Math in Music II: Music Theory and Symmetry

Why do we care about music theory? Performing musicians will have a different answer from composers, who will have a different answer from academic music theorists. Generally, we want to understand why some music works as it does, and how those ideas can be used to generate new music. Academic music theorists are primarily concerned with the first question, of explaining existing music.

Traditional music theory that you might learn in a course like AP Music Theory, centers around the musical language of composers like Bach or Mozart. Adam Neely, referencing work by Philip Ewell of Hunter College, pointedly noted that the term “music theory” in most contexts can be replaced with “the harmonic style of 18th century European musicians.” Ewell and Neely, among others, make a really convincing criticism of the “whiteness” of music theory scholarship that I don’t want to ignore, and I plan to eventually write about the topic in this blog. “Music theory” should be a much broader term than it’s usually made out to be.

Nevertheless, I'm interested in music theory which looks at different kinds of symmetries in music made out of the chromatic scale. Two prominent of these theories, pitch-class set theory and neo-Riemannian analysis, come from this world of music theory academic to examine primarily Western European music. 

I hope to provide historical context where I find it noteworthy, but I'm also interested in arriving at these ideas from scratch, independent of any specific musical pieces. Analyzing music using group theory is, I hope, interesting as a standalone idea. 

So let's talk about musical symmetries. Expect letter note names, then numbers, then letters again but this time no longer note names. Math!

Transformations: Inversion and Transposition

My ultimate goal is to highlight different ways of thinking about music based on underlying mathematical structures. This section will provide one way of musical thinking, pitch-class theory, which I hope will be an approachable introduction to thinking about music as more than just letter notes. 

A major chord can be defined by its intervals. Begin with the root, say C. Go up a major third to E, then up a minor third to G. This holds for any major chord; indeed, a major chord is defined by these intervals. A minor chord is defined by reversing the two thirds. Start with C, go up a minor third to Eb, then a major third to G. 

We can see that a minor chord is in some sense an inversion of a major chord. Take C, and this time move down a major third, then a minor third. This gives C, Ab, and F, which makes up F minor. So reflecting C major across the note C gives F minor, in some sense. The intervals are preserved in a different order. If you’ve made it this far, you may have heard about negative harmony before, which has to do with this idea of reflecting chords across a certain pitch axis. 

You can think of inversion as a transformation performed on one chord to get another chord. Neo-Riemannian theory, discussed later, is also all about transformations, though different ones.

I’ve been using C major as an example, but we can be more general. Indeed, assuming 12-TET, enharmonic equivalence (C# equals Db, etc.), and octave equivalence (C4 equals C5), C major is not different from D major or E major in any significant perceptible sense. They’re transpositions of the same chord. Transposition is another type of transformation. 

Instead of thinking about note names, we can think about numbers. There are twelve notes in the chromatic scale, so we can label them 0 through 11. If you’d like, you can assign C=0, making C#=1, D=2, etc. But the usefulness of the numerical labels is that 0 can represent any note, and what’s important is the distance between notes. 

In this framework, moving up or down a certain interval is akin to adding or subtracting a number. A minor third is made up of 3 semitones, so a minor third up from the note 2 is 2+3=5. This says that F=5 is a minor third above D=2 assuming C=0. Because we assume octave equivalence, our addition and subtraction operations have to loop back around like a clock (modular arithmetic for the math people). So 10+3=11+2=0+1=1 (a minor third up from Bb is Db).

So the transposition transformation, mathematically speaking, is modular addition or subtraction. What about inversion (reflection)? As we’ve seen, we want E=4 to map to Ab=8 and G=7 to map to F=5. Note that the modularity of the chromatic scale (“clockness”) gives negative values meaning too. In fact, Ab=8=-4 and F=5=-7, which helpfully demonstrates that inversion across 0 is the negation of the original number.

Pitch-Class Set Theory

Pitch-class theory extends these ideas. One of its ideas is to think about chords and scales as the same thing: a set of notes. We can represent these notes as a set of numbers. For example, (0,4,7) is a major chord because it contains a major third above the root and a minor third above the next note. (1,5,8) is a transposition of (0,4,7) and thus also a major chord. We can represent all major chords (n,n+4,n+7) with modular arithmetic as being equivalent to (0,4,7) under transposition. The major scale is also a set: (0,2,4,5,7,9,11).

