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Chapter 1: Lerdahl's Model of Tonal Pitch Space LERDAHL’S SURFACE TENSION RULE: VALIDATION OR MODIFICATION DEBORAH LENORE HENRY A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN MUSIC YORK UNIVERSITY TORONTO, ONTARIO April 2017 © Deborah Lenore Henry, 2017 ii ABSTRACT Lerdahl’s Tonal Pitch Space (2001) combines music theory with current understanding of music perception and cognition creating of model of tonal pitch space. Lerdahl’s goals include quantification of areas of tension and relaxation perceived by listeners experienced in Western tonal music. Tension is associated with instability, distance, uncommon tones, and weak attractional force; relaxation with stability, proximity, common tones, and strong attractional force. Quantification requires creation of a time-span segmentation derived from the metrical grid and grouping analysis of the score. The time-span segmentation is necessary for creating the time-span reduction. The time-span reduction removes structurally less significant elements from the musical surface through a series of steps not unlike the layers of Schenkerian analysis. The ultimate goal is the prolongational reduction accompanied by prolongational tree. Global tension is quantified by summing values obtained when considering the region in which an event occurs, distance between successive chords revealed by their position on the chordal circle-of-fifths, number of distinct pitch classes between successive chords, tension inherited by subordinate chords from superordinate chords, melodic and harmonic attraction, and surface dissonance. Lerdahl’s Surface Tension Rule assigns tension added values due to chord Inversion, chord note in the top voice (Melody), and nonharmonic chord tones. This study tested the validity of assigned tension added values for Inversion and Melody asking 82 participants familiar with Western tonal music to rate perceived tension of Major and minor four-note chords heard devoid of tonal and musical contexts. Results showed Lerdahl’s tension added values required modification. Root position chords and chords with the root in the Melody require a tension added value greater than 0. Tension due to First Inversion is not the same as tension due to Second Inversion. Tension due to First and Second Inversion is greater than tension due to the third or fifth of a chord in Melody. Tension due to Second Inversion is not different from tension due to root in Melody. A new category, chord Quality, needed to be added. Expertise did not play a role. Lerdahl’s model and these results provide insight for performers, teachers, listeners, and composers. iii DEDICATION To my family, past and present. iv ACKNOWLEDGMENTS I did not write this dissertation alone. I had support and encouragement— from family, friends, and colleagues— whose interest in my work helped sustain me. While this group of well-wishers is too numerous to name individually, I would be remiss if I did not identify a few by name. I am thankful to have had Dr. Jay Rahn, Dr. Melody Wiseheart, and Dr. Dorothy de Val as my committee members. Their guidance offered me the opportunity to think deeply and creatively. Jay’s questions and comments lead me in new and exciting directions. Melody introduced me to wonderful new methods of data analysis, and the members of her lab responded willingly and patiently to all my questions about statistics. I am grateful to Dorothy for her attention to detail and her thoughtful, thought-provoking observations. v TABLE OF CONTENTS ABSTRACT ii DEDICATION iii ACKNOWLEDGMENTS iv TABLE OF CONTENTS v LIST OF TABLES x LIST OF FIGURES xv INTRODUCTION 1 CHAPTER 1: LERDAHL’S MODEL OF TONAL PITCH SPACE AND 5 ITS BASIS IN MUSIC THEORY Introduction 5 Music Theorists 5 Tonal Pitch Space – the book 9 The Four Hierarchical Structures 11 Tonal Pitch Space – the model 14 The Pitch Class Space 16 The Chordal Space 19 The Regional Space 23 Tonal Tension – Sequential and Hierarchical 28 Sequential and Hierarchical Tension 29 Melodic Tension 33 The Melodic Attraction Rule 34 The Harmonic Attraction Rule 36 Finding the Tonic 38 Conclusions 38 CHAPTER 2: LERDAHL’S MODEL OF TONAL PITCH SPACE AND 41 EMPIRICAL EVIDENCE Introduction 41 Spatial Distance and Cognitive Distance 41 Stratified Hierarchy 41 Single Model 42 Empirical Findings 42 Tonal Pitch Space – the model 42 Pitch Class Space 42 Chordal Space 46 Regional Space 49 Lerdahl’s Four Goals Revisited 53 Pitch Class Space 53 vi Chordal Space 54 Regional Space 54 Second Goal 54 Third Goal 55 Conclusions 57 CHAPTER 3: AN EXPERIMENT TESTING ELEMENTS OF 60 LERDAHL’S SURFACE TENSION RULE Introduction 60 Evidence from Music Theory and Music Perception 60 Scale Degree 61 Inversion 63 Nonharmonic Tones 64 Devising an Experiment 66 