tuning/gws/sevlat.htm

> Er, no, sorry. It's just a cubic lattice. The BCC lattice
> he talks about is (octave-equivalent) valspace.

It is interesting, anyway. Euclidean distances on the FCC lattice use
a quadratic norm:

(2 1 1)
(1 2 1)
(1 1 2)

which is like a variance but adding instead of subtracting. So <x2> +
<x>2 instead of <x2> - <x>2 in a crude ASCII form. He then identifies
the dual as this BCC metric:

(-1 1 1 )
( 1 -1 1)
( 1 1 -1)

Is it what he used for geometric complexity? I never did work that
out but I don't think I tried this. I'm not sure if it's equivalent
to a standard-deviation type metric. I think there are some
equivalences but I forget how it works out.

Anyway, my scalar badness is the standard deviation of the weighted,
octave-specific mappings. That can work in octave-equivalent terms if
you make the number of steps to the octave a special case ... somehow
or other. I don't know if there's an approximate mapping for this
kind of space that re-constructs the missing octave mapping. A
straight standard deviation of the octave-equivalent parts doesn't
work, anyway.

I don't know how to find these dual metrics. Maybe Gene does, or
maybe he happens to know that FCC and BCC are dual. I'd hope that the
dual of an odd-limit metric would look something like the FCC lattice.
I haven't worked out odd-limit metrics yet, but I think they can be
done, similar to the Farey limits. For that matter, it'd be nice to
know what dual lattice the Farey limits or standard deviation or
errors imply.

Graham

2008/12/5 Paul H <phjelmstad@msn.com>:
<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

--- 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

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

--- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> 2008/12/9 Paul H <phjelmstad@...>:
>
> > 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.

I also perused Wikipedia, and it appears at least that infinite
groups can be continuous, of course, the obvious case of the
infinite circle of fifths...

> A discrete group can be infinite, can't it? Like the integers
under addition?

Right.

> > 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
>

Well, its just another lattice, I have SPLAG, so all these
lattices are talked about in there, its based on M24, it
would be fun if there was a lattice based on M12, which I am in
love with:) I wish I had more time for all of this...anyone
want to pay me to study this stuff?: Somehow I don't think
grad schools are putting a lot of money into mathematical music
theory these days, given the state of the economy.....

PGH

--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@> wrote:
> >
> > 2008/12/9 Paul H <phjelmstad@>:
> >
> > > 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.
>
> I also perused Wikipedia, and it appears at least that infinite
> groups can be continuous, of course, the obvious case of the
> infinite circle of fifths...
>
> > A discrete group can be infinite, can't it? Like the integers
> under addition?
>
> Right.
>
> > > 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
> >
>
>
> Well, its just another lattice, I have SPLAG, so all these
> lattices are talked about in there, its based on M24, it
> would be fun if there was a lattice based on M12, which I am in
> love with:) I wish I had more time for all of this...anyone
> want to pay me to study this stuff?: Somehow I don't think
> grad schools are putting a lot of money into mathematical music
> theory these days, given the state of the economy.....
>
> PGH

Here is a connection from Wikipedia:

Thus, compact connected Lie groups have been completely classified.
There is a fruitful relation between infinite abstract groups and
topological groups: whenever a group à can be realized as a lattice
in a topological group G, the geometry and analysis pertaining to G
yield important results about Ã. A comparatively recent trend in the
theory of finite groups exploits their connections with compact
topological groups (profinite groups): for example, a single p-adic
analytic group G has a family of quotients which are finite p-groups
of various orders, and properties of G translate into the properties
of its finite quotients.

2008/12/10 Paul H <phjelmstad@msn.com>:

> I also perused Wikipedia, and it appears at least that infinite
> groups can be continuous, of course, the obvious case of the
> infinite circle of fifths...

Where are these defined? "Continous group" takes me to the Lie
algebra page and "infinite group" takes me to Group Theory. Neither
actually defines the term they link from.

If a spiral of fifths is continuous, then so is any regular
temperament in octave equivalent space if it isn't tuned as an equal
temperament.

>> > 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.
>
>
> Well, its just another lattice, I have SPLAG, so all these
> lattices are talked about in there, its based on M24, it
> would be fun if there was a lattice based on M12, which I am in
> love with:) I wish I had more time for all of this...anyone
> want to pay me to study this stuff?: Somehow I don't think
> grad schools are putting a lot of money into mathematical music
> theory these days, given the state of the economy.....