Additionally, we can also treat sets of pitches as equivalent under inversion, too. We’ve seen that C major (0,4,7) can be inverted to F minor (5,8,12=0). (Note that really we should write the 0 first in F minor, giving (0,5,8), but I wrote it as I did for ease of understanding.) And since F minor is equivalent to C minor (0,3,7) under transposition, C major is equivalent to F minor under transposition and inversion. 

Pitch-class theory thus treats all major and minor chords the same. It says all major and minor chords have the same prime form—essentially, the most compact version of a set that begins with 0—of (0,3,7). Mathematically, you could say they are invariant under transposition and inversion; in some sense, they’re all the same.

Major and minor scales are also the same in this framework. This is because they are modes of each other. Every major scale contains a minor scale if you start on the sixth note of the scale. C major (CDEFGAB) contains the same notes as A minor (ABCDEFG), just offset. There are seven modes of the major scale, corresponding to its seven notes. 

Another way to explain modes is that they contain the same distances between notes. Counting semitones between notes, C major has 2 semitones between each note except between the third and fourth and between the seventh and tonic—2212221. Minor is 2122122, the same intervals just offset.

So major and minor scales (and all their modes) have the same prime form—(0,1,3,5,6,8,10)—which describes the Locrian mode, or the same notes as a major scale beginning on the seventh scale degree (e.g. BCDEFGA). 

Pitch-class theory, by assigning a single prime form to each set of chords which are invariant under inversion and transposition, nicely highlights certain symmetries between scales and chords.

If you want to explore more sets of notes, I would be remiss if I didn't recommend Ian Ring's scale finder. There are 2^12=4096 possible sets of unique notes within an octave, and you can find info on all of them.

I do want to take a second to pause and acknowledge that it seems wrong to group major and minor chords (or scales)  into the same category. In most of western music, they're treated very differently. The idea to treat inversions equivalently emerged out of a desire to treat atonal music, rather than tonal music. 

From a math perspective, I find it interesting to simply consider what happens if you define inversions to be in some sense equivalent. Whether that definition stands when listening to music is a separate question entirely. But I hope at least some of the ideas in this section, like transposition being a type of symmetry which does not change the character of a set of pitches, are valuable.

What is Tonality?

Tonal music is music that centers around one note, called the tonic. Atonal music, therefore, is untethered to any particular note. Unless you’re a big fan of Schoenberg, almost all the music you probably listen to is tonal. If you can pinpoint a home key or I chord (“one chord” using the Roman numeral I to denote the first note in the scale) in a section of music, that section is tonal. 

Introductory music theory classes explain chords in terms of the tonic chord. The major chord built from the fifth note in the major scale—G in C major—is the dominant chord; F, the subdominant; etc. Each of these chords have functions which relate to the tonic chord. 

But tonal music isn’t limited to this traditional analysis. Sometimes you can’t easily pick which key a song is in using traditional theory, but even that doesn’t necessarily make music atonal. Some music theorists who study pop and rock music, like Philip Tagg, provide an entirely different framework to analyze four-chord loops common in those genres, with terms like “outgoing,” “medial,” and “incoming” replacing the “dominant” and “subdominant” in traditional classical theory. This is still tonal music because each chord is still related to some sort of overarching key center. 

Arnold Schoenberg (1874-1951) is often considered the poster boy of atonal music. In many ways, he was a conservative composer attempting to build on a long tradition of Western European music, but his compositions are considered radical for jettisoning with any tonal center. 

His twelve-tone technique attempts to give each note equal importance by requiring that the twelve notes in the chromatic scale are each played in a line before any is repeated. Harmony emerges from the interaction between multiple lines being played simultaneously, called counterpoint (a technique present in all classical music, tonal and atonal). His music is atonal in the sense that none of the notes are related to any tonal center; they’re only related to each other.

