The operations discussed in chapter 3 are generalized representations of compositional techniques whereby PC sets are derived from each other. However, not all relations between PC sets are based on derivation. This chapter and the following ones will deal with other relations, seen within a historical framework. In this chapter I will discuss the evolution of the concept of PC set equivalence.

Like the term “operation,” the term “relation” evokes the world of mathematics. In mathematics, more specifi cally in algebra, a relation is commonly conceived as an open sentence, designated P, connecting the elements of two collections, S and T. This open sentence is true or not true for each ordered pair of these elements. If s is an element of S, and t is an element of T, a relation can be defi ned as the collection of ordered pairs (s,t) for which the open sentence P is true. This is a subcollection of the entire collection of these pairs. A relation like this is called a relation “from S to T.”

We will deal here with the special case in which S = T. Then, the relation is called “a relation in S.” In chapter 2, we defi ned the relation “is congruent modulo 12 with” for the collection of pitch values under twelve-tone equal temperament (PITCH). This relation singles out specifi c ordered pairs of pitch values from this collection, such as (14, −10), (5,29), or (10, −2). For these ordered pairs the relation obtains: 14 is congruent modulo 12 with −10, 5 is congruent modulo 12 with 29, and 10 is congruent modulo 12 with −2.

Definitions of Equivalence

The relation “is congruent modulo 12 with,” when it is defi ned on PITCH, is what algebra refers to as an “equivalence relation.” In chapter 2, we have seen that this relation satisfi es all three conditions of equivalence: refl exivity, symmetry, and transitivity. The concept of equivalence helps us bring large collections of musical elements down to reasonable proportions. PITCH can be mapped onto PITCHCLASS; and the collection of PC sets (PCSET) can be compressed by defi ning equivalences on the basis of certain transformations or common properties, as this chapter will show.