Today’s reading is a more complex application of the product network described by Richard Cohn and Douglas Dempster. Here I start with the network of signature transformations created by Jay Hook and set it into alignment with other readings. Whether the reading in the graphic below is sufficiently organized to qualify as a product network is moot, but the alignments are certainly suggestive in any case. The second system adds the familiar network of LP transformations, along with their "undoing" in measure 30. The remaining systems gather three Schenkerian analyses and align them with the two transformation networks.
Just as the C# minor triad is conceptually necessary but not literally present in the move from the first strain to the second, at measure 30 C# minor can be understood only indirectly in terms of the A7/G+6: E# becomes E-natural and the traditional "home" of the augmented sixth chords is the minor mode, not the major. This use of mixture in P allows the Schenkerian reading from ^5 in (c) to work best with the network readings; note, however, that the graph in (c) appends the final cadence, which is not needed in the transformational readings. Neither (d) nor (e) fit well, at least at this level, since they delegate the "correction" of the mixture to the foreground, where E# "splits" into E-natural in a lower voice and F# in the principal voice (in the background/middlegrounds depicted, E# acts as a chromatic passing tone between E and F#).
In principle, we could combine a wide variety of readings (pairs, as here, or larger groups) into product networks. In practice, there would probably always remain a substantial gap between the formalisms of Lewin's networks and such combinations of informal components. (Given his dogged insistence on hierarchies, it is not surprising that Fred Lerdahl (31) rejects Cohn and Dempster's product network model out of hand.)
References:
Cohn, Richard, and Douglas Dempster. "Hierarchical Unity, Plural Unities: Toward a Reconciliation." In Bergeron, Katherine and Phillip V. Bohlman, eds. Disciplining Music: Musicology and its Canons, 156-81. Chicago: University of Chicago Press, 1992.
Hook, Julian. "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.
Lerdahl, Fred. Tonal Pitch Space. New York: Oxford University Press, 2001.
Lewin, David. Generalized Music Intervals and Transformations. New Haven: Yale University Press, 1987.
