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A Hierarchical Approach to the Analysis of Intermediary Structures Within the Modified Contour Reduction Algorithm

Robert Morris’s (1993) Contour-Reduction Algorithm—later modified by Rob Schultz (2008) and hereafter referred to as the Modified Contour Reduction Algorithm (MCRA)—recursively prunes a contour down to its prime: its first, last, highest, and lowest contour pitches. The algorithm follows a series of steps in two stages. The first stage prunes c-pitches that are neither local high points (maxima) nor low points (minima). The second stage prunes pitches that are neither maxima within the max-list (pitches that were maxima in the first stage) nor minima within the min-list (pitches that were minima in the first stage). This second stage is repeated until no more pitches can be pruned. What remains is the contour’s prime. By examining how the reduction process is applied to a given c-seg, one can discern a hierarchy of levels that indicates new types of relationships between them. In this thesis, I aim to highlight relationships between c-segs by analyzing the distinct subsets created by the different levels obtained by the applying the MCRA. These subsets, or sub-csegs, can be used to delineate further relationships between c-segs beyond their respective primes. As such, I posit a new method in which each sub-cseg produced by the MCRA is examined to create a system of hierarchical comparison that measures relationships between c-segs, using sub-cseg equivalence to calculate an index value representing degrees of similarity. The similarity index compares the number of levels at which two c-segs are similar to the total number of comparable levels. I then implement this analytical method by examining the similarities and differences between thirteen mode-2 Alleluias from the Liber Usualis that share the same alleluia and jubilus. The verses of these thirteen chants are highly similar in melodic content in that they all have the same prime, yet they are not fully identical. I will examine the verses of these chants using my method of comparison, analyzing intermediary sub-csegs between these 13 chants in order to reveal differences in the way the primes that govern their basic structures are composed out.
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