Yahoo Tuning Groups Ultimate Backup tuning-math The Topos of Music

I've checked this monstrosity out. It is a massive 1335 pages, and is
full of music theory, mathematics, and philosophy--which happen to be
three subjects I know a lot about. Even so, I'm in trouble; this is
because a lot of it is simply bullshit in the classic European style,
but fortunately or perhaps unfortunately it isn't 100% bullshit,
meaning wading through it and sorting the sense from the nonsense is
not as easy as it might be.

One thing Monz might be interested to know is that Mazzola is a fan,
in effect, of his rational exponents. He first remarks that the real
numbers are a module over the rationals, which is true, but very
annoying to a mathematician to hear. This is because the rationals are
a field, and a free module over a field you are supposed to call a
vector space. He claims the additive structure of R as a vector space
over Q is the "basis of thinking in pitch distance"; this he calls the
"pitch module" and the real numbers here can be thought of as cents,
though saying something like that is just not Mazzola's style. He then
introduces the vector space of dimension three over the rationals,
which he calls the "Euler module", where the components are thought of
as rational exponents of 2, 3 and 5. These are, in other words, monzos
allowed to have rational rather than merely integer coefficients. He
remarks then that the logarithm map embeds this into the so-called
"pitch module".

He then gives a slightly obfuscated discussion of commas, and says
absurdly that these are important only in just intonation.

🔗monz <monz@attglobal.net>

hi Gene,

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

> One thing Monz might be interested to know is that Mazzola
> is a fan, in effect, of his rational exponents.

wow, there *is* someone else out there!

> He first remarks that the real numbers are a module over
> the rationals, which is true, but very annoying to a
> mathematician to hear. This is because the rationals are
> a field, and a free module over a field you are supposed
> to call a vector space. He claims the additive structure
> of R as a vector space over Q is the "basis of thinking
> in pitch distance"; this he calls the "pitch module" and
> the real numbers here can be thought of as cents, though
> saying something like that is just not Mazzola's style.
> He then introduces the vector space of dimension three
> over the rationals, which he calls the "Euler module",
> where the components are thought of as rational exponents
> of 2, 3 and 5. These are, in other words, monzos allowed
> to have rational rather than merely integer coefficients.

hmm ... i've *always* (well, since joining the tuning list
in 1998, when Paul drummed into me the historical importance
of meantone) intended for my monzos to have any real numbers
for the exponents, or perhaps even pi, etc.

i realize that your definition of monzo in my Encyclopaedia
states that they must be integers ... and my rational monzos
have always been a thorn in Paul's side, but i never really
understood why.

to me, visualizing pitches in a prime-space is a
valuable thing, and if certain linear tunings have
pitches whose prime-factors need rational exponents,
what's the big deal? they allow the tuning to be
latticed as a nice straight line, which is how a
linear temperament should look on a lattice.

i realize that using rational exponents means that
there is an infinite variety of different combinations
of rational exponents which will all equal the same
ratio, but in the case of fraction-of-a-comma meantone
temperaments, these lattice do show exactly what happens.

see, for examples:

http://tonalsoft.com/monzo/meantone/lattices/lattices.htm

http://tonalsoft.com/monzo/meantone/lattices/PB-MT.htm

http://tonalsoft.com/enc/19edo.htm#salinas

> He remarks then that the logarithm map embeds this into
> the so-called "pitch module".
>
> He then gives a slightly obfuscated discussion of commas,
> and says absurdly that these are important only in just
> intonation.

i wonder what his reasoning is behind that ... maybe
there is some logic to it.

at any rate, sounds like a book i should read.
in fact i think i'd like to correspond with Mazzola.
i suspect we'd have a lot to discuss.

thanks for the heads-up.

-monz

🔗monz <monz@attglobal.net>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> i realize that using rational exponents means that
> there is an infinite variety of different combinations
> of rational exponents which will all equal the same
> ratio, but in the case of fraction-of-a-comma meantone
> temperaments, these lattice do show exactly what happens.

