Yahoo Tuning Groups Ultimate Backup tuning-math The upshot

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Thanks for summarizing, Gene.
>
> >(1) From both the Minkowski block and the hobbit, I got the {16/15,
> >15625/15552} block.
>
> Can you post a scala file for your solution?

! starling11.scl
Starling[11] hobbit <11 18 26 31| in <135 214 314 379| tuning
11
!
80.00000
231.11111
311.11111
391.11111
577.77778
622.22222
808.88889
888.88889
968.88889
1120.00000
1200.00000
!
!! prestarling11.scl
!!
! Starling[11] transversal <11 18 26 31|
! 11
!
! 25/24
! 144/125
! 6/5
! 5/4
! 25/18
! 36/25
! 8/5
! 5/3
! 125/72
! 48/25
! 2/1

I don't see mean variety when I run "show data", which is pretty sad for the most important statistic of them all. If I run "show intervals", I get that the average number of intervals per class is 3.2. Some statistics I consider important are that it has two major and two minor tetrads, and four major and four minor triads.

🔗Carl Lumma <carl@lumma.org>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Carl Lumma <carl@lumma.org>

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> At 03:32 PM 10/27/2010, you wrote:
> >
> >
> >--- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> >
> >> It looks like you win, but worryingly, prestarling hits
> >> the same number...
> >
> >Of course it does. Why does that worry you?
>
> Maybe if I knew what a traversal was I'd worry less.

A transversal is a mathematical term:

http://en.wikipedia.org/wiki/Transversal

I'm using it to characterize JI scales which when tempered give you the tempered scale you are looking for. For instance, the famous Ptolemy-Zarlino Intense Diatonic Syntonon is a transversal for Meantone[7] and also 7et.

One way to look at the situation here is that while the minimax tuning for starling, with eigenmonzos 2, 3 and 7 gives a much better tuning, it has rows with fractional monzsos; whereas the 5-limit tuning, with 2, 3, and 5 eigennmonzos, gives an integral matrix. And we like that, since we like to work with JI>

🔗Carl Lumma <carl@lumma.org>

Gene wrote:

>A transversal is a mathematical term:
>
>http://en.wikipedia.org/wiki/Transversal
>
>I'm using it to characterize JI scales which when tempered give you
>the tempered scale you are looking for. For instance, the famous
>Ptolemy-Zarlino Intense Diatonic Syntonon is a transversal for
>Meantone[7] and also 7et.

It seems you'd need something more than the concept described
on that page, but no doubt you've explained it already. I wish
you would occasionally fly low and say what things like the
pseudoinverse do, or at least what made you think of using them.

>One way to look at the situation here is that while the minimax tuning
>for starling, with eigenmonzos 2, 3 and 7 gives a much better tuning,
>it has rows with fractional monzsos; whereas the 5-limit tuning, with
>2, 3, and 5 eigennmonzos, gives an integral matrix. And we like that,
>since we like to work with JI.

Yes I get that, though I'm not able to manipulate such things
myself yet.

-Carl

🔗genewardsmith <genewardsmith@sbcglobal.net>

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

> >I'm using it to characterize JI scales which when tempered give you
> >the tempered scale you are looking for. For instance, the famous
> >Ptolemy-Zarlino Intense Diatonic Syntonon is a transversal for
> >Meantone[7] and also 7et.
>
> It seems you'd need something more than the concept described
> on that page, but no doubt you've explained it already.

I've just put up a transversal page on Xenwiki; I hope that explains it:

http://xenharmonic.wikispaces.com/Transversal

I wish
> you would occasionally fly low and say what things like the
> pseudoinverse do, or at least what made you think of using them.

They become interesting if we have a Euclidean space, and in particular are great devices for finding orthogonal projections. For TOP-RMS tuning, for instance, we want to find the projection othogonal the commas, which means orthogonal to the kernel of the mapping matrix; that is, projecting to the orthogonal complement of the kernel.

