Dual interval space in twentieth-century music

A dual interval space (DIS) is a two-dimensional array of pitch classes in which each dimension corresponds to a unique (non-zero) interval class. Given some pitch-class collection, the members of that collection can be interpreted as residing in various locations of a DIS.These locations can then be translated within the space or flipped about some axis.The flipping operations in particular offer new ways to relate set classes,even set classes of different cardinalities.This essay develops the concept from a theoretical standpoint, exploring the effects of the operations on pitch-class sets, and demonstrates its relevance for analysis by examining music of Ruggles, Schoenberg, and Webern.