Pitch Class Sets Part the Second

First Read: Pitch Class Sets Part the First

How does one assign numbers to pitches? Easy. There are twelve chromatic pitches to cover: A, A#, B, C, C#, D, C#, E, F, F#, G, and G#. We could get all silly and assign 1 to A, 18 to A#, and 6,655,321 to C but that's not really all that useful. Instead, we arbitrarily pick C to be 0 and go up from there. So: C=0, C#=1, D=2, D#=3, E=4, and so on and so forth until we get to B=11. Calling the pitches the numebrs through 11 is convenient because it equalizes their importance. There is no "key" of A where A is important. A will always be 9 so just live with it. The numbers also take advantage of an existing arithmetic system, namely arithmetic (har har har). Figuring out the interval between A and D# takes some thought and often involves, for many, visualizing a keyboard. With pitch numbers, it's easy once you know the numbers well. D=2 and A#= 10 so from A# to D is 10 to 2 is 4 (remember that the pitches wrap around so 2 is in essence 14) and from D to A# is 8.

The wrapping nature of pitches is the reason why they are called pitch class. For most people, hearing a C really on a cello sounds similar to hearing the same C high on a violin. They're both Cs but one sounds low and sounds high. This is the concept of octave equivalence. So the numbers stand for not a single pitch but a pitch class, regardless of the octave (this is the same as calling a note a C rather than a C3, for example).

So we've got pitch classes, now we need to get sets in there. Fundamentally, chords are bunches of notes played at the same time. Given enough importance, they become entities in their own right and retain identity through deformation, separation of its constituents, and other tranformations. A C major chord sounds like a C Major chord whether you play C, E, and G all at the same time or one after another. The sonority is the same. But after old Ahnold (the Schoenberg one) comes along, it's pointless to call something C Major because that harmony is assoicated with the old notion of a scale. Instead, we could have chords like C, C#, E, G, and Ab all played at the same time. What would you call that? A C-15 chord with a raised 15th, a flat 13th, and missing 7th, 9th, and 11th? Seems silly. Here's were the sets of pitch class set theory comes in.

Continued in the next and final installment of this exciting new series!