# Networks in transformational music theory (3)

We continue with musical networks of musical objects and their transformations, and this time we are going to explore how networks themselves can be transformed. Previously (here and here), we have reviewed the definition of a Klumpenhouwer network (K-net) which encodes network of musical objects and transformations between them. For example, a chord is made up of notes with specific transformations between them: a major triad consists by definition of a stacked major third and a minor third. Different networks can bear similarities either in their vertices (musical objects) or their arrows (musical transformations), and we therefore need a framework to compare/relate networks. You may recall that in the very first post on networks, I have mentionned the notion of network isographies, in particular positive and negative isographies. These are the historical notions developed by Klumpenhouwer and Lewin to relate K-nets. In the second part of this series, we have reviewed the definition of a Klumpenhouwer network and we have seen that they admit categorical generalizations, namely poly-Klumpenhouwer networks (PK-nets).  It’s time we organize everything and formally define morphisms of PK-nets.

Let’s recall first the categorical definition of a poly-Klumpenhouwer network which we introduced in the last post.

Def. 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 $SF$.

In this post, we will assume that the category $\Delta$ and the functor $R$ remain the same, i.e. the skeleton of the network is identical though the labelling of vertices and the arrows may change. In the formalism of ordinary Klumpenhouwer networks, and in the papers of Klumpenhouwer and Lewin, this is always the case. However, nothing prevents us from changing the structure of our network, and I will present the more general case at the end of this post. Keeping $Delta$ and $R$ makes the introduction of PK-nets morphisms simpler for now.

That said, assume we have two PK-nets $(R,S,F,\phi)$ and $(R,S',F',\phi')$, with two different musical contexts $S \colon \mathbf{C} \to \mathbf{Sets}$ and $S' \colon \mathbf{C'} \to \mathbf{Sets}$, as shown diagrammatically below. How do we relate them, or equivalently, how do we define possible morphisms between them ?

Before we give the general definition, it is useful to recall how traditional Klumpenhouwer networks are compared. Klumpenhouwer networks label arrows with elements of the $T\text{/}I$ group. As is well known from group theory, the automorphisms of the $T\text{/}I$ group are given by the pairs $(k,p)$ with $k \in \{1,5,7,11\}$ and $p \in \mathbb{Z}_{12}$. The action of an element $(k,p) \in \text{Aut}(T\text{/}I)$ on the elements of $T\text{/}I$ is given by

• $(k,p)(T_n) = T_{kn}$, and
• $(k,p)(I_n) = I_{kn+p}.$

In the literature on transformational music theory, the usual notation for automorphisms of the form $(1,p)$ is $\langle T_p \rangle$, whereas the notation for automorphisms of the form $(11,p)$ is $\langle I_p \rangle$. Automorphisms of the form $(5,p)$ or $(7,p)$ are rarely seen. With this mind, Klumpenhouwer networks are compared using the notion of network isography:

• Two Klumpenhouwer networks are said to be in positive isography, if the edge labels of the second are obtained through the transformation of the edge labels of the first by an automorphism of the type $\langle T_p \rangle.$
• Two Klumpenhouwer networks are said to be in negative isography, if the edge labels of the second are obtained through the transformation of the edge labels of the first by an automorphism of the type $\langle I_p \rangle.$
• The strong isography relationship is a particular case of positive isography, in which the applied automorphism is $\langle T_0 \rangle$, i.e. the identity on transformations.

These definitions bring the following three remarks.

• We have considered the specific case of the $T\text{/}I$ group, but in the general case of a group $G$ all that counts is that the transformations are consistently transformed by an automorphism $N$ of $G$.
• Since group automorphisms are functors from a categorical point-of-view, and since we are not restricted to groups but are considering categories $\mathbf{C}$, we can consider functors from $\mathbf{C}$ to $\mathbf{C'}$ in the general case.
• Klumpenhouwer and Lewin considered automorphisms, but we may generalize this to any group homomorphism, or in other words any functor from $\mathbf{C}$ to $\mathbf{C'}$ and not just the invertible ones.

It should be clear by now that our definition for a morphism of PK-net should include the need for a functor $N$ from $\mathbf{C}$ to $\mathbf{C'}$ such that $F'=NF$, so that the labelling of the arrows of the network is consistently maintained by the functor $N$. In addition, we need to ensure that the labelling of the vertices is also consistently maintained. Without further ado, here is the general definition of PK-net homographies.

