Yahoo Tuning Groups Ultimate Backup tuning-math General references needed...

I'm tired of feeling stupid when I try to delve into the deeper math
behind this stuff. Would someone be so kind as to point me to some
good basic references that are prerequisite to understanding what goes
on around here? My mathematical education is very lopsided at the
moment - I've taken up to Calc III in college, did only some basic
stuff with vectors and all that, but then took I years and years of
engineering math courses that are based mostly around signal
processing without ever taking linear algebra. So I'm all off balance
here.

It seems like the fields that most pertain to what everyone is
discussing here, and in which I am unfortunately most deficient, are

- Linear algebra
- Group theory
- Number theory

Is it an accurate assessment to say that those are the most important
prerequisites for really understanding what's going on around here?
And, if so, can someone point me towards some helpful resources in
developing my understanding of this stuff? Internet resources would be
nice, but the Wolfram site is often too complicated and Wikipedia is
often of too uneven quality for me to make sense of it.

Thanks, I would appreciate it greatly.

-Mike

🔗Carl Lumma <carl@lumma.org>

Mike wrote:
>I'm tired of feeling stupid when I try to delve into the deeper math
>behind this stuff.

I fell asleep from that in 2002 and still haven't woken up. :)

>It seems like the fields that most pertain to what everyone is
>discussing here, and in which I am unfortunately most deficient, are
>- Linear algebra
>- Group theory
>- Number theory
>Is it an accurate assessment to say that those are the most important
>prerequisites for really understanding what's going on around here?

Linear and Grassman algebra. Very little group and number theory
as such. Graham and Gene have recommended textbooks. People from
other realms have recommended these as superior to average textbooks

http://www.axler.net/DwD.html

http://shoup.net/ntb

I haven't done much with them.

>And, if so, can someone point me towards some helpful resources in
>developing my understanding of this stuff? Internet resources would be
>nice, but the Wolfram site is often too complicated and Wikipedia is
>often of too uneven quality for me to make sense of it.

Mathworld tends to read like a trivia list. Wikipedia is generally
better, but yeah... uneven.

-Carl

🔗Mike Battaglia <battaglia01@gmail.com>

On Tue, Feb 15, 2011 at 10:14 PM, Carl Lumma <carl@lumma.org> wrote:
>
> Linear and Grassman algebra. Very little group and number theory
> as such. Graham and Gene have recommended textbooks. People from
> other realms have recommended these as superior to average textbooks
>
> http://www.axler.net/DwD.html
>
> http://shoup.net/ntb
>
> I haven't done much with them.

Wow, those look like awesome resources. I just started reading the
down with determinants link and this is exactly what I need. Hopefully
this will lead me towards a better understanding of things. Thanks.

> >And, if so, can someone point me towards some helpful resources in
> >developing my understanding of this stuff? Internet resources would be
> >nice, but the Wolfram site is often too complicated and Wikipedia is
> >often of too uneven quality for me to make sense of it.
>
> Mathworld tends to read like a trivia list. Wikipedia is generally
> better, but yeah... uneven.

And sometimes, unfortunately, inaccurate.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

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

> Linear and Grassman algebra. Very little group and number theory
> as such. Graham and Gene have recommended textbooks. People from
> other realms have recommended these as superior to average textbooks
>
> http://www.axler.net/DwD.html

Very slick proofs, coordinate free and not using determinants, of some basic theorems. But is Mike looking for proofs? And really, we want to be happy with coordinates and determinants also, not just eigenvalues and eigenvectors. I would have thought linear algebra would have come up in all of that engineering math.

> http://shoup.net/ntb

Computational number theory; I don't know if anything is particularly relevant.

An undergrad textbook on linear algebra, and reading the chapter, if there is one, in an abstract algebra book on abelian groups, would be a good start.

