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Apr 16, 2006, 3:32:36 PM4/16/06

to

While searching for a decent online ebook, or course notes,

on category theory I came across the above. Talk about

category theory getting everywhere these days! Anyone

studied this?

Reading the table of contents, or just the book's title,

I'd be inclined to suspect it's a lot of pretentious

twaddle. But then Birkhuaser isn't noted, AFAIK, for

publishing vacuous vapourings, and I believe category

theory has been applied in computer science and even

linquistics. So it probably is sound after all. Might

even order a copy and take a closer look.

The best online material I've so far found on category theory

is http://math.berkeley.edu/~gbergman/245/ "An Invitation to

General Algebra and Universal Constructions", which from what

I've read so far is a leisurely and well-motivated introduction.

Cheers

John Ramsden

Apr 16, 2006, 8:12:40 PM4/16/06

to

jhnr...@yahoo.co.uk wrote:

> While searching for a decent online ebook, or course notes,

> on category theory I came across the above. Talk about

> category theory getting everywhere these days! Anyone

> studied this?

>

> Reading the table of contents, or just the book's title,

> I'd be inclined to suspect it's a lot of pretentious

> twaddle. But then Birkhuaser isn't noted, AFAIK, for

> publishing vacuous vapourings, and I believe category

> theory has been applied in computer science and even

> linquistics. So it probably is sound after all.

> While searching for a decent online ebook, or course notes,

> on category theory I came across the above. Talk about

> category theory getting everywhere these days! Anyone

> studied this?

>

> Reading the table of contents, or just the book's title,

> I'd be inclined to suspect it's a lot of pretentious

> twaddle. But then Birkhuaser isn't noted, AFAIK, for

> publishing vacuous vapourings, and I believe category

> theory has been applied in computer science and even

> linquistics. So it probably is sound after all.

Speaking as a mathematician who, at one time, was also majoring in music

theory and composition, I'd also "be inclined to suspect it's a lot of

pretentious twaddle." Mind you, I'm not saying that there's likely to be

anything truly unsound mathematically. Rather, I suspect that it's "full

of [mathematical] sound and fury, signifying nothing [of interest in music

theory]."

> Might even order a copy and take a closer look.

If you do, please give us a report!

Cheers,

David Cantrell

Apr 17, 2006, 3:10:51 AM4/17/06

to

jhnr...@yahoo.co.uk wrote:

>

> While searching for a decent online ebook, or course notes,

> on category theory I came across the above. Talk about

> category theory getting everywhere these days! Anyone

> studied this?

Oops - I forgot to include the link: http://www.encyclospace.org/

(Scroll two thirds down to reach the link to the table of contents.)

Apr 17, 2006, 6:34:02 AM4/17/06

to

David W. Cantrell wrote:

> Speaking as a mathematician who, at one time, was also majoring in music

> theory and composition, I'd also "be inclined to suspect it's a lot of

> pretentious twaddle." Mind you, I'm not saying that there's likely to be

> anything truly unsound mathematically. Rather, I suspect that it's "full

> of [mathematical] sound and fury, signifying nothing [of interest in music

> theory]."

Aren't you being rather narrow-minded?

Music used to be one of the four (?) parts of Mathematics,

and I'm sure Pythagoras would have approved of using topos theory

in the study of music.

I looked at this book, and couldn't really make sense of it,

but it didn't strike me as "pretentious twaddle".

--

Timothy Murphy

e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie

tel: +353-86-2336090, +353-1-2842366

s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland

Apr 18, 2006, 2:07:43 AM4/18/06

to

In article <1145215955.9...@i40g2000cwc.googlegroups.com>,

jhnr...@yahoo.co.uk wrote:

jhnr...@yahoo.co.uk wrote:

> While searching for a decent online ebook, or course notes,

> on category theory I came across the above. Talk about

> category theory getting everywhere these days! Anyone

> studied this?

Not I. Here's Math Reviews:

MR1938949 (2004a:00013)

Mazzola, Guerino(CH-ZRCH-MMI)

The topos of music. (English. English summary)

Geometric logic of concepts, theory, and performance. In collaboration

with Stefan Göller and Stefan Müller. With 1 CD-ROM (Windows, Macintosh

and UNIX).

