Author: Ilomäki, Tuukka
Title: On the Similarity of Twelve-Tone Rows
Institution: Sibelius Academy
Begun: September 2002
Completed: April 2008
The relations of twelve-tone rows are of theoretical, analytical, and compositional interest. While relations based on the properties and transformations of rows have been widely studied, less at- tention has been paid to relations based on similarity. Formal similarity measures can be used to explicate ways of being similar.
This study presents an analysis and categorization of 17 similarity measures for twelve-tone rows. Nine of them are new. The categorization of the similarity measures suggests the notion of different conceptions of twelve-tone rows. Five such conceptions are identified and explicated: vector, ordered pairs, subsegments, subsets, and interval contents. Similarity measures could thus be grouped into families based on the conception that they suggest.
The similarity of twelve-tone rows allows two interpretations: comparison of the properties of the rows and the measurement of their transformational relations. The latter could be conveniently formalized using David Lewin’s Generalized Interval Systems as the framework. This allows the linking of the discussion on permutations in mathematics and computer science because the mea- surement of the complexity of a transformation coincides with the notion of presortedness of per- mutations.
The study is in three parts. The first part gives an overview of the types of relations between twelve-tone rows, and presents a formalization of twelve-tone rows and row operations in terms of group theory.
The second part focuses on the properties of similarity. By way of background a review and criticism of the literature on similarity in music theory is presented. The transformational approach and the metric are promoted. It is shown that transformational similarity measures create perfectly symmetrical spaces since every row is related to the other rows by precisely the same set of trans- formations. Since most of the similarity measures discussed in this study are dissimilarity measures of the distance between rows, the mathematical concept of the metric is applicable; many similarity measures define a metric. One of the main findings is that any metric for twelve-tone rows that is transformationally coherent under the operations generating row classes also defines a metric for those row classes.
The third part discusses the similarity measures and the respective conceptions in detail. While the study focuses on the similarity of twelve-tone rows, the possibilities of extending the measures to the examination of other ordered pitch-class sets are also discussed. The work concludes with some examples of their analytical application.
Keywords: similarity, twelve-tone row, metric, transformation, GIS, permutations, posets
Part I: Background
Definitions and conventions
2 Twelve-tone rows and row operations
Part II: Similarity
3 Similarity and its properties
4 Similarity measures and the metric
5 The transformational approach to similarity
6 Topologies of twelve-tone rows
Part III: Similarity measures
7 An introduction to twelve-tone-row similarity measures
8 Similarity measures based on the vector approach
9 Similarity measures based on ordered pairs
10 Similarity measures based on subsegments
11 Similarity measures based on subsets
12 Similarity measures based on intervals
13 Partial orders, graph theory, and similarity
C Partially ordered sets
E Similarity measures for twelve-tone rows
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