Def. A PK-homography $(N, \nu) \colon K \to K'$ between PK-nets $K=(R,S,F,\phi)$ and $K=(R,S',F',\phi')$ consists of a functor $N \colon \mathbf{C} \to \mathbf{C'}$ and a natural transformation $\nu \colon SF \to S'F'$ such that $F' = NF$ and $\phi' = \nu \phi$. A PK-homography is called a PK-isography if $N$ is an isomorphism and $\nu$ is an equivalence.

Thus, we obtain the category of PK-nets of fixed form functor $R \colon \Delta \to \mathbf{Sets}$, whose objects are PK-nets $(R,S,F,\phi)$ and whose morphisms are PK-homographies $(N,\nu)$ between them.

Among all PK-homographies, it is possible to distinguish particular cases which we now introduce

Complete PK-homographies

We give directly the definition of complete PK-homographies and detail the consequences.

Def. A PK-homography $(N, \nu)$ between two PK-nets $K$ and $K'$ is called a complete homography if the natural transformation $\nu$ can be expressed as $\nu = \tilde{\nu}F$, where $\tilde{\nu}$ is a natural transformation from $S$ to $S'N$.

Diagrammatically, these complete PK-homographies can be represented as follows.

The reason these homographies are called complete is because they do not depend of the choice of functor $F$. Roughly speaking, whatever the transformation $f$ in $\mathbf{C}$ which relates two elements $x \in S(X)$ and $y \in S(Y)$, their images by $\tilde{\nu}$ are related by the image transformation $N(f)$.

By considering only complete PK-homographies, we get a subcategory which is the coslice category $R \downarrow (\mathbf{Cat} \downarrow \mathbf{Sets})$ for a given functor $R$, which has all small limits and all connected colimits. In fact, I already talked about complete homographies in a previous post. At the time, I only talked about the morphisms themselves, and not about their connection to networks, and I focused on the case where $\mathbf{C}$ is actually a group.

This brings us to the following question: given two functors $S \colon \mathbf{C} \to \mathbf{Sets}$ and $S \colon \mathbf{C'} \to \mathbf{Sets}$, can we explicitly calculate the PK-homographies ? And what if $\mathbf{C}$ and $\mathbf{C'}$ are groups ? Or if $\mathbf{C}=\mathbf{C'}$ ?

After all, if we are to do some music analysis, we will need to explicitly transform our networks. It turns out that the general case cannot easily be answered, so we are going to focus on particular cases, namely when $\mathbf{C}=\mathbf{C'}=\mathbf{G}$ where $G$ is a group (I use the notation $G$ for the group, and $\mathbf{G}$ for the group-as-single-object-category).

Complete isographies of representable functors

First, assume that the functor $S \colon \mathbf{G} \to \mathbf{Sets}$ is representable. In transformational music theory terms, this means equivalently that we consider a Generalized Interval System (GIS). Then we have the following nice result regarding the automorphism group of the functor $S \colon \mathbf{G} \to \mathbf{Sets}$, i.e the group of complete PK-isographies of $S$ (note that from now one and by an abuse of notation, I will speak of pairs $(N,\tilde{\nu})$ as complete PK-isographies).

Prop.  The automorphism group of the functor $S \colon \mathbf{G} \to \mathbf{Sets}$ is isomorphic to the holomorph of $G$, i.e. the semidirect product $G \rtimes \text{Aut}(G)$.

This is a direct consequence of the Yoneda lemma: the complete PK-isographies $(N,\tilde{\nu})$ are completely determined by the choice of an automorphism $N$ of $G$, and the choice of an element $g \in G$ which determines the natural transformation $\tilde{\nu}$.

Observe as a corollary that this group of PK-isographies contains a copy of $G$ which is a normal subgroup, corresponding to all identity automorphisms. By definition of the natural transformation, the action $\tilde{\nu}$ of this normal subgroup commutes with the action of $G$ given by the functor $S$. In other words, we have found a very general case of duality in the sense of Lewin, which I presented a long time ago in the case of the $T\text{/}I$ and the $PRL$ groups acting on the set of the 24 major and minor triads.