But we need not go as far as Schoenberg. Neo-Riemannian music theory (Hugo, not Bernhard Riemann from calculus) emerged as a way to study music that came in the familiar triadic harmony—root, third, fifth—without any tonal center. Like in Schoenberg’s music, the chords are only related to each other. Neo-Riemannian analysis doesn’t concern itself with explaining an entire piece of music; rather, it explains the relationships between successive chords in passages that seem to defy traditional tonal explanation. It’s been used to study Wagner (1813-1883) and Liszt (1811-1886), but also sections from earlier composers’ works, like Mozart (1756-1791).

Transformations in Neo-Riemannian Music Theory

Say you have a C major chord in root position (C E G ascending), and you want the next chord to be A minor (A C E). You could shift all three notes down a third, where C goes to A, E to C, and so on. But it is more efficient to move the G up to an A, getting an A minor chord arranged C E A ascending—the same chord arranged differently. Notice that only one note had to move a whole step.

The name of the game is voice leading, or the way individual notes move to change chords. We call the voice leading which minimizes the amount of half steps moved parsimonious or smooth.

Beginning with a major chord, there are only three ways to voice lead into a major or minor chord while only moving one note. Equivalently, you can think of each voice leading as preserving two of the notes. C E G can become C E A as we’ve seen, called the relative minor (or major in version). C E G can become C Eb G, where the third moves down a half step to get the parallel minor (major). Lastly, C E G can become B E G, where the root moves down a half step, called leading-tone exchange. 

These three operations—C major to A minor, to C minor, and to E minor—are the three fundamental transformations in Neo-Riemannian theory, denoted R, P, and L respectively. I’ve used C major as an example, but the three operations work the same on any major chord. For example, G major maps to E minor under R, G minor under P, and B minor under L.

These operations map minor chords back to major chords in the opposite way. That is, C maps to A minor under R, so A minor maps to C major under R. Holding C minor constant, we get Eb major under R, C major under P, and Ab major under L. 

Any consonant triad (major or minor chord) can map to any other consonant triad under a combination of those three operations. Because we’ve already established that inversion and transposition are sufficient to map consonant triads to each other, it suffices to prove that PLR can invert chords and transpose chords by any number of scale degrees.

C major ⇒ A minor (R) ⇒ F major (L) gives a transposition up a fourth under RL. We know repeatedly transposing up a fourth will generate the entire chromatic scale (along the circle of fourths), so we can clearly transpose from any major key to any other major key with repeated RL operations.

Likewise, C minor ⇒ Ab major (L) ⇒ F minor (R) transposes minor chords.

Inversion is given under RLP by C major ⇒ A minor (R) ⇒  F major (L) ⇒ F minor (P). Correspondingly, F minor maps to C major under PLR.

So because some combination of the transformations PLR move invert chords and transpose them by any interval, PLR is sufficient to generate any chord from a single starting chord. 

There’s a nice visualization of each of these operations called a Tonnetz diagram. 

Each triad is represented by a triangle, with each corner being one of the notes in the triad. Major triads point down, while minor triads point up. Recall that each operation (P, L, or R) preserves two notes in the triad; this is part of smooth voice leading. So we visually represent each transformation by moving to an adjacent triangle which shares an edge. 

Look at the C major chord in the middle (CEG in dark red). The parallel operation P goes to C minor, which is akin to reflecting across the C-G edge. I say reflection rather than simply "moving up" because the inverse operation, which moves C minor back to C major, reflects back over the same edge.

Likewise, the relative operation R moves C major to A minor, reflecting across the C-E edge. Leading-tone exchange L reflects across the E-G edge, giving E minor. 

Repeated transformations, like C major to C minor (P) to Ab major (L), basically looks like marching along the grid, in this case up and then left. 

The Tonnetz diagram could be seen as an infinite plane, but the assumption of 12-TET nicely allows it to close the loop. In fact, it ends up forming a torus, or a donut shape. I just think that's neat.

Why do we care about any of these transformations? Personally, in my experience composing, I'm often interested in how to relate successive chords to one another with smooth voice leading. Neo-Riemannian theory nicely formalizes this notion of successive parsimonious transformations between chords—not strictly necessary for making anything work, but a deceptively intuitive way to think about chord progressions that don't have some home key. 

Music theorists who work in the field probably have different answers, but in any case I hope you find it at least passively interesting. There's much more to neo-Riemannian analysis than this, but by now you've seen some of the basics.

