I know it’s been a while since I’ve talked about either of these topics, but I’ve always been meaning to point something funny out. I thought I might formally work it out and write it up to submit to a music theory journal, but no one would probably accept it anyway. So I’ll sketch the idea now. Back here I talked about stacks as a useful way to generalize what we mean by a “space.” Back here I talked about the math behind the idea of pitch class sets.
I know Mazzola wrote a whole book on using topos theory in music, but I’ve never dug into it very deeply. I fully admit this is probably just a special case of something from that book. But it’s always useful to work out special cases.
Recall that a pitch set (or chord) is just converting notes to numbers: 0 is C, 1 is C#, 2 is D, etc. A given collection of pitches can be expressed in a more useful notation when there isn’t a key we’re working in. For example, a C major chord is (047).
A pitch class set is then saying that there are collections of these we want to consider to be the same. For one, our choice of 0 is completely arbitrary. We could have set 0 is A, and we should get the same theory. This amounts to identifying all pitch sets that are the same after translation.
We also want to identify sets that are the same after inversion. In the previous post on this topic, I showed that if we label the vertices of a dodecagon, this amounts to a reflection symmetry. The reflections together with the translations generate the dihedral group , so we are secretly letting act on the set of all tuples of numbers 0 to 11, where each number only appears once and without loss of generality we can assume they are in increasing order.
Thus a pitch class set is just an equivalence class of a chord under this group action. It is not the direction I want this post to go, but given such a class, there is always a unique representative that is usually called the “prime form” (basically the most “compact” representative starting with 0).
Here’s where we get to the part I never really worked out. The set of all “chords” should have some sort of useful topology on it. For example, (0123) should be related to (0124), because they are the same chord except for one note. I don’t think doing something obvious like defining a distance based on the coordinates works. If you try to construct the lattice of open sets by hand based on your intuition, a definition might become more obvious. Call this space of chords .
Now we have a space with a group action on it. One might want to merely form the quotient space . This will be 24 to 1 at most points, but it will also forget which chords were fixed by elements of the group. Part of the “theory” in music theory is to remember that information. This is why I propose making the quotient stack . It seems like an overly complicated thing to do, but here’s what you gain.
You now have a “space” whose points are the pitch class sets. If that class contains 24 distinct chords, then the point is an “honest” point with no extra information. The fiber of the quotient map contains the 24 chords, and you get to each of them by acting by the elements of (i.e. it is a torsor under ). Now consider something like the pitch class set [0,2,4,6,8,10]. The fiber of the quotient map only contains elements: (02468T) and (13579E). The stack will tag these points with , which is the subgroup of symmetries which send this chord to itself.
Now that I’ve drawn this, I can see that many of you will be skeptical about the simplicity. Think of it this way. The bottom thing is the space I’m describing. Each point in the space is tagged with the prime form representative together with the subgroup of symmetries that preserve the class. That’s pretty simple. Yet it remembers all of the complicated music theory of the top thing! If the topology was defined well, then studying this space may even lead to insights on how symmetries of classes are related to each other. Let me know if anyone has seen anything like this before.