Birkhäuser Verlag, Basel, 2002. xxx+1335 pp. ISBN 3-7643-5731-2

00A69 (18B25)

The author transposes the Pythagorean theory of harmonies into

commutative algebra because, on one hand, linearity is the modern

version of additivity and proportion, and on the other hand there are

standard means for programming with modules as data types. This working

Pythagoreanism includes a CD with software for analyzing scores and

preparing them for electronic performance, and recordings of some

examples. Indeed the CD includes a pdf of the entire book, nicer than

the printed copy as it has color graphics plus active links to the

bibliography and to cross-references in the text.

There is an extensive mathematical analysis of musical structures

operationalized as categorical semantics for the programs. And the book

gives a very great deal of more cognitive scientific, semiotic, and

literary music theory. The author is known as a composer and a jazz

pianist, and has other publications in scheme theory, such as \ref[J.

Algebra 78 (1982), no. 2, 292--302; MR0680361 (84d:16039)].

A key use of commutative algebra is given as "alterations are

tangents". For example, F-sharp and G-flat name the same tone and the

same key on a piano keyboard. The two names denote two different

structural roles in composition---it matters that F-sharp is not just a

note, but an alteration of F-natural. The author regards the space of

tones as an abelian group $\Bbb{M}$, so any single note is an element

of the group, which the author treats as a $\Bbb{Z}$-element, a

homomorphism $\Bbb{Z}\rightarrow\Bbb{M}$ from the group of integers.

Then an altered note can be considered as a pair of notes, a base plus

an altered value, and so as a $\Bbb{Z}\oplus\Bbb{Z}$-element of

$\Bbb{M}$. In the traditional alterations, to sharps or flats, the

altered value is near the base note, and the author models these by

infinitesimal deformations, or tangent vectors. That is, he uses the

ring of dual numbers $\Bbb{Z}[\epsilon]$, and all of this is applied

over other ground rings than $\Bbb{Z}$, and thus brings us to scheme

theory.

Symmetries within scores, and structural relations between scores,

drive the mathematics up to sheaves, and very briefly to toposes and

Grothendieck topologies. The author candidly states he is unsure

whether this musicological perspective can use topos cohomology (p.

436).

The remaining 800 pages of the book are more concrete (until the

100-page appendix surveying concepts from group theory through schemes

to vector fields and differential equations). They deal with major

semioticians, philosophers, music critics and music theorists,

especially computational music theorists. They apply algebraic and

geometric notions of symmetry, along with the physiology of perception,

to analyze harmony, cadence, motifs, tempo and counterpoint. Examples

analyzed at length focus on work of Bach and Beethoven, but include

Mozart and Debussy, Glenn Gould's eccentric performances, and the

author's works on the CD. All is operationalized as computational

musicology, in the software.

Reviewed by Colin McLarty

--

Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

Apr 20, 2006, 9:29:36 AM4/20/06

to

Timothy Murphy <t...@birdsnest.maths.tcd.ie> wrote:

> David W. Cantrell wrote:

>

> > Speaking as a mathematician who, at one time, was also majoring in

> > music theory and composition, I'd also "be inclined to suspect it's a

> > lot of pretentious twaddle." Mind you, I'm not saying that there's

> > likely to be anything truly unsound mathematically. Rather, I suspect

> > that it's "full of [mathematical] sound and fury, signifying nothing

> > [of interest in music theory]."

>

> Aren't you being rather narrow-minded?

> David W. Cantrell wrote:

>

> > Speaking as a mathematician who, at one time, was also majoring in

> > music theory and composition, I'd also "be inclined to suspect it's a

> > lot of pretentious twaddle." Mind you, I'm not saying that there's

> > likely to be anything truly unsound mathematically. Rather, I suspect

> > that it's "full of [mathematical] sound and fury, signifying nothing

> > [of interest in music theory]."

>

> Aren't you being rather narrow-minded?

I like to think that I am not, that I have an open mind. Indeed, I would be

absolutely delighted if the mathematics presented turned out to be so

significant for music theory that mathematics departments around the world

would need to offer "service courses" in category theory for music

theorists. But I doubt that will come to pass. Yes, I'm _guessing_. But my

suspicion is based on my experience with works which claim to

use "advanced" mathematics in the service of music theory, when the only

thing really served seems to be the author's publication list.

