Pitch-class set theory is a musical application of mathematical set theory. As with the latter, it defi nes a set as a collection of things, and it refers to these things as the elements of the set. Sets are usually denoted by capital letters, like A, B, C, etc. If not specifi ed, the elements are denoted by lower-case letters: a, b, c, etc. In pitch-class set theory, too, a set is defi ned by its elements only, not by a particular arrangement of these elements or by their quantities. When two sets A and B consist of the same elements, they are considered equal (A = B). What is special about pitch-class set theory is the type of element it deals with: the pitch class. Therefore, I will start this discussion of the theory with defi nitions of the terms “pitch” and “pitch class.”

Pitch and Pitch Class

Pitch is one of the four distinguishing attributes of a musical tone (the others being loudness, duration, and timbre). Broadly speaking, pitch is the position of a tone on a spectrum that runs from low to high. In pitch-class set theory, the term “pitch” designates a particular value assigned to that position. Thus, it refers to what is often expressed by a letter name (C, D, E, etc.) together with an indication of the octave range. Pitch-class set theory does not usually apply these letter names. Under the postulate of equal temperament—the division of the octave into (twelve) equal parts—it associates pitches with integers. It arbitrarily assigns the number 0 to C4 (middle C), so that, by the equal distances between all successive pitches, C♯4 = 1, B3 = −1, D4 = 2, B♭3 = 2, E♭4 = 3, and so on (ex. 2.1). Thus, each pitch can be associated with a pitch number, which is a positive or negative integer. Of course, it is possible to assign the number 0 to a pitch other than C4. This may help to clarify specifi c musical contexts. However, unless otherwise stated, the pitch number 0 will represent C4 in this study. Enharmonic notes (e.g., B♭ and A♯, or E and F♭) are given the same pitch numbers.