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Well, I really thought this post would be about the Z-relation in music, but a series of recent discussions with Andrée Ehresmann, a mathematician specialized in category theory, led to some nice results in transformational music theory which I thought would be interesting to share here. By the way, I’m renaming those posts as “Transformational Music Theory” as it is becoming more general than neo-Riemannian theory (I’m still tagging them with it, so you can easily browse the whole set of posts).

Before I delve into mathematical theory, let me present you two small musical examples to introduce you to the problems at hand.

In the previous posts (here, here, and here), we have seen various groups of musical transformations acting on various sets of musical objects. For example, you should be familiar now with the cyclic group \mathbb{Z}_{12} = \langle z \mid z^{12}=1 \rangle, acting on the set of the twelve pitch classes. You are probably familiar too with the infinite cyclic group \mathbb{Z} = \langle t \rangle acting on the set of equally distributed beats (for example every quarter, or every eigth note).

You could then wonder: what if we defined a musical object not by only one characteristic (for example, pitch or position in time), but by two ? For example, we could say that a note is a pitch class at a precise position in time, which could be represented by a pair (n,k), where n represents the pitch class (between 0 and 11), and k represents the k-th beat. Then, we can simply imagine a group which would be the direct product \mathbb{Z} \times \mathbb{Z}_{12}, and which would act on the set of pairs (n,k). Indeed, it turns out, as we will see below, that the framework of group and group actions allows you to do so in any situation.

For the second example, look at the following progression:

Beats

As you can notice, the initial note is raised by two semitones every beat. It brings us the idea that there is some kind of “transfer” of the group action of \mathbb{Z} on the set of beats, to the group action of \mathbb{Z}_{12} on the set of pitch classes, wherein the action of the generator t would be “mapped” to that of the group element z^2.

Incidentally, we have an example of such a progression at the end of Chopin’s Sonata 3 in B minor, op. 58, wherein the initial motive is raised by two semitones every half-bar:

Chopin

 

Let’s look at the mathematical details now.

Categories of musical actions

Previously, we have seen that a group acting on a set can also be understood as a functor from the group viewed as a single-object category (where all the morphisms are invertible) to the category \mathbf{Sets}. Among such functors, we find those who are representable. Remember that having a representable functor from a group to \mathbf{Sets} is equivalent to having a simply transitive group action on the corresponding set, which is in turn equivalent to having a Generalized Interval System. This had been all worked out in a paper by Oren Kolman :

  • “Transfer Principles for Generalized Interval Systems”, O. Kolman, Perspectives of New Music, 42(1), 2004, pp. 150-190,

and more recently by Fiore, Noll, and Satyendra

  • “Morphisms of Generalized Interval Systems and PR-Groups”, T.M. Fiore, T. Noll, R. Satyendra, Journal of Mathematics and Music, 7(1), 2013, pp. 3-27

Now, we are going to consider all groups and all group actions and form a category. This is hinted at in the paper of Kolman, and formally introduced, though only in the case of GIS, in the paper of Fiore et al. So, we consider the category \mathbf{MusGrAct}, wherein

  • objects are the pairs (G,F), where where G is a group (viewed as a single-object category), and F is a functor  G \to \mathbf{Sets},
  • morphisms between objects (G_1,F_1) and (G_2,F_2) are pairs (L,\lambda), where L is a functor G_1 \to G_2 (in other words, a group homomorphism), and \lambda is a natural transformation from F_1 to F_2 \circ L.

Note that this is not the category of functors [G, \mathbf{Sets}] for a given group G, as we allow the groups to vary from one functor to another. The category \mathbf{MusGrAct} has a sub-category \mathbf{MusGrActRep}, whose objects have representable functors. This is the category of GIS studied by Fiore et al. (though curiously, they never mention natural transformations, but simply equivariant maps…which is equivalent).

The second example I have given above is a diagram in \mathbf{MusGrActRep}, composed of two objects (\mathbb{Z},F_1) and (\mathbb{Z}_{12},F_2) (the functors F_1 and F_2 are the natural group actions on the set of beats and pitch classes respectively), and a morphism (L,\lambda) between them, where L is the homomorphism which sends the generator t to z^2, and where \lambda maps the beat k on the pitch class (2k+n_0) \pmod{12}, where n_0 is the pitch class corresponding to the initial beat (in our example, F, i.e. n_0=5).

