jitsi.pebibits.com/9908-cell-number.php Secondly, there is an effect of online learning of this motivic sequence during the course of the first eight measures. While this effect can be simply accounted for in terms of online learning and a short-term model of pitch-class distribution or melodic-harmonic bigram structure, this and the previous example illustrate the strong contribution of online-learning to listening and the interaction of long-term and short-term knowledge to musical experience. Effects of expectancy, expectancy violation, ambiguity, and revision continue to have an albeit weaker effect even over the course of multiple listening.
Such effects and their emotional correlates that remain after multiple listening would be difficult to account for considering the ongoing implicit learning and, particularly, the learning of the veridical structure of a piece. Altogether, the examples above illustrate how closely musical expectancy is linked to implicit learning and implicit knowledge both in long-term enculturation and short-term musical listening. Automatic expectancy formation, effects of retardation, anticipation, expectancy violations, deceptive structures, ambiguity, musical garden-path phenomena and revision: Such effects in musical listening and the emotional experience  result from the operation of an ongoing parsing mechanism that processes the musical stream, generates likely parses and continuations, and matches them with the ongoing stream of musical events.
Generally, perspectives of cognition and modeling may provide a number of contributions to the field of music theory: Apart from demonstrating the necessity of precise specification of covert assumptions and characterizing constraints of theoretical description that arise from problems such as sparsity or overfitting, they illustrate the insight that theoretical models of music and expectancy are intrinsically linked to implicit or explicit underlying formal, computational assumptions. After all, musical expectancy is intrinsically linked to cognitive accounts of predictive processing.
It provides a constructive case for the mutual interaction of music theory and music cognition  and illuminates ways in which concepts from music cognition, computational modeling, and neuro psychology may help to address music-theoretical issues from a different perspective. They provide ways to support, adapt, and revise music-theoretical concepts, to clarify theory formation in music analysis and to take into account music-theoretical insights in the formation of cognitive theory.
I owe special thanks to Markus Neuwirth for many inspiring discussions about this text and the ongoing exchange about ways of bridging the gap between music theory and music cognition. I am also very grateful to Taiga Abe, Christian Utz, and Jan Philipp Sprick for their numerous suggestions that improved the article to a great deal.
It is important to note that a well-defined, formal computational model is by no means equivalent with a statistical corpus analysis in general. Both requirements are not trivial at all. Schenker , and also the remarkable level of complexity encountered by Masatoshi Hamanaka et al. Communication arises through emergence, autopoietic stabilization and reproduction.
Individual competence is a product of interactive social and cognitive adaptation processes. See also related arguments by Luhmann , , or Polth Moreover, computational models that infer their parameters from exposure are capable of expressing individual variation. Even though the method of deriving at such estimates may be debatable as are decisions of human analysts and may be revised by improved methods, the computed numbers are internally consistent by being computed using the same algorithm whereas human analyses of such a large corpus may be prone to inconsistencies across different pieces.
Already low-level sensory neural processing is capable of dealing with such regularities without requiring higher-order knowledge-driven processes, see Koelsch Although veridical evaluation is avoided in modern computational modeling because of the problems discussed in the following paragraphs, this case was included to exemplify the effect sizes of overfitting. See Temperley for a contrasting viewpoint regarding the relevance of nonlocal models. Note that already the first F-minor chord is sufficient for almost unambiguously establishing the key of F minor.
An n-gram model with a padding symbol marking initial silence or a Bayesian model would support this result in straightforward ways based on the distribution of piece beginnings in a corpus. The latter could be understood as a special case of nonlocal dependencies with no intervening material; hence a process that is able to capture nonlocal dependencies will naturally also capture local dependencies. The occurrence of such crossed patterns of dependency would provide evidence for the necessity of an even more complex model of dependency structure and associated forms of expectancy.
If the probabilities for the different options of continuation were not skewed but rather similar, we would hear them as two or more equally possible or plausible continuations. Agawu, Kofi. Edited by Anthony Pople. Cambridge: Cambridge Universtiy Press: 86— Bharucha, Jamshed J.
Borges, Jorge L. Caplin, William E. Oxford: Oxford University Press. Chomsky, Noam. Conklin, Darrell. Cross, Ian. Cuthbert, Michael S. Edited by J. Stephen Downie and Remco C. Davidson, Donald. Edited by Michael Krausz. Notre Dame, Ind. Subjectivity, Intersubjectivity, Objectivity. DeBellis, Mark.