> Music used to be one of the four (?) parts of Mathematics,

Perhaps you're thinking about the Quadrivium: arithmetic, geometry, music,

astronomy.

> and I'm sure Pythagoras would have approved of using topos theory

> in the study of music.

Maybe. We don't know very much, with certainty, about Pythagoras. (Cf. the

recent thread "Pythagoras and beans".)

> I looked at this book, and couldn't really make sense of it,

> but it didn't strike me as "pretentious twaddle".

It didn't strike me as anything, since I haven't looked at it. I was merely

guessing, based on my experience, that looking at it would be a waste of my

time. I hope my guess is wrong.

And thanks to Gerry Myerson for posting the review.

Regards,

David

Apr 21, 2006, 7:04:43 AM4/21/06

to

David W. Cantrell wrote:

>> Music used to be one of the four (?) parts of Mathematics,

>

> Perhaps you're thinking about the Quadrivium: arithmetic, geometry, music,

> astronomy.

>

>> and I'm sure Pythagoras would have approved of using topos theory

>> in the study of music.

>

> Maybe. We don't know very much, with certainty, about Pythagoras. (Cf. the

> recent thread "Pythagoras and beans".)

Well, everyone seems agreed that Pythagoras thought

there was a close link between music and mathematics,

which is not surprising since he (or his cult) seem to have discovered

the connection between harmony and ratios.

I must say I would take the opposite tack to you;

breaking the link between mathematics and music

may well have damaged both.

Apr 21, 2006, 8:04:50 PM4/21/06

to

Timothy Murphy wrote:

>

>> I must say I would take the opposite tack to you;

> breaking the link between mathematics and music

> may well have damaged both.

>

>

Finally someone explained why we have both James Harris *and* Eminem.

Whew! I owe you a pint of Guiness, dude.

Apr 23, 2006, 10:13:41 AM4/23/06

to

Timothy Murphy <t...@birdsnest.maths.tcd.ie> wrote:

> David W. Cantrell wrote:

>

> >> Music used to be one of the four (?) parts of Mathematics,

> >

> > Perhaps you're thinking about the Quadrivium: arithmetic,

> > geometry, music, astronomy.

> >

> >> and I'm sure Pythagoras would have approved of using topos theory

> >> in the study of music.

> >

> > Maybe. We don't know very much, with certainty, about Pythagoras.

> > (Cf. the recent thread "Pythagoras and beans".)

>

> Well, everyone seems agreed that Pythagoras thought

> there was a close link between music and mathematics,

> David W. Cantrell wrote:

>

> >> Music used to be one of the four (?) parts of Mathematics,

> >

> > Perhaps you're thinking about the Quadrivium: arithmetic,

> > geometry, music, astronomy.

> >

> >> and I'm sure Pythagoras would have approved of using topos theory

> >> in the study of music.

> >

> > Maybe. We don't know very much, with certainty, about Pythagoras.

> > (Cf. the recent thread "Pythagoras and beans".)

>

> Well, everyone seems agreed that Pythagoras thought

> there was a close link between music and mathematics,

Certainly. (That argument is "diluted" a bit, however, when you think of

the fact that the Pythagoreans thought there was a close link between

mathematics and _everything_.)

> which is not surprising since he (or his cult) seem to have

> discovered the connection between harmony and ratios.

In modern terms, that's the connection between sensory consonance and

frequency ratios of small whole numbers. (I can't help but wonder if

Pythagoras really discovered it himself or if he merely brought the concept

back to Greece, having been shown it on his travels, perhaps while in

India.)

> I must say I would take the opposite tack to you;

> breaking the link between mathematics and music

> may well have damaged both.

I don't understand your comment. I too think that there is a close link

between music and mathematics. And the link, as a whole, is profound. (That

little part of the link known to the Pythagoreans, while clearly very

important in some sense, is trivial.)

David

Apr 23, 2006, 6:03:23 PM4/23/06

to

jhnr...@yahoo.co.uk wrote:

> While searching for a decent online ebook, or course notes,

> on category theory I came across the above. Talk about

> category theory getting everywhere these days! Anyone

> studied this?