As an example complete PK-isographies, here is Gesualdo’s motet Deus refugium et virtu.

The passage which interests us starts at 0:51 and ends at 1:19. Below is the corresponding score.

Each “pietatis” consists in a succession of three triads, which can be analyzed using transformations in the $PRL$ group. Below are the triads for each “pietatis”, along with the corresponding successive transformations between them.

• A major to C minor to G major – $(RL)^6R$$(RL)^4R$
• E major to C minor to D major – $(RL)^7R$$(RL)^5R$
• E major to G minor to D major – $(RL)^6R$$(RL)^4R$
• B major to G minor to A minor – $(RL)^7R$ – (…)

If we omit the last A minor chord, there is a particularly nice symmetrical structure behind all these chords. The first and third successions are related by the same transformations, so the corresponding networks are related by the identity automorphism. From the discussion above, this means that there is a corresponding complete isography between these networks which corresponds to the action of an element of the $T\text{/}I$ group, in our case the transposition $T_7$. Next, observe that the transformations in the second succession of chords can be obtained from the first by applying the automorphism $N$ of the $PRL$ group which sends $L$ to $R$ and $R$ to $RLR$. There is an associated a PK-isography which describe the transformation between the corresponding networks where $\tilde{\nu}$ sends a major chord to a major chord a fifth above, and which is the identity on the minor chords. The same goes between the third and fourth succession. Overall we have the following diagram corresponding to the above Gesualdo passage.

Complete isographies of faithful functors

Assume now that the functor $S \colon \mathbf{G} \to \mathbf{Sets}$ is not representable, but is nevertheless faithful. Then, we have the following proposition.

Prop. The automorphism group of the faithful functor $S \colon \mathbf{G} \to \mathbf{Sets}$ contains a subgroup isomorphic to $G$, and a normal subgroup isomorphic to the center $Z(G)$ of $G$.

Let $X$ be the image set of the single object of $\mathbf{G}$ by $S$. The above proposition means that any element of $G$ can not only be considered as a musical transformation of the set $X$, but it can also be uniquely identified to a complete PK-isography transforming any network of elements in this set. To see this more clearly, we can study the specific case where $G$ is the group $T\text{/}I$ acting on the set of the 12 pitch classes, and in this case we can actually compute explicitly the group of complete PK-isographies.

Prop. The automorphism group of $T\text{/}I \to \mathbf{Sets}$, where $S$ is given by the standard action of the $T\text{/}I$ group on the set of pitch classes $\mathbb{Z}_{12}$, is isomorphic to $\text{Aut}(T\text{/}I)$.

The elements of $\text{Aut}(T\text{/}I \to \mathbf{Sets})$ can be bijectively identified with pairs $(\langle k_l\rangle, z)$, where $k \in \{1, 5, 7, 11\}$, $l$ is even, and $z \in Z$, where $Z = \{0, 6\}$ is the additive subgroup of order 2 of the cyclic group $\mathbb{Z}_{12}$. They correspond to PK-isographies $N=\langle k_l\rangle$ and $\tilde{\nu} = kx+l/2+z$.

In particular the group of PK-isographies of $T\text{/}I \to \mathbf{Sets}$ contains a copy of $T\text{/}I$. Roughly speaking, this means that not only an element of $T\text{/}I$ is a transformation between a bunch of pitch classes, it also corresponds to a unique PK-isography transforming any network underlying them. For example, let’s consider the pitch classes $C\sharp$, $F$, and $B$. Their image by the inversion $I_8$ of the $T\text{/}I$ group are $G$, $E\flat$, and $B$.

Following the above proposition, the inversion $I_8$ uniquely corresponds to a PK-isography, namely $(\langle I_4\rangle, 6)$, so that any network describing the initial three pitch-classes $C\sharp$, $F$, and $B$ gets transformed into a negatively isographic (in the sense of Klumpenhouwer and Lewin) network corresponding to the automorphism $N=\langle I_4\rangle$, as shown below.

We see here the double nature of the $T\text{/}I$ group, either as an extension $1 \to \mathbb{Z}_{12} \to T\text{/}I \to \mathbb{Z}_{2} \to 1$ which corresponds to the usual transpositions and inversions, or as an extension $1 \to \mathbb{Z}_2 \to T\text{/}I \to D_{12} \to 1$ which corresponds to the possible PK-isographies.

