The n-Category Café

I tried once to read the Mazzola book. I sincerely hope that my impressions of the book are inaccurate, as that would restore some of my lost hopes for the use of topoi in interesting areas.

I found the book to be ghastly - to fluffy for mathematicians, and probably too mathematical for music theorists - and wrote a review for the Jena-based mathematics magazine that said as much. The review ended up not being published, because it was too harsh, but being widely circulated among the magazine staff, because it was amusingly so.

I work at UC and while I am neither a professional mathematician nor musician, I have some training in both fields, probably just enough that this talk won’t go completely over my head. I’m planning on attending and will post a summary here. Thanks for mentioning this, I probably would have missed it otherwise!

I am a composer working in algorithmic composition.

First, a thanks to John Baez for his prior “This Weeks Find” week 234 on music theory, which was a gold mine for me.

Second, I have put some of the papers referenced by Professor Baez to practical use in algorithmic composition. If you are interested in the details, check out my personal Web site at http://www.michael-gogins.com.

I presume other composers are doing similar work, but the only one I personally know of is my friend Drew Krause.

Musically, I have found most useful the work of Tymoczko, but also neo-Riemannian transformations. I have tried to penetrate The Topos of Music, but I suspect that my self-taught mathematical background is not (yet?) adequate; it may also be that the concepts are not as useful for my style of composition as the ones mentioned above.

Regards,
Mike Gogins

I do not know enough to make a qualified mathematical comment, but since you mention Pythagoras and admit to being a Pythagorean of sorts, I take this as an opportunity to point out a wiki-entry that I have written:

a href = “http://en.wikipedia.org/wiki/Pythagoreans”

(I’m responsible for the subsections “Natural Philosophy” and “Cosmology.”) The section “Natural Philosophy” discusses some Pythagorean music philosophy. The history of Pythagoreanism is one of my long-time hobbies.

I’ve always thought there should be Riemannian geometry associated with a sort of rubato.

There is a sort of rubato where the left hand keeps strict time, and the right is free. I imagine the strict time is like flat space.

There is another sort of rubato where both hands are free - perhaps some call these “agogic distortions”. I imagine these to be like curved space, where each little bit is approximately flat, but large portions. If there is an “Einstein Field Equation for this”, maybe the stress-energy tensor would be some of Mazzola’s “gestures”?

OK, I will try to read the Mazzola book.

Well, I’m about 14 years late to this discussion, but have some thoughts on Mazzola’s work.

I’m currently a graduate student in music composition, so much of my work consists of just writing music. But my compositional thinking has been for many years influenced by mathematics. One way that I see the difference between mathematics and music is that if mathematics provides a view of structure as “eternal”, then music is the temporalization of such structures. It’s like bringing the Platonic heavens down to earth.

Moving on to Mazzola. Much of the theories in Topos of Music are based on Mazzola’s form and denotator theories. Forms constitute the basic kind of object wherein lie musical “facts”. So a form may consist of (say) four parameters such as Onset, Duration, Pitch, and Loudness, and a point in in this space is like a note event. Hence points in this space consist of an onset coordinate, a duration coordinate, a pitch coordinate, and a loudness coordinate. Once you have such a form, call it $Notes$, you can then take the “power set” $\mathcal{P}(Notes)$, and this space consists of all subsets of note events. Hence you may conceive of a musical score as a point in $\mathcal{P}(Notes)$.