I'm unhappy that we're not attracting mathematicians. The old
dismissal is that we weren't attracting musicians, and the mathematics
wasn't interesting. I think the mathematics is interesting now even
if it's old. But still Gene's disappeared and that's that. It's
possible that all this is old work, and so it wouldn't get a research
grant, but I still can't find the references to prove that. If
there's anything original it should be a perfectly valid project.
This is real mathematics.

Then again, I don't think looking at interesting lattices will lead
anywhere. The actual lattices that come out are too simple. There
are some open problems:

- What is the parametric badness formula in tone-space? I tried
inverting the tuning space formula and it doesn't work. I can get
scalar badness from exterior algebra but not the parametric measure.

- What's the geometric model for regular temperament errors in tone-space?

- How many equal temperaments do you need to cover the best n rank 2 classes?

- What is the best formula for finding temperament classes?

- What complexity should go with TOP-max in higher ranks?

- How do we find a torsion-free set of unison vectors?

- Proof for TOP-RMS being the limit to infinity of equal weighted
Tenney limits, and other things like that.

Graham

--- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> 2008/12/10 Paul H <phjelmstad@...>:
>
> > I also perused Wikipedia, and it appears at least that infinite
> > groups can be continuous, of course, the obvious case of the
> > infinite circle of fifths...
>
> Where are these defined? "Continous group" takes me to the Lie
> algebra page and "infinite group" takes me to Group Theory. Neither
> actually defines the term they link from.

Some holes in Wikipedia I guess....

> If a spiral of fifths is continuous, then so is any regular
> temperament in octave equivalent space if it isn't tuned as an equal
> temperament.

Well, I work intuitively of course, but I have read hundreds of
articles. But I need a rudder, which is of course a teacher which
means of course going to school. I think it is exciting both
that rationals fill the space but not as densely as the reals
(I think this is Cantor stuff) and of course that high primes
approach irrationals etc but I venture into territory where
I have not the expertise...

> >> > 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.
> >
> >
> > Well, its just another lattice, I have SPLAG, so all these
> > lattices are talked about in there, its based on M24, it
> > would be fun if there was a lattice based on M12, which I am in
> > love with:) I wish I had more time for all of this...anyone
> > want to pay me to study this stuff?: Somehow I don't think
> > grad schools are putting a lot of money into mathematical music
> > theory these days, given the state of the economy.....
>
> I'm unhappy that we're not attracting mathematicians. The old
> dismissal is that we weren't attracting musicians, and the
mathematics
> wasn't interesting. I think the mathematics is interesting now even
> if it's old. But still Gene's disappeared and that's that. It's
> possible that all this is old work, and so it wouldn't get a
research
> grant, but I still can't find the references to prove that. If
> there's anything original it should be a perfectly valid project.
> This is real mathematics.

Oh those of little faith!!!!!!!!! Yes it is real. Maybe now
that pure math is being respected in physics again it will be
respected here as well.

> Then again, I don't think looking at interesting lattices will lead
> anywhere. The actual lattices that come out are too simple. There
> are some open problems:
>
> - What is the parametric badness formula in tone-space? I tried
> inverting the tuning space formula and it doesn't work. I can get
> scalar badness from exterior algebra but not the parametric measure.
>
> - What's the geometric model for regular temperament errors in tone-
space?
>
> - How many equal temperaments do you need to cover the best n rank
2 classes?
>
> - What is the best formula for finding temperament classes?
>
> - What complexity should go with TOP-max in higher ranks?
>
> - How do we find a torsion-free set of unison vectors?
>
> - Proof for TOP-RMS being the limit to infinity of equal weighted
> Tenney limits, and other things like that.
>
>
> Graham

Yes all great stuff. I feel that I need to focus on M12 a bit longer
and then dive into these things. My problem is that I read a lot
and then don't remember where I read certain things. Anyway the
Universe is all Music, although a person can get a little idealistic
and lose his focus:

According to the thinkers of the East, there are five different
intoxications:

Of beauty, youth and strength

Then the intoxication of wealth

The third is power, command, the power of ruling

And there is the fourth intoxication, which is the intoxication of
learning, of knowledge.

