Anyone read The Topos of Music by Guerino Mazzola?
jhnr...@yahoo.co.uk
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
David W. Cantrell
> 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
jhnr...@yahoo.co.uk
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.)
Timothy Murphy
> 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
Gerry Myerson
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)
David W. Cantrell
> 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
Timothy Murphy
>> 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.
porky_...@my-deja.com
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.
David W. Cantrell
> 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
Gene Ward Smith
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.
Gene Ward Smith
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.
kibby
> 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
Gerry Myerson
"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.
Gene Ward Smith
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?
Gene Ward Smith
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
Gerry Myerson
"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.
Gene Ward Smith
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.
Timothy Murphy
>> 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?
David W. Cantrell
> 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