> >One way to look at the situation here is that while the minimax tuning
> >for starling, with eigenmonzos 2, 3 and 7 gives a much better tuning,
> >it has rows with fractional monzsos; whereas the 5-limit tuning, with
> >2, 3, and 5 eigennmonzos, gives an integral matrix. And we like that,
> >since we like to work with JI.
>
> Yes I get that, though I'm not able to manipulate such things
> myself yet.

What's hanging you up?
> -Carl
>

🔗Carl Lumma <carl@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

"genewardsmith" <genewardsmith@sbcglobal.net> wrote:

> I've just put up a transversal page on Xenwiki; I hope
> that explains it:
>
> http://xenharmonic.wikispaces.com/Transversal

Ah, that's good. I think I remember these from a group
theory book. A lot of the concepts relating to periodicity
blocks were in there under different names.

> I wish
> > you would occasionally fly low and say what things like
> > the pseudoinverse do, or at least what made you think
> > of using them.

Carl! I mentioned a while back about books for linear
algebra, and also the Wikipedia page on orthogonal
projections. This is one case where libraries are useful.
I've always been able to find accessible books on linear
algebra in college libraries. I don't know if you've done
this or not. But projection matrices, orthogonal
complements, and least squares optimizations will all be in
there. Maybe the pseudoiverse won't be mentioned by name,
but it'll be implicit in the formula for least squares
solutions.

There's a whole book on generalized inverses that I found
in a library once. It's still in the bibliography of
primerr.pdf although I took out the references to it: Wang,
Wei, and Qiao. I don't think it had anything useful that I
couldn't find in a simpler text. I can't promise to have
read it all the way through.

> They become interesting if we have a Euclidean space, and
> in particular are great devices for finding orthogonal
> projections. For TOP-RMS tuning, for instance, we want to
> find the projection othogonal the commas, which means
> orthogonal to the kernel of the mapping matrix; that is,
> projecting to the orthogonal complement of the kernel.

This is exactly what the text books say.

Note: scalar badness is an orthogonal complement with
respect to the JI line. That also makes it a projection.
It projects all points on the JI line to the unison.

What Gene says about projecting orthogonal to the commas
means the commas have to be tempered out. They're projected
to unisons. So the temperament class is defined by the
projection matrix that makes this so, and which happens to
be produced by the pseudoinverse.

There's another projection matrix that sends vals belonging
to the temperament class to the origin. You can get it
from the pesudoinverse of the weighted unison vectors. The
one relates to the other as P = I - Q.

Note: what is the logic behind the P = I - Q relationship
of dual metrics? The same thing happens with scalar
badness, where the badness of an interval comes out as its
size in cents (or related units). But that doesn't
generalize to matrices of unison vectors and I want to know
why.

Graham

🔗Carl Lumma <carl@lumma.org>

>> I wish
>> you would occasionally fly low and say what things like
>> the pseudoinverse do, or at least what made you think
>> of using them.
>
>Carl! I mentioned a while back about books for linear
>algebra, and also the Wikipedia page on orthogonal
>projections. This is one case where libraries are useful.
>I've always been able to find accessible books on linear
>algebra in college libraries. I don't know if you've done
>this or not. But projection matrices, orthogonal
>complements, and least squares optimizations will all be in
>there. Maybe the pseudoiverse won't be mentioned by name,
>but it'll be implicit in the formula for least squares
>solutions.

On my hard drive, I've got...

Blake - Whitehead's Geometric Algebra.pdf
Browne - Grassmann Algebra.pdf
Hefferon - Linear Algebra.pdf

and there's an interesting-looking chapter on matrices in

Shoup - Computational Intro to Number Theory and Algebra.pdf

Those were all free from the authors, not even pirated.
(See anything you want?)

Problem is, I don't really have the time or spare neurons
to plow through this to where I could build stuff with it
like you and Gene do. I've always had very broad interests
and this shows no sign of stopping. For instance over the
past year I've been posing as a nuclear engineer, going to
conferences and studying reactor design. It's all quite
simple compared to this tuning business; just memorize some
facts and apply basic arithmetic. But it takes neurons
and time. I'm actually to the point where learning new
things pushes out old ones. I know the feeling because
learning Dvorak in 2001 caused me to forget QWERTY which
I'd used since I was 12. That's because the same neurons
were repurposed.

