A General Theory of Comparative Music Analysis by Richard R. Randall Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Supervised by Professor Robert D. Morris Department of Music Theory Eastman School of Music University of Rochester Rochester, New York 2006 Curriculum Vitæ Richard Randall was born in Washington, DC, in 1969. He received his un- dergraduate musical training at the New England Conservatory in Boston, MA. He was awarded a Bachelor of Music in Theoretical Studies with a Distinction in Performance in 1995 and, in 1997, received a Master of Arts degree in music theory from Queens College of the City University of New York. Mr. Randall studied classical guitar with Robert Paul Sullivan, jazz guitar with Frank Rumoro, Walter Johns, and Rick Whitehead, and solfege and conducting with F. John Adams. In 1997, Mr. Randall entered the Ph.D. program in music theory at the University of Rochester’s Eastman School of Music. His research at the Eastman School was advised by Robert Morris. He received teaching assistantships and graduate scholarships in 1998, 1999, 2000. He was adjunct instructor in music at Northeastern Illi- nois University from 2002 to 2003. Additionally, Mr. Randall taught music theory and guitar at the Old Town School of Folk Music in Chicago as a private guitar instructor from 2001 to 2003. In 2003, Mr. Randall was ap- pointed Visiting Assistant Professor of Music in the Department of Music and Dance at the University of Massachusetts, Amherst. i Acknowledgments TBA ii Abstract This dissertation establishes a kind of comparative music analysis such that what we understand as musical features in a particular case are dependent on the music analytic systems that provide the framework by which we can contemplate (and communicate our thoughts about) music. We value how different analytic systems allow us to create different perceptions of a piece of music and argue that the identity of a musical work is in fact dependent on the combination of a work and an analytic system. Comparing musical works, then, necessarily compares such combinations. Therefore we assert that comparing pieces ought to be reframed as comparing the interpretation of pieces under specific analytic systems. Comparative analyses are carried out on local and global levels. Local music analytic systems map one set of musical events to another a set of musical events—the entirety of which comprises a piece of music. This is our description of what is normally thought of as analysis. Global music analytic systems analyze (or comment on) pieces of music determined by different types of local analyses. The theory of comparative music analysis is divided into two parts. The iii ABSTRACT iv first part defines a geometry of global music analytic systems, by far the most non-traditional aspect of the research. The geometric model covers all global systems. The second part defines a comparison of two local mu- sic analytic systems. Specifically, these local systems are tonal models, one defined by Lerdahl and the other based on experimental data of Bharucha and Krumhansl. Comparing local systems requires the establishment of a contextual equivalence between the two systems. In the case presented be- low, the equivalence is the formal structure of the metric space. These two approaches (global and local) are not alternatives to each other. They are in fact comparisons of musical pieces on different, yet interrelated levels. Comparing local music analytic systems is complicated by arbitrary design choices inherent in each system. Like comparisons of local systems them- selves, solutions to this “design choice problem” are contextual. One such solution is presented as is a discussion of a meta-analytic ramification of comparing local systems. Contents Curriculum Vitæ i Acknowledgments ii Abstract iii 1 Preliminaries 1 1.1 Pieces of Music . 3 1.1.1 Music as an Intentional Object . 4 1.2 Commentary . 11 2 History of Comparative Music Analysis 14 2.1 Leonard B. Meyer . 20 2.2 Jan LaRue . 24 2.3 Eugene Narmour . 27 2.4 Alan Lomax . 30 v CONTENTS vi 2.5 The Information Theorists . 35 2.5.1 Joseph Youngblood . 38 2.5.2 Barry S. Brooks . 42 2.5.3 Frederick Crane and Judith Fiehler . 43 2.5.4 James Gabura . 44 2.5.5 Leon Knopoff and William Hutchinson . 44 2.5.6 John L. Snyder . 46 3 Global Music Analytic Systems 50 3.1 Mathematical Underpinnings . 50 3.1.1 Introduction . 50 3.1.2 Pieces of Music . 52 3.1.3 Global Music Analytic Systems . 53 3.1.4 A Vector Space . 55 3.1.5 lp(F )............................ 57 3.1.6 lp-norms . 