What is a Pitch Class Set?

This is the first in a series of posts introducing pitch class set theory at a very basic level. In this post I'll say a few things about what the theory's for and why it's useful, and give some of the most basic definitions.

Why Should I Learn It?

To be honest, I always hate it when students ask me this question about anything, which they very often do. Why not learn it? Why does learning have to have a purpose? All knowledge is worth having, and the process of learning is valuable in itself; at least, that's what I say if I'm feeling grouchy.

Not all knowledge, though, is worth the effort of getting it, and I do understand that our time on the Earth is finite and we might not want to expend a lot of energy learning some theory that's not going to be of any use to us.

It's my view that pitch class set theory is valuable primarily because it's beautifully simple and clear. It can be applied to almost any music, and it makes certain things very easy to understand. It make some practical activities, such as inversion and transposition, so easy that they can be done on the fly; more importantly, perhaps, it provides a very general language with which we can think about our own musical ideas and deveolop them far beyond anything we could accomplish intuitively.

I personally believe that this theory should be taught from the beginning, because it makes a lot of sense for musicians in the jazz and rock traditions as well as more avant garde music. It strips away a lot of the confusing complexity that only really makes sense in the context of ninteenth century Romanticism.

It's time to have a look at what the most basic elements of pitch class set theory are.

Some Basic Definitions

In Western music you can think of a "pitch" as pretty much meaning "a piano key" -- if you like, consider the piano keyboard extended up and down so it can play the very lowest and highest audible notes. I'll build up definitions of the three words in the name one by one: "pitch", "pitch class" and finally "pitch class set".

"A pitch" is actually a sightly odd thing. After all a note, a specific sound you get from hitting a piano key, has pitch as one of its properties, along with a timbre, a dynamic (volume), a duration and so on. The idea of "a pitch" is an abstraction of just one of those properties.

Imagine a saxophone and a piano both playing, say, middle C. Their timbres will be different, and the sax may be louder than the piano. It may also sustain the note longer in time as the piano note dies away. There can be lots of variations, but we can still say that they are playing "the same pitch" (assuming they're in tune). This thing that's the same is the thing pitch class set theory is interested in.

For our purposes, a "class" is a collection of things that we consider to be somehow equivalent. Let's imagine that the sax player is now performing a duo with a bassist. The bass player plays the root of a C major chord, producing a low C, while the sax plays the root high up. These are different pitches -- one is low and one is high. Yet they are both C, and in a sense they're both "the same", just separated by a few octaves.

If two pitches are the same in this way they have the same name, like C or F# or Ab. You know, for example, that the sixth and first strings on the guitar are both tuned to E, although they're certainly not the same pitch as each other. This "sameness" is called the "pitch class".

There are 12 pitch classes in standard Western music: C, C#, D, D#, E, F, F#, G, G#, A, A# and B. Every pitch that can be called "an F", say, is collected together into the pitch class that we just call "F". As you can imagine from the name of the theory, pitch class set theory is a theory about pitch classes; so as well as timbre and dynamics and duration and all the rest, it doesn't care about octave differences either. An F is an F, no matter how high or low you play it.

The final word is "set" and, like "class", we can think of this word as meaning "collection". But where a "pitch class" is a collection of pitches that are all in a sense "the same", a "pitch class set" is a collection of several different pitch classes, and the differences between them will be important.

To sum up, imagine you have an old piano and some bags. You remove all the "C" keys and put them into one bag, labelled "C". You remove all the "C#" keys and put them into the next bag, labelled "C#", and continue until you've removed all the keys. You'll find you've filled twelve bags, and each of these bags is what we call a "pitch class". Pitch class set theory considers sets of several of these bags -- say, just the bags labelled C, E and G. This set contains three pitch classes, and it's what we usually call the "C major triad".

In future posts, where we'll look at how this simple object -- a pitch class set -- can give us powerful insights into the workings of many different kinds of Western music.