Using real numbers would do this, but the point of using rational
numbers is that they do not do this. The real numbers are an
(uncountably) infinite-dimensional vector space over the rationals,
but only a one-dimensional vector space over the reals. Using real
numbers there are an infinite number of ways of writing the 1/4 comma
meantone fifth as a 3D monzo, but using rational numbers only one way,
|0 0 1/4>.

> at any rate, sounds like a book i should read.
> in fact i think i'd like to correspond with Mazzola.
> i suspect we'd have a lot to discuss.

This is a book everyone on this list should read, since they might
then quit claiming I am too obscure. Unfortunately there is a huge
amout of bullshit in it, which is harder to sort through than the
bullshit in The Myth of Invariance, for example.

🔗monz <monz@attglobal.net>

hi Gene,

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

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i realize that using rational exponents means that
> > there is an infinite variety of different combinations
> > of rational exponents which will all equal the same
> > ratio, but in the case of fraction-of-a-comma meantone
> > temperaments, these lattice do show exactly what happens.
>
> Using real numbers would do this, but the point of using
> rational numbers is that they do not do this. The real
> numbers are an (uncountably) infinite-dimensional vector
> space over the rationals, but only a one-dimensional vector
> space over the reals. Using real numbers there are an
> infinite number of ways of writing the 1/4 comma meantone
> fifth as a 3D monzo, but using rational numbers only one
> way, |0 0 1/4>.

so then there *is* some utility to what i've been doing?
just as i suspected.

> > at any rate, sounds like a book i should read.
> > in fact i think i'd like to correspond with Mazzola.
> > i suspect we'd have a lot to discuss.
>
> This is a book everyone on this list should read, since
> they might then quit claiming I am too obscure. Unfortunately
> there is a huge amout of bullshit in it, which is harder
> to sort through than the bullshit in The Myth of Invariance,
> for example.

hmm ... well, you might recall that i also really disagree
with you over that one.

at any rate, if Mazzola's book will help me understand
what you post, i'll read it. i've already enrolled in
what will be a 2- or 3-year course in math, for the
same purpose.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> so then there *is* some utility to what i've been doing?
> just as i suspected.

You might find it useful. If q is a rational number, q-comma meantone
will be |-1 1 0> - q*|-4 4 -1> = |4q-1 -(4q-1) q>, which defines a
line; maybe that you would find useful, for example.

> at any rate, if Mazzola's book will help me understand
> what you post, i'll read it.

I don't suppose it will do that; I'd guess your best bet would be to
decypher what you can while pretty well ignoring any difficult math;
it's a big book with lots in it, including many illustrations, so who
knows what you'd discover.

i've already enrolled in
> what will be a 2- or 3-year course in math, for the
> same purpose.

I hope no one goes out and studies algebraic geometry in the hope
Mazzola will then make perfect sense, because it won't.

🔗monz <monz@attglobal.net>

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > so then there *is* some utility to what i've been doing?
> > just as i suspected.
>
> You might find it useful. If q is a rational number,
> q-comma meantone will be |-1 1 0> - q*|-4 4 -1> =
> |4q-1 -(4q-1) q>, which defines a line; maybe that
> you would find useful, for example.

i believe that i figured out exactly that same algorithm
by actually using numbers on a spreadsheet.

if you look at any of my webpages about meantones you'll
see monzos with rationals in them ... or whatever we're
going to call them now, if monzos have to have integers.

> > i've already enrolled in what will be a 2- or 3-year
> > course in math, for the same purpose.
>
> I hope no one goes out and studies algebraic geometry in
> the hope Mazzola will then make perfect sense, because
> it won't.

well, the main reason i've enrolled in my math course
is to be able to understand what *you* wrote, that's
lying around in the archives of this list.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> well, the main reason i've enrolled in my math course
> is to be able to understand what *you* wrote, that's
> lying around in the archives of this list.

