## Dissertation Index

Author: Yust, Jason D
Title: Formal Models of Prolongation
Institution: University of Washington
Begun: September 2005
Completed: November 2006
Abstract: Prolongation is a fundamental concept for Schenkerian analysis, and formal modeling clears up the potential ambiguities of prolongational claims in musical analysis. Previous models of prolongation have borrowed the idea of a phrase-structure tree from linguistics. Phrase-structue trees can take many different forms, but the most useful phrase-structure models for musical prolongation are the “stratified” model, which asserts a fixed set of reductions of an event sequence, and the binary model, which maximally constrains possible sets of reductions without implying a single fixed set. I propose a model of prolongation based on maximal outerplanar graphs (MOPs) that is similar to the phrase-structure model but views prolongation as a relationship of motions defined by events rather than a relationship between the event themselves, as in a phrase-structure model. The MOP model better reflects the Schenkerian idea that passing motion is a fundamental form of prolongation. This model extends nicely to a method for contrapuntal analysis that combines MOP analyses for individual voices into a complete harmonic analysis taking the form of a 2-tree (“2-dimensional tree”), a class of graphs that includes MOPs. This complete harmonic analysis includes consonant groups of events and dissonant events, and is constrained only by the order of events in each voice, so that it can assert consonant relationships between both simultaneous and non-simultaneous event pairs. A number of different ways of defining the MOP class correspond to different semantic aspects of the MOP model of prolongation. In the last two parts of the paper I prove the equivalence of twelve different graph-theoretic characterizations of MOPs. Keywords: prolongation, Schenker, analysis, formalization, networks, mathematical models, counterpoint, tonality, harmony, grammar.
TOC: Introduction: Formalization and Schenkerian Analysis 1 Part 1: The MOP Model of Prolongation The Concept(s) of Prolongation 10 Prolongations as Passing Events 31 Some Conceptual Problems in Theories of Prolongation 39 Graphs and Digraphs as Analytical Models 44 David Lewin’s Node-Arrow Systems 46 Maximal Outerplanar Graphs 52 Refinements of the MOP Model 66 A Comparison of Analyses Using the MOP Model 75 Minimality and Chordality 82 Part 2: Phrase-Structure Models of Prolongation The General Phrase-Structure Model of Prolongation 86 Comparison of Chomsky’s Phrase-Structure Grammar to the Phrase- Structure Model of Prolongation 97 Comparisons of the MOP and Phrase-Structure Models of Prolongation 100 The MOP Model of Prolongation as a Binary Phrase-Structure Model 102 Combinatorial Comparisons of MOPs and Binary Phrase-Structure Trees 105 Relative Backgroundness in Phrase-Structure Analyses 111 Unstratified Phrase-Structure Models 116 Backgroundness Partial Orderings for Phrase-Structure Trees 122 Backgroundness Partial Orderings for MOPs 130 Comparing MOPs and Phrase-Structure Analyses through Reduction-Lists 133 Semantics of the Mapping from Phrase-Structure Trees to MOPs 149 Prolongational Models and Musical Intuition 152 Part 3: Formal Models of Contrapuntal Analysis Criteria for a Contrapuntal Model 155 The Representation of Counterpoint in Smoliar’s Model 158 The Representation of Counterpoint in Rahn’s Model 162 The Extension of the MOP Model of Prolongation to Contrapuntal Analysis 175 A Formal Definition of the Complete Harmonic Prolongational Analysis 186 Analytical Decision-Making in the Contrapuntal Model 191 Ambiguity and Formalization: A Summary 195 Part 4: Mathematical Characterizations of MOPs Properties of MOPs 198 Basic Terms and Definitions 199 Statement of Theorem 1 (Characterizations of MOPs) 202 (1) Unary 2-Trees 203 (2) Maximal Cliques, 2-Overlap Clique Graphs, and Clique Trees 206 (3) Maximal Outerplanar Graphs, First Definition 208 (4) Maximal Outerplanar Graphs, Second Definition 211 (5) Chordality 214 (6) Minimality 214 (7) Cycle-Connectedness 215 (8) Confluence 218 (9) Chordality and Confluence 220 (10) H2-Intrasymmetry 221 (11) HOP-Intrasymmetry 225 (12) HC-Intrasymmetry 225 An Overview of the Characterizations of Theorem 1 226 Part 5: Proof of Theorem 1, Characterizations of MOPs Outline of the Proof 228 Part 1 229 Part 2 230 Part 3 237 Part 4 244 Part 5 247 Part 6 249 Part 7 250 Part 8 250 Part 9 251 Part 10 253 Part 11 254 Part 12 255 Bibliography 258 Appendix: Proofs of Propositions 262 Contact: 3828 Meridian ave. N, Seattle, WA, 98103 (206) 914-0611 (206) 632-2147 |