I’m back to the topic of networks in transformational music theory (continuing this post), after the last post about wreath products. In fact, there was one reason I wanted to talk about wreath products first, and the reason is that they are intricately linked to some groupoids that arise in considering networks and poly-Klumpenhouwer networks. This is some work that was published with co-authors Andrée Ehresmann and Moreno Andreatta in the MCM proceedings of last year:

- Popoff A., Andreatta M., Ehresmann A. (2019) Groupoids and Wreath Products of Musical Transformations: A Categorical Approach from poly-Klumpenhouwer Networks. In: Montiel M., Gomez-Martin F., Agustín-Aquino O. (eds) Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science, vol 11502. Springer, Cham, available here.

But let me explain all this in a simplified way.

In previous posts, we have seen networks describing musical objects and transformations between them. For example, if we consider pitch-classes and transformations of the group between them, we can draw networks like the one below.

It describes specific relationships between C sharp, F, and B flat using the transformations (group elements) and , the transformation being the result of composing the latter with the former. Here is another network.

These are different pitch classes, but the relationships between them are the same as in the first network. In fact, we can consider the following abstract skeleton

in which the nodes can be filled with pitch classes, provided that the relationships between them are respected. In mathematical terms, and this is where category theory kicks in, this skeleton is described using a functor , where is in our case a poset with three objects and is a category, in our case a single-object category with invertible morphisms, i.e. a group. The functor labels the arrows of with morphisms of . Now, if we want to fill the nodes (either with single pitch-classes, i.e. singletons, or more), we would have to add some functors to or and throw in some natural transformations for consistency. You can check the previous posts (here, here, here, and here) to get all the details about that, but in this post we are going to focus on the functors .

Having defined networks, we are obviously interested in how they evolve, how they are transformed. In previous posts, we have seen that one way to consider relationships between networks is to consider how they are transformed by a morphism of the category to another category (and some additional stuff, but let’s keep it simple). Historically, this was considered as a specific case in which the category was the group, and in which its automorphisms were studied, hence the notion of *network isography*.

But as one can immediately remark, this limits number of possibilities by which networks can be transformed. In other words, if one has two functors , and , there is no guarantee that we can find an isomorphism such that .

So let’s relax that condition completely by considering transformations of another type. We fix the category (we’ll add more conditions in a minute) and we do not consider isomorphisms anymore. Instead, if we have two functors and , we will look at the transformation that takes one to the other, or in category-theory terms, a natural transformation . This is a well-known case in category theory: we are in fact looking at the category of functors which has

- functors as objects, and
- natural transformations between functors and as morphisms.

The interesting thing about the morphisms (the natural transformations ) in this category is that, from a musical point of view, they have the potential to bridge two different angles at which we can consider collections of musical objects, namely the internal relations and the inter-collection relations. If we consider pitch-classes for example, we could have on one side internal transformations of pitch classes, and on the other side voice leadings. For example, the diagram below represents in graphical form two different functors and and one possible natural transformation between them. The natural transformation is defined by components for each object of which are elements of , and which we represent by the labelled purple arrows.

In aparte, these two angles are in fact exchangeable: if we consider diagrams in the category , i.e. functors , then the category of functors is isomorphic to (a result which had been observed in another math/music context by Mazzola).

Back to our category : what can we say about it ? It turns out that if is a (finite) poset with a bottom (or top) element, and is a groupoid, then it can immediately be observed that is also a groupoid: any natural transformation between two objects of is entirely determined by its component on the bottom element, and since morphisms of are invertible, so is the natural transformation. In the particular case in which is a group, for any two objects of , the hom-set between them can be bijectively identified with the set of elements of the group.

Notice that if we have the constitutive functors of a whole poly-Klumpenhouwer network, and in particular the functor , then a morphism of can be extended to a morphism of networks. For example, in the network above, if we fill the nodes of the network on the left as follows,

then we can extend the components of the natural transformation (the labelled purple arrows) through the action of (in this case, the usual action of the group on the twelve pitch classes) to obtain the resulting network on the right, as follows.

In fact, thanks to this, we get an action of the groupoid on sets of networks, i.e. a functor where the images of the objects of are sets of poly-Klumpenhouwer networks built on a given functor , and which are transformed through the extension the of natural transformations by .

The interplay between internal relations and voice-leadings is best explained through an example found in Berg Op. 5/2, at bars 5-6 of the piano right hand. You can listen to Berg’s Op. 5/2 below.

And here’s the reduction of the piano right hand at bars 5-6:

We can represent the pitch-classes of the different chords and the transformations of these pitch classes between chords as follows. We split the analysis in two at the second C minor chord (the fifth chord from the beginning) for reasons which will soon become obvious.

Notice that the progression of voices is similar in the two segments: , , then for the top note in our diagram, and the same for the other ones.

Now let’s add internal relations for each one of these chords. Note that this is not the only possible choice, but one that highlights a particular aspect of this progression.

The interesting thing is that in each segment, only two functors are used, and though they are different from one segment to another, they result in the same voice leadings.

Now, at the beginning of this post, I’ve said that there is a link between groupoids and wreath products. The link is the notion of groupoid bisection which we define as follows. We consider a finite groupoid and we index the objects of by , where is the number of objects in . Then, a bisection of the groupoid is the data of a permutation and a collection of morphisms of for . We notate a bisection as .

We can easily compose bisections as

and they obviously form a group . In a sense, bisections are a way to ‘package’ individual transformations of the groupoid into a single big transformation.

The main result we have is that, if we define the group of endomorphisms of any object of , then the group is isomorphic to the wreath product (the demonstration is a bit technical).

Now, obviously we want to take the groupoid to be our groupoid (or any subgroupoid of) . If is a group then the group of endomorphisms of any object of is isomorphic to the group , so that is isomorphic to a wreath product (with a suitable , since is a finite poset).

Recall that we can define a functor defined by the action of on sets of networks. From there, we can obtain a group action of the wreath product on the disjoint union of all these sets. By ‘packaging’ groupoid morphisms in bisections, and by ‘packaging’ the different sets into a single large one, we recover a traditional group-theoretic approach to transformational music theory. In fact, in the case of the group, we have shown that, through an appropriate categorical construction described in the above paper, we can recover Hook’s group of Uniform Triadic Transformations (UTT), which we have seen previously to be isomorphic to . I won’t detail the construction here, which is a bit technical: it consists in considering a subgroupoid of such that the group of endomorphisms of its objects is , so that we obtain wreath products of the form instead of the more complex wreath products .

I realize this post was quite technical, so I’ll stop there. Feel free to leave comments or contact me for explanations !