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On the Similarity of Twelve-Tone Rows Tuukka Ilomäki On theSimilarityofTwelve-ToneRowsStudiaMusica30SibeliusAca Tuukka Ilomäki On the Similarity of Twelve-Tone Rows ISBN 952-5531-32-5 (paperback) ISBN 952-5531-33-3 (pdf) ISSN 0788-3757 demy Yliopistopaino Helsinki 2008 Studia Musica 30 2234856_Ilomaki_kansi.indd34856_Ilomaki_kansi.indd 1 220.02.20080.02.2008 118:34:058:34:05 Tuukka Ilomäki On the Similarity of Twelve-Tone Rows Studia Musica 30 Sibelius Academy c Tuukka Ilomäki and Sibelius Academy 2008 ISBN 952-5531-32-5 (paperback) ISBN 952-5531-33-3 (pdf) ISSN 0788-3757 Yliopistopaino, Helsinki 2008 SIBELIUS ACADEMY DocMus Department Studia Musica 30 Ilomäki, Tuukka On the Similarity of Twelve-Tone Rows 274 + 26 pages ABSTRACT 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. Acknowledgments This work was carried out in the DocMus Department at the Sibelius Academy during the period between 2002 and 2008. During the academic years 2004–2005 and 2005–2006 I studied at the Eastman School of Music under the auspices of the Fulbright program. In all these years I had the privilege to work with wonderful people who have given me a prodigious amount of help. First of all, I wish to express my gratitude to my supervisors Prof. Marcus Castrén and Prof. Dave Headlam, both of whom have followed the project closely from its inception to its end. Their supervision and support have made the experience delightful. Prof. Robert Morris functioned as both a consulting supervisor and a reviewer of the final manuscript. His insight, suggestions, and criticism have shaped much of the work. Prof. Robert Peck, the second reviewer of the final manuscript, gave valuable criticism and inspiring suggestions. Prof. Eero Saksman read some critical portions of the manuscript and was also an inexhaustible source of encouragement. I am particularly grateful to him for his refusal to accept hand-waving as a method of proving theorems. The numerous Friday- afternoon discussion sessions with Prof. Ciro Scotto were inspiring and helped me clarify my ideas. The discussions with Dr. Tuire Kuusi were also helpful. My colleague Yki Kortesniemi gave me numerous lessons in the art of scientific writing, and provided solid advice on various issues during the various stages of this work. Joan Nordlund revised the language of the final manuscript. I am deeply grateful to the teachers and my fellow students at the DocMus Department, the Pythagoras Graduate School, and the Eastman School of Music, who provided an inspiring and challenging working environment. I wish to express my gratitude to the following institutions for their financial support: the Academy of Finland through the Pythagoras graduate school, the Sibelius Academy Innovation Centre, the DocMus department, the Fulbright Center, and the Leonora and Yrjö Paloheimo foun- dation. Finally, I would like to thank my family and friends for copious things, most of which have abso- lutely nothing to do with either twelve-tone rows or similarity, and which could never be expressed in terms of group theory. Table of Contents Part I: Background 1 Definitions and conventions 3 1 Introduction 6 1.1 Rows and their relations . 6 1.2 Row spaces . 7 1.2.1 The formalization of row spaces . 7 1.2.2 Row spaces by operation . 8 1.2.3 Row spaces by property . 10 1.2.4 Row spaces by similarity . 10 1.3 Methodology . 11 1.4 On the objectives of this study . 12 2 Twelve-tone rows and row operations 15 2.1 Formalizing twelve-tone rows . 15 2.1.1 Representations of rows . 15 2.1.2 The duality of pitch classes and order numbers . 16 2.1.3 Applying the theory of permutations . 17 2.2 Formalizing row operations . 