As Graham pointed out, if you go directly for what you'd need to
understand wedgies, it isn't really too complicated. If you went
directly for what you need to understand "the Galois theory of
concepts", you would need years of hard work on very abstruse
mathematics, at the end of which you'd discover it was bullshit. It
would be interesting to go through the Topos of Music and sort out all
the sense from all the nonsense, but it would be a huge and difficult
job, which the authors should have already done for us by not writing
so much nonsense. Even so, there's actually a lot in it!

🔗hstraub64 <hstraub64@telesonique.net>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > at any rate, sounds like a book i should read.
> > in fact i think i'd like to correspond with Mazzola.
> > i suspect we'd have a lot to discuss.
>
> This is a book everyone on this list should read, since they might
> then quit claiming I am too obscure. Unfortunately there is a huge
> amout of bullshit in it, which is harder to sort through than the
> bullshit in The Myth of Invariance, for example.

The book is definitely a monster, a very frightening one. I got to say
I really hesitate to recommend it. The jargon is one point; another
one is the fact that the book has a lot of authors (there are 20
mentioned on the cover alone) and contains not one but actually a lot
of different theories that do not always blend together well. Many are
above my head, too.
Anyway, as one of the guys whose names appear on the cover (one of the
minor ones, however!), I would be interested what exactly you see as
bullshit. Could you give some details?
--
Hans Straub
http://home.datacomm.ch/straub

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:

The jargon is one point; another
> one is the fact that the book has a lot of authors (there are 20
> mentioned on the cover alone) and contains not one but actually a lot
> of different theories that do not always blend together well.

That's part of what I mean; there really is a lot of stuff in there,
if you can dig it out.

Many are
> above my head, too.
> Anyway, as one of the guys whose names appear on the cover (one of the
> minor ones, however!), I would be interested what exactly you see as
> bullshit. Could you give some details?

One of the reasons we are told in the introduction that a
mathematician might want to read this book is to learn the Galois
theory of concepts. Speaking as someone who has published in the area
of Galois theory, I was interested to see what in the world this could
possibly be; but a check in the index fails to locate it. A web search
shows the only people talking about this are the Mazzola gang.

So, what in the world is it? This is unfortunately typical; large,
pretentious sounding words are tossed about, but they don't always
seem to actually mean anything. Extremely abstruse notions such as
"scheme" and "topos" are tossed around, but we don't ever seem to see
an honest example. The book has a huge mathematical appendix, in which
all kinds of mathematics is discussed; if we are going to define
"scheme", which the authors, consulting I presume standard textbooks,
proceed to do, and if we are going to talk about schemes as if they
were actually being used; on would expect to find real algebraic
geometry in play. But where is it? How can it possibly be used?

As a grad student I learned about schemes, functors, categories,
injective and projective modules, Lie algebras, Jacobson radicals,
local rings, Zariski topologies, simplical complexes, and on and
on--in other words, the math the book claims to use. But mostly, it
doesn't seem to actually use it. While there is a lot of material in
the book and some of it is heavily mathematical, it's very pretentious
in claiming to use more math, and at a higher level, than it actually
does.

My impression is that often math is being used in this book as a
weapon to beat people over the head and make them feel stupid; it
doesn't work very well on me because I actually know a lot of math,
and am in a position to ask, where's the beef? What, for instance, is
the Galois theory of concepts? How do you use schemes in music theory?
On a less abstruse level, what in the world does the free abelian
group on strings of ascii characters have do with anything? I can add
two cows and a banjo, and subtract Boris Yeltsin, and get
2*cow + banjo - Boris - Yeltsin, and because of commutivity I can see
I won't be able to make sentences this way, even if I wanted to. Now what?

What almost makes it worse is that the book is far from being complete
nonsense from cover to cover, so one can't simply say throw it in the
trash.

