Dissertation Index

Author: Yust, Jason D

Title: Formal Models of Prolongation

Institution: University of Washington

Begun: September 2005

Completed: November 2006


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.


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


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