In a lecture at Harvard University in 1943, Bartók acknowledged his discovery and use of a transformation that maps musical entities back and forth between diatonic and chromatic modular systems. But Bartók's transformation need not be limited to these; one can find examples of mappings to and from other modular spaces in Bartók's own music. This article formalizes Bartók'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 then demonstrates that the generalized operation is not linked to any one compositional style in the twentieth century by showing its use in works by Debussy, Stravinsky, and Schoenberg. Finally, it defines a new equivalence class, the modular set type, which groups together those set classes that may be connected via the generalized transformation, and uses the new equivalence class and generalized transformation in analyses of Webern's op. 5, no. 3 and Stravinsky's Concerto in D.