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Feb 16, 2011 at 1:20 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> > Linear and Grassman algebra. Very little group and number theory
> > as such. Graham and Gene have recommended textbooks. People from
> > other realms have recommended these as superior to average textbooks
> >
> > http://www.axler.net/DwD.html
>
> Very slick proofs, coordinate free and not using determinants, of some basic theorems. But is Mike looking for proofs? And really, we want to be happy with coordinates and determinants also, not just eigenvalues and eigenvectors. I would have thought linear algebra would have come up in all of that engineering math.

I'm at the point where I don't understand what the point of a
determinant is to begin with. It's a stupid little algorithm that you
can perform on a matrix to get a scalar and I had to learn it to pass
my AP Calc BC test. I also don't really understand why matrices exist,
either, except that they serve some mysterious function in linear
algebra. This is what happens when you take engineering math, they
dump stuff on you without giving you any insight into why it exists -
they just show you how it can be used to solve problems.

I put my foot down and took special time to understand the concepts
behind the Laplace and Fourier transforms and related stuff, mostly
because my involvement on these lists gave me some impetus to do so.
Now I need to repeat all of that with this stuff, I think. It's hard
for me to understand Cangwu badness when I still don't understand why
matrices were ever invented to begin with, short of giving me more
homework to do in high school with no clear application.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> It's hard
> for me to understand Cangwu badness when I still don't understand why
> matrices were ever invented to begin with, short of giving me more
> homework to do in high school with no clear application.

Well, Axler tries to avoid them. The free linear algebra textbook by Hefferon I googled up seems more standard in approach. If you understand that a determinant is a hypervolume, that's a good start, which may be a problem with Axler's approach, which must use an algebraically closed field such as the complex numbers. See what you make of Hefferon's "Linear Algebra".

Googling on "Free X textbook" seems to work pretty well if you want to learn about X these days.

🔗Carl Lumma <carl@lumma.org>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Carl Lumma <carl@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

Mike Battaglia <battaglia01@gmail.com> wrote:

> - Linear algebra

One I found very helpful is:

B. Kolman and D. R. Hill, 2003. ``Introductory Linear
Algebra: An Applied First Course.'' Eighth edition.
Prentice Hall.

It covers least squares geometry well, which makes it
directly relevant. But I've since found other books that
do much the same. The best thing is to walk into your
local academic library or bookshop and take a few Linear
Algebra books off the shelf to see how they look. You can
also try online books. I haven't, so I can't speak of them.

> - Group theory

Shaum's Outline.

> - Number theory

I have a book called something like "The Adventurer's Guide
to Number Theory." It's a good read but not that
applicable to tuning. There are graduate level texts that
cover algebraic geometry (I think that's the term: things
like linear diophantine equations) but they're pretty
dense. Also worth a mention:

J.~von zur Gathen & J. Gerhard, 1999.
Modern Computer Algebra.
Cambridge University Press.

Not an easy read but pretty good. Explicitly covers tuning
theory!

> Is it an accurate assessment to say that those are the
> most important prerequisites for really understanding
> what's going on around here? And, if so, can someone
> point me towards some helpful resources in developing my
> understanding of this stuff? Internet resources would be
> nice, but the Wolfram site is often too complicated and
> Wikipedia is often of too uneven quality for me to make
> sense of it.

Could be. Grassman Algebra's been mentioned as more
relevant than number theory. There's an online book we all
used for that:

http://www.grassmannalgebra.info/grassmannalgebra/book/index.htm

I think statistics is also a relevant field and there may
be discoveries waiting for somebody who understands it
better. It's also useful for daily life.

I did, in fact, slog my way through this:

Statistical Decision Theory and Bayesian Analysis
Series: Springer Series in Statistics

Berger, James O.

Originally published with the title: Statistical Decision
Theory

2nd ed. 1985. Corr. 3rd printing, 1985, XVII, 624 p. 23
illus., Hardcover

ISBN: 978-0-387-96098-2

http://tinyurl.com/4rcoeng

I bought a much cheaper Chinese reprinting. It gets very
difficult, but the early bits are the most important, and I
think I could understand them if I went through them
again. You don't need it for tuning as such but it would
be useful for adaptive tuning engines, and algorithmic
composition, and so on.

