Defining modular transformations

In a lecture at Harvard University in 1943, Bartok acknowledged his discove ry and use of a transformation that maps musical entities back and forth be tween diatonic and chromatic modular systems. But Bartok's transformation n eed not be limited to these; one can find examples of mappings to and from other modular spaces in Bartok's own music. This article formalizes Bartok' s transformation, as well as generalizes it to map musical entities to and from any one of five different spaces: chromatic (mod12), octatonic (mod8), diatonic (mod7), whole-tone (mod6), and pentatonic (mod5). The article the n demonstrates that the generalized operation is not linked to any one comp ositional style in the twentieth century by showing its use in works by Deb ussy, Stravinsky, and Schoenberg. Finally, it defines a new equivalence cla ss, the modular set type, which groups together those set classes that may be connected via the generalized transformation, and uses the new equivalen ce class and generalized transformation in analyses of Webern's op. 5, no. 3 and Stravinsky's Concerto in D.