# Networks in transformational music theory (4)

I’m going to continue the thread on musical networks and transformational music theory (last post here) and at the same time merge the ideas that we have seen in the last post on the $\mathcal{P}_{m,n}$ relation of Douthett and Steinbach between chords. This post will clarify the ideas we published in Journal of Mathematics and Music in 2018:

• Popoff, A., Andreatta, M., Ehresmann, A.: “Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis“, Journal of Mathematics and Music, Vol. 12(1), 2018, pp. 35-55, available here.

Recall from the previous post that Douthett and Steinbach introduced the $\mathcal{P}_{m,n}$ relation between chords as follows: two chords are $\mathcal{P}_{m,n}$-related if they have $m$ notes which differ by a semitone, and $n$ notes which differ by a whole tone. They also introduced the definition of the parsimonious graph for a $\mathcal{P}_{m,n}$ relation on a set $H$ of chords as the graph whose set of vertices is $H$ and whose set of edges is the set $\{(x,y) | x \in H, y \in H, \text{and } x\mathcal{P}_{m,n}y\}$. One important case is the $\mathcal{P}_{1,0}$ relation between major, minor, and augmented triads, which give rise to the “Cube Dance” graph we have seen previously.

With these notions in place, and pursuing the idea of writing musical networks wherein nodes are labelled with musical objects and arrows are labelled with transformations, we could be tempted to use the $\mathcal{P}_{m,n}$ relations as labels for arrows. For example, the opening chords in Muse’s Take a Bow (see the previous post) could be represented by the following network.

However, there are two immediate problems with this naive attempt :

• Recall that, in our conception of musical networks, the underlying diagram is a category so we expect arrows to be composed. Here we have an arrow from $D_M$ to $G_{\text{aug}}$ and an arrow from $G_{\text{aug}}$ to $D_m$, but we have not defined what should be the arrow from $D_M$ to $D_m$. In fact, with just the $\mathcal{P}_{1,0}$ relation in hand, we cannot.
• Writing a musical network in this form is ok as long as labels are transformations, since musical objects would have a unique image by these transformations. However, in this case we have a relation, which means that multiple musical objects (or none at all) could be related to a single initial musical object. In the network above, the augmented chord $G_{\text{aug}}$ is related to six other chords by the $\mathcal{P}_{1,0}$ relation, the minor chord $D_m$ being one of them. How can we formalize the fact that we are selecting only one of these six possibilities ?

The first problem is easily solved, since relations can be composed : just as we have a category $\mathbf{Sets}$ of sets and functions between them, there is a category $\mathbf{Rel}$ of sets and relations between them (functions are a particular case of relations). For clarity, let’s call $\mathcal{S}$ the $\mathcal{P}_{1,0}$ relation restricted to the set of the 24 major and minor triads, plus the four augmented triads. We can consider the relation $\mathcal{S}^2$, $\mathcal{S}^3$ obtained by the iterated composition. In fact, we can consider $\mathcal{S}$ as a generator and determine the monoid generated by this element. It turns out that this is a small monoid $M_{\mathcal{S}}$ of seven elements with the following presentation.

$M_{\mathcal{S}} = \langle \mathcal{S} \mid \mathcal{S}^7=\mathcal{S}^5 \rangle$

So, what we get really is a monoid $M_{\mathcal{S}}$ and a functor $S \colon M_{\mathcal{S}} \to \mathbf{Rel}$, wherein the image of the single object of $M_{\mathcal{S}}$ is the set of the 24 major and minor triads, plus the four augmented triads.

As for the second problem, let’s recall the definition of a poly-Klumpenhouwer network (PK-Net) which we studied in the previous post on musical networks (see here).

Definition:
Let $\mathbf{C}$ be a category, and $S$ a functor from $\mathbf{C}$ to the category $\mathbf{Sets}$. Let $\Delta$ be a small category and $R$ a functor from $\Delta$  to $\mathbf{Sets}$ with non-void values. A poly-Klumpenhouwer network of form $R$ and of support $S$ is a 4-tuple $(R, S, F, \phi)$, in which $F$ is a functor from $\Delta$ to $\mathbf{C}$, and $\phi$ is a natural transformation from $R$ to $SF$.

We represent a PK-Net by the following diagram.

In this diagram, the right part (the category $\mathbf{C}$ and the functor $S \colon \mathbf{C} \to \mathbf{Sets}$) represent the musical context of our network: this is the possible transformations and the sets of musical objects they act upon. For example, the category $\mathbf{C}$ could be the famous $PRL$ group and the functor $S$ could represent its action on the set of the 24 major and minor triads. The left part (the category $\Delta$ and the functor $R \colon \Delta \to \mathbf{Sets}$) represents the structure of our network: the category $\Delta$ describes its skeleton (nodes and arrows) to which we associates sets of elements via the functor $R$. The arrows of this skeleton are labelled with transformations (morphisms) in $\mathbf{C}$ via the functor $F$, and the natural transformation $\phi$ labels the elements of the sets as musical objects coming from the functor $S$. Let’s consider again the diagram of the PK-Net I have introduced in my previous post, which describes the transposition of the C-major triad to the E-major triad subset of the dominant seventh E7 chord.