18 2.2.1 Background . 18 2.2.2 The group structure of row operations . 20 2.2.3 Operations acting on rows . 21 2.2.4 Twelve-tone rows, row operations, and permutations of rows . 24 2.2.5 Pitch-class operations and order-number operations . 25 viii Table of Contents Part II: Similarity 29 3 Similarity and its properties 31 3.1 Introduction . 31 3.2 Terminological issues . 32 3.3 Twelve-tone rows and similarity . 33 3.4 Properties of similarity measures . 35 3.5 Similarity and dissimilarity . 35 3.6 Previous studies . 36 3.6.1 Domain . 36 3.6.2 Range . 38 3.6.3 Internal coherence and aspects of similarity . 40 3.6.4 The role of algorithms . 43 3.6.5 Transitivity . 49 3.7 Transformational coherence . 52 4 Similarity measures and the metric 54 4.1 Similarity measures as distances . 54 4.2 The definition of the metric . 55 4.3 The metric and similarity measures . 56 4.4 The metric and set-class similarity measures . 58 4.5 Scaling the values . 61 5 The transformational approach to similarity 65 5.1 Transformational theory . 65 5.2 Transformation theory and similarity . 66 5.3 Pitch-class transformations and order-number transformations . 67 5.4 Rows and transformations as a GIS . 67 5.5 From a GIS to a similarity measure . 70 5.6 Left invariance . 72 5.7 Interval-preserving transformations . 73 6 Topologies of twelve-tone rows 75 6.1 Symmetries of row spaces . 75 6.2 The similarity of row classes . 76 6.3 Distribution . 82 6.4 Relative similarity . 84 Part III: Similarity measures 87 7 An introduction to twelve-tone-row similarity measures 89 Table of Contents ix 7.1 Previous research . 89 7.1.1 Twelve-tone-row similarity measures . 89 7.1.2 The permutational approach . 90 7.1.3 Related similarity measures . 90 7.2 The approach in this study . 91 7.2.1 Two stages of defining a similarity measure . 91 7.2.2 Transformational and non-transformational readings . 92 7.2.3 Rows used as examples . 93 8 Similarity measures based on the vector approach 97 8.1 The twelve-tone row as a vector . 97 8.2 DERANGEMENT ................................... 99 8.3 DISPLACEMENT ................................... 110 8.4 PITCH-CLASS DISPLACEMENT . 119 8.5 Transformations in the vector approach . 126 9 Similarity measures based on ordered pairs 127 9.1 The twelve-tone row as a set of ordered pairs . 127 9.2 ORDER INVERSIONS ................................. 129 9.3 BADNESS OF SERIAL FIT .............................. 138 9.4 Transformations in the ordered-pairs approach . 146 9.4.1 Exchanging adjacent pitch classes . 147 9.4.2 Exchanging non-adjacent pitch classes . 148 9.5 CAYLEY DISTANCE .................................. 151 9.6 CORRELATION COEFFICIENT ............................ 155 10 Similarity measures based on subsegments 162 10.1 The twelve-tone row as a set of subsegments . 162 10.2 GENERALIZED ORDER INVERSIONS . 163 10.3 SUBSEGMENT CONTENT DIFFERENCE . 169 10.4 ULAM’S DISTANCE .................................. 173 10.5 ORDER-INTERVAL INVARIANT N-TUPLES . 177 10.6 DIVISIONS ...................................... 183 10.7 FRAGMENTATION ................................... 186 10.8 Transformations in the subsegments approach . 192 11 Similarity measures based on subsets 196 11.1 Nestings . 197 11.1.1 Background . 197 11.1.2 Creating a nesting from rows . 198 11.1.3 Creating rows from a nesting . 207 11.1.4 Nesting and uniqueness . 215 x Table of Contents 11.1.5 Nestings of more than two rows . 217 11.2 The NESTINGS similarity measure . 218 11.3 SCATTERING ..................................... 222 12 Similarity measures based on intervals 229 12.1 The twelve-tone row as a succession of ordered pitch-class intervals .
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