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A Theory of Pitch and Duration Hierarchies for Post-Tonal Music

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A Theory of Pitch and Duration Hierarchies for Post-Tonal Music CONTOUR REDUCTION ALGORITHMS: A THEORY OF PITCH AND DURATION HIERARCHIES FOR POST-TONAL MUSIC by MUSTAFA BOR B.A., Bilkent University, 2001 M.Mus., The University of Alberta, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Music) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2009 © Mustafa Bor, 2009 ii ABSTRACT This dissertation takes work on contour by Robert Morris as a point of departure and develops a set of contour reduction algorithms, called window algorithms. These involve the notion of a hypothetical window or frame of a specific width (i.e. number of events) through which the contour succession in a given melody is experienced temporally (much like the way a landscape is experienced visually through the side window of a moving automobile or train). Certain normative principles relevant to windows of various widths are devised and represented with the help of symbolic logic and flowcharts. Reiterative application of the window algorithms on a melody to “prune” pitches at a series of successive levels, introduces notions of melodic contour hierarchy that are explored in various ways throughout the dissertation. The application of the algorithms is demonstrated on a variety of 20th century musical excerpts reflecting a wide range of melodic archetypes, thereby enabling observation of the behavior of the algorithms in different musical contexts. Phenomenological and cognitive implications of the algorithms are discussed from the perspective of a listener implementing the algorithm on the fly. An analysis focusing on the Hauptstimmen in the first movement of Schoenberg’s Third String Quartet explores how intervallic features of the reduced contours can form the basis for a tonal-formal reading of the movement. The possibility of extending the theory to the duration domain is also introduced; following a preliminary analysis involving instrumental gestures from the opening of Webern’s Variations for Orchestra, and a discussion of analytical, conceptual, and methodological inconsistencies arising from the application of the algorithms on duration contours, an alternative approach that attaches appropriate durations to (reduced) pitch contours is iii developed, examined, and advocated. The relationships and interactions between the pitch and duration contours in Berio’s Sequenza I are examined in the light of the proposed theory. iv TABLE OF CONTENTS Abstract................................................................................................................................ ii Table of Contents .................................................................................................................iv List of Tables .......................................................................................................................vi List of Figures .................................................................................................................... vii Acknowledgements ..............................................................................................................xi CHAPTER 1: Introduction ....................................................................................................1 Definition and Context............................................................................................1 Outline of Chapters.................................................................................................2 Theoretical Concepts and Terminology...................................................................5 CHAPTER 2: Theoretical Context: Selected Recent Theories of Contour in Music .............12 Origins..................................................................................................................12 Contour Perception in Non-Tonal Context ............................................................15 Foundations of Contemporary Contour Theory .....................................................17 Extensions to the Theory.......................................................................................26 Robert Morris’s Contour Reduction Algorithm .....................................................29 Recent Developments ...........................................................................................33 Rob Schultz’s Refinements to the Contour Reduction Algorithm ..........................37 Summary ..............................................................................................................46 CHAPTER 3: Window Algorithms and Contour Reduction Functions.................................48 The Contour Reduction Algorithm by Morris........................................................48 The 3-Window Algorithm.....................................................................................55 The 5-Window Algorithm.....................................................................................67 Segments with Repeating cps................................................................................78 Formal Definitions of the Window Algorithms .....................................................85 Contour Reduction Functions................................................................................97 A Contour Representation of Depth Levels ...........................................................99 Comparing the Window Algorithms and Morris’s Reduction Algorithm .............101 CHAPTER 4: Applying the Algorithms: A Gallery of Examples.......................................111 Elliott Carter’s A 6 Letter Letter (1996)...............................................................112 Pierre Boulez’s First Piano Sonata (1946)...........................................................116 Iannis Xenakis’s Herma (1961)...........................................................................119 Alois Hába’s Nonet, op. 40 (1931) ......................................................................121 v Milton Babbitt’s Soli e Duettini for Flute and Guitar (1989)................................131 Harrison Birtwistle’s Verses for Clarinet and Piano (1965) .................................134 Igor Stravinsky’s Septet (1953)...........................................................................135 Elliott Carter’s Steep Steps (2001).......................................................................136 Anton Webern’s “Schatzerl klein,” op.18, no.1 (1925)........................................142 CHAPTER 5: The Listener’s Window: Reduction Algorithms, Memory, and the Listening Experience..................................................................................147 A Model for the 3-Window.................................................................................147 Short-Term Memory ...........................................................................................151 Alternative Models for the 3-Window.................................................................154 Models for the 5-Window ...................................................................................159 Successive Windows...........................................................................................163 Hearing Depth Levels .........................................................................................169 CHAPTER 6: An Analytical Application: The Hauptstimmen in Schoenberg’s String Quartet No.3, Op. 30, I................................................................................172 Introduction ........................................................................................................173 Analytical Perspective 1: Intervallic Content at Successive Depth Levels............177 Analytical Perspective 2: Tonal Residues in the Contour Frameworks ................209 CHAPTER 7: Duration Contours: A Cumulative Approach...............................................232 Introduction ........................................................................................................232 Duration Contours ..............................................................................................233 Webern, Variations for Orchestra: A Preliminary Analysis.................................238 Paucity in the Duration Domain as an Analytical Obstacle..................................253 Conceptual and Methodological Inconsistencies Arising from the Application of the Window Algorithms to the Duration Domain ............................................255 Two Solutions to the Inconsistencies...................................................................261 CHAPTER 8: Analysis: Interacting Pitch and Duration Contours in Berio’s Sequenza I ....270 Introduction ........................................................................................................270 Segmentation ......................................................................................................272 Pitch and Duration Contour Frameworks ............................................................280 A Transformational Approach.............................................................................307 CHAPTER 9: Conclusions and Future Directions..............................................................321 BIBLIOGRAPHY .............................................................................................................332 vi LIST OF TABLES Table 4.1. Contour Reduction Functions with Different Cardinalities ................................124 Table 5.1. 3-Window
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