Iterated automorphisms and isographies

The automorphism group of any functor $S \colon \mathbf{G} \to \mathbf{Sets}$ is the set $\{(N,\tilde{\nu})\} with$latex N\$ an automorphism of $G$ and $\tilde{\nu}$ is an equivalence. Observe that $\tilde{\nu}$ has $X$ for domain and codomain, so that we readily obtain a new functor $S' \colon \text{Aut}(S \colon \mathbf{G} \to \mathbf{Sets}) \to \mathbf{Sets}$. We can then wonder what are the automorphisms of this functor, and repeat again the operation.

Elements of the automorphism group of $S'$ transform PK-isographies, but by a similar argument as above, they can also be considered as transformations of the underlying networks, and as transformations of the underlying pitch classes. Let’s look at this in the familiar case where $G$ is the $T\text{/}I$ group. We have seen that the automorphism group of the usual functor $S \colon T\text{/}I \to \mathbf{Sets}$ is isomorphic to $\text{Aut}(T\text{/}I)$, which defines a new functor $S' \colon \text{Aut}(T\text{/}I) \to \mathbf{Sets}$. It is not complicated to show that the automorphism group of $S'$ is also isomorphic to $\text{Aut}(T\text{/}I)$, which contains in particular a copy of the $T\text{/}I$ group.

So, any element of $T\text{/}I$ can be viewed as either a transformation of pitch classes, a PK-isography of networks, a PK-isography of PK-isographies, etc. Let’s take the example above, and apply the inversion $I_3$ on each network, as shown below.

The inversion $I_3$ acts on the pitch classes of each network; it is also a PK-isography corresponding to the automorphism $N=\langle I_6 \rangle$. What we see now is that it also corresponds to a transformation of the PK-isography $(N=\langle I_4 \rangle,\tilde{\nu}=I_8$ into the PK-isography $(N=\langle I_8 \rangle,\tilde{\nu}=I_{10}$.

Topoi and PK-homographies

It is well known that the category of presheaves $\mathbf{Sets}^{\mathbf{C}}$ on a small category $\mathbf{C}$ is a topos. This means that given a functor $S \colon \mathbf{C} \to \mathbf{Sets}$ and a subobject $A \colon \mathbf{C} \to \mathbf{Sets}$ of $S$, we have a characteristic morphism $\chi$ to the subobject classifier $\Omega \colon \mathbf{C} \to \mathbf{Sets}$. This can also be considered as a PK-homography, where $N$ is the identity on $\mathbf{C}$ and $\tilde{\nu}=\chi$. Thus topoi and characteristic morphisms are also a great source of PK-homographies.

Revisiting some structures in mathematical music theory

The theory of complete PK-homographies or PK-isographies also sheds new light on some classic structures or objects in mathematical music theory.

For example, I am occasionally asked “But what are these inversions $I_p$ ? Do they have any musical meaning ?”. By looking at a piano keyboard, transpositions $T_p$ are immediately understandable, which is not really the case for inversions. They are in fact the final aspect of the answer to a more complex question which deals with intervals rather than single pitch classes. To see this, consider the group $G=\mathbb{Z}_{12}$ of transpositions, acting on the set $\mathbb{Z}_{12}$ of the twelve pitch-classes, which defines the functor $S$. Note that we deal here with transpositions only. It is quite easy to see that inversions arise as elements of the automorphism group of the functor $S$. This means that given a PK-net $(R,S,F,\phi)$, the corresponding PK-isography will invert the intervals. For example, $D$ and $F\sharp$ are related by a transposition $T_4$, which can be described by a simple PK-net $(R,S,F,\phi)$ where the category $\Delta$ is the category with two objects and only one non-trivial morphism between them, and where the functor $R$ is representable. Applying the PK-isography corresponding to the inversion $I_4$ results in the new pitch classes $D$ and $B\flat$, with the $T_8$ transformation between them, i.e. the inverted $T_4$ transposition.