It strikes me as Mathematics Made Difficult applied to music. I've got

the relevant background in both math and music, and I don't see that

there is much of interest in it, and a lot of it really boils down to

obfuscation. By far the most interesting and sophisticated discussion

of mathematics and music, it seems to me, is on the Yahoo group

tunin...@yahoo.com, which is an offshoot of the old Mills College

mailing list. The relevant algebra for a great deal of music theory is,

I'm afraid, boring old finitely generated abelian groups. Multilinear

algebra gets used a lot on tuning-math.

Mazzola and company, by the way, are starting up a new music/math

journal, and Mazola does know math.

Apr 23, 2006, 6:50:23 PM4/23/06

to

Gerry Myerson wrote:

> A key use of commutative algebra is given as "alterations are

> tangents". For example, F-sharp and G-flat name the same tone and the

> same key on a piano keyboard. The two names denote two different

> structural roles in composition---it matters that F-sharp is not just a

> note, but an alteration of F-natural. The author regards the space of

> tones as an abelian group $\Bbb{M}$, so any single note is an element

> of the group, which the author treats as a $\Bbb{Z}$-element, a

> homomorphism $\Bbb{Z}\rightarrow\Bbb{M}$ from the group of integers.

> Then an altered note can be considered as a pair of notes, a base plus

> an altered value, and so as a $\Bbb{Z}\oplus\Bbb{Z}$-element of

> $\Bbb{M}$. In the traditional alterations, to sharps or flats, the

> altered value is near the base note, and the author models these by

> infinitesimal deformations, or tangent vectors. That is, he uses the

> ring of dual numbers $\Bbb{Z}[\epsilon]$, and all of this is applied

> over other ground rings than $\Bbb{Z}$, and thus brings us to scheme

> theory.

I would say the correct explanation is a lot more lowbrow, which is

typical of the problem with the book.

We have a free rank three abelian group, the group of "5-limit just

intonation",

J = {2^a 3^b 5^c}. As an abstract group this is isomorphic to Z^3. This

maps to a free rank two group, the group M of "5 limit (logical)

meantone", meaning the group generated by 2 and some tuning of a flat

fifth, which we regard as log-linearly independent from 2 over Q. That

is, as an abstract group, it is a*O + b*F, where a and b are integers,

and O is the octave and F is the (flattened) fifth, written additively

(logarithmically.) This is the group of common practice musical tones

from the Renaissance through the seventeenth century and well in the

eighteenth, and forms the historical basis of our understanding of

musical tones and tonality. In this group, the "5" of Z^2, [0,0,1], is

represented by 4*F, and there is a map with kernel [-4, 4, -1], or

81/80, which sends J-->M, "tempering out" the syntonic comma 81/80.

In M, enharmonic notes are distinct; for example C# is -4*O+7*F, and

Db is

3*O-5*F. This is exactly what they were in the medieval Pythagorean

tuning, with pure fifths and octaves, except now the fifth is flat,

even flatter than it is in 12-equal temperament, and hence C# rather

than being sharper than Db (2187/2048 > 256/243) is now flatter. This

is the tuning that the functional harmony of the earlier common

practice period is based on, and the tuning which even later for the

most part (W. A. Mathieu describes some exceptions in his "Harmonic

Experience") explains what is going on in common practice music. I've

had a lot of experience lifting 12 equal music to meantone, and my

conclusion is that most of the time, meantone explains what is going on

even in the likes of Wagner or Brahms.

Now take the rank one group generated by 2^(1/12), which is the group

of 12-equal music and isomorphic as a group to Z. We now have maps

J-->M-->Z, and music in Z (you can think of these as midi note

numbers.) But to *understand* the music in Z, a lot of times you should

lift it to M, which is where so much of Western musical tone

vocabularly historically comes from.

No dual numbers need apply. It's just J-->M->Z, rather than merely

J--Z. It factors through meantone.

Apr 23, 2006, 7:35:44 PM4/23/06

to

Gene Ward Smith wrote:

> jhnr...@yahoo.co.uk wrote:

>

> > While searching for a decent online ebook, or course notes,

> > on category theory I came across the above. Talk about

> > category theory getting everywhere these days! Anyone

> > studied this?