Scale Degree 66 Inversion 67 Nonharmonic Tones 67 The Experiment 68 Method 69 Participants 69 Materials 70 Procedure 71 Results 71 Discussion 75 Inversion 76 Melody 77 Chord Quality 77 Expertise 78 Two-way Interactions 80 Melody and Expertise 80 Inversion and chord Quality 81 Inversion and Melody 83 Three-way Interactions 85 Inversion, Melody, and chord Quality 85 Inversion, chord Quality, and Expertise 88 Conclusions 88 CHAPTER 4: AN ALTERNATIVE ANALYSIS USING CONFIDENCE INTERVALS 91 Introduction 91 Results and Discussion 92 vii Expertise 92 Inversion 94 Melody 95 Chord Quality 96 Interaction of Inversion and Chord Quality 97 Major Quality 97 minor Quality 98 Interaction of Melody and chord Quality 99 Major Quality 99 Minor Quality 100 Interaction of Melody and Expertise 101 Interaction of Inversion and Melody 102 Conclusions 103 CHAPTER 5: CONTRIBUTIONS OF MELODIC AND HARMONIC ATTRACTION, 104 SPECTRAL ANALYSIS, ROUGHNESS, AND INSTABILITY Introduction 104 Disruptor Sequences 105 Melodic Attraction 107 Harmonic Attraction 109 Psychoacoustics 111 Auditory Roughness 112 Vassilakis and Fitz’s Spectral and Roughness Analysis (SRA) 112 SRA and chord Quality 114 SRA and Inversion 115 SRA and Melody 116 SRA Conclusions 116 Cook Seeing Harmony (SH) 117 SH and chord Quality 119 SH and Inversion 121 SH and Melody 123 SH and 47 Chord Formations 124 SH Conclusions 127 Lahdelma and Eerola 127 Conclusions 134 CHAPTER 6: MODIFICATIONS TO LERDAHL’S SURFACE TENSION RULE 137 Introduction 137 Implication of Results 138 Methods of Modifying Tension Added Values 140 viii Ranking 140 Ratio of Means of Medians 141 Bivariate Linear Regression 142 Portion of Total Perceived Tension 146 Seven Aspects of Lerdahl’s Tension Added Values Requiring Modification 146 Surface Tension, Local Tension, and Global Tension 148 Surface Tension 149 Local Tension 151 Global Tension 152 Influence of Surface Dissonance on Global Tension 152 Expertise 156 Conclusions 156 BIBLIOGRAPHY 159 APPENDIX A: MUSICAL EXAMPLES OF LERDAHL’S FOUR HIERARCHIES 165 APPENDIX B: CHORD FORMATIONS AND ASSOCIATED DISRUPTOR SEQUENCES, 169 PLUS CHORD FORMATIONS IN ALL MAJOR AND MINOR KEYS APPENDIX C: EXPLANATION OF RANDOMISATION PROCESS 176 APPENDIX D: EXPLANATION OF STATISTICAL TERMS AND PROCEDURES 181 Factors and Levels 175 Statistical Tests 175 Descriptive Statistics 175 Three Graphical Depictions of Data Distribution 176 Two Statistical Procedures 179 Two Calculations of Effect Size 180 Mauchly’s Test of Sphericity 180 APPENDIX E: OUTLIERS, SKEWNESS, AND THE VIOLATION OF ASSUMPTION 188 OF NORMAL DISTRIBUTION APPENDIX F: RESULTS OF ANOVA PERFORMED ON MEANS OF MEDIANS 192 OF DATA WITHOUT OUTLIERS APPENDIX G: TABLES OF DATA FOR FIGURES IN CHAPTER 4 198 APPENDIX H: EXPLANATION AND EXAMPLE OF MEAN ROUGHNESS FOR 202 F MINOR CHORD FORMATIONS USING VASSILAKIS AND FITZ’S SRA (SPECTRAL AND ROUGHNESS ANALYSIS) APPENDIX I: COOK’S SEEING HARMONY (SH) PROGRAMME FOR 205 MEASURING TENSION, DISSONANCE, AND INSTABILITY IN FOUR-NOTE CHORDS APPENDIX J: TABLES FOR FIGURES 6.2 TO 6.5 AND RATIO TO MINOR 209 APPENDIX K: DETERMINING MUSICAL EXPERTISE 218 ix APPENDIX L: THE ATTRACTIONAL CONTEXT AND EVENT GOVERNANCE RULES 223 APPENDIX M: FINDING THE TONIC 225 APPENDIX N: FUNCTION AS PROLONGATIONAL POSITION 229 x List of Tables INTRODUCTION: 1 Table Intro.1 Entries in databases by category 3 Table Intro.2 Itemisation of some of the Music and Psychology categories of citations in 3 Music Index, Humanities and Social Science Index Retrospective and Humanities International Index, and Psychinfo CHAPTER 1: LERDAHL’S MODEL OF TONAL PITCH SPACE AND 5 ITS BASIS IN MUSIC THEORY no Tables CHAPTER 2: LERDAHL’S MODEL OF TONAL PITCH SPACE AND 41 EMPIRICAL EVIDENCE Table 2.1 Comparison of chord hierarchy between Lerdahl's calculations and 49 Krumhansl (1990), and Bharucha and Krumhansl (1983) Table 2.2 Comparison between Krumhansl's correlations of regions to C major and 51 Lerdahl's calculation of distance between C major and its closely related regions Table 2.3 Comparison between Krumhansl's correlations of regions to c minor and 51 Lerdahl's calculation of distance between c minor and its closely related regions Table 2.4 Comparison between Krumhansl's correlations of regions to C major and 52 to c minor with Lerdahl's calculation of distance between C major and its closely related regions, and between c minor and its closely related regions CHAPTER 3: AN EXPERIMENT TESTING ELEMENTS OF 60 LERDAHL’S SURFACE TENSION RULE Table 3.1 Means of medians, Standard Deviation, and Margin of Error for all chord formations 72 Table 3.2 Analysis of Variance for Perceived Tension in Chord Formations 74 Table 3.3 Paired sample t-tests (two-tailed) for differences between 75 levels of Between subjects
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