Musical Set Theory

I play music and I’ve always been good at maths. It seems whenever anyone puts these two facts together, I’ll then be told “I think there’s a link between maths and music”. I was skeptical of this statement, partially because I don’t consider myself to be a good musician, but mainly because I just didn’t see the link between being good at sums and being expressive with your instrument. However when I came to study music theory, the links became very clear. Here was where my naturally analytic mind really came through. This was the link between maths and music that my anecdotal evidence supported.

So I wasn’t surprised when I found out that music is being analysed using some relatively complex mathematics. Musical set theory is much more abstract than other music theory I’ve learnt before. The concepts behind different types of scales, the circle of fifths, and intervals are relatively easy to pick up, you can actually hear them. Musical set theory takes these ideas and builds on them.

Take the keyboard below, it is showing one octave of notes in which there are 12 distinct pitches. This octave pattern repeats all the way up the piano.

It should be noted that although, for example, a low pitched F and a high pitched F are different notes, the human ear interprets them as sounding essentially ‘the same’. This property allows us to only consider the relationships between these 12 pitches in musical set theory. As such we can rename each of the lettered pitches as a number between 0-11, these are known as pitch classes. We will take C to be equal to 0, although this is not always the case.

• C = 0
• C# = 1
• D = 2
• D# = 3
• E = 4
• F = 5
• F# = 6
• G = 7
• G# = 8
• A = 9
• A# = 10
• B = 11

A musical set is an unordered collection of these pitch classes. The set { 3, 5, 10 } refers to the notes D#, F, & A#.

Now we have defined what a musical set is, it is possible to look at some operations we can perform. The two basic operations we can perform are Transposition Inversion. A transposition moves a collection of notes up or down in pitch by a constant interval. Using pitch class notation a transposition T can be defined as follows, take a number n as the constant interval we are transposing by, and take x as the pitch class.

T(x) = x + n (mod 12)

“(mod 12)” refers to modular arithmetic, here’s the wiki article, though we can simply think of it as how we would add numbers on a clock. The following example illustrates the Transposition operation.

Given the musical set { 2, 5, 9 } let us transpose by 5 semitones, i.e. let n = 5. For the cases where x = 2 and x = 5 it is obvious to see that T(2) = 7 and T(5) = 10 respectively, however when x = 9 we obtain the following: T(9) = 9 + 5 = 14. 14 is not a pitch class, but as we are working in mod 12, we shall subtract 12 to obtain the pitch class of T = 2. Therefore, after transposition, the new musical set is { 2, 7, 10 }.

Inversion is similarly defined using modular arithmetic, and can thought of as a reflection around some pitch class. Indeed given any n, the Inversion of the pitch class x is defined to be:

I(x) = n – x (mod 12)

These are just the very basic operations and ideas of Musical set theory, this post could not have covered it all. There’s a whole field to explore, one that has had applications in atonal music. Indeed, Musical set theory has aided certain musicians in the way they think about the music they create.

EDIT: Here’s a link to a follow up post, that goes into some more depth about musical set theory.