Author: Buchler, Michael H.
Title: Relative Saturation of Subsets and Interval Cycles as a Means for Determining Set-Class Similarity
Institution: Eastman School of Music
Begun: September 1995
Completed: October 1997
The issue of how one should or might compare two or more events in atonal music has been a major concern--either explicitly or implicitly--in much music theoretical literature this century. While musical entities related by certain transformations (most notably transposition and inversion) have commonly been understood as equivalent, there is to date no commonly accepted method for comparing non-transformationally-related musical entities.
In an attempt to bridge this gap, many theorists and composers have developed similarity measures or other constructs that are useful for relating two collections of notes (typically pitch classes). The first chapter of the dissertation provides a brief history and critique of various works in this field. The second chapter, which forms the theoretical heart of the dissertation, introduces several new similarity indices which take into account the degree of "saturation" that each element (interval class, subset class, or subset of an interval cycle) enjoys in a given set class. In order to examine the degree to which set X is saturated with element x, we compare the number of x's in X to the minimal and maximal amount of x that is possible in any set of the same size.
The third and fourth chapters provide examples of how similarity indices can inform analyses of some post-tonal music. Chapter 3 is a comparative analysis of the first movement of Igor Stravinsky's Three Pieces for Clarinet Solo (1920) using eight different similarity indices. In Chapter 4, the second movement of Olivier Messiaen's Petites esquisses d'oiseaux (1985) is analyzed through the lens of one of the similarity indices defined in chapter 2.
The final chapter of the dissertation addresses some criticisms of similarity indices in particular and pitch-class based atonal analysis in general. It also provides the groundwork for extending the idea of elemental saturation into other domains, most notably "pitch-space," in which octave-related notes are not considered equivalent.
Keywords: Similarity, relations, indices, measures, set theory, Messiaen, Stravinsky, p-space, pc-space
Chapter 1: A brief history and critique of interval/subset class vectors and similarity functions
Chapter 2: Saturation vectors and associated similarity indices
Chapter 3: A comparative analysis of Stravinsky's Three Pieces for Clarinet Solo, first movement (1920) using similarity indices and pcset networks
Chapter 4: Similarity and dissimilarity of set types in Olivier Messiaen's Petites esquisses d'oiseaux, second movement ('Le Merle noir')
Chapter 5: Extensions and Conclusions
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