Poster Session 1: 20th/21st-Century Compositional Strategies

This proposal contributes to the field of post-tonal rhythm studies by demonstrating the rhythmic structure in a chamber work by the Chinese composer Guoping Jia (b. 1963)—The Wind Sound in the Sky (for cello, percussion, and a Chinese wind instrument sheng; 2002). I show how Jia organizes his rhythmic structure based on strings of integers—which indicate durations—derived from the poem September (1986) written by the Chinese poet Haizi.

The Wind Sound contains seven short movements. My presentation focuses on Mvt. I, which contains two parts. The duration of a rhythmic segment in each instrument is generated by one of the two numerical parameters derived from Haizi’s poem: 1) the number of strokes to write each Chinese character, forming string 1 <14, 11, 10, 11, 10, 10, 14, 14, 10, 7>; and 2) the number of characters in each line, forming string 2 <14, 11, 10, 11, 10, 10, 14, 14, 10, 7>.

Counting one eighth as a beat, the durations of the rhythmic segments in the sheng are represented by string 1. String 2 characterizes the segments in the cello (sixteenth = 1 beat) and percussion (eighth = 1 beat). While segments in the outer voices perfectly align with the two-part formal division, those in the middle smoothly flow through the division, blurring the formal boundary. During my presentation, I will lead the audiences to listen to the music, experiencing the two-part form narrated by the three intricate layers of different durations of the rhythmic segments. 

• Handout

This poster presents a generalization and consolidation the of lineage of approaches to metrical inclusions put forward by Cohn (1992, 2001, 2019), Leong (2007), Murphy (2009), and Guerra (2018, 2019) that model metrical states in which pulses are related by combinations of the powers of 2 and 3. As oft noted, current methodologies could adequately address the substitution of higher primes for either 2 or 3. However, the inclusion of more (co-)prime generators necessitates a reexamination of “ski-hill graphs” and “metric cubes”—recasting them as k-dimensional ski-hill lattices and k-ary n-cubes, respectively. Expanded thusly, the ski-hill affords a new category of hemiola (k-Hemiola) that captures up to k hemiolas on the same metrical level, as distinct from “complex” hemiolas that occur across different levels. Isochronous k-ary n-cubes form a GIS as the group action of the n-fold Cartesian production of Z_k: (Z_k)ⁿ.

A generalized space is proposed that encompasses all possible isochronous and non-isochronous metrical states and their hemiolic relations. As hemiolas are here defined in reference to the space’s abstract structure, I present a three-tier conception of hemiolas based on the properties of the chosen generators: dissonances arising from the powers of 2 and 3; those arising from powers of higher primes; and those arising from co-prime generators. Illustrated by Ligeti’s Études, King Crimson, and Animals as Leaders, the generalizations presented here offer a means of evaluating abstractions of hemiolas and metrical dissonance, while providing a foil against which their traditional conceptions can be heard anew.

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Besides electronic sound, another field of post-war technological innovation was the explosion of innovative music notation (graphic scores of Cage, Feldman, Oliveros, Ligeti, Busotti, and Stockhausen). Some of the same composers were forging integral serialism (deriving multiple features from one numeric series) and were simultaneously tantalized by the spontaneity afforded by aleatoricism (or open form). The aural result of integral serialism is often so kaleidoscopically fluid that it sounds derived by chance and vice versa. Boulez writes that “fluidity of form must integrate fluidity of vocabulary.” Yet beyond these two-fold connections, the multiple temporal unfoldings of Stockhausen’s Refrain (1959) (for piano, percussion, and celeste) uniquely synthesize all three of these innovational strands: notation, serialism, and aleatoricism in one entanglement. 

Refrain’s striking visual presentation is known as a circularly shaped score with a rotating transparent strip of additional noteheads. Radial trajectories at varying distances from the center share features with spiral motion, which is what generates (on chromatic pitch space) the composition’s hidden all-interval 12-tone series unveiled at its centerpoint. Yet the pitch motives on the rotating strip arise from another row derived through poetic sestina permutation, which itself is spiral. Rotation of the strip expands or contracts the temporal distance of these motives, creating indeterminacy in the pacing of varied repetition heard in performance. In this way Refrain’s indeterminacy and serialism are multiply wrapped together through this concept of varied-distance radial motion, such that the technical operation of the score’s visual form mediates between, thereby encompassing, two opposites of Cold-war music. 