57 3.1.7 l2-norms . 58 3.2 Representations in l2(F )..................... 60 3.2.1 Norms . 65 3.2.2 Comparison of two functions . 66 CONTENTS vii 3.2.3 Implications of Concatenated Sequences . 68 <∞ 3.2.4 Comparing Subsets of N . 70 4 Local Music Analytic Systems 73 4.1 Comparing the Tonal Models of Lerdahl and Bharucha, et. al. 73 4.1.1 Rationale for Comparing L and BK . 76 4.2 Graphical Models and Associated Metric Models . 77 4.2.1 L and BK as Intra-Regional Metric Models . 80 4.3 A General Theory for Distance Measures Between Models . 86 4.3.1 Results of δ(L, BK) ................... 90 5 Normalization and Canonical Representation of Metric Models 92 5.1 Normalizing Metric Models . 93 5.2 Canonical Representatives . 96 5.3 Canonical Representatives for [L] and [BK] . 98 5.4 Considering L and BK as Global Systems . 100 5.4.1 Geometry of L and BK . 102 5.4.2 Commentary . 104 6 A Meta-Analytical Ramification of the General Theory of Comparative Music Analysis 105 CONTENTS viii 6.0.3 Reducing λR using a new model F . 109 6.0.4 Recursive Metric Models . 109 6.1 Multidimensional Global Geometry . 113 7 Conclusion 114 7.1 Summary . 114 7.2 Suggestions for Future Research . 118 List of Tables 2.1 Sample of Lines from Lomax (1968). 32 2.2 Line 1: The Vocal Group from Lomax (1968). 33 3.1 Analysis of N under j and k. .................. 62 3.2 Sums, differences, and inner product of j and k. 69 3.3 Analysis of N under j1 and j2................... 71 4.1 Theoretical Harmonic Relations from Lerdahl (2001). 84 4.2 Perceived Harmonic Relations from Bharucha-Krumhansl (1983). 84 4.3 Symmetrized Harmonic Relations derived from Bharucha-Krumhansl. 85 4.4 L (M1) and BK (M2)...................... 91 5.1 Canonical Representatives L and BK . 99 5.2 Analysis of N under L and BK . 103 6.1 Canonical Representatives L and BK . 107 ix LIST OF TABLES x 6.2 Functional Harmonic Distances. 110 6.3 Canonical Representatives F and BK . 111 6.4 The Metric Space ([MR], λR). 112 List of Figures 1.1 Pieces p and q as members of the set of all pieces P (E). 7 1.2 Type I comparative analysis. 7 1.3 Type II comparative analysis. 8 1.4 Type III comparative analysis. 8 2.1 Adler’s Historical Musicology from Mugglestone (1981). 16 2.2 Adler’s Systematic Musicology from Mugglestone (1981). 17 2.3 Meyer’s constraint hierarchy. 22 2.4 LaRue’s SHMeRG schema (2001). 26 2.5 Four reductive levels from Narmour (1999). 29 2.6 First-order transition probabilities from Youngblood (1958). 40 2.7 Snyder’s Table 3 comparing results under rpc/ksd and sd/st (1990). 47 3.1 The set of all music analytic functions F and its members . 60 xi LIST OF FIGURES xii 3.2 Segments Ij and Ik and angle θj,k . 66 3.3 Opinions on disjoint sets . 67 3.4 The arc of agreement for f and g . 68 3.5 A “single” music analytic system’s interaction with “two” pieces of music. 72 4.1 Lerdahl’s L model (2001). 82 5.1 Canonical representation as the unique function that maps min to 1 and max to n. .................... 97 6.1 W. A. Mozart’s Don Giovanni, “Madamina” Aria. Mm. 85-92 106 6.2 Madamina Progression Under L and BK. 108 6.3 Analysis of the Madamina Progression Under L, BK, and F . 110 6.4 Three-dimensional geometry of F , BK , and L . 113 Chapter 1 Preliminaries This dissertation presents a general theory of comparative music analysis. We analyze pieces of music, we compare pieces of music, and we often do this to group or distinguish them.1 There is no doubt that in some quarters of the community of musical scholars there is a high degree of intersubjective agreement about what it means to “compare music analytically.” In fact, the second chapter reviews numerous projects that appear to put forth a rigorous treatment of the subject. My goal is to establish a kind of comparative music analysis such that what we understand as musical features in a particular case are dependent on the music analytic systems that provide the framework by which we can contemplate (and communicate our thoughts about) music. We value how different analytic systems allow us to create different perceptions of a piece of music and argue that the identity of a musical work is in fact dependent 1There are a variety of reasons for such groupings including historical, stylistic, and ethnographical—to name a few. 1 CHAPTER 1. PRELIMINARIES 2 on the combination of a work and an analytic system. Comparing musical works, then, necessarily compares such combinations. I assert, therefore, that comparing pieces ought to be reframed as comparing the interpretation of pieces under specific analytic systems.