Dennett, Daniel C. Kinds of Minds: Toward an Understanding of Consciousness. Eerola, Tuomas. Farbood, Morwaread M. Edited by Reiss, Joshua D. Reiss, and Geraint Wiggins. London: University of London: — Hansen, Niels C. Heidelberg: Springer: — The Effect of Tessitura on Melodic Structure. Huron, David. Jackendoff, Ray. Juslin, Patrik N. New York: Oxford University Press.
Discussion: — Keiler, Allan. Koelsch, Stefan. Krumhansl, Carol L. Cognitive Foundations of Musical Pitch. Leman, Marc. Heidelberg: Springer. A Generative Theory of Tonal Music. Luhmann, Niklas. Art as a Social System. Stanford: Stanford University Press. Mandelbrot, Benoit. Manning, Christopher D. Foundations of Statistical Natural Language Processing. Margulis, Elizabeth H. Marsden, Alan.
Meyer, Leonard B. Emotion and Meaning in Music. Chicago: University of Chicago Press. Music, The Arts, and Ideas. Narmour, Eugene. Edited by Diana Deutsch. Second edition. San Diego: Academic Press: — Neuwirth, Markus. PhD Diss. University of Leuven. Parncutt, Richard. Harmony: A Psychoacoustical Approach. Pearce, Marcus T. Department of Computing, City University, London. Polth, Michael. Quinn, Ian. Rabiner, Lawrence R.. Published by the IEEE doi: Rameau, Jean-Philippe.
Language and Music as Cognitive Systems. Rohrmeier, Martin. Towards Modelling Movement in Music.
Athens: National and Kapodistrian University of Athens: 82— Athens: National and Kapodistrian University of Athens: 97— Sapporo, Japan: Hokkaido University: — In press. Schellenberg, E. Schenker, Heinrich. Vienna: Universal Edition. Schmalfeldt, Janet. Schmuckler, Mark. Smoliar, Stephen W. Steedman, Mark J. Edited by Alan Garnham and Jane Oakhill. NJ: Psychology Press: — Oxford: Oxford University Press: 14— Swain, Joseph P. Temperley, David. Tillmann, Barbara. Tymoczko, Dmitri. Voss, Richard F. Wiggins, Geraint. Oxford: Oxford University Press: — Bologna: University of Bologna: — Zipf, George Kingsley.
The Psycho-Biology of Language. Boston: Houghton Mifflin. Human Behaviour and the Principle of Least Effort. Cambridge, MA: Addison-Wesley. This is an open access article licensed under a Creative Commons Attribution 4. Rohrmeier, Martin : Musical Expectancy. Introduction A mind is fundamentally an anticipator, an expectation-generator. Relating Theoretical, Psychological and Computational Perspectives Fundamentally, expectancy is a core cognitive process i. Example of veridical expectancy The Problems of Overfitting and Sparsity It is a commonplace in computational modeling that descriptions and models do not necessarily get better by adding more information.
Local and Hierarchical Structure The discussion above has largely focused on only one specific type of expectancy models, namely local models such as Markov or n-gram models, and to some extent, regular grammars which share the common assumption that event prediction is only characterized by the local context, consisting of the immediately preceding events. Hierarchical Structure and Expectancy One linguistic example will first illustrate the issues involved in hierarchical processing before turning back to music.
Figure 5. Nested syntactic patterns of expectancy This case bears a musical analogue. Crossing patterns of expectancy that are predicted to be impossible in musical contexts Once a hierarchical model of music is involved, the notion of expectancy becomes less straightforward as outlined in the previous example.
Without cookies your experience may not be seamless. Holt, L. Bollis, P. Lerdahl, F. Entropy as a measure of style: The influence of sample length. Intervals, scales, and tuning. Her book, Cognitive Foundations of Musical Pitch Krumhansl, has been reviewed by David Huron and is a standard resource for teachers and students of music psychology and one of the discipline's most cited sources.
A very simple example of this is provided by the following harmonic progression: 5 I ii V [ vi ii V ] I This sequence sets up a V—I implication which is interrupted by a deceptive progression V—vi which in turn initiates two further predictive dependencies to reestablish and potentially strengthen the predictive effect of the initial dominant context as indicated by the brackets.
Conclusion Generally, perspectives of cognition and modeling may provide a number of contributions to the field of music theory: Apart from demonstrating the necessity of precise specification of covert assumptions and characterizing constraints of theoretical description that arise from problems such as sparsity or overfitting, they illustrate the insight that theoretical models of music and expectancy are intrinsically linked to implicit or explicit underlying formal, computational assumptions.
Notes 1 I owe special thanks to Markus Neuwirth for many inspiring discussions about this text and the ongoing exchange about ways of bridging the gap between music theory and music cognition. Huron Huron ; Koelsch See, for instance, Piantodosi in press for a recent discussion. References Agawu, Kofi. Brain and Music. Hoboken, NJ: Wiley-Blackwell.
