Table of Contents of my doctoral dissertation

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Enrique Moreno


Embedding Equal Pitch Spaces and The Question Of Expanded Chromas: An Experimental Approach

TABLE OF CONTENTS

Abstract iv
Dedication v

Foreword and Acknowledgements vi


Chapter One: Theoretical Introduction to Expanded Tunings
1.1 General Structure of Expanded Tunings 1

1.2 Definition of An Equal Tuning T 3
1.3 Primary Interval Lemma 4
1.4 Harmonic Location Theorem 5
1.5 Redundancy Theorem 8
1.6 Intervals of Equivalence 9
1.7 Equivalence Intervals and Their Terminology 12
1.8 Definition of Fully Expanded Tunings Te 16
1.9 Expanded Tunings Not as nth roots of k 17
1.10 Arbitrary Non-Octave Intervals of Equivalence 18
1.11 Cubic Equivalence Inferred from Square 19
1.12 Intervals of Equivalence Justified by Prescribed Overtone Structure 20

Chapter Two: Theoretical Introduction to Expanded Chromas
2.1 The Chroma Phenomenon 23
2.2 Theories of Octave Equivalence 26
2.3 Coincidence of Partials or Total Consonance 27
2.4 Observation to 2.3 27
2.5 Repetition Rate 29
2.6 Observation to 2.5 29
2.7 Activity of a Central Pitch Processor 30
2.8 Observation to 2.7 30
2.9 Uniqueness 30
2.10 Observation to 2.9 30
2.11 Initial Hypotheses 31

Chapter Three: Practical Introduction to Expanded Tunings
3.1 Implementation of Expanded Tunings 35
3.2 Selecting One Tuning among Many Other Possible Tunings 38
3.3 The Music Notation Problem 42
3.4 Expanded Notation 44
3.5 Rules for The Use of Color in Music Notation 47

Chapter Four: Practical Introduction to Expanded Chromas
4.1 Practical Meaning of Expanded Chroma 49
4.2 Musical Examples: Example One 53
4.3 Musical Examples: Examples Two, Three and Four 55
4.4 Finding Optimal Tunings for Expanded Chroma 58
4.5 Avoiding The Interference of Foreign Chromas 63
4.6 Discussion: Is Perception of Expanded Chroma Possible? 68

Chapter Five: Experimental Introduction to Expanded Chromas
5.1 Introduction 71
5.2 Pitch Dimensions 72
5.3 Definition of Morenoctaves 73
5.4 The Three Hypothesized Dimensions of Pitch on The Helix 74
5.5 Revised Model of The Fundamental Pitch space 82
5.6 Overtone Structure and Expanded Chroma 83
5.7 Some Definitions of Experimental Terms 86
5.8 Experimental Pre-Requisites for Subjects 87
5.9 General (Formal) Hypothesis 88
5.10 Design of The Main Experiment 89
5.11 Ancillary Experiments 90

Chapter Six: Experiment One
6.1 Design 93
6.2 Methodology 96
6.3 Description of The Computer Program Interface for Experiment One 98
6.4 Musical Fragments (Stimuli) 110
6.5 Subjects 112
6.6 Randomization in The Order of Presentation of Musical Stimuli 114
6.7 Data Collected 114
6.8 Expected Theoretical Distribution 115
6.9 Main Experimental Hypothesis 118
6.10 Results Per Subjects 118
6.11 Results Per Trial 126
6.12 Other Related Results 135
6.13 Discussion on Consonance/Dissonance and The Validity of Results 137
6.14 Experiment One: Summary and Conclusions 139

Chapter Seven: Experiments Two, Three, and General Conclusions
7.1 Design of Experiment Two 141
7.2 Results of Experiment Two 143
7.3 Design of Experiment Three 145
7.4 Results of Experiment Three 152
7.5 General Conclusions 157

References 161

Appendices 167 (Includes cassette tape with original examples)

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