Abstract
Two kinds of extensions to David Lewin's transformation theory are proposed. First, cross-type transformations transform one sort of object to another: for instance, mappings from triads to seventh chords. Special cases include GIS homomorphisms and interscalar transpositions and inversions. The second extension relaxes Lewin's requirement that the products of transformations along two paths from one node to another in a transformation graph must match. It is argued that this path consistency condition is unnecessarily restrictive; several alternatives are considered. Examples from Bach to Bartók illustrate musical relationships that can be most effectively modeled through the use of cross-type transformations and non-path-consistent graphs.
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