Dissertation Index

Author: Moreno, Enrique I.

Title: Embedding Equal Pitch Spaces and The Question Of Expanded

Institution: Stanford University

Begun: September 1994

Completed: December 1995


Traditional tunings are based on the assumption that tones separated by one or more octaves belong to the same pitch class. Here, the concept of equal temperaments is generalized to intervals given by k^(1/m), where k = 3, 5 ,6, 7... N, and n, and m, are natural numbers not equal to x, y (for x, y = 1,2,3 ... N). The resulting tunings, called "expanded tunings" are organized according to the periodicity of k.

However, the perceptual organization of such tunings may seem unclear in the absence of powers of two (octaves). k-type intervals (termed "morenoctaves" in the context of an expanded tuning) seem theoretically similar to octaves. Furthermore, a review of the literature reveals that psychoacoustically, there is no reason to reject the similarity of morenoctaves _ a phenomenon termed "expanded chroma" _ provided that certain conditions hold _ for example, that a tuning based on the similarity of morenoctaves does not contain close approximation to octaves.

Solutions to practical compositional problems are proposed, and problems concerning notation of expanded tunings are examined. The use of color is proposed as a solution to the dissociation between pitch class and pitch name in tunings with more or less than twelve classes. Several originally composed musical examples are examined in relation to these problems. The dimensionality of pitch is discussed in the light of current models, and a new geometrical, basic three-dimensional model of pitch _ instead of the traditional two-dimensional one _ is proposed as a better means to represent expanded chromas. A paradigm of empirical research, based on this model is explained in relation to the problem of obtaining experimental evidence for the cognitive reality of expanded chroma.

The results of three experiments show that, among trained musicians, some subjects (at least 24%) have a strong spontaneous relative sense of expanded chroma because they are capable of recognizing the similarity of harmonic functions of a fragment of music in expanded tunings under the operation of rigid transposition, and _ with much more difficulty _ under the operation of rigid chord inversion. This means that perception of the dimension of chroma in music _ hitherto unexplored _ is a capability of the human mind, and that expanded tunings can be coherent.

Keywords: Theoretical Intonation, Tuning Theory, Music Cognition, Music Perception


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)


Enrique I Moreno
724 Arastradero Apt. 206
Palo Alto, CA 94306
Tel. (415) 813 9750
Fax (415) 723-8468
e-mail: eig@ccrma.stanford.edu

     Return to dissertations