I've also never been great with very abstract stuff -- my
brain works better if it can see practical results at
each turn. That's why I like Paul's recipe-style approach
where each step is justified as you go. I'll never forget
how to do that now but it's really just one corner of this
linear algebra universe. Understanding projections better
would be the next thing, I think you're right about that.

>What Gene says about projecting orthogonal to the commas
>means the commas have to be tempered out. They're projected
>to unisons. So the temperament class is defined by the
>projection matrix that makes this so, and which happens to
>be produced by the pseudoinverse.

Ok, that's a start. I didn't trust the determinant algorithm
for 3x3 matrices until I saw that it's equivalent to computing
the area of a parallelogram given the coordinates of the
corners. Some peg or derivation as to why the pseudoinverse
produces such a projection would help. But I recognize at
some point that these derivations get more and more
complicated and you're supposed to start taking things on
faith. Or take them on faith until they're very good with
them and then go back and derive them.

>There's another projection matrix that sends vals belonging
>to the temperament class to the origin. You can get it
>from the pesudoinverse of the weighted unison vectors. The
>one relates to the other as P = I - Q.

What are P, I & Q? I've seen you write them a half dozen
times before but I'll be damned if I know now.

>Note: what is the logic behind the P = I - Q relationship
>of dual metrics?

There, that's the kind of thing that helps. Except in this
case it didn't because I don't even know what it's about.
But don't let that discourage you.

-Carl

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Graham Breed <gbreed@gmail.com>

Carl Lumma <carl@lumma.org> wrote:

> On my hard drive, I've got...
>
> Blake - Whitehead's Geometric Algebra.pdf
> Browne - Grassmann Algebra.pdf

I've got the incomplete version Browne. Did he finish it?
Geometric algebra is similar as I understand it, involving
wedge products, and a lot of what we do can be expressed
that way. (Not everything -- I haven't got Cangwu badness
working with it.)

> Hefferon - Linear Algebra.pdf

I don't know it but it's likely to cover linear algebra.

> and there's an interesting-looking chapter on matrices in
>
> Shoup - Computational Intro to Number Theory and
> Algebra.pdf

That could indeed be interesting because temperaments do
relate to number theory. Maybe it explains lattices. And
Pari is a package aimed at number theorists that does a lot
of what I want.

> Those were all free from the authors, not even pirated.
> (See anything you want?)

Wouldn't they become pirated when you sent them to me?

> Problem is, I don't really have the time or spare neurons
> to plow through this to where I could build stuff with it
> like you and Gene do. I've always had very broad
<snip>

How much free time and neurons do you think I have? You
have a habit of making demands on my time, particularly
internet time, that I don't have. I'm slowly making
progress though.

> I've also never been great with very abstract stuff -- my
> brain works better if it can see practical results at
> each turn. That's why I like Paul's recipe-style approach
> where each step is justified as you go. I'll never forget
> how to do that now but it's really just one corner of this
> linear algebra universe. Understanding projections better
> would be the next thing, I think you're right about that.

Maybe asking short, simple questions would help you build
the recipes.

> >What Gene says about projecting orthogonal to the commas
> >means the commas have to be tempered out. They're
> >projected to unisons. So the temperament class is
> >defined by the projection matrix that makes this so, and
> >which happens to be produced by the pseudoinverse.
>
> Ok, that's a start. I didn't trust the determinant
> algorithm for 3x3 matrices until I saw that it's
> equivalent to computing the area of a parallelogram given
> the coordinates of the corners. Some peg or derivation
> as to why the pseudoinverse produces such a projection
> would help. But I recognize at some point that these
> derivations get more and more complicated and you're
> supposed to start taking things on faith. Or take them
> on faith until they're very good with them and then go
> back and derive them.

