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University Mierxjftlms International 300 N INFORMATION TO USERS This was produced from a copy of a document sent to us for microfilming. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help you understand markings or notations which may appear on this reproduction. 1. The sign or “target” for pages apparently lacking from the document photographed is “Missing Page(s)”. If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure you of complete continuity. 2. When an image on the Him is obliterated with a round black mark it is an indication that the film inspector noticed either blurred copy because of movement during exposure, or duplicate copy. Unless we meant to delete copyrighted materials that should not have been filmed, you will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., is part of the material being photo­ graphed the photographer has followed a definite method in “sectioning” the material. It is customary to begin filming at the upper left hand comer of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again—beginning below the first row and continuing on until complete. 4. For any illustrations that cannot be reproduced satisfactorily by xerography, photographic prints can be purchased at additional cost and tipped into your xerographic copy. Requests can be made to our Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases we have Aimed the best available copy. University Mierxjftlms International 300 N. ZEEB ROAD, ANN ARBOR, Ml 48106 18 BEDFORD ROW, LONDON WC1R 4EJ, ENGLAND 8107334 H a h n , Ju n e Ir e n e A MODEL OF OCTAVE STRETCH WITH IMPLICATIONS FOR THE SUBJECTIVE REPRESENTATION OF PITCH The Ohio Stale University PH.D. 1980 University Microfilms International300 N. Zeeb Road, Ann Arbor, MI 48106 A MODEL OF OCTAVE STRETCH WITH IMPLICATIONS FOR THE SUBJECTIVE REPRESENTATION OF PITCH DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By June Irene Hahn, B.S., M.A. The Ohio State University 1980 Reading Committee: Approved By Mari R. Jones Paul D. Isaac Thomas E. Nygren James R. Leitzel Department of Psychology To my mother and the memory of my father ii ACKNOWLEDGMENTS The author is indebted to Dr. Mari R. Jones for the guidance and encouragement she provided both during the writing of this dissertation and during the years of graduate study that preceded it. As an adviser and teacher her knowledge, enthusiasm, and insight were always freely given and greatly appreciated. The author also wishes to thank the members of her reading committee, Drs. Paul Isaac, Thomas Nygren, and James Leitzel for their time and their helpful comments and suggestions. Thanks go to Robin Wetzel for her help in scheduling musicians for the two experiments, and to the musicians themselves for their participation. Finally, the author wishes to thank her friends and fellow graduate students, June Baird and Lorraine Normore who understood better than anyone else what the writing of this dissertation entailed. They provided support, meals, and entertainment at all the right times. iii VITA July 7, 1951 Born - St. Louis, Missouri 1973 B.S., Applied Mathematics, University of Missouri, Rolla, Missouri 1975-1978 Teaching Associate, Department of Psychology, The Ohio State University, Columbus, Ohio. 1977 M.A., Psychology, The Ohio State University, Columbus, Ohio. 1978-1980 Statistical Consultant, Department of Psychology, The Ohio State University, Columbus, Ohio. PRESENTATIONS "Robustness of INDSCAL and ALSCAL with Respect to Violations of Metric Assumptions". Presented at Psychometric Society Meeting, August 27, 1978, McMaster University. PUBLICATIONS Jones, M.R., Kidd, G., and Hahn, J. "Space-Time Expectancies in Auditory Pattern Memory". OSURF Technical Report No. 2, Columbus, Ohio: Ohio State University, 1978. FIELDS OF STUDY Major Field: Quantitative Psychology Studies in Mathematical Models and Perception. Professor Mari R. Jones Studies in Statistics and Methodology. Professors Paul Isaac, Thomas Mygren, and Robert MacCallum. iv TABLE OF CONTENTS Page DEDICATION ii ACKNOWLEDGMENTS iii VITA iv LIST OF TABLES vii LIST OF FIGURES viii CHAPTER I INTRODUCTION 1 II LITERATURE REVIEW 7 Early Representations of Pitch 7 Experimental Studies of Pitch 11 Scaling Studies of Pitch 20 Mathematical Representations 27 Summary 34 III A MODEL OF OCTAVE STRETCH 35 Introduction 35 Octave Stretch 36 A Threshold Model of Octave Stretch 47 A Subjective Representation Based upon the Logarithmic Spiral: A Psychophysical Model with Implications for Multidimensional Scaling 52 Summary 55 v Page IV EXPERIMENT ONE 57 Method 5 7 Results 59 Discussion 65 V EXPERIMENT TWO 70 Method 70 Results 72 Discussion 80 VI GENERAL DISCUSSION 84 Introduction 84 Experiment One 86 Experiment Two 92 Implications for Further Research 96 Summary and