>

> It strikes me as Mathematics Made Difficult applied to music. I've got

> the relevant background in both math and music, and I don't see that

> there is much of interest in it, and a lot of it really boils down to

> obfuscation. By far the most interesting and sophisticated discussion

> of mathematics and music, it seems to me, is on the Yahoo group

> tunin...@yahoo.com, which is an offshoot of the old Mills College

> mailing list. The relevant algebra for a great deal of music theory is,

> I'm afraid, boring old finitely generated abelian groups. Multilinear

> algebra gets used a lot on tuning-math.

> jhnr...@yahoo.co.uk wrote:

>

> > While searching for a decent online ebook, or course notes,

> > on category theory I came across the above. Talk about

> > category theory getting everywhere these days! Anyone

> > studied this?

>

> It strikes me as Mathematics Made Difficult applied to music. I've got

> the relevant background in both math and music, and I don't see that

> there is much of interest in it, and a lot of it really boils down to

> obfuscation. By far the most interesting and sophisticated discussion

> of mathematics and music, it seems to me, is on the Yahoo group

> tunin...@yahoo.com, which is an offshoot of the old Mills College

> mailing list. The relevant algebra for a great deal of music theory is,

> I'm afraid, boring old finitely generated abelian groups. Multilinear

> algebra gets used a lot on tuning-math.

I'm not sure I completely agree. To be honest, my music theory is

probably stronger than my algebra, but I don't think I'm lacking any

grasp on the author's thesis or the point of this book.

The author is merely trying one way to abstract music into mathematics.

In fact, by the end, it looks like he tries to do it in dozens of

different ways. While I agree this is a far cry from being a standard

either in the mathematical or the musical world, I don't agree that it

lacks interest. People have attempted to link the two fields in every

conceivable way--everything from statistical analysis of popular music

to converting algebraic groups to violin solos.

To the best of my knowledges, this has led to few major advances in

mathematics, other than some occasional very interesting applications

of algebra, analysis, and probability theory. On the other hand, it has

sparked a lot of creativity in the musical world. Composers have found

ways to transform these links into musical pieces of all times, finding

homes everywhere from coffee shops to concert halls.

Case in point: In the 50's (give or take some years), a (mostly) Greek

composer Iannis Xenakis develops methods for converting formal

stochastic processes into music, partly in reaction to the state of the

serial music sweeping Europe at the time. Entirely as a result of his

innovative composition methods, within only a few years, he's being

conducted in America by Aaron Copland, awarded the Grand Prize from the

French Recording Academy, invitied to teach at University of Indiana

and Bloomington,....

While I agree most any attempt to formalize music will be arbitrary and

not terribly contributory to the field of math, I don't agree it's

pointless. Anything that can create new, inspired, and even substantial

music has certainly benefitted the musical community. I think Mazzola's

inquiry has the potential to do just that. If even one composer

somewhere, including Mazzola himself, is inspired by his abstraction of

tuning and creates music as a result, then his book was not in vain.

This is by NO means unlikely. If many do, then a new Xenakis may come

out of it.

Richard

Apr 23, 2006, 8:10:58 PM4/23/06

to

In article <1145832623....@u72g2000cwu.googlegroups.com>,

"Gene Ward Smith" <genewa...@gmail.com> wrote:

"Gene Ward Smith" <genewa...@gmail.com> wrote:

> Gerry Myerson wrote:

>

> > A key use of commutative algebra is given as "alterations are

> > tangents". For example, F-sharp and G-flat name the same tone and the

> > same key on a piano keyboard. The two names denote two different

> > structural roles in composition---it matters that F-sharp is not just a

> > note, but an alteration of F-natural. The author regards the space of

> > tones as an abelian group $\Bbb{M}$, so any single note is an element

> > of the group, which the author treats as a $\Bbb{Z}$-element, a

> > homomorphism $\Bbb{Z}\rightarrow\Bbb{M}$ from the group of integers.

