| 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. |