Piston, Walter. New York: Norton. See Cross and also the discussions in Wiggins a und b. See also similar distinctions by Huron ; Margulis For this distinction, see Bharucha ; Eerola ; Huron Zipf , Frequency is approximately proportional to the inverse rank, i. The neuropsychology of anxiety: an enquiry into the functions of the septo-hippocampal system Jeffrey A. Gray 2. Elements of episodic memory Endel Tulving 3.
Conditioning and associative learning N. Mackintosh 4. Visual masking: an integrative approach Bruno G. Breitmeyer 5. The musical mind: the cognitive psychology of music John Sloboda 6. Elements of psychophysical theory C. Falmagne 7. Animal intelligence Edited by L. Weiskrantz 8. Response times: their role in inferring elementary mental organization R. Duncan Luce 9. Mental representations: a dual coding approach Allan Paivio Memory, imprinting, and the brain Gabriel Horn Working memory Alan Baddeley Blindsight: a case study and implications L.
Weiskrantz Profile analysis: auditory intensity discrimination David M. Green Spatial vision R. DeValois and K. DeValois The neural and behavioral organizations of goal-directed movements Marc Jeannerod Visual pattern analyzers Norma Graham Cognitive foundations of musical pitch Carol L. First published in by Oxford University Press, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press.
Includes index. ISBN X; pbk. Music—Psychological aspects. Musical pitch. Cognitive psychology. K76 '. Ward This page intentionally left blank Preface Cognitive Foundations of Musical Pitch considers the problem of how listeners encode, organize, and remember pitch patterns in music. The work seeks to explicate the nature of listeners' knowledge of how pitch structures are formed, identify musical properties that shape this knowledge, and characterize the process through which sequences of sounds become coherent, memorable, and meaningful.
The approach taken is that of cognitive psychology, in which laboratory methods examine the nature of mental representations and processes. Previous publications have described a number of the studies. They are summarized here together with new results, allowing richer connections to be drawn between the empirical findings. Theoretical and methodological issues surrounding the laboratory studies are also examined.
The experiments focus primarily on pitch structures in traditional Western music. This choice of focus is based on the large corpus of literature in music theory that deals with this style.
This literature has been important for designing the experimental materials and interpreting the results. A number of studies extend the methods to music outside this tradition. Throughout, care has been taken to provide adequate background in experimental methods and music theory so that no special background is needed to follow the major arguments. However, the reader will naturally discover certain topics to be of greater interest than others depending on his or her special expertise. This project has evolved over a period of more than a decade, during which there has been remarkable progress in the study of music perception and cognition.
No attempt has been made here to summarize this rapidly developing field; only the literature of most direct relevance to the present investigations is reviewed. The project owes a tremendous debt to many individuals and institutions. My greatest debt, of course, is to my numerous collaborators on the studies reported here. I would especially like to acknowledge the contributions of Roger Shepard, who encouraged the work from the beginning and was responsible for crucial methodological innovations, and Jamshed Bharucha, who formulated the theoretical framework for understanding the results on musical harmony.
The research has been gener- viii Preface ously supported by grants from both the National Science Foundation and the National Institute of Mental Health. The first draft of the monograph was written while I was a fellow at the Center for Advanced Studies in the Behavioral Sciences in Stanford, California.
Both leaves were supported by the National Science Foundation. I am deeply thankful for the encouragement and wealth of constructive suggestions made by my readers for Oxford University Press: John Sloboda, W. Jay Dowling, and Fred Lerdahl.
Ithaca, New York April C. Contents 1. Objectives and methods Listening to music, we hear the sounds not as isolated, disconnected units, but integrated into patterns. Our perceptual experience goes beyond sensory registration of single musical events. Sound elements are heard in context, organized in pitch and time, and are understood in terms of their functions within that context. The absolute pitch of a particular tone is less important to the listener than the intervals it forms with surrounding pitches.
Each pitch participates in extended melodic and harmonic patterns. Similarly, the absolute duration of a musical event is less important than its relationship to neighboring events with which it combines to form metrical and rhythmic units. In taking context into account, the listener apprehends increasingly larger temporal groupings. The listener appreciates the organization of a musical work by assigning significance to the sounded elements according to their roles in the musical context, and by integrating them into the broader pattern. Certain musical relationships are explicit in the music; these are features or properties of the music that can be derived more or less directly from the sounded sequence.
