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I’d like to start a thread about a branch of mathematical music theory called neo-Riemannian theory. My goal is to investigate the mathematical aspects of this theory, beginning with the work of Lewin up to the more recent developments we’ve seen in the last few years (including some of my research). I’ll be less focused about the applications for music analysis and/or composition, though I’ll try to provide references if needed.

Now, before we start looking at the details, some preliminary remarks.Since I said neo-Riemannian theory has a mathematical background, some of you may think it takes its name from Bernhard Riemann, the famous mathematician who had a large influence on modern mathematics and physics. Well, it’s not that Riemann.It actually takes its name from Hugo Riemann, a German (music) theorist and composer. Riemann (Hugo) lived in a time when the style of classical music was changing quite rapidly, and which became difficult to analyze using traditional tools. He therefore developed a theory (Riemannian theory) based on transformations of chords from one to another. I’m not going to expose his theory in this post, since I’m mainly concerned about neo-Riemannian theory, which, as it turns out, has only loose connections to the original Riemannian theory. However you can find some interesting material in these references :

Riemann’s developments were forgotten during most of the XXth-century, until the work of David Lewin first, Richard Cohn, Henry Klumpenhouwer, Julian Hook (among others) then. I’ll expose the details of Lewin’s work in a next post, but let’s say for now that his main contribution was to create a new musical theory based on a transformational point of view. In other words, instead of being interested in musical objects by themselves, one looks at the possible transformations between them. In doing so, Lewin tied together two domains which apparently had nothing to do with each other: music and mathematical group theory.

Well, have those domains really nothing in common ? Let’s do a simple experiment. Imagine you’re sitting in front of your favorite piano (or using your favorite fretted guitar, that works also). Now, your piano is a bit particular since instead of a finite keyboard, it has an infinite one, i.e an infinite number of keys on the left and on the right
(For those who prefer guitar, imagine an infinite guitar neck… ok that’s close but weird). Now pick your favorite note and play it. Good. Not much of a tune, though. We’ll need another note for that (at least one, though one would be enough, at least according to Ligeti).

So what can you do ? You can actually pass a certain number of keys (not just the white ones but all of them, in the order given by your keyboard), say $n$, until you end up on your next note. So you actually have here an operation, or transformation, which is “passing $n$ keys”, which we’ll denote by $T_n$. Notice that $n$ might be negative or positive depending if you’re going to the left or to the right of your actual note.

All that seems rather obvious, but notice the following facts :

• You can compose (in the sense of applying successively) operations, and this will always give another operation. For example, passing 2 keys, then 3 keys, is actually the operation of passing 5 keys. If we use the symbol $\circ$ for composing operations, then we have in mathematical terms $T_n \circ T_m = T_{n+m}$
• You have a special operation, which is “passing 0 keys”. You’ll end up on the same note you began with. We’ll call this operation the “identity operation”, $T_0$ and we can notice that, quite obviously, for any $n$ we have $T_n \circ T_0 = T_0 \circ T_n = T_n$.
• For any operation, we always have an inverse operation which takes us back exactly where we began. For example, if we pass 5 keys, then -5 keys, that’s equivalent to applying the identity operation. This works in the other way too: if we pass -5 keys, then 5 keys, that’s $T_0$.
• Say, we have three operations: $T_p$, $T_q$ and $T_r$. Observe that if we pass first $(q+r)$ keys, then $p$ keys, that’s exactly the same operation as passing first $r$ keys, then $(p+q)$ keys (in the end, we always end up passing $(p+q+r)$ keys). So we have $T_p \circ (T_q \circ T_r) = (T_p \circ T_q) \circ T_r$. We say that the composition of operations is associative.

It turns out that these facts are exactly what defines a mathematical group. Our infinite keyboard (or fretted guitar) comes with a natural group structure, in which case the group is isomorphic to $\mathbb{Z}$, the additive group of the integers. Group theory is therefore not a total stranger in the world of music theory. We’ll see in the next post how other groups, such as the cyclic groups and dihedral groups, naturally arise in this world.

PS: And what about an infinitely long bass guitar, which has no frets at all ? Well, you can always move your finger by any amount of space, even an infinitely small one. What’s the group structure associated with that guitar ? That would be the additive group of the reals $\mathbb{R}$.