To derive forms, Mazzola starts with the category $\mathbf{Mod}$ of modules for objects and affine transformations for morphisms. The idea is that many music-theoretic constructions are grounded in modules. For instance, the set of pitch-classes in Western music is the module $\mathbb{Z}_{12}$ with multiplication ring $\{ 1, 5, 7, 11\}$, the space of onset times is $\mathbb{Q}$, and so on. However, $\mathbf{Mod}$ is not really an adequate category to work in for a number of reasons. One of the most obvious reasons is that in music theory, we are often interested in taking power sets of certain sets. For instance, we have the set $\mathbb{Z}_{12}$ of pitch-classes, but if we want the set of pitch-class sets, then we need to take the power set $\mathcal{P}(\mathbb{Z}_{12})$. But $\mathbf{Mod}$ cannot accommodate such power set constructions. Therefore Mazzola takes instead the category of set-valued presheaves over $\mathbf{Mod}$, which he calls $\mathbf{Mod}^{[at]}$, and this enables much more general constructions with modules, which are necessary for music-theoretical purposes. So (roughly) the presheaves in $\mathbf{Mod}^{[at]}$ are called forms, and this category allows taking arbitrary limits, colimits, and power objects. (Although those familiar with size issues may be wary of this construction…)

Denotators are then the generalized elements of forms. So now we can do a whole bunch of music theory working with forms and denotators. For instance, what is usually called a set-class in music theory is a set of pitch-class sets that is invariant under the set of transpositions and inversions, otherwise known as the $T/I$ group. For instance, the pitch class set $(037)$ is the set of all major and minor triads, since any major/minor triad can be transformed into any other major/minor triad via some action in $T/I$. Hence the set of set-classes is a subobject of $\mathcal{P}(\mathcal{P}(\mathbb{Z}_{12}))$, and the points in this space are the set-classes.

While Mazzola’s theories work wonderfully for a great deal of music-theoretic constructions, the world of actually composing music tends to be much more complex. Compositional ideas very often do not comply with the simplistic algebraic nature of modules. Even the gestures that have been mentioned above are not accommodated by Mazzola’s form and denotator theories, which is why he had to invent a brand new gesture theory for such situations. Because of these inadequacies, and because I’m a composer, I thought it would be really interesting if Mazzola’s formalisms could be generalized to accommodate just about any kind of musical thought conceivable. The difficult thing is that music can really be anything whatsoever, so long as it’s sonified! I can’t imagine any mathematical structure, for instance, that couldn’t be sonified in some way. Not that all these would be interesting, of course, but in principle it would seem that a fully universal framework for conceiving of musical phenomena would have to provide a universal definition of structure as such. Because of all this, I developed my own theories that generalize Mazzola’s. Mazzola’s forms and denotators are replaced by what I call structures and compositions (the term “compositions” here is not specifically “musical compositions”). I devise a category $\mathbf{Rel}^{[at]}$ that I claim consists of all structured sets for objects, and allowing any kind of (underlying) set function for morphisms. It is my belief that this category is a “universal concrete category”, in the sense that any structured set (such as e.g. topological spaces, groups, vector spaces, partial orders, and anything else that is called a “structured set”) exists in this category. Anyway, it subsumes Mazzola’s forms because modules exist in $\mathbf{Rel}^{[at]}$, and it accommodates colimit, limit, and power constructions. It also subsumes his gesture theories since topological spaces exist in $\mathbf{Rel}^{[at]}$ as well. For what it’s worth, so far I have found it very useful in my own compositional thinking.

Anyway, if anyone is interested in this, I can send you my writings. Otherwise, I hope I was able provide some helpful information on Mazzola’s work!

Well, I’m about 14 years late to this discussion, but have some thoughts on Mazzola’s work.

I’m currently a graduate student in music composition, so much of my work consists of just writing music. But my compositional thinking has been for many years influenced by mathematics. One way that I see the difference between mathematics and music is that if mathematics provides a view of structure as “eternal”, then music is the temporalization of such structures. It’s like bringing the Platonic heavens down to earth.

Moving on to Mazzola. Much of the theories in Topos of Music are based on Mazzola’s form and denotator theories. Forms constitute the basic kind of object wherein lie musical “facts”. So a form may consist of (say) four parameters such as Onset, Duration, Pitch, and Loudness, and a point in in this space is like a note event. Hence points in this space consist of an onset coordinate, a duration coordinate, a pitch coordinate, and a loudness coordinate. Once you have such a form, call it $Notes$, you can then take the “power set” $\mathcal{P}(Notes)$, and this space consists of all subsets of note events. Hence you may conceive of a musical score as a point in $\mathcal{P}(Notes)$.