🔗hstraub64 <hstraub64@telesonique.net>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> This is unfortunately typical; large,
> pretentious sounding words are tossed about, but they don't always
> seem to actually mean anything. Extremely abstruse notions such as
> "scheme" and "topos" are tossed around, but we don't ever seem to
> see an honest example. The book has a huge mathematical appendix,
> in which all kinds of mathematics is discussed; if we are going to
> define "scheme", which the authors, consulting I presume standard
> textbooks, proceed to do, and if we are going to talk about
> schemes as if they were actually being used; on would expect to
> find real algebraic geometry in play. But where is it? How can it
> possibly be used?
>

Well, OK - I can only speak for the parts I know (which is, I
admit, not the major part) so I cannot say whether there is more
presented than actually used. In any case, I did not specifically
notice til now (maybe I will after this...) Concerning the
mathematical appendix, I cannot imagine any reason to put stuff
there that is not used at least once - the book is far too big
anyway, and the appendix is, after all, mostly standard stuff that
is there mainly so that the reader does not constantly need a whole
pile of other books to look things up. So if nowhere else, at least
there I would expect every opportunity to shorten to be taken.

>
> My impression is that often math is being used in this book as a
> weapon to beat people over the head and make them feel stupid; it
> doesn't work very well on me because I actually know a lot of math,
> and am in a position to ask, where's the beef?

Well, there is a certain amount of polemics, e.g. against certain
music theorists who are against the use of mathematics (or even
strict scientific methods) in their field. About other secret
intentions, I do not know, of course.

> What, for instance, is the Galois theory of concepts? On a less
abstruse level, what in the world does the free abelian group on
strings of ascii characters have do with anything? I can add two
cows and a banjo, and subtract Boris Yeltsin, and get 2*cow + banjo -
Boris - Yeltsin, and because of commutivity I can see I won't be
able to make sentences this way, even if I wanted to. Now what?
>

I am not familiar with this special one right now, but it looks like
it was done with computer programming in mind - a considerable part
of the formalism is that way - it comes out quite clearly in the
concept of denotators, for example. The whole thing is not just
music and mathematics, but also informatics.

But what is true for sure is that the index is unsatisfactory. More
than once, I have been in the situation that there was a reference
(e.g. Carey/Clampitt) and I tried to find the place in the book
where it is used - no chance! So it is quite possible that you get
the impression there is "no beef" not because it is not there but
because it cannot be found... Which is, of course, a serious point -
maybe that would be an improvement idea to communicate...

I will check whether I can find the Galois theory thing (I did not
even know about that one!)
--
Hans Straub
http://home.datacomm.ch/straub

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > at any rate, sounds like a book i should read.
> > in fact i think i'd like to correspond with Mazzola.
> > i suspect we'd have a lot to discuss.
>
> This is a book everyone on this list should read, since
> they might then quit claiming I am too obscure. Unfortunately
> there is a huge amout of bullshit in it, which is harder
> to sort through than the bullshit in The Myth of Invariance,
> for example.

well, i hope i can sort thru the bullshit and find
something useful, because i've just spent over $100 on it.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:

Concerning the
> mathematical appendix, I cannot imagine any reason to put stuff
> there that is not used at least once - the book is far too big
> anyway, and the appendix is, after all, mostly standard stuff that
> is there mainly so that the reader does not constantly need a whole
> pile of other books to look things up. So if nowhere else, at least
> there I would expect every opportunity to shorten to be taken.

It doesn't seem to have been. It would be interesting to see an actual
use of schemes, for instance. About the simplest possible use would be
to use Spec(Z), the affine scheme of the integers, since as Mazzola
points out we use integers a lot, and real numbers are a field so
scheme-theoretically they are trivial. If we had a piece of music
represented using integers (what we could get out of a midi file, for
instance) we have something over the scheme as a whole. Now how do we
get it to localize to the sheaf of rings? We have something in Z, and
we want to define it as something in Z_(p) for some prime p in a
natural way which is useful. We use abelian groups a lot, which are
Z-modules, so we could extend coefficients when we do that, and
consider them as Z_p-modules, which is on the way to the quotient
field of Z, namely Q. If we use Z-module monzo elements and if with
Joe we use Q-vector space monzo elements, we could certainly use them
for any open set of Spec(Z), so we could have scheme-theoretic monzos,
but what does that buy you? It sounds like serious math overkill.

