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In my previous post, I promised we would look at a Generalized Interval System (GIS) for chords, but I’d like to give some more details first about the concept of GIS and how it can be defined mathematically. I’m assuming here that you’re familiar with basic group theory, and also basic category theory (with such things as objects, morphisms, functors, representable functors, etc…). If not, I encourage you to look up these subjects but you can skip this part as it won’t be needed for the next post.

When we defined Generalized Interval Systems in the previous post, we did it using two approaches. I denote here by $S$ the set of musical objects, and by $G$ the group of generalized intervals. The first approach is the Lewin one, using a function $int$ which maps a couple of musical objects $s_1, s_2 \in S$ to a group element $g \in G$ which corresponds to the generalized interval between $s_1$ and $s_2$. The second one defines a $G$-torsor on $S$, i.e equips $S$ with a simply transitive group action of $G$, and the interval between $s_1$ and $s_2$ is the unique $g \in G$ such that $s_2=g \cdot s_1$, or $s_2=s_1 \cdot g$ depending if you’ve considered a left- or right- group action. We have also seen that using $G$-torsors, we can identify musical objects, the elements of $S$ with the elements of $G$, as soon as we have identified one particular element of $S$ as the identity element.

Let’s put it in another way: assume you have a friend who only knows about the note E, and who has never heard of any other notes. How would explain to him what is a F ? or a G ?

The first way of doing it is to actually show him a piano keyboard, and show him where are all the notes. Then, given two notes, you can measure the number $n$ of semitones between them and therefore assign the group element $z^n \in \mathbb{Z}_{12}$ for the generalized interval between these notes. This is the Lewin approach.

The second way is to tell him “The note F is the transposition (of E) by one semi-tone”. We’re assuming here that your friend knows what is a transposition by semitones, even though he’s never heard a F. This is the “group action” approach. Notice that we’re implicitly assuming that E is our reference point, i.e we have identified E as our identity element. And consequently, we are directly identifying notes with group elements from $\mathbb{Z}_{12}$.

The fact that elements of a set equipped with a simply transitive group action can be put in bijection with group elements is rather obvious, but there is a bigger picture here, and it involves category theory.

A group $G$ can be considered as a one-object ($\bullet$) category $\mathcal{G}$, with morphisms the elements of $G$, i.e $Hom(\bullet,\bullet) = G$. What is, then, a functor $F: \mathcal{G} \to \mathfrak{Set}$, from the group-as-category $\mathcal{G}$ to the category of sets ? This is in fact a set $S=F(\bullet)$ equipped with a group action of $G$ (if you’re not convinced, try to prove it). If you have chosen a covariant functor $F$, you’ll end up with a left group action, whereas if you’ve chosen a contravariant functor, you’ll obtain a right group action. It may be useful to think of functors $F: \mathcal{G} \to \mathfrak{Set}$ as “sets modelled after the category $\mathcal{G}$“. Beware, though ! Our set is just equipped with a group action, not (yet) a simply transitive one.

Among all functors $F: \mathcal{G} \to \mathfrak{Set}$, two of them are of particular importance, the covariant $Hom(\bullet,-)$ functor and its contravariant counterpart, the $Hom(-,\bullet)$ functor. They each send the only object $\bullet$ of $\mathcal{G}$ to the set of morphisms of $\bullet$, i.e we obtain the set of group elements of $G$. The group action is given by the left- or right- action of $G$ on itself, i.e a morphism $h \in Hom(\bullet,\bullet)$ is mapped to a function $F(\bullet) \to F(\bullet)$ which sends the (set) element $g$ to the (set) element $h \cdot g$ (covariant), or $g \cdot h$ (contravariant). This is beginning to look like a GIS…

We say that a functor $F: \mathcal{G} \to \mathfrak{Set}$ is representable if it is naturally isomorphic to a $Hom$ functor. It is mathematical folklore that a functor $F: \mathcal{G} \to \mathfrak{Set}$ is representable iff the corresponding set is equipped with a simply transitive group action of $G$. And therefore, we see that the GIS are in fact the representable functors $F: \mathcal{G} \to \mathfrak{Set}$.

But there is more. Suppose we are given a representable functor $F: \mathcal{G} \to \mathfrak{Set}$ (where we denote the set of musical objects by $S=F(\bullet)$): it is naturally isomorphic to a $Hom: \mathcal{G} \to \mathfrak{Set}$ functor. By definition of a natural isomorphism that means there is a particular bijection between elements of $S$ and morphisms of $\mathcal{G}$, i.e elements of $G$. What is this bijection ? This is where the Yoneda lemma is of crucial importance: it states that the natural isomorphisms are in one-to-one correspondence with the elements of $Hom(\bullet,\bullet)$, the one-to-one correspondence being given by the identification of the identity element with an element of $S$. This is exactly what we have been saying about G-torsors, except from a categorical point of view !

This approach may seem far too complex for such things as groups and Lewin’s GIS, but we’ll see that categories (in particular, groupoids) do appear in music theory, and the representable functor framework allows to easily define GIS for these generalizations.