Graham

🔗Graham Breed <gbreed@gmail.com>

Mike Battaglia <battaglia01@gmail.com> wrote:

> I'm at the point where I don't understand what the point
> of a determinant is to begin with. It's a stupid little
> algorithm that you can perform on a matrix to get a
> scalar and I had to learn it to pass my AP Calc BC test.
> I also don't really understand why matrices exist,
> either, except that they serve some mysterious function
> in linear algebra. This is what happens when you take
> engineering math, they dump stuff on you without giving
> you any insight into why it exists - they just show you
> how it can be used to solve problems.

One thing I've learned from teaching is that people like
stories. So now I'll tell mine.

It goes back to Maths for Physicists lectures. I think
these are like engineering math. What I worked out is that
they spent a lot of time on proofs and examples to explain
what were actually fairly simple equations. The more I
tried to follow the examples the more confused I got. This
was particularly true with matrices. My solution was to sit
there reading Schoenberg's Theory of Harmony (on loan from
the university library) and when an important theorem or
definition came up, I'd write it down. I passed the exam.

What got me actually interested in matrices was quantum
mechanics. Our lecturer seemed to be a fan of Heisenberg's
matrix mechanics (as against the generally more popular
wave mechanics of Schrödinger) and that's what led me to
understand what the matrices were actually doing. It's
these principles that I carried into music theory, when I
tried to do for tuning what Gell-Mann did for nuclear
physics.

Then I found myself going back to the notes I made in those
lectures where I wasn't paying attention and found they
were pretty useful after all. The trouble with how
matrices (and other mathematical subjects) are usually
taught is that you learn a lot of seemingly unconnected
facts and not the reasons why they're useful. I can see
Mike suffered from this as well.

I tried to do better in a page on my website explaining
matrices. That got picked up by the search engines, and for
a while was far and away my most popular page. Now there
are much better resources on the web and hopefully you'll
find them useful.

Another field that got me a practical understanding of
matrices was computer graphics. Matrix
transformations are basic concepts and obviously useful.
That's something you could read up on.

I happened to learn of the matrix solution of least squares
problems in a computer vision lecture. This was related to
my master's degree, by which time I'd been transformed into
a model student. I sat in on this lecture even though I
wasn't registered on the course! And as Paul Erlich had
already created a stir with TOP error, I knew immediately
that it was the way to solve the equivalent problem using
RMS of weighted errors. That's now called TE error and was
for a while called TOP-RMS. It took some time to re-produce
the equation, because I hadn't been taking notes. (It came
up suddenly in a context that wasn't otherwise relevant to
me.) But I got there.

Graham

🔗Carl Lumma <carl@lumma.org>

Graham wrote:

>The trouble with how
>matrices (and other mathematical subjects) are usually
>taught is that you learn a lot of seemingly unconnected
>facts and not the reasons why they're useful.

You noticed!

>I happened to learn of the matrix solution of least squares
>problems in a computer vision lecture. This was related to
>my master's degree, by which time I'd been transformed into
>a model student. I sat in on this lecture even though I
>wasn't registered on the course! And as Paul Erlich had
>already created a stir with TOP error, I knew immediately
>that it was the way to solve the equivalent problem using
>RMS of weighted errors. That's now called TE error and was
>for a while called TOP-RMS. It took some time to re-produce
>the equation, because I hadn't been taking notes. (It came
>up suddenly in a context that wasn't otherwise relevant to
>me.) But I got there.

Great story. I didn't know (or remember properly) you were
still in school when TOP broke. -Carl

🔗Graham Breed <gbreed@gmail.com>

🔗Mike Battaglia <battaglia01@gmail.com>

On Thu, Feb 17, 2011 at 3:10 PM, Graham Breed <gbreed@gmail.com> wrote:
> My solution was to sit there reading Schoenberg's Theory of Harmony (on loan from
> the university library) and when an important theorem or
> definition came up, I'd write it down. I passed the exam.

