In analyzing atonal music, we often pay no attention to the sequence in which pcs are represented in the analyzed segments; we deal with unordered pc sets. At times, however, we may notice that the order in which pcs appear is itself a structural feature. For example, this order may be subject to patterned manipulation. If so, we would likely want to draw attention to any patterns we perceive. Although the word "set" is sometimes used for such ordered patterns, they are more often referred to as series (or rows); and music in which series are a prominent structural aspect is serial music.
Sets of any size can be manipulated serially, but the most significant kind of serial music has been twelve-tone music, in which the 12-pc aggregate set is so treated There is, of course, only one 12-pc set class, so structural designs cannot be created here on the basis of pc-content or ic-content alone. Only order manipulations can distinguish one use of the aggregate from another (and hence relate one to another). It was Arnold Schoenberg who, in the early 1920s, developed the basic operations used in 12-tone composition: the common order permutations of retrograde, inversion, and retrograde-inversion; and such secondary features as complement relations among serial subsets. Analysis of 12-tone music is beyond the scope of the present guide. In recent years, however, pc set analysis has proved quite useful in revealing structural aspects of serial music other than those dealt with by serial analysis alone.