This is what the relevant chapter of a linear algebra book
would tell you. And it would all be laid out in a way that
students have been able to follow. If you like practical,
something aimed at engineers would be best.

> >There's another projection matrix that sends vals
> >belonging to the temperament class to the origin. You
> >can get it from the pesudoinverse of the weighted unison
> >vectors. The one relates to the other as P = I - Q.
>
> What are P, I & Q? I've seen you write them a half dozen
> times before but I'll be damned if I know now.

P and Q are the letters Gene used for two different
projection matrices. P comes from multiplying the
pseudoinverse of the weighted mapping with itself the
interesting way round. It sends weighted unison
vectors to zero. Q comes from doing the same thing with
weighted unison vectors, and sends weighted mappings for
the temperament class to zero. I is the standard symbol for
the identity matrix.

A temperament class is defined by mapping unisons vectors
to unisons. It shouldn't be surprising that a projection
matrix that does this defines the temperament class.

> >Note: what is the logic behind the P = I - Q relationship
> >of dual metrics?
>
> There, that's the kind of thing that helps. Except in
> this case it didn't because I don't even know what it's
> about. But don't let that discourage you.

Well, it was a real question, not a rhetorical one. But it
has something to do with orthogonal projections. If Q
defines an orthogonal projection, I-Q is the orthogonal
component, from what I remember.

Graham

🔗Carl Lumma <carl@lumma.org>

>> Those were all free from the authors, not even pirated.
>> (See anything you want?)
>
>Wouldn't they become pirated when you sent them to me?

No. Or you could download them yourself. The Browne I
have says "incomplete draft 2001". I also have a few papers
that may be of interest

Beaver et al - A 2-D Minkowski Question Function.pdf
Donaldson - Minkowski Reduction of Integral Matrices.pdf
Peterson - Unshackling Linear Algebra from Linear Notation.pdf

>How much free time and neurons do you think I have? You
>have a habit of making demands on my time, particularly
>internet time, that I don't have.

Do I?

I was alluding to the fact that you voluntarily got linear
algebra textbooks, studied them, obtained a foundational
understanding, wrote your PDFs, and so on.

>> >There's another projection matrix that sends vals
>> >belonging to the temperament class to the origin. You
>> >can get it from the pesudoinverse of the weighted unison
>> >vectors. The one relates to the other as P = I - Q.
>>
>> What are P, I & Q? I've seen you write them a half dozen
>> times before but I'll be damned if I know now.
>
>P and Q are the letters Gene used for two different
>projection matrices. P comes from multiplying the
>pseudoinverse of the weighted mapping with itself the
>interesting way round. It sends weighted unison
>vectors to zero. Q comes from doing the same thing with
>weighted unison vectors, and sends weighted mappings for
>the temperament class to zero. I is the standard symbol for
>the identity matrix.

Well that helps, thanks. What's minus?

>A temperament class is defined by mapping unisons vectors
>to unisons. It shouldn't be surprising that a projection
>matrix that does this defines the temperament class.

Sure.

-Carl

🔗Chris Vaisvil <chrisvaisvil@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗chrisvaisvil@gmail.com

I believe 17 but don't hold me to that. The hand posiitions are a bit difficult so my new axis *might* help.

Chris
*

-----Original Message-----
From: "genewardsmith" <genewardsmith@sbcglobal.net>
Sender: tuning-math@yahoogroups.com
Date: Fri, 29 Oct 2010 20:18:17
To: <tuning-math@yahoogroups.com>
Reply-To: tuning-math@yahoogroups.com
Subject: [tuning-math] Re: The upshot

--- In tuning-math@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> in the archive
>
> http://micro.soonlabel.com/hobbit_scales/
>
> BTW I've been playing the semi-marvelous dwarf quite a bit - I think
> I'll have something interesting in it the near future.
> It seems to lend itself to extended chords which is rather nice.

I'm looking forward to it! How many notes to the octave does this semimarvelous dwarf have?