But all these four intoxications fade away just like stars before the
sun in the presence of the intoxication of music. The reason is that
it touches that deepest part of man's being. Music reaches farther
than any other impression from the external world can reach. And the
beauty of music is that it is both the source of creation and the
means of absorbing it. In other words, by music was the world
created, and by music it is withdrawn again into the source which has
created it.

* * * *

So its funny how important this work is and also how almost nobody
thinks that it is important at all!

PGH

--- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> 2008/12/4 Paul H <phjelmstad@...>:
> > Hi Carl,
> >
> > I don't usually top-post but you did:) So I am going to read this
> > lecture and try to get back in the loop here. I need to ask a
really
> > naive question, though, to you or Graham, to ask what the
connection
> > is between these kinds of lattices and group-theoretical kinds
> > of lattices, such as the Leech Lattice, and whether or not those
> > lattices have any value in a tuning-math framework, (what the
heck,
> > why not E8 while we are at it?)
>
> What's a Leech Lattice?-)
>
> These are the "lattices" as defined in group theory, although you
can
> also see them as linear algebra with integers. A group with a
> metric/norm is the simplest definition. The norm could give you the
> pitch difference of an interval or its complexity in ratio space.
> That complexity relates to the kind of error you want to measure.
>
> This "val space" as Gene called it, where the lattice points are ET
> mappings or vals, is the dual of ratio space. But, as a continuous
> space, each point is a rank 1 temperament class with a tuning, AKA
> "tuned temperament" AKA "equal temperament". The simplest metric is
> the usual Euclidean distance which gives the mean squared error.
You
> add weighting by using a rectangular instead of square lattice. A
> tuned temperament's error is the distance from its point in this
space
> (probably not on the lattice) to a point (never on the lattice for
> harmonic timbres) representing just intonation: the JI point.
>
> The full temperament class, including octave stretches, is a line
> drawn through the origin and the defining val point. In linear
> algebra terminology, the "span" of the val. Calling this a
> "temperament" may raise hackles because most (an infinite
proportion)
> of the points along the line are useless temperaments. The
complexity
> of a temperament class is the distance from the val to the origin.
> The optimal error is the smallest distance from the temperament
class
> line to the JI point. This also happens to be the sine of the angle
> between the temperament class line and the span of the JI point (the
> JI line).
>
> A rank r temperament class is defined by r equal temperament
mappings
> (vals). The set of possible tunings is the span of those vals. The
> complexity is the distance of the line (plane, etc) joining the vals
> from the origin. The optimal error is the distance from the plane
> (hyperplane) spanning the vals to the JI point or the angle between
> the (hyper)plane and the JI line.
>
> The scalar badness I defined is the distance from a val to the JI
> line. For a Euclidean metric, it's equal to the error times the
> complexity. You can model this geometrically by projecting the val
> parallel to the JI line onto a (hyper)plane that includes the origin
> and is perpendicular to the JI line. Then the badness is the
distance
> to the origin. The parametric badness is a generalization of this,
> giving an arbitrary balance between error and complexity. It's
> defined by a quadratic form.
>
> I got the geometric model for higher rank temperaments in badness
> space wrong before. I'm not sure what it comes out as, but the span
> of a set of vals still defines a temperament class, and the distance
> to the origin is the badness.
>
> The trouble with LLL reduction is that it I believe it doesn't work
> with arbitrary quadratic forms. Only with a "Euclidean metric"
which
> seems to be the Euclidean distance on a rectangular lattice. For
> these badness spaces, the metric can still be Euclidean, but the
> lattice gets skewed, or something. A lattice basis reduction
> algorithm for quadratic norms would be nice as we could feed in the
> prime intervals and get out the optimal equal temperament mapping.
>
> LLL reduction can be used for finding the simplest set of ET
mappings
> or unison vectors for a given temperament class. However, it
doesn't
> remove torsion. I'm wondering if a Gram-Schmidt orthogonalization
can
> be guaranteed to reveal torsion...
>
>
> Graham

Thanks, this is finally starting to sink in. And funny enough,
I am pretty good with Gram-Schmidt orgonalization/orthonormalization,
but my knowledge overall is pretty patchy...