Yes, I Know It Sounds Like Group Theory

Math-minded readers who have made it this far (I appreciate it) have undoubtedly noticed that there are existing mathematical definitions that fit these musical concepts really nicely. I’m talking about group theory. (Note: while pitch-class theory is a branch of musical set theory, there is no inherent connection to mathematical set theory.)

To avoid belaboring the point, I'll just consider the PLR operations in neo-Riemannian theory as I've presented it, but obviously you can also use the tools and language of abstract algebra on pitch-class theory as well (e.g. there's a bijection between the chromatic scale and the integers mod 12, etc.).

Consider the set of all major and minor triads. Assume all the necessary assumptions (12-TET, enharmonic and octave equivalence) so that one element represents every possible C major chord, and another every C minor, etc. So the set contains 24 elements.

A group, as you may know, is a set that has some associative operation which takes in two elements in the set and outputs a third element in the set. The set must contain an identity, where taking the operation on any element with the identity returns the original element. Each element must also have an inverse, which returns the identity when operating on an element and its inverse.

One way to define our set as a group is as follows. Let X and Y be arbitrary chords in the set. Denote x to be the operation. We showed earlier that C major can reach any chord Y by some combination of the neo-Riemannian transformations PLR. Let X x Y be the chord you get when you apply those same transformations in the same order, but start at X instead. 

For example C major becomes Ab major under PL, so Dmin x Abmaj = F#min because D minor becomes F# minor under PL. 

I claim that this operation makes C major the identity. Cmaj x X = X obviously by the operation's construction. To show X x Cmaj = X, note that C major is already itself, i.e. the identity transformation takes C major to C major, X will not be changed. 

Associativity occurs because the composition of functions is always associative. In this case, each of P, L, and R are functions on the set, and repeatedly applying them is the same as composing them, making PLR associative. Clearly the group operation is thus associative.

Usefully, the inverse of P was defined to be P, and likewise for L and R. So if you want to transform a chord by, say,  PLPLR (C major to Db minor), the operation RLPLP is a perfectly adequate inverse. This operation will certainly be represented by some chord Y in the set, so all inverses will exist as required.

It's probably easier to more rigorously prove all of these statements by treating each chord as a set of numbers from 0 to 11, and then defining each operation using arithmetic.

It might be easier to think of PLR as generators for the aforementioned PLR-group (or neo-Riemannian group). In fact, because P = RLRLRLR, the generating set contains L and R alone. (As I defined it, L is Emin and R is Amin.)

And I'm not going to prove it—this is getting long enough—but it turns out that the group in question is isomorphic to the dihedral group D12 of order 24. 

(I have to confess: I've actually not yet taken an abstract algebra course in college, so it's very possible I've made a mistake somewhere. If it reads confusingly, it's because the way I defined the group is my own construction—sorry! But the internet has a wealth of information if you want to learn more, and I found a PhD thesis which covers the topic to start you off.)

Parting Thoughts

What's the point? Even most neo-Riemannian theorists probably do not care about the group-theory version of their PLR framework. Neo-Riemannian theory is most often used to explain why certain music works, and it's not evident that adding the rigor of group theory tells music theorists anything they don't already know. 

I have two thoughts on that. The first is that drawing connections to group theory brings all the tools of group theory to this side of music theory, and that opens the door for brilliant, creative people to think of cool ways to apply those tools. It might be a cool composition, or simply a new way of analyzing a piece of music. I don't know, but I hope it's out there.

It's the same point I made about Jacob Collier in part 1. It's all esoteric nonsense until someone with a deep understanding comes around to turn it into something we haven't seen before. That's how innovation works: first ideas float around, then someone finds something to do with them. But you have to brave the parts that are hard, with no clear application in sight, to reach that point.

Second, and more important to me, the fun of making these connections is enough. Applications are cool, but there's something beautiful about seeing deep mathematical concepts, or indeed deep concepts in any field, appear in other fields even if we can't see an apparent use. Perhaps you don't think so, in which case: I appreciate that you made it this far, and trust that there will be content which has to do more with applications. But I hope you see where I'm coming from.

So what's the point? It's kinda cool. That's enough for me.