Dissertation Index

Author: Vojcic, Aleksandra

Title: Rhythms as Form: Rhythmic Hierarchy in Later Twentieth-Century Piano Music

Institution: Graduate Center CUNY

Begun: January 2009

Completed: January 2009


This study focuses on structurally salient rhythmic structures in piano works by composers active in the later twentieth century, and develops a conceptual framework adequate to the problems the repertoire poses. It develops a hierarchical concept of rhythm, sta rting at the level of pulse and culminating in large-scale formal divisions. Rhythmic hierarchy is defined in terms of three perceptible strata: the foreground, the middleground, and the background.

Since the focus of this study is on rhythmic form, the discussion of the rhythmic foreground is largely incorporated into the conceptual framework developed for the middleground and the background. However, as part of this interaction between hierarchical levels, taleas, beat trains, and arithmetical patterns of inter-onset intervals between beats and beat-groups appear as examples of organized rhythmic impulses that can be transferred from the foreground to the middleground.

The rhythmic middleground is largely examined via recurring composite metric patterns I term meta-measures. Meta-measures organize tactus beats and beat groups into coherent middleground periodicites, and I demonstrate the types of structural boundaries identified specifically for composite metric patterns. The formal significance of meta-measure patterns is summarized via an original analytical representation called the time signature map.

Additionally, this study identifies different modes of interaction between meta-measures and other significant rhythmic processes such as polyphony and tempo modulation, and my tempo maps summarize these interactions as well. The concept of flexible cycles, which is often limited to rhythmic figures, is also extended to meta-measure patterns.

The rhythmic (and formal) background is identified with rhythmic scaffolding. Rhythmic scaffolding comprises a series of structural markers that represent structurally salient rhythmic events in one or more pulse streams. In this study, rhythmic scaffolding is summarized by original graphic representations such as a pulse stream graph or a tempo map.

Keywords: rhythmic form, rhythm, meter, phrase structure, pulse streams, meta-measures, twentieth century, piano music



1. Structural Levels

2. Patterns of Rhythmic Variation

3. Structural Markers in Pulse Streams

4. Structural Boundaries of Composite Meters

5. Interaction of Pitch and Rhythm

6. Proportional Relations

Chapter 1: Rhythmic Hierarchy and Structural Functions of Pulse Streams

1.Pulse Stream Graph of Harmonic Rhythm in Colin Matthews's "Study 1"

2.Pulse Streams in Open and Closed Rhythmic Processes

a)Pulse Streams in Open Rhythmic Processes
b)Pulse Streams in Closed Rhythmic Processes
c)Flexible and Closed Processes Combined:
Elliott Carter's "90+"

Chapter 2: Beat Patterns and Structural Boundaries in Composite Meters

1.Beat Hierarchy

2.Time Signatures in Compound Meters

3.Beat Complement in Compound Meters

4.Middleground Periodicity of Composite Time
Signature Patterns

Chapter 3: Meta-Measure Prototypes and Categories

1.Meta-Measure Preliminaries

2.Meta-Measure Prototypes

3.Meta-Measure Antecedents

Chapter 4: Basic Meta-Measure Patterns

1.Simple Meta-Measure Patterns

a)Composite Aksak Meters in György Ligeti's "Capriccio"

b)A Labyrinth Metaphor in Alexander Goehr's
"Metric Maze"

c)Episodes of Metric Stability in Ralph Shapey's "Passacaglia"

2.Complex Meta-Measure Patterns

a)Meta-Measure Semi-Patterns in Robert Di Domenica's "Sonatina"

b)Foreground Variation of the Metric Pattern in Messiaen's
"Le Corps glorieux"

c)Formal Significance of Composite Time Signature Patterns in Anton Webern's String Trio Op.20/ I

Chapter 5: Meta-Measures as Flexible Cycles

1.Types of Flexible Cycles

2.Metric Rhyming and Flexible-Cycle Meta-Measures

a)Stravinsky's "Danse sacrale"

b)Messiaen's "L'ange aux parfums"

3.Duplication of Metric Groups in Goehr's "Moto Perpetuo"

4.Structural Markers as Meta-Measure Boundaries in Matthews's "Study 5"

5.Meta-Measure Hybrid in Schoenberg's Gigue from
Suite Op.25

Chapter 6: Meta-Measure Prototypes and Rhythmic Polyphony

1.Categories of Rhythmic Polyphony

2.Meta-Measures and Rhythmic Polyphony in Matthews's
"Study 4"

3.Transfer of Meta-Measure Pattern from the Foreground to the Middleground in Ligeti's "L'escalier du diable"

4.Meta-Measures and Polymeter in Ligeti's "En suspens"

Chapter 7: Meta-Measures and Tempo Modulation

1.Tempo Modulation

2.Meta-Measure Hybrids Containing Tempo Modulation

a)Matthews, "Study 5": Foreground-to-Middleground Time-Stretching

b)Carter, "Penthode": Proportional Relations of Tempi and Meters

c)Peter Maxwell Davies, Scherzo: Tempo Modulation on
the Level of Tactus

Chapter 8: Tempo Schemas and Formal Design

1.Foreground Tempo Shaping in Thomas Adès's "Aetheria"

2.Tempo Schemas

a)Nesting of Tempi and Dynamics in Jean Barraqué's Piano Sonata

b)Sectional Demarcation in Joan Panetti's "Cavata"

c)Tempo Schema as a Metaphor of Time Stretching in Harrison
Birtwistle‚s "Harrison's Clocks," IV

3.Beat Trains

a)Birtwistle, "Clock IV"

b)David Lumsdaine, "Kelly Ground"

4.Musical Mechanisms

a)Arithmetic Expansion in Birtwistle's "Clock IV"

b)Arithmetic Contraction in Berg's "Lyric Suite," V



Aleksandra Vojcic

University of Michigan
School of Music, Theater & Dance
1100 Baits Drive
Ann Arbor, MI 48109-2085


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