Is that perhaps a good place to start as well then?

> It's these principles that I carried into music theory, when I
> tried to do for tuning what Gell-Mann did for nuclear
> physics.

What did Gell-Mann do for nuclear physics? Wikipedia is not helpful.

> Then I found myself going back to the notes I made in those
> lectures where I wasn't paying attention and found they
> were pretty useful after all. The trouble with how
> matrices (and other mathematical subjects) are usually
> taught is that you learn a lot of seemingly unconnected
> facts and not the reasons why they're useful. I can see
> Mike suffered from this as well.

The best application for matrices that I've found is to use them to
pass math exams involving matrices. They were also brutal about
teaching the Fourier and Laplace transforms. The latter is usually
taught as a neat trick to solve differential equations. It's actually
an expansion of the Fourier transform to deal with damped harmonic
oscillators as basis waveforms instead of just simple harmonic
oscillators. That's a pretty simple sentence that is uttered far too
often when teaching that stuff.

> I tried to do better in a page on my website explaining
> matrices. That got picked up by the search engines, and for
> a while was far and away my most popular page. Now there
> are much better resources on the web and hopefully you'll
> find them useful.

I forgot about that - I'll go back through it and check it out.

> And as Paul Erlich had already created a stir with TOP error, I knew immediately
> that it was the way to solve the equivalent problem using
> RMS of weighted errors. That's now called TE error and was
> for a while called TOP-RMS. It took some time to re-produce
> the equation, because I hadn't been taking notes. (It came
> up suddenly in a context that wasn't otherwise relevant to
> me.) But I got there.

Tangentially, what's the rationale behind using RMS there? I've always
been curious about that.

-Mike

🔗Carl Lumma <carl@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

Carl Lumma <carl@lumma.org> wrote:

> Hey, don't know if you saw it on tuning, but do you
> remember where you saw the Wilson stuff with
> fractional-octave periods? I remember looking once and
> not seeing it, but it's hard to go through it all. I ask
> because I'm expecting Paul to challenge me on it at some
> point. :)

Yes, I remember it, but I haven't found the references.
Kraig posted them recently. Possibly part of the
discussion about the definitions.

Okay, let's have a look at my Wilson folder.

Here we go, there's something called "13.pdf" that I
downloaded 28 Nov 2010. He talks about a "generic octave"
of 13/12 octaves. It's dated 1994.

Graham

🔗Carl Lumma <carl@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

Mike Battaglia <battaglia01@gmail.com> wrote:
> On Thu, Feb 17, 2011 at 3:10 PM, Graham Breed
> <gbreed@gmail.com> wrote:
> > My solution was to sit there reading Schoenberg's
> > Theory of Harmony (on loan from the university library)
> > and when an important theorem or definition came up,
> > I'd write it down. I passed the exam.
>
> Is that perhaps a good place to start as well then?

Schoenberg's not a good place to start for mathematics, or
even tuning. I believe there are better books on harmony
but I'm not an expert on that subject.

I was also reading Chekhov short stories at the same time.
They have nothing to do with tuning.

University libraries are a good place to start if you
happen to have access to one.

> > It's these principles that I carried into music theory,
> > when I tried to do for tuning what Gell-Mann did for
> > nuclear physics.
>
> What did Gell-Mann do for nuclear physics? Wikipedia is
> not helpful.

Gell-Mann "discovered" quarks. That is, he took a load of
particles, some of which make up nuclei, and showed that
their properties seemed to be described by combinations of
three out of a set of more fundamental particles. This
happens to involve group theory of a nature I still don't
understand. You can also write a matrix with quarks on
one side and hadrons (as they're called) on the other. I
wrote out that matrix and thought it looked really neat.