Conclusions 97 APPENDIXES A Observed FrequencyAdjustments and Values Predicted by Threshold Model 99 B Tone Pairs Used in Scaling Study 108 C Instructions for Scaling Study 113 REFERENCE NOTES 116 REFERENCES 117 vi LIST OF TABLES Page Table 1. Harmonic Ratios Generated by a Spiral Analysis 32 2 . Ward's 1954 Data on Octave Stretch 37 3. Number of Times that the Variable Tone Was Set Higher or Lower Than an Octave 60 4. Means and Standard Errors for Octave Adjustment Task 61 5. Root Mean Square Deviations and Average Absolute Deviations of Observed Settings from Values Predicted by Galilean Model 63 6 . Predictions and Fit of Terhardt Model 64 7. Root Mean Square Deviations, Average Absolute Deviations, and Estimated k Values for Threshold Model 66 8. Cumulative Proportions of Ratings Differing by 0, 1, or 2 Points Across the Two Sessions 74 9. Mean Ratings for Musical Intervals 75 10. Subject Weights for the 2 and 3-dimensional Scaling Solutions 81 11. Rank Ordering of Dissimilarity Ratings by Octave 94 vii LIST OF FIGURES Figure Page 1. Two Pictorial Representations of Pitch 10 2. Two Psychophysical Scales 13 3. Scaling Solution of Levelt, Van de Geer, and Plomp 23 4. Krumhansl's Chroma Cone Configuration 26 5. Pitch Spiral 30 6. Two-dimensional Scaling Solution 77 7. Three-dimensional Scaling Solution Dimension 1 (horizontal) vs Dimension 2 (vertical) 78 8. Three-dimensional Scaling Solution Dimension 1 (horizontal) vs Dimension 3 (vertical) 79 viii CHAPTER ONE INTRODUCTION A question that has recurrently fascinated psychologists for many years concerns the manner in which individuals perceive events or sequences of events. At the heart of this question is the fact that an individual's perception of an event, while depending on aspects of the event that can be physically defined, does not necessarily simply mirror these physical aspects. Subjectively, the event may be somewhat distorted. What psychologists have sought is a way to quantify the relationship between these physically defined events and their subjec­ tive representation. One area in which there have been attempts to develop such quantitative relationships is auditory perception, in particular with musical stimuli comprising the events. Music is generally composed of long sequences of complex sounds. An ultimate goal of any psychological study of musical events is to understand more about how people interact with or learn these complex sequences. But to arrive at this goal it is necessary to begin very simply. First the physical stimuli must be adequately described. Traditionally the description of musical stimuli has relied upon a one-dimensional frequency continuum. However, this linear representa­ tion, though reflecting tone height, may fail to reflect other important 1 2 relations which are also inherent in the tones themselves. In light of this, alternative representations may have some appeal. One that is especially useful has been proposed by Hahn and Jones (Note 1). This representation is a bi-dimensional description of frequency in which both pitch height and tone chroma are represented. Given that the stimuli can be objectively described, a related question concerns how these physical sitmuli are subjectively represent­ ed. Many attempts at a pictorial representation have been made, from Drobisch's spiral shaped configuration in 1946 to Shepard's five dimensional double helix in 1979 (cf. Krumhansl, 1979). A necessary next step is to relate these two representations — the physical and the subjective. In this area many questions remain unanswered. One of these questions concerns the relationship between frequency and pitch. Musical tones can be specified as definite physical quantities and measured using physical instruments. For example, middle C on a piano can be identified as that sound having a fundamental frequency of 261.63 Hz. Pitch, on the other hand, is the subjective evaluation of the frequency of a sound (Backus, 1969). This subjective property is what allows a tone to be compared to other tones, e.g., is it higher or lower than another tone. Furthermore, frequency does not appear to be the sole determinant of pitch, as the pitch of a pure tone, or tone with no harmonics, is a function not only of the frequency of the tone but also of its intensity (Stevens, 1935). Traditionally psychologists have been most concerned with describ­ ing the relationship between frequency and subjective pitch of a single tone. This tradition is best represented by classical psychophysics. 3 However, this approach often ignores the fact that musical relations are defined by frequencies between two tones. That is, an interval refers not to the difference in frequency between two notes, but rather to the ratio of the frequencies of the two notes involved.
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