Though the z-relation—the relationship whereby two sets not related by transposition and/or inversion have the same interval-class content—has been one of the core concepts of pitch-class set theory since its inception, the principles underlying the relationship have to a large extent remained obscure. However, new information is emerging. First, recent work on the Fourier transform for pitch-class-set analysis (in particular, by Ian Quinn, which builds on work by David Lewin) provides new information concerning the subset structure of z-related sets. Second, as recognized by Clifton Callender and Rachel Hall, the z-relation is an instance of what crystallographers call homometry, which has been written about extensively. The aim of this study is to utilize these new means to present a comprehensive description of the z-relation that addresses the criteria upon which the z-relation relies as well as the ways in which z-related sets interrelate.
Building upon algebraic formulas suggested by crystallographers, I develop an approach that describes z-related pairs in terms of formulas that correspond to subset properties illustrated by the Fourier transform. In addition to devising my own formulas, I show that the formulas can be extended with a method called ‘pumping’ (O'Rourke, et alii), which adds pitch-classes to preformed formulas. Together, the formulas and pumping lead to a general theory of the z-relation that not only describes the relationships between the sets of a z-related pair, but also those between z-related pairs of different cardinalities.
Since it exposes transformational relationships, the algebraic approach lends itself well to the analysis of music that involves z-related sets, whether twelve-tone music based on rows with discrete hexachords that are z-related, or other non-serial music that prominently features z-related sets as motivic harmonies. In my own work I use the theory to analyze twentieth-century works by composers including Schoenberg, Berg, Carter and Berio. Overall, the theory exposes that transformational networks involving z-related sets are not only possible, but are altogether relevant to the analysis of music that involves such harmonies.
|Commitee:||Hasegawa, Robert, Quinn, Ian|
|School:||University of Rochester|
|School Location:||United States -- New York|
|Source:||DAI-A 74/03(E), Dissertation Abstracts International|
|Keywords:||All-interval tetrachords, Crystallography, Fourier transform, Homometric, Homometry, Pitch class, Z-relation, Z-transformations|
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