As we can see, the natural transformation $\phi$ is picking out elements of the set of the twelve semitones as labels (notes) for the elements in the sets $R(X)$ and $R(Y)$.

This is the key insight for solving our second problem : out of the many possibilities given by a relation, the natural transformation $\phi$ will pick out some of them.

Relational poly-Klumpenhouwer networks

In fact, to define relational poly-Klumpenhouwer networks (rel-PK-Nets), we have very little to change in our above definition. Instead of dealing with musical contexts of the form $S \colon \mathbf{C} \to \mathbf{Sets}$, we will now consider the more general case of functors $S \colon \mathbf{C} \to \mathbf{Rel}$, and we get the following definition.

Definition:
Let $\mathbf{C}$ be a small 1-category, and $S$ a lax functor from $\mathbf{C}$ to the category $\mathbf{Rel}$. Let $\Delta$ be a small 1-category and $R$ a lax functor from $\Delta$ to $\mathbf{Rel}$ with non-empty values. A relational PK-net of form $R$ and of support $S$ is a 4-tuple $(R,S,F,\phi)$, in which $F$ is a functor from $\Delta$ to $\mathbf{C}$, and $\phi$ is a lax natural transformation from $R$ to $SF$, such that, for any object $X$ of $\Delta$, the component $\phi_X$ is left-total.

There are three changes which are technicalities needed for rel-PK-Nets to work:

• The functors $S$ and $R$ are lax functors instead of strict functors, which gives more possibilities for networks. We use a slightly different definition of lax functors than the usual one : it preserves identities exactly, and composition up to 2-morphisms, which in $\mathbf{Rel}$ is equivalent to saying that the relation $S(g)S(f)$ should be included in $S(gf)$.
• The natural transformation $\phi$ should also be a lax one. A lax natural transformation $\phi$ between functors $F \colon \mathbf{C} \to \mathbf{D}$ and $G \colon \mathbf{C} \to \mathbf{D}$ is the data of a collection of relations $\{\phi_X \colon F(X) \to G(X)\}$ for all objects $X$ of $\mathbf{C}$, such that, for any morphism $f \colon X \to Y$, the relation $\eta_Y F(f)$ is included in the relation $G(f) \eta_X$ (instead of requiring it to be identical). This is precisely what will allow us to pick out some of the elements in the possibilities offered by a relation.
• Finally, all the components of the natural transformation $\phi$ should be left-total, i.e. every elements in the image sets by $R$ relates to some element in the image sets by $SF$. Since the components of $\phi$ are now relations as well, we need this condition to ensure that all the nodes in our musical network are effectively labelled.

Below is an example of what a complete rel-PK-Net might look like, which shows how the lax natural transformation is selecting only one possible element out of the three elements related to $u$.

Keeping in mind the machinery behind rel-PK-Nets, we can simplify their representation when there is no ambiguity. The following example shows a rel-PK-net describing the $T_2$ transposition of the dominant seventh $C^7$ chord to the dominant seventh $D^7$ chord and the successive $I_3$ and $I_5$ inversions of its underlying $C$ major triad . It shows the advantage of dealing with lax functors when describing transformations between sets of varying cardinalities.

Coming back to our first example, the new definition of rel-PK-nets allows us to write down the following network,

keeping in mind that we are working with the functor $S \colon M_{\mathcal{S}} \to \mathbf{Rel}$ and a lax natural transformation which allows us to select only one chord at each node among the possible chords related by the $\mathcal{S}$ relation.

New monoids for music analysis

Considering functors $S \colon M \to \mathbf{Rel}$ to do music analysis opens up a boulevard of possibilities. In this part, I’m going to give a few examples.

In the previous post, in addition to the $\mathcal{P}_{1,0}$ relation, we have also introduced the $\mathcal{P}_{2,0}$ relation, which gives rise to the “Weitzmann Waltz” graph. The Tarnhelm motif in Wagner’s Tetralogy and the Imperial March in Star Wars are famous examples wherein chords are related by the $\mathcal{P}_{2,0}$ relation (at the time, we analyzed it in terms of the neo-Riemannian $PL$ transformation). For clarity, let’s call $\mathcal{T}$ the $\mathcal{P}_{2,0}$ relation restricted to the set of the 24 major and minor triads, plus the four augmented triads. In this case, the monoid $M_{\mathcal{T}}$ generated by $\mathcal{T}$ has four elements and the following presentation.