We can also revisit the concept of time-spans, which I have introduced previously. Introduced by Lewin, times-spans consist in a pair $(t,\delta)$ of a real number $t \in \mathbb{R}$ and a strictly positive real number $\delta \in \mathbb{R}_{>0}^{+}$, which encode the data of a point in time and a duration. We can define a group of transformations of time-spans, namely the semidirect product $G = (\mathbb{R},+) \rtimes (\mathbb{R}_{>0}^{+},\times)$. It turns out that this group of transformations can also be considered as PK-isographies. Consider the additive monoid of the strictly positive real numbers $(\mathbb{R}_{>0}^{+},+)$ acting on the real line $\mathbb{R}$, which defines the functor $S$. A time-span can be described by a simple PK-net $(R,S,F,\phi)$ where the category $\Delta$ is the category with two objects and only one non-trivial morphism between them. The image of this morphism by $F$ indicates the duration of the time-span. We have automorphisms of $(\mathbb{R}_{>0}^{+},+)$ of the form $N(x)=kx$ for $k$ a positive real number. It can be easily checked that the corresponding PK-isographies $(N,\tilde{\nu})$ are such that $\tilde{\nu}(x)=kx+l$, where $l$ is a real number. We thus recover the group of transformations of time-spans introduced by Lewin.

Local PK-homographies

Complete PK-homographies do not however cover all cases of PK-homographies. Consider for example the two networks of pitch classes below. They are clearly isographic by the $\langle T_1 \rangle$ isography, but as we have seen above complete PK-isographies in the $T\text{/}I$ group only cover the PK-isographies $\langle k_l \rangle$ where $l$ is even. Thus there is no complete PK-isography which can describe a transformation of the first network into the second.

We can nevertheless describe such a transformation using the concept of local PK-homography.

Def. A PK-homography $(N, \nu)$ between two PK-nets $K$ and $K'$ of identical form $R$, is called a local PK-homography if they have the same support $S$ and if there is a natural transformation $\hat{\nu} \colon F \to F' = NF$ such that $\nu= S\hat{\nu}$ . It is a local PK-isography if $N$ is an isomorphism and $\nu$ is an equivalence.

The reason we call these PK-homographies local is because it attributes to each object $U$ of $\Delta$ a morphism of $\mathbf{C} = \mathbf{C'}$ which describes the local transformation $\hat{\nu}_U$ at this node. For the networks above, we will thus look for each node for an element of the $T\text{/}I$ group which defines a natural transformation $\hat{\nu} \colon F \to NF$ where $N$ is the $\langle T_1 \rangle$ automorphism of $T\text{/}I$, as shown below.

Look at my previous post on networks: the Webern example which I described is also an example of a local PK-isography.

Conclusions

We have seen how to describe network transformations from a categorical point-of-view, with details about the different types of network homographies/isographies. The subject is quite vast, and there are many other points that deserve exposition. I will briefly mention them here.

• So far we have consider PK-nets $(R,S,F,\phi)$ where the functor $R$ remains identical, but we can imagine PK-homographies in a more general setting between PK-nets $(R,S,F,\phi)$ and $(R',S',F',\phi')$. In that case, we would need to impose two functors $E \colon \Delta \to Delta'$ and $N \colon \mathbf{C} \to \mathbf{C'}$ such that $F'E=NF$, and two natural transformations $\epsilon \colon R \to R'E$ and $\nu \colon SF \to S'F'E$ such that $\nu \phi = (E\phi')\epsilon$.
• The group of complete PK-isographies is in fact also a 2-group, since there can be 2-morphisms between functors $N \colon \mathbf{C} \to \mathbf{C}$ and $N' \colon \mathbf{C} \to \mathbf{C}$. They can also appear in the general case of PK-homographies.
• Many examples throughout this post call out for a notion of “network of networks”. There are many ways to define this formally (and the iterated versions, “networks of networks of …”), but this still an unclear issue for me. In my previous post, I mentionned the paper of Andreatta and Mazzola where this notion is mentionned. We have also seen how iterated automorphism groups of functors to $\mathbf{Sets}$ can give rise to networks of networks.

Finally, I’d like to recall that this framework of musical networks and transformations can readily be studied from a computational point-of-view. This was the motivation behind the Opycleid Python package which I developed (and which has been accepted recently in the Journal of Open Source Software).