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• Stockhausen Refrain MP4

Olivier Messiaen's early compositions often feature his second mode of limited transposition—the octatonic scale—a collection whose symmetrical properties have long fascinated theorists. Analyses of these second-mode pieces, including analyses by Messiaen himself, are often content to identify which octatonic collection contain a particular series of chords. In this poster presentation, I show that this music can be analyzed more effectively with the algebraic tools of neo-Riemannian theory, in a manner inspired by Leah Frederick's recent research into diatonic voice-leading spaces.

In particular, I define an octatonic-step transformation, in which multiple voices move by successive parallel steps within an octatonic scale. Messiaen calls such progressions "parallel successions of chords," and they are a common surface feature of his music. Using a model that employs a mod-8 voice-leading space, I show that this transformation operates at more abstract levels as well, though its presence is often hidden by oblique or contrary contrapuntal motion on the surface. This model provides a fuller explanation of Messiaen's characteristic octatonicism than previous analyses.

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In his work on Lutosławski, Charles Bodman Rae has explored a crucial trait: namely, the composer often focuses on a pair of interval classes, treating them as building blocks of structure. Dubbed “interval pairing” by Bodman Rae, this technique most commonly occurs in the melodic dimension of Lutosławski’s music, but sometimes influences his approach to harmony as well. This presentation aims to carry forward Bodman Rae’s work by harnessing the Tonnetz to investigate interval pairing in Lutosławski’s music. The presentation examines numerous excerpts from Lutosławski’s music of the 1950s through the 1980s, including a detailed look at his Grave for Cello and Piano (1981). 

Of particular interest, Lutosławski’s music typically switches among two or more different pairings of interval classes. This begs the question: does Lutosławski treat his various interval pairings in a similar way? In other words, can we discern any general principle governing his approach to interval pairing? As I will demonstrate, such a principle does in fact underlie most of Lutosławski’s passages involving interval pairing. Specifically, these passages tend to operate within a 2xN region of a Tonnetz—or to put it another way, they tend to inhabit two parallel “tracks” of a Tonnetz. Though Lutosławski’s various choices of interval pairings often result in passages that outwardly sound very different from each other, this principle serves as a common thread uniting them. By exploring this principle in a number of his works, this presentation sheds new light on a fundamental aspect of Lutosławski’s music across the decades.

Musical excerpts can consist of some unknown combination of in-scale and out-of-scale pitches with indeterminate underlying scale structure. A notable example comes from Temperley and de Clercq’s (2013) corpus analysis of 200 rock songs, which identified ten commonly occurring pitch-class collections that mostly do not correspond to commonly cited musical scales. Drawing on perceptual studies that suggest pitch-class occurrence frequency as a cue of scale membership, I propose a simple heuristic for separating in-scale and out-of-scale pitch classes across musical excerpts sharing a pitch-class collection with unknown scale structure. I then apply this heuristic to de Clercq and Temperley’s rock corpus, finding that the ten common pitch-class collections reduce to four underlying scales: two reduce to the major diatonic, four reduce to the major pentatonic, three reduce to the “pentatonic union” identified by Temperley and de Clercq, and one reduces to Schuller’s proposed nonatonic blues scale. All four collections are exceptional in terms of a property previously highlighted as a cross-stylistic predictor of scale candidacy: they are as densely packed as possible with hierarchically nested, repeating stepwise patterns. While the pentatonic union and blues nonatonic collections may at first seem like surprising candidates to function as independent scales, their use is not unique to blues-influenced music, with transpositions appearing as klezmer scales. This approach to separating in-scale and out-of-scale pitch classes may be useful in analyzing other repertoire with ambiguous scale structures and in understanding cross-stylistic patterns in pitch-class collections’ scale candidacy.

• Handout