HOW DISSONANT IS THE AUGMENTED TRIAD ? By JOSHUA CLEMENT BROYLES B.A. Music, California State University, Hayward, 1992 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (School of Music, Music Theory) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1999 © Joshua C. Broyles, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) 11 ABSTRACT: HOW DISSONANT IS THE AUGMENTED TRIAD ? Throughout the centuries, music theorists have consistently designated the augmented triad as dissonant, but not for entirely consistent reasons. In one interpretation of this "dissonant nature," an interpretation with which this thesis is concerned, the augmented triad is less harmonically "stable" than the major and minor triads in root position or in first inversion and, at most, only as stable as the second inversions of the major and minor triads. The various arguments against the stability of the augmented triad have largely been of the three basic types: acoustic/numerological, psychoacoustic/perceptual, and cognitive/tonal-syntactic. A small number of theorists, from very early on, have not been entirely committed to the intrinsic instability of the augmented triad as compared to major and minor triads. In recent decades research in music perception has drawn into question the absolute validity of this designation, but has stopped short of demonstrating specific conditions under which an augmented triad would actually be likely to sound more harmonically stable than a major or minor triad. This thesis documents a perceptual experiment and its results which statistically support the claim that conditions exist under which listeners may perceive an augmented triad as more harmonically stable than a major triad. These conditions are specific but they are not abnormal in twentieth-century music, and they are not totally absent in earlier Western music. iii TABLE OF CONTENTS ABSTRACT , ii LIST OF ILLUSTRATIONS iv ACKNOWLEDGEMENTS v Chapter 1. INTRODUCTION 1 2. SOME RATIONAL PERSPECTIVES OF MUSIC THEORISTS 8 3. SOME EMPIRICAL PERSPECTIVES '. 27 4. THE SPECIAL PERSPECTIVE OF HELMHOLZ 46 5. SUMMARY OF EXPERIMENT 51 6. THE STIMULI 56 7. THE DATA 65 8. CONCLUDING REMARKS 74 Bibliography 76 Appendix 1. EXAMPLE RESPONSE FORM 80 2: TEST ENGINE 82 3: DATA TABLES 83 4: THE PARTICIPANTS 86 iv ILLUSTRATIONS Figure 1. The Experimental Response Form, Side 1 80 2. The Experimental Response Form, Side 2 81 3. The Experimental Test/Treatment Engine 82 Table 1. Simplified Pre-test and Post-test Table for Groups 1 and 2 Combined 66 2. Data Table for Treatment Group 1 84 3. Data Table for Treatment Group 2 and Table for Incomplete Responses 85 Example 1. Notation of Audio Example 1 53 2. Pitch Content of Pretest/Post-test Examples 61 3. Pitch Content of Treatment Examples 64 ACKNOWLEDGEMENTS I wish to thank Dr. John Roeder, Dr. William Benjamin and Dr. Richard Kurth for the excellence they have shown me as teachers, and for the special efforts each of them has made which have enabled me to get to this point in my education. I wish to thank Dr. Eugene Narmour, with whom I have neither met nor spoken, but whose book, The Analysis and Cognition of Basic Melodic Structures, showed me that music theory need not continue to be so much like doing crossword puzzles, or worse, the reading of tea leaves. It was this new view of music theory which prevented me from entirely abandoning music theory upon receipt of my B.A. in music composition. I wish to thank my previous employers, Christina and James Bennett, for giving me the job flexibility I needed in order to pursue graduate school, and for encouraging me to move forward even when it meant certain difficulties for them. I wish to thank my grandparents, Robert and Arlene Dart, for their continued financial assistance, for their apparently blind faith in me as a scholar of music, and for their amazing patience. Most of all, I wish to thank my partner Heather O'Connor-Shull for encouraging me to return to higher education, for every possible type of support imaginable, which she has given me most unselfishly throughout my years of graduate school, and most of all for believing in my scholastic ability even in the face of sometimes overwhelming counterevidence. 1 Chapter 1: Introduction In this thesis, the word "normal" will be used primarily in the statistical sense. That is, something like the opposite or complement of "abnormal" (such as in the expression "psychologically normal")!; plausibly representing a much larger population, the study of which should be of general rather than of special interest. In this case the population intended to be represented includes types of humans. Although certain sample biases must be acknowledged, neither was the study deliberately confined to participants who might be considered poorly representative (such as criminals, or the mentally ill), nor were any participants intentionally excluded in order to maintain an artificial standard of normality. It is not my intention, through the frequent use of the word "normal," to glorify normality. However, I will assume that normal listeners are the people whose perceptions are of greatest interest to music theory for three reasons: 1) because music theorists I have met and those whose works I have read so frequently invoke the authority of normal listeners (usually without identifying them, or verifying their normality), 2) because the perception of normal listeners has formed the core point of interest of so much scientific research already, and 3) because, while abnormal listening does interest me personally, I have my doubts about the immediate or foreseeable usefulness of abnormal listening behaviours to analysis or to composition. For the purposes of this thesis, the terms "dissonance" and "harmonic instability" will be used interchangeably, not because I believe they are exactly the same thing, but because this interchangeability is a widespread convention in the literature to which I will refer, and among the numerous theory teachers who have made a point over the years of defining for me the augmented triad as having such qualities. I suspect that if one considers these Music theorists will not be considered to be normal listeners. 2 two terms not to be interchangeable, one will also see in my conclusion a partial clarification of one way in which the definitions of these terms should differ. Indeed, the lack of clarity between these definitions is perhaps part and parcel of the main problem I intend to address in the pages to follow. That is, while "roughness" and "lack of resolution" may correlate musically, this does not necessarily mean that they are one thing, or even that the correlation is completely dependable. For the purposes of this thesis, the following terms will also be used somewhat interchangeably; augmented triad (or {C,E,G#}, etc.) and 3-12 (or {0,4,8}, etc.). Other pitch- class structures will receive similar treatment. It is not my intention to use this interchangeability to execute semantic sleights of hand, but only to emphasize, in each case, the immediate context in which the object is to be best understood at that point of the discussion. The reader may assume that any of these terms may apply to any spelling of the object to which it refers, and to any inversion or system of intonation unless otherwise specified. I understand that standard classification system for pc sets assumes equal temperament, but both this system and more traditional tonality-oriented vocabularies have been widely used both in the analysis of equal-tempered music, and in the analysis of music not necessarily adhering to equal temperament, such as string music and vocal music. Therefore I ask the reader to accept that both the terms "augmented triad" and "3- 12" currently imply equal temperament, but that the older one simply implies this less strongly. Before the advent of equal temperament, verticalities that could be octave-transposed to two "stacked" major thirds were regarded as awkward-sounding at least for the reason that they contained an augmented fifth or diminished fourth, neither interval of which were expressions of the simple mathematical ratios defined as consonant by divisions of a string on the monochord, or arising from the parrials of overblown wind instruments. ^ 2 Theorists referring to arithmetic ratios as representative of musical intervals, even merely in concept, will be referred to hereinafter as "monophonists", alluding to the 3 Later claims by tonal theorists to the effect that the harshness of the augmented triad may be softened by using it in first inversion seem to deduce the treatment of the augmented triad from treatment of the diminished triad, but may also derive in part from (or, at least converge with) the grammars of much earlier music (not using equal temperament) in which such verticalities did not occur, such as free organum in three parts. That is, to the monophonist, the pitch collections {B,D,F} and {C,F,G} are similar in that they sound most consonant when the lowest voice is consonant with both upper voices, though these upper voices are dissonant with each other (the perfect fourth was considered a consonance until the Renaissance) .