Systems of musical analysis are always built upon certain axioms. An axiom is a "given": a statement accepted as true  in this case for the purpose of developing an analytical method. (Of course, we can always try not accepting an axiom, to see what different analytical insights might result.) Our introduction to pitchclass set analysis can begin with two underlying axioms.
Axiom 1: Octave equivalence As in tonal music, the octave seems to play a special role in nontonal music. Pitches distant from each other by one or more octaves can be treated as equivalent. We must be clear about what the word "equivalent" means. It does not mean "identical"; pitches an octave apart are different pitches, and a composer would never use one when he means the other. Rather, "equivalent" means "conceptually equal to each other" for the purpose of our analysis. The axiom of octave equivalence underlies the concept of pitch class (pc): all pitches that are separated by one or more octaves are given the same name and are considered to belong to the same class. This is a familiar concept among musicians. When we speak, for instance, about a piece of music in D major, it's not any particular pitch D that we think of as the tonal centre, but the pitch class D. Notice that pitch class is not itself a primary musical experience  we hear pitches, not pitch classes  but a concept abstracted from that experience.
Axiom 2: Enharmonic equivalence In tonal music, pc Asharp is not equivalent to pc Bflat; pitches belonging to these two classes appear in different tonal contexts and with different structural meanings. In nontonal music, however, such differences no longer seem to matter. Here, enharmonically paired pcs usually seem equivalent, and we shall assume such equivalence from now on. Enharmonic equivalence reduces the number of pcs from at least twentyone (actually more, counting doublesharps and doubleflats) to just twelve. To reflect enharmonic equivalence, we shall give the pcs new names, ones that carry no tonal biases. Using integer notation, we shall simply number the pcs from 0 to 11. You may find that the new names take a bit of getting used to, but they will simplify some of the operations we shall later use in analysis.
