Two types of transposition operators may be applied to diatonic objects such as chords or melodic fragments: the familiar mod-12 transposition operators (which may be understood to transpose the underlying diatonic scale along with the object itself); and the diatonic, or mod-7, transposition operators (which shift the original object within a fixed diatonic scale). Both types of transposition are expressible in terms of signature transformations. A signature transformation reinterprets any diatonic object in the context of a different key signature. With an appropriate understanding of octave and enharmonic equivalence, the signature transformations can be shown to generate a cyclic group of order 84, of which both the mod-12 and mod-7 transposition groups are subgroups. Signature transformations therefore hold considerable theoretical potential in unifying chromatic and diatonic structures, and relate to a number of established constructions in transformation theory and diatonic set theory. Direct applications of signature transformations may be observed in the works of many composers, as illustrated by examples from composers as diverse as Schubert, Debussy, and Michael Torke.
Hook applies the signature transformations not only to the obvious case of the abrupt shift to C# major in the contrasting middle but also to the succession of four-element eighth-note motives passed back and forth between the upper voices.
Reference:
Julian L. Hook. "Signature Transformations." In Jack Douthett, Martha Hyde, and Charles J. Smith, eds. Music Theory and Mathematics: Chords, Collections, and Transformations, pp. 137-160. University of Rochester Press, 2008. Link to book page on the UR Press site.
