Set-Type Saturation among Twelve-Tone Rows

Abstract

A twelve-tone row may be defined as a string of 12 pitch-classes (pcs) without replication, but it is also important to regard the row as determining a family of pcsets [1]-each pcset derived from the content of a unique adjacent segment of the row. Although Schoenberg himself considered only the segments of 3 to 9 pcs in length to generate (functional) pcsets, [2] there are 78 segments in any row-from the 12 one-element segments (the pcs themselves) to the entire row. Two pcsets are deemed equivalent if they are related by TTOs. Let us call the equivalence-classes set up by this definition set-types. Thus a row's family of pcsets represents a family of set-types. Each member of every settype represented by a row's family will be found in at least two members of the row's row-class and each row in a row-class shares exactly the same set-type family with all other members of the class. While the twelve-tone literature has largely focused on the content and relations of a row's non-overlapping segments (especially the terminal hexachords) often within the context of a theory of unordered pcsets such as Allen Forte's, [3] this paper, among other things, will examine the sonorous and structural potential of rows which have multiple representations of certain set-types in their pcset families. The first example is a particularly complete instance of this kind of row.