Having these categories, Andrée Ehresmann and I have studied their structure. It turns out we have this very simple but nice result:

Theorem 1: The category \mathbf{MusGrAct} is complete and cocomplete.

Remember that “complete” means that the category has all small limits, and that “cocomplete” means it has all small colimits. In particular, it therefore means that the category has all products ! Therefore, for any groups acting on sets of musical objects, you can form the direct product acting on the direct product of sets.

What about the category \mathbf{MusGrActRep} ? Alas, it is not as well-behaved as the previous one. To begin with, it is not complete. Take for example G_1=\mathbb{Z}_2 acting on the two element set \{p_1,p_2\}, and G_2=\mathbb{Z}_4 acting on the four element set \{q_1,q_2,q_3,q_4\}, the non-trivial homomorphisms L_1=L_2 \colon \mathbb{Z}_2 \to \mathbb{Z}_4 and the natural transformations \lambda_1 which sends p_1 to q_1 and p_2 to q_3, and \lambda_2 which sends p_1 to q_2 and p_2 to q_4. You can easily check that this system has no limit in \mathbf{MusGrActRep}.

Neither is this category cocomplete. For example, consider the discrete system wherein G_1=G_2=\mathbb{Z}_2 acting on two-element sets, with no morphisms between these GIS. This system has no colimit in \mathbf{MusGrActRep}. You can also check that the category has no initial object.

So what can we do, then ? Still, we have some partial structure in this category.

Theorem 2: The category \mathbf{MusGrActRep} has products, and filtered colimits.

So we can still consider products of GIS ! Hence the example which I mentionned at the beginning, in the case of pitch classes and beats. Some colimits can also be considered.

But what is the musical meaning of limits and colimits ? Let’s consider again the second example, with its two objects (the two GIS) and the morphism between them. It turns out that this system has both a limit and a colimit in \mathbf{MusGrActRep}. The limit is the group \mathbb{Z} acting on the set of pairs (k,2k+n_0). The first coordinate of these pairs represents the position in time, i.e. the k-th beat, while the second coordinate represents the associated pitch class (it is not free, since we have a precisely defined morphism of GIS). So, we get a “supra”-group of transformations acting on a set of musical objects which contain information about time and information about pitch. The colimit is the group \mathbb{Z}_{12} acting on a set of twelve elements which represents the different classes of the original musical objects, in terms of position in time and pitch. For example, we have the element F_{\{0,6,12,18,\ldots\}}, since the pitch class F will be found on the initial beat, as well as on the 6-th, 12-th, 18-th, etc.

Now let’s try to be even more general. Why should we limit ourselves to group actions ? After all, we have seen previously (here and here) some examples of monoids acting on a set of musical objects (remember that monoids can be viewed as single-object categories, where the morphisms are not necessarily invertible). We could also envision groupoids acting on multiple sets of musical objects, or even very general categories.

In fact we have very little to modify in the definitions above. Consider the category \mathbf{MusAct} of musical actions, where

  • objects are the pairs (\mathcal{C},F), where \mathcal{C} is a small category, and F is a functor \mathcal{C} \to \mathbf{Sets},
  • morphisms between objects (\mathcal{C}_1,F_1) and (\mathcal{C}_2,F_2) are pairs (L,\lambda), where L is a functor \mathcal{C}_1 \to \mathcal{C}_2, and \lambda is a natural transformation from F_1 to F_2 \circ L.

This category has a subcategory \mathbf{MusActRep}, where the objects have representable functors. Let’s look at their structure.

Theorem 3: The category \mathbf{MusAct} is complete and cocomplete.

Good news ! We can also consider limits, products, colimits… in this much more general case. It should be clear, however that the subcategory \mathbf{MusActRep} is neither complete, nor cocomplete.

We thus see that the framework of category theory can provide us with a precise formalization of some musical concepts. This was the last post of 2014 ! I wish everybody a very happy new year 2015 !

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