What I wanted to do is find the fundamental "particles"
that musical intervals are built out of. And what I
concluded is that there are always three for 5-limit JI,
but it doesn't matter which three. If you understand the
fundamental theorem of arithmetic that's obvious, but it
wasn't originally to me. Then I wrote the matrices as a way
of converting from one choice of intervals (generators as we
now call them) to another. And I worked out that meantones
(as I now call them) are always generated by two intervals,
and wrote the matrices for them, and relating them to just
intonation. That's roughly where I was when I joined the
tuning list.

> The best application for matrices that I've found is to
> use them to pass math exams involving matrices. They were
> also brutal about teaching the Fourier and Laplace
> transforms. The latter is usually taught as a neat trick
> to solve differential equations. It's actually an
> expansion of the Fourier transform to deal with damped
> harmonic oscillators as basis waveforms instead of just
> simple harmonic oscillators. That's a pretty simple
> sentence that is uttered far too often when teaching that
> stuff.

Too little, rather. That does help me understand what
Laplace transforms are. They're still useful for solving
differential equations, though.

> > And as Paul Erlich had already created a stir with TOP
> > error, I knew immediately that it was the way to solve
> > the equivalent problem using RMS of weighted errors.
> > That's now called TE error and was for a while called
> > TOP-RMS. It took some time to re-produce the equation,
> > because I hadn't been taking notes. (It came up
> > suddenly in a context that wasn't otherwise relevant to
> > me.) But I got there.
>
> Tangentially, what's the rationale behind using RMS
> there? I've always been curious about that.

It's the easiest thing to optimize for. It's the obvious
thing to choose when the direction of errors doesn't matter
and it comes up in contexts like electronics. It models the
basic property of harmonic entropy and related curves that
they're curvy and not piecewise-linear.

There's a more high falutin' argument, that you can model
dissonance as quadratic basins around JI points. If you
then compare the average dissonance of different tunings,
the inherent dissonances of the JI points drops out, and
what you're left with is a weighted RMS error. Unless you
get outside a basin, but it's only a model.

TOP(-max) is different. You can construct a
piecewise-linear dissonance function, and find the tuning
giving minimax difference as a linear program. The result
will depend on the inherent dissonances. It won't be TOP.

Graham

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> What did Gell-Mann do for nuclear physics? Wikipedia is not helpful.

Lots of things. One of them was applying the theory of the representations of Lie algebras to physics. Gell Mann tells the story himself, and it's most amusing. He was in Paris, and used to pal around with various people including Serre. Serre is a world class mathematician, and an expert on the representations of Lie algebras among many other things. You would think, of course, that Gell Mann would have taken the opportunity to ask Serre about them. You would be wrong. Back then, physicists and mathematicians did not talk to each other, other than about topics unrelated to physics or mathematics, such as tennis. Gell Mann had no idea Serre was an expert on the relevant mathematics, and did not think to ask, since physicists and mathematicians had given up trying to communicate. Nor did it occur to Serre to ask his American friend about his work. It was only years later that he found out that his current algebras had long been well-known to mathematicians and that Serre was a world-class expert on them. He figured out the Eight Fold Way and quarks anyhow.

🔗Mike Battaglia <battaglia01@gmail.com>

On Fri, Feb 18, 2011 at 3:07 AM, Graham Breed <gbreed@gmail.com> wrote:
>
> There's a more high falutin' argument, that you can model
> dissonance as quadratic basins around JI points. If you
> then compare the average dissonance of different tunings,
> the inherent dissonances of the JI points drops out, and
> what you're left with is a weighted RMS error. Unless you
> get outside a basin, but it's only a model.

Where was this argument made? One of the things I was working on, if
you remember, was to model an interval's field of attraction as a
multivariate Gaussian-shaped basin around its JI point on the lattice.
This stemmed out of all of the work with HE and convolution that I was
doing (which I still have yet to finish). I thought that it would then
be nice to explore a "fuzzy" group structure where every interval had
a Gaussian-shaped field of existence on the lattice instead of just a
single point, such that five "fuzzy" 10/9's can get you to a 4/3 or
four "fuzzy" 3/2's can get you to a 5/1. Are we describing similar
things?

-Mike