Two readings after Cohn and Dempster, no. 2

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.

Schubert's "Riemannian Hand"

The modulation to C# major is an LP transformation (Hook 139): A major moves to c# minor moves to C# major. Here is that change from the first to second strain, from A major to C# major, as a direct move in the left hand (thanks to Steve Rings for pointing this out):


While thinking about improvisation, about Schubert sitting at the piano playing while his friends danced, I realized that the piano permitted the sound of the waltz that would have been most familiar to people in Vienna about 1800 -- two violins and bass -- to be transferred from tavern or restaurant to the home. The three-layer texture of melody (first violin), bass, and accompanimental chords (second violin) became right hand, left-hand accents, and the offbeat "oompahs", respectively. (Link to D790n3 played apparently on a period instrument: note the timbral differences in the three registers.) In the heat of improvisation, the latter could serve Schubert well as voice leading stabilizers -- and, as in this case, enablers of modulations. Indeed, we might speak of his "Riemannian hand" and visualize it, as below, where a simple shift of thumb, middle finger, or pinky would effect a particular transformation.

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. Also see these posts: (1); (2).

more to signature transformations

This is actually just an addendum to the post with abstract and publication information for Jay Hook's article on signature transformations. I have finally acquired a copy of Music and Mathematics and want to explain "Neumeyer (forthcoming)" in footnote 1 (159). That was a book project that ultimately ran afoul of reviewers with strongly opposed views (that is, opposed to one another). The nub of chapter 1 is in the MTS article, chapters 2 & 3 will become articles when I get around to it, and chapter 4 has by now migrated in great part to this blog. Jay was generous in writing a section for chapter 4 at my request; I'm glad that the work sparked some serious thought and has not only resulted in a substantial publication for him but also in a construct with real potential for music analysis.

Signature transformations and D779n13

Jay Hook has published an essay that uses the A-Major waltz as its principal example from the era of traditional European major-minor tonality. Here is the abstract:

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.