> > Then an altered note can be considered as a pair of notes, a base plus

> > an altered value, and so as a $\Bbb{Z}\oplus\Bbb{Z}$-element of

> > $\Bbb{M}$. In the traditional alterations, to sharps or flats, the

> > altered value is near the base note, and the author models these by

> > infinitesimal deformations, or tangent vectors. That is, he uses the

> > ring of dual numbers $\Bbb{Z}[\epsilon]$, and all of this is applied

> > over other ground rings than $\Bbb{Z}$, and thus brings us to scheme

> > theory.

>

> I would say the correct explanation is a lot more lowbrow, which is

> typical of the problem with the book.

Please note that I didn't write what Gene says I wrote - I merely

quoted it, and gave the source in my original post.

Apr 23, 2006, 8:20:48 PM4/23/06

to

Gerry Myerson wrote:

> Please note that I didn't write what Gene says I wrote - I merely

> quoted it, and gave the source in my original post.

Sorry. Have you thought at all about music theory stuff lately, BTW?

Apr 24, 2006, 12:46:44 AM4/24/06

to

kibby wrote:

> People have attempted to link the two fields in every

> conceivable way--everything from statistical analysis of popular music

> to converting algebraic groups to violin solos.

>

> To the best of my knowledges, this has led to few major advances in

> mathematics, other than some occasional very interesting applications

> of algebra, analysis, and probability theory.

Music, like many other fields, has occasionally been a source of

interesting theorems and conjectures. An example is Størmer's theorem,

which says that if you take the set S_p integers factorizable into

primes no larger than p, then there are only a finite number of

integers n such that n and n-1 are both in S_p. For this problem

Størmer gives an effective method for computing all such n.

http://en.wikipedia.org/wiki/Stormer%27s_theorem

Here are some conjectures relating to music theory about the Riemann

zeta function:

http://www.research.att.com/~njas/sequences/A117536

http://www.research.att.com/~njas/sequences/A117537

http://www.research.att.com/~njas/sequences/A117538

Apr 24, 2006, 12:55:47 AM4/24/06

to

In article <1145838048.7...@t31g2000cwb.googlegroups.com>,

"Gene Ward Smith" <genewa...@gmail.com> wrote:

"Gene Ward Smith" <genewa...@gmail.com> wrote:

No, but I have sent my daughter to get a degree in music composition,

so maybe the next generation will have something to say. My own musical

endeavors have all been applied, as I try to sing the right notes in

the choir I've joined.

Apr 24, 2006, 2:24:40 AM4/24/06

to

Gerry Myerson wrote:

> No, but I have sent my daughter to get a degree in music composition,

> so maybe the next generation will have something to say.

Tell her microtonality rocks. :)

> My own musical

> endeavors have all been applied, as I try to sing the right notes in

> the choir I've joined.

Mine are here:

I actually got mentioned recently on the blog of the New Yorker's music

critic, Alex Ross, so now I'm wondering if I'm a real composer.

Apr 24, 2006, 7:27:42 AM4/24/06

to

Gene Ward Smith wrote:

>> My own musical

>> endeavors have all been applied, as I try to sing the right notes in

>> the choir I've joined.

>

> Mine are here:

>

> http://www.xenharmony.org

Interesting.

You seem to me to be exactly the right person

to tell us if The Topos of Music is wise or foolish.

I wonder if music and mathematics involve the same region of the brain?

Apr 24, 2006, 7:33:06 AM4/24/06

to

Timothy Murphy <t...@birdsnest.maths.tcd.ie> wrote:

> Gene Ward Smith wrote:

>

> >> My own musical

> >> endeavors have all been applied, as I try to sing the right notes in

> >> the choir I've joined.

> >

> > Mine are here:

> >

> > http://www.xenharmony.org

>

> Interesting.

> You seem to me to be exactly the right person

> to tell us if The Topos of Music is wise or foolish.

> Gene Ward Smith wrote:

>

> >> My own musical

> >> endeavors have all been applied, as I try to sing the right notes in

> >> the choir I've joined.

> >

> > Mine are here:

> >

> > http://www.xenharmony.org

>

> Interesting.

> You seem to me to be exactly the right person

> to tell us if The Topos of Music is wise or foolish.

I thought he'd already done that:

"It strikes me as Mathematics Made Difficult applied to music. I've got

the relevant background in both math and music, and I don't see that

there is much of interest in it, and a lot of it really boils down to

obfuscation."

which is what I'd guessed.

David

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