$M_{\mathcal{T}} = \langle \mathcal{T} \mid \mathcal{T}^4=\mathcal{T}^3 \rangle$

As can be seen on the “Weitzmann Waltz” graph, there are chords which are not related by $\mathcal{T}$ (the graph is disconnected). So, we could try to mix both the $\mathcal{S}$ and the $\mathcal{T}$ relation. The monoid generated by these two elements has eight elements and the following presentation

$M_{\mathcal{ST}} = \langle \mathcal{S}, \mathcal{T} \mid \mathcal{TS}=\mathcal{ST}, \hspace{0.2cm} \mathcal{S}^3=\mathcal{ST}, \hspace{0.2cm} \mathcal{T}^4=\mathcal{T}^3, \hspace{0.2cm} \mathcal{TS}^2=\mathcal{T}^2, \hspace{0.2cm} \mathcal{ST}^3=\mathcal{ST}^2 \rangle$

And here is its Cayley graph.

For example, $A$ minor chord and the $F\sharp$ major chord are related by the element $\mathcal{TS}$ of this monoid, as can be shown more explicitly below.

Similarly, let’s call $\mathcal{K}$ the restriction of the $\mathcal{P}_{2,1}$ relation to the set of major, minor, and augmented triads. Here is the Cayley graph of the monoid generated by $\mathcal{S}$ and $\mathcal{K}$.

Or why not considering $\mathcal{W}$, the restriction of the $\mathcal{P}_{1,2}$ relation to the same set of triads ?. Here is the Cayley graph of the monoid generated by $\mathcal{S}$ and $\mathcal{W}$.

There are plenty more relations you could study. For example, Reenan and Bass have studied the $\mathcal{P}_{3,0}$ relation in late 19th-century music:

• Reenan, S., Bass, S., “Types and Applications of P3,0 Seventh-Chord Transformations in Late Nineteenth-Century Music“, Music Theory Online, Vol. 22(2), available here.

Let’s do something different. As we have remarked before, a major or minor triad and its image by the neo-Riemannian operations $P$ and $L$ are also related by $\mathcal{S}$ (but not the neo-Riemannian operation $R$). So, instead of considering $\mathcal{S}$ as a whole, we can “separate” it in

• The relation $\mathcal{P}$ which is the symmetric relation such that we have $n_M \mathcal{P} n_m$ for $0 \leq n \leq 11$, and $n_\text{aug} \mathcal{P} n_\text{aug}$ for $0 \leq n \leq 3$ (assuming a semitone encoding, $n_M$ is the notation for a major triad of root $n$, and so on for minor and augmented triads). This is the relational analogue of the neo-Riemannian $P$ operation.
• The relation $\mathcal{L}$ which is the symmetric relation such that we have $n_M \mathcal{L} (n+4)_m$ for $0 \leq n \leq 11$, and $n_\text{aug} \mathcal{L} n_\text{aug}$ for $0 \leq n \leq 3$. This is the relational analogue of the neo-Riemannian $L$ operation,
• The relation $\mathcal{U}$ which is the symmetric relation such that we have $n_M \mathcal{U} (n \pmod 4)_\text{aug}$ for $0 \leq n \leq 11$, and $n_m \mathcal{U} ((n+3) \pmod 4)_\text{aug}$ for $0 \leq n \leq 11$.

With these three separate relations, the “Cube Dance” graph can be represented as follows.

The monoid $M_{\mathcal{UPL}}$ generated by these three elements has 40 elements. Here is its Cayley graph.

Armed with this new monoid, we can go back to the example of Muse’s Take a Bow, and write down a rel-PK-Net describing the chord progression in the beginning of the piece.

But there is more ! In the previous post, we studied how PK-Nets can be transformed by applying complete or local homographies. The same notions can be defined for rel-PK-Nets. In particular, we can study the automorphism group of the functor $M_{\mathcal{UPL}} \to \mathbf{Sets}$ we have just defined, or in other words the complete isographies. I won’t go into the details, but I’m fairly confident that this group is of order 7776 and is isomorphic to $(({\mathbb{Z}_3}^4 \rtimes \mathbb{Z}_2) \rtimes D_8) \rtimes D_6$. In this group of complete isographies, there is the element $(N,\tilde{\nu})$ where $N$ is the identity automorphism of $M_{\mathcal{UPL}}$, and $\tilde{\nu}$ is the natural transformation which transposes all triads by five semitones higher. And thus, we can propose a second description of this progression of chords using the iterated application of this complete isography.

In other words, this second analysis exactly describes the progression by fourth of the initial three-chord motive.

Computational aspects

For some time, I was convinced that transformational music theory was the study of functors $\mathbf{C} \to \mathbf{Sets}$, but I now think that it’s really all about functors $\mathbf{C} \to \mathbf{Rel}$. Maybe we’ll see more and more applications of such functors to music analysis in the future. Quite often, the number of elements in monoids generated by relations can become quite large, and maybe this is a reason why they seldom appeared in the math/music literature of the 90s. With the powerful enough computers we have now, this is not a problem. I’d like to point out that the Python package for transformational music theory I’ve developed (Opycleid, see here for previous discussions) is capable of handling relations and rel-PK-Nets. I will probably show more examples using Opycleid in future posts.