Saturday, 9:00 am–12:00 pm
Monroe 213
Getting RICH: Chain Transformations, Row Areas, and a Twelve-Tone Space in Webern's op. 22/ii
The amorphous qualities of the second movement of Webern’s op. 22 Quartet for Violin, Clarinet, Saxophone, and Piano have often been defined in contrast to the more controlled handling of the twelve-tone technique in the first movement. This paper reevaluates the second movement of op. 22, not in an attempt to explain difficult passages as more straightforward than previously thought, but in order to understand a (pre-)compositional environment in which the movement operates. To construct such an environment, the paper begins by considering the properties and byproducts of row elision—one of Webern’s favorite compositional techniques. Properties of elided, or chained, rows are used to create row families that are distinguished by chain type, invariance, and inversional potential. Ultimately, this paper constructs a robust, hierarchical “twelve-tone space” structured by these families and their inherent properties. Along the way, the paper shows that while the musical surface betrays little of the order and tidiness associated with the first movement, its substructure is neatly constructed. The twelve-tone space and resulting analysis do not explain away the movement’s surface turbulence. They do, however, give some insight into the background structure within which this tumult operates.
K-Nets, Inversion, and Gravitations Balance
Klumpenhouwer networks have recently been the source of controversy. Michael Buchler has voiced concerns about the ability of K-nets to reflect musically pertinent events. This paper proposes a type of situation in which a K-net approach carries musically meaningful implications. Strongly isographic networks can be used in instances involving pairs of sets moving in contrary motion in pitch space; the isography highlights the constant axis around which the pairs balance themselves. This paper calls such an axis a gravitational center. While gravitational centers can be calculated with mathematical precision, they also reflect the degree to which given passages of music are balanced according to specific pitch-space partitions. Certain non-isographic K-nets can share gravitational centers, and can thereby be meaningfully related. The concept also allows for the comparison of partitioned sets of differing set class and cardinality, differences for which neither traditional K-net theory nor dual transformations can account. Excerpts from works by Messiaen, Bartók, Schoenberg, Harbison, and Schnittke are discussed with reference to gravitational balance. Ultimately, the paper offers a new perspective on what K-nets are able to model.
Aligned Cycles in Thomas Adès's Piano Quintet, op. 20
This paper focuses on aligned cycles in Thomas Adès’s Piano Quintet. An aligned cycle occurs when two (or more) voices move together in the same direction by different interval cycles in a note-against-note alignment—for example, a rising whole-tone scale (an interval 2-cycle) simultaneously with a rising chromatic scale (an interval 1-cycle). Composed in 2000, Adès’s quintet is a single-movement composition that is structured as a sonata form. In addition, three-voice aligned cycles play a significant role in the structural design of the quintet. Every aligned cycle in the piece is a combination of interval cycles 2, 3, and 4, and though Adès does not use any other interval cycles (i.e., 0, 1, 5, and 6) he constantly changes the registral ordering to generate all six permutations: <2,3,4>, <2,4,3>, <3,2,4>, etc. No matter which combination Adès uses, he aligns the three interval cycles so that non-functional major triads, members of set-class [037], are prominently featured. In addition to discussing how the non-functional major triads of the aligned cycles interact with the sonata-form structure of the piece, I explore how Adès varies and transforms his aligned-cycle patterns. This paper concludes that the aligned cycles in Adès’s Piano Quintet play a significant role in the structural design of the composition. The non-functional major triads of all the aligned cycles intersect with the form of the piece to create a twenty-first-century commentary on tonality and sonata form.
A Transformational Approach to Harmony and Voice Leading in Elliott Carter's Recent Music
Elliott Carter’s recent harmonic practice has often been characterized as an effort to achieve maximum variety within the confines of a limited harmonic vocabulary. In the preface to his Harmony Book (2002), Carter remarks that “From about 1990, I have reduced my vocabulary of chords more and more to the 6-note chord no. 35 and the 4-note chords nos. 18 and 23, which encompass all the intervals.” The six-note chord no. 35 is better known as the all-trichord hexachord (012478)—a hexachord that embeds all twelve of the trichordal set classes. The four-note chords nos. 18 and 23 are better known as the all-interval tetrachords (0146) and (0137) respectively. Given Carter’s use of a relatively limited set-class vocabulary in which the degree of common-tone retention is of particular importance, this paper explores ways that transformational techniques, especially those inspired by neo-Riemannian theory, are able to model the dynamism of Carter’s recent harmonic practice. Compositional spaces generated by the complement-union property (CUP) are reconfigured to create Tonnetze; contextual transformations are then defined on these spaces. A toroidal geometric space that models transformations among all-interval tetrachords is explored in detail. The paper concludes by suggesting a space to model transformations among all-trichord hexachords. Works to be considered include Scrivo in Vento for solo flute (1991), Figment for solo cello (1994), Shard for solo guitar (1997), and Two Diversions for solo piano (1999).