Simple Pitch Class Set Transformations

We've looked at basic definitions and set out a numbering system for pitch classes that does what we want. Now it's time to see how powerful these ideas can be from an analytical perspective, and to develop some more ideas and techniques along the way.

Ordered and Unordered Sets

We've been describing "pitch class sets" as groups of pitch classes that have no real structure to them; a set in this case is just a kind of bag of things, and in particular there's no ordering (i.e. sequence) to the pitch classes in a set. So there's a pitch class set that contains the classes C, E and G -- that's the C major triad. It doesn't matter whether we play them C-E-G or G-C-E or whatever: we're still playing a C major triad, and it's all the same from a pitch class set perspective.

Following the mathematical convention we'll write a pitch class set like this in curly braces: {0, 4, 7} for example represents the major triad built on the root note. This is useful in many situations where we want to talk about structures like major triads without caring how they're played.

In other cases it will matter what order a set of notes is played in. In these cases we write the notes between square brackets, like this: [0, 4, 7]. This means play pitch classes 0, 4 and 7 in that order. In this case it makes sense sometimes to repeat pitch classes: [0, 4, 7, 4] is a pattern of pitch classes in sequence, whereas {0, 4, 7, 4} doesn't work because all we have a a set of three pitch classes and it hardly makes any sense to have the same pitch class in an unordered set twice.

Simple Transformations

Now we have that bit of notation out of the way we'll look at three simple transformations that can be applied to pitch class sets. The first two will apply to both ordered and unordered sets and the last only makes sense in relation to ordered sets.

The first is transposition. To transpose a pitch class set up n semitones, simply add n to each pitch class number. So, if we're playing a major triad at the root note we have {0, 4, 7}, and if we move it up two semitones we get {2, 6, 9}. To transpose down n semitones it's easier if we transpose up 12 - n instead. You may want to think about why this works, remembering that pitch classes don't care what octave they're in.

We do have to take into account cases where adding gives you a number that isn't a pitch class number, because it's more than eleven. For example, if we move {2, 6, 9} up another two whole tones (4 semitones) we get {6, 10, 13}. This is no good, because 13 isn't one of our pitch classes. It's easy to get around this, though: if any number is bigger than 11 simply subtract 12 from it until it's not. Since 13 - 12 = 1, the result of our transposition is {6, 10, 1}. If the root note is C then this gives us the pitch classes F#-A#-C#, which make the F# major triad. Since we got here by moving a C major triad up a total of six semitones, this is what we'd expect.

The second transformation is inversion. Now, "inversion" has different meanings in different contexts, but for us it means simply this: replace each pitch class x in the set with the pitch class 12 - x. When x is zero we have to apply the rule just given in the previous paragraph, which makes 0 its own inversion (check this for yourself). So if we invert the major triad {0, 4, 7} we get the pitch class set {0, 8, 5}. If 0 is C again then this is C-F-Ab, which is an F major triad.

Inversions are very frequently-used in atonal music because when a pitch class set is inverted the result is a set that sounds similar yet different, and this enables the atonal composer (or improviser!) to use inversions to create variation without too much chaos. We'll have more to say about inversions in later instalments.

The last transformation only works with ordered sets and is usually called "retrograde". All this means is playing the ordered set of notes backwards, so that for instance the pattern [0, 4, 7, 4] becomes [4, 7, 4, 0]. No doubt you've used this many times yourself without really thinking about it, perhaps when constructing the matching descending pattern for an ascending run, or vice versa. With a little practice you can learn to read music backwards, and it's extremely interesting to hear how, as with inversion, you end up with something both different and very similar to what you started with.

This is all very abstract at this point, but then pitch class set theory is a bit like that. If you intend to follow this series make sure you understand the principles involved in the transformations above, especially transposition and inversion, and try out a few examples for yourself. Calculate inversions of note-groups you already use in your playing, for example, and try introducing them as variations. If you're playing in a tonal context you may have to transpose them as well, to avoid very strong dissonances, but that's OK.