Each combination of tones can be identifi ed on the basis of its PC set. However, for such a combination to be considered of structural interest—that is, worth identifying at all—it is a necessary (though not a suffi cient) condition that it bear a relation to other combinations. A relation between two combinations of tones can sometimes be conceived of as a transformation. PC set theory has defi ned several operations that transform one PC set into another, the most important of which are transposition (T), inversion (I), and multiplication (M). No doubt, transposition and inversion are backed by the longest history. The discussion of them in this chapter will reveal the strong bonds that tie PC set theory to the history of music theory. Multiplication is considerably younger as a concept of a musical transformation, but it, too, antedates PC set theory and is rooted in compositional practice.

Notes on the Term “Operation”

The term “operation” has been borrowed from mathematics. Some confusion may arise over its correct use. A mathematical operation is defi ned with respect to a collection of elements. Often, it is a protocol (a “function” or “mapping” in mathematical language) assigning to each pair of elements of a collection one element of the same collection. This is called a “binary operation.” If we think of the collection of integers, ordinary addition, subtraction, and multiplication are examples of binary operations. Under addition, for example, 7 is assigned to the pair consisting of 3 and 4.

There are similar protocols for collections consisting of sets. For example, the union of two sets A and B (denoted A ∪ B) is a set consisting of all elements belonging to A and all elements belonging to B. The intersection of A and B (denoted A ∩ B) is a set containing the elements that A and B have in common. The difference of A and B (denoted A − B) is the set consisting of those elements of A that do not belong to B.

A special case of the difference of two sets is the complement. This has been an important concept in PC set theory, for reasons to be discussed in chapters 4 and 6. Given a set of things that I shall call ELEMENT, ℘(ELEMENT) is the set of subsets of ELEMENT. One member of ℘(ELEMENT) is the set that equals ELEMENT.