Numbering Systems for Pitch Classes

This post -- the second in our series on pitch class set theory -- looks at three different ways to number pitch classes. These numbering systems are alarmingly similar, so they can get confusing, but an understanding of them is essential for what follows, so hold onto your hat.

Basic Pitch Class Numbering

Remember that a pitch class is a collection of all the notes that have the same name -- all the Cs make one pitch class, all the C#s make another and so on, giving us twelve in all. One obvious thing to do is to just give them numbers. By convention (and for mathematical reasons we won't get into here) we use the numbers from zero up to eleven rather than from one to twelve:

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

Notice how the numbering goes back to zero when we get to the C an octave above where we started: because these are both Cs, they're the same pitch class so they get the same number.

This is a very standard way to number pitch classes, and it's useful for many applications. It's not, however, the method we will use in these posts. Why not? Because when we learn something on the guitar we usually want it to work in any key we like simply by moving it up or down the neck.

Using this system we would identify the sequence C-D-E as the set {0, 2, 4} and D-E-F# as the set {2, 4, 6}. These lists of numbers don't look similar, but in a sense the latter is just the former moved up the neck two frets. As guitarists this method might be useful some of the time but it doesn't quite fit with our usual and most natural way of working. As a result, as guitarists we don't usually want the sequence C-D-E to be described in a completely different way from D-E-F#.

This numbering system definitely has its uses, but it's not quite what we want for our purposes as guitar players seeking to understand and explore musical resources. Let's move on to a second, and completely different, numbering system instead.

Numbering Based on the Major Scale

Another very standard way to number pitch classes is based on the Major scale, and it avoids the issue just mentioned. It's important not to get confused at this point: this system is completely different from the one just described. We're going to be combining them in the next section, but for now you need to keep the two separate in your head.

In this second system we fix any starting note we choose, which we call the "root note". If we number this "1" (not zero, notice) then, choosing choose the note "G" as the root, the numbers will come out like this:

G G# A A# B C C# D D# E F F# G ...
1 2 3 4 5 6 7 1 ...

This numbering system can be extended by filling in the "gaps" between the notes of the major scale. It's logical to use sharps or flats for this, so that going up one semitone from the root gives us either a #1 or a b2 (here "b" means the flat sign). This gives us a complete numbering system for the twelve pitch classes -- here's an example using G as the root note again:

G G# A A# B C C# D D# E F F# G ...
1 #1 2 #2 3 4 #4 5 #5 6 #6 7 1 ...
b2 b3 b5 b6 b7

This numbering system is very popular among jazz and rock musicians. Notice that in the previous system a number referred to a specific pitch class: 0 was C, 1 was C# and so on. In this system a number can refer to any pitch class, and which one will be determined by what the root note is. This is in a sense equivalent to "which key you're in", or at least what the root of the underlying chord is. Since we like to be able to move our musical materials around and play them in different keys, this system is obviously very appealing.

A Hybrid Pitch Class Numbering System

There's another reason why people like the system just described: the numbers mean something. If a scale contains a b3, for example, we might be inclined to think of it as a "minor" sound, since it contains a minor third. This is excellent if you're working with music that uses traditional tonal harmonies, including most jazz, rock and pop music.

If you're not, though, those names like "b3" and "#5" carry too much baggage. If you want to start thinking outside the box of conventional tonality then they'll always pull you back in the wrong direction; choosing "C" as your starting-note doesn't mean you want to think of Eb as the flat third, or minor third, if you're trying to get away from functional harmony and the triads that form its basis.

So we're going to mix the aspects of both systems that fit our purpose. We'll use the numbers 0-11 just as in the first system, but we'll allow 0 to refer to any pitch class just as 1 can in the second system. This means our numbers will be "neutral" and won't imply any particular relagtionship with triads or the major scale, but they'll still enable us to move the ideas we develop around on the guitar neck.