To derive forms, Mazzola starts with the category $\mathbf{Mod}$ of modules for objects and affine transformations for morphisms. The idea is that many music-theoretic constructions are grounded in modules. For instance, the set of pitch-classes in Western music is the module $\mathbb{Z}_{12}$ with multiplication ring $\{ 1, 5, 7, 11\}$, the space of onset times is $\mathbb{Q}$, and so on. However, $\mathbf{Mod}$ is not really an adequate category to work in for a number of reasons. One of the most obvious reasons is that in music theory, we are often interested in taking power sets of certain sets. For instance, we have the set $\mathbb{Z}_{12}$ of pitch-classes, but if we want the set of pitch-class sets, then we need to take the power set $\mathcal{P}(\mathbb{Z}_{12})$. But $\mathbf{Mod}$ cannot accommodate such power set constructions. Therefore Mazzola takes instead the category of set-valued presheaves over $\mathbf{Mod}$, which he calls $\mathbf{Mod}^{[at]}$, and this enables much more general constructions with modules, which are necessary for music-theoretical purposes. So (roughly) the presheaves in $\mathbf{Mod}^{[at]}$ are called forms, and this category allows taking arbitrary limits, colimits, and power objects. (Although those familiar with size issues may be wary of this construction…)

Denotators are then the generalized elements of forms. So now we can do a whole bunch of music theory working with forms and denotators. For instance, what is usually called a set-class in music theory is a set of pitch-class sets that is invariant under the set of transpositions and inversions, otherwise known as the $T/I$ group. For instance, the pitch class set $(037)$ is the set of all major and minor triads, since any major/minor triad can be transformed into any other major/minor triad via some action in $T/I$. Hence the set of set-classes is a subobject of $\mathcal{P}(\mathcal{P}(\mathbb{Z}_{12}))$, and the points in this space are the set-classes.

While Mazzola’s theories work wonderfully for a great deal of music-theoretic constructions, the world of actually composing music tends to be much more complex. Compositional ideas very often do not comply with the simplistic algebraic nature of modules. Even the gestures that have been mentioned above are not accommodated by Mazzola’s form and denotator theories, which is why he had to invent a brand new gesture theory for such situations. Because of these inadequacies, and because I’m a composer, I thought it would be really interesting if Mazzola’s formalisms could be generalized to accommodate just about any kind of musical thought conceivable. The difficult thing is that music can really be anything whatsoever, so long as it’s sonified! I can’t imagine any mathematical structure, for instance, that couldn’t be sonified in some way. Not that all these would be interesting, of course, but in principle it would seem that a fully universal framework for conceiving of musical phenomena would have to provide a universal definition of structure as such. Because of all this, I developed my own theories that generalize Mazzola’s. Mazzola’s forms and denotators are replaced by what I call structures and compositions (the term “compositions” here is not specifically “musical compositions”). I devise a category $\mathbf{Rel}^{[at]}$ that I claim consists of all structured sets for objects, and allowing any kind of (underlying) set function for morphisms. It is my belief that this category is a “universal concrete category”, in the sense that any structured set (such as e.g. topological spaces, groups, vector spaces, partial orders, and anything else that is called a “structured set”) exists in this category. Anyway, it subsumes Mazzola’s forms because modules exist in $\mathbf{Rel}^{[at]}$, and it accommodates colimit, limit, and power constructions. It also subsumes his gesture theories since topological spaces exist in $\mathbf{Rel}^{[at]}$ as well. For what it’s worth, so far I have found it very useful in my own compositional thinking.

Anyway, if anyone is interested in this, I can send you my writings. Otherwise, I hope I was able provide some helpful information on Mazzola’s work!