2 research outputs found
Fuzzy Family Ties: Familial Similarity Between Melodic Contours of Different Cardinalities
All melodies have shape: a pattern of ascents, descents, and plateaus that occur as music moves through time. This shape—or contour—is one of a melody’s defining characteristics. Music theorists such as Michael Friedmann (1985), Robert Morris (1987), Elizabeth Marvin (1987), and Ian Quinn (1997) have developed models for analyzing contour, but only a few compare contours with different numbers of notes (cardinalities), and fewer still compare entire families of contours. Since these models do not account for familial relations between different-sized contours, they apply only to a limited musical repertoire, and therefore it seems unlikely that they reflect how listeners perceive melodic shape.
This dissertation introduces a new method for evaluating familial similarities between related contours, even if the contours have different cardinalities. My Familial Contour Membership model extends theories of contour transformation by using fuzzy set theory and probability. I measure a contour’s degree of familial membership by examining the contour’s transformational pathway and calculating the probability that each move in the pathway is shared by other family members. Through the potential of differing alignments along these pathways, I allow for the possibility that pathways may be omitted or inserted within a contour that exhibits familial resemblance, despite its different cardinality.
Integrating variable cardinality into contour similarity relations more adequately accounts for familial relationships between contours, opening up new possibilities for analytical application to a wide variety of repertoires. I examine familial relationships between variants of medieval plainchant, and demonstrate how the sensitivity to familial variation illuminated by fuzzy theoretical models can contribute to our understanding of musical ontology. I explain how melodic shape contributes to motivic development and narrative creation in Brahms’s “Regenlied” Op. 59, No. 3, and the related Violin Sonata No. 1, Op. 78. Finally, I explore how melodic shape is perceived within the repetitive context of melodic phasing in Steve Reich’s The Desert Music. Throughout each study, I show that a more flexible attitude toward cardinality can open contour theory to more nuanced judgments of similarity and familial membership, and can provide new and valuable insights into one of music’s most fundamental elements
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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