Anyway, Mazzola doesn't actually do anything like this from what I
have seen; it would be interesting to see an actual application of
schemes, which to be non-trivial really ought to be over something
more than just Z anyway. Let's see music over an elliptic curve scheme!

> I am not familiar with this special one right now, but it looks like
> it was done with computer programming in mind - a considerable part
> of the formalism is that way - it comes out quite clearly in the
> concept of denotators, for example. The whole thing is not just
> music and mathematics, but also informatics.

I don't see how a formalism which counts occurances of character
strings, and allows one to be cancelled by another, but which cannot
put the strings into any order, could be useful in informatics.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Anyway, Mazzola doesn't actually do anything like this from what I
> have seen; it would be interesting to see an actual application of
> schemes, which to be non-trivial really ought to be over something
> more than just Z anyway. Let's see music over an elliptic curve scheme!

The only nontrivial schemes I can think of in connection with music
theory are the projective varieties of wedgies, which if expressed in
the modern language of algebraic geometry would be a scheme, which
like any scheme is a scheme over Z (as Mazzola would be quick to say,
Spec(Z) is the final object in the category of schemes.) If I was to
try to apply this formalism in a nontrivial way, I'd start here, but
nothing suggests itself to me as useful just now. In any case, it is
sort of nice to know that it would be *possible* to use schemes in
music theory, so putting them in the appendix need not be merely an
act of random lunacy. By the way, am I talking to myself or am I right
in assuming that Hans at least is following this?

🔗monz <monz@attglobal.net>

🔗hstraub64 <hstraub64@telesonique.net>

I have looked up the Galois theory thing now. It's the theory of
Denotators (chapter 6) that is compared to Galois theory - and I got
to say I could not see the connection, too. So you may be right in
this aspect - the naming may not always be appropriate. Which does,
of course, not mean that the theory of Denotators is a fake itself.
Just look at it without the expectations the word "Galois theory of
concepts" evokes - it's math, after all, so everybody who knows
about math can see for himself what it is good for and what not.
Chapter 9 gives some more hints of what the idea is. There may be no
direct applications in this case - after all, chapter 9 is mainly
philosophical, and in philosophy there is often not much "beef"
anyway. But IMHO, the thoughts are interesting of their own, and the
parallels between category theory and music are, in any case,
stimulating.

But I see another problem you addressed - there are just so many
different things. Philosophical stuff of no "direct" use like in
chapter 9, and then again, very concrete applications like the
modulation model (chapter 27.1), yielding a quite amazing analysis
of Beethoven's Hammerclavier Sonata (chapter 28.2) as well as
providing a working system for modulation in some microtonal
contexts. (There is, however,another naming problem there: The
claimed connection to quantum mechanics in physics I have not been
able to see yet. But apart from this, the model works, as I have
tried out myself.)

Hope that helps. I will have a look at schemes and specs next...
--
Hans Straub
http://home.datacomm.ch/straub

🔗Carl Lumma <ekin@lumma.org>

🔗hstraub64 <hstraub64@telesonique.net>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >very concrete applications like the
> >modulation model (chapter 27.1), yielding a quite amazing
> >analysis of Beethoven's Hammerclavier Sonata (chapter 28.2)
>
> Ok, what's the citation for this book?
>

The above chapter numbers are from "The Topos of Music" - I do not
understand the question?

> >But apart from this, the model works, as I have
> >tried out myself.)
>
> Can you share any examples?
>

My composition "On-To-Sû, So-Na-Ta", which is explained on my
homepage
(http://home.datacomm.ch/straub/mamuth/modul/ontosu_e.html), is
based on it. What is decribed for now, however, is not the full
model but merely the concept of the cadence-set, which is,
mathematically, quite trivial.
Adding the full formalism results in some quite interesting
insights - I have been planning for a long time to add a (hopefully
easy) explication of this to my homepage but have not found the time
yet. Maybe I will finally take the time now...