Examples are the ratios of pitch frequencies and durations, and the repetition of melodic and rhythmic patterns. To say that such aspects of musical structure are explicit in the music, however, does not necessarily imply they are perceived. To understand the listener's response to music, it is necessary to specify which of the potentially available features of the music are realized by the listener, how they are processed and remembered, and how they contribute to an appreciation of the overall plan of the piece of music. It cannot be assumed that this experience is a direct mapping of the external musical events, although some objective musical features may have direct psychological correlates.
The listener's experience of a particular musical sequence is driven and constrained by the sounds registered by auditory mechanisms and processed by available mental resources. Furthermore, the construction of the music itself must respect the listener's capacities for appreciating structured auditory information. That is, the way musical materials are formed must take into consideration certain inherent limitations and special capabilities of the listener.
It is possible, for instance, that particular pitch combinations predominate in music because they are perceived as consonant or pleasing, and this may be a consequence of peripheral auditory mechanisms for registering pitch. Simple ratios of durations, such as 2 : 1 and 3 : 1 , may be more readily perceived and more accuratelyj 3 4 Cognitive foundations of musical pitch remembered, and thus more commonly employed in the formation of metrical and rhythmic units.
Or, to take another example, the duration of rhythmic and pitch patterns may reflect limitations in memory capacity. All this suggests that music psychology must seek to understand the interdependencies between the structured sound material of the music and the listener's capacity for apprehending and remembering relations among the sounded events.
In discussing this problem, it may be useful to distinguish the objective external musical stimulus from the subjective internal experience of music as indicated in Figure 1. The objective musical attributes can be divided into those that the listener does and does not perceive. This classification would, presumably, depend on the listener's training and experience. Some attributes of music are probably accessible even to naive listeners because they fit with general principles of psychological organization.
For example, there may be natural tendencies to group elements on the basis of similarity of pitch or timbre, locate phrase boundaries at relatively long pauses, and notice periodic repetitions of temporal patterns. More extensive experience may be required to perceive other attributes. This experience might enable the listener to hear a melody as a variation of another, recall the lyrics that accompany it, or associate it with a sociocultural context in which it is conventionally played. More important, however, the listener may interpret the sounded events in terms of a system of knowledge that specifies the ways in which musical passages tend to be constructed within a relevant musical style system.
In addition, there are undoubtedly features of music that can be objectively specified but cannot be perceived. This may be due to the listener's limited ability to discriminate along objective auditory dimensions, to limitations in memory, or to a mismatch between the underlying organizational scheme of a composition and natural tendencies to perceive auditory information in a particular way. These limitations may in some cases be overcome through perceptual learning or by special instruction or additional descriptions. However, there are inherent bounds to the ways in which music can be perceived due to the nature of the human psychological system for perceiving music.
In a similar way, the subjective experience can be divided into aspects that do and do not correspond to objective musical structures. Various considerations suggest a rather direct mapping between certain objective properties of music and the listener's organized internal representation. The construction of music reflects certain intuitions about the listener's capacities for internalizing relations existing between the sounded events.
It takes into account predispositions to organize pitch and time in particular ways. For example, pitch materials may tend to be categorized in terms of a limited number of pitch classes, and periodically regular metrical units may be preferred. Simple patterns of melodic contour and rhythmic grouping may facilitate the formation of larger perceptual units.
Objectives and methods 5 Fig. Objective musical attributes are distinguished from subjective musical experiences. In some cases there is a fairly direct mapping between attributes that can be objectively specified and attributes that are perceived, indicated by the arrow. Other objective musical attributes may not be perceived, and some aspects of the listener's experience may be difficult to trace to objective musical attributes. Moreover, the listener may have abstracted and internalized structural regularities from experience with a musical idiom.
This knowledge may include the pitch intervals in musical scales, probable combinations of simultaneous and successive pitches, and characteristic rhythmic patterns. Once acquired, this knowledge may serve as a framework for precisely encoding, organizing, and remembering new musical sequences written in the musical idiom. The present investigations focus on that part of the musical experience that can be traced quite directly to objective musical structures.
Other aspects of the subjective experience, such as emotional responses, personal associations, or visual imagery evoked by the music, may be quite indirectly related, or unrelated, to objective musical attributes. Consequently, they are difficult to study systematically using the methods of cognitive psychology to which we now turn.
The approach of cognitive psychology The methods and theoretical orientation of the research to be presented are those of cognitive psychology. Cognitive psychology is a subarea of experimental psychology concerned with describing human mental activity. A diverse collection of activities has been considered by investigators in the field, including encoding and interpreting perceptual information such as visual patterns and speech, learning and remembering perceptual and linguistic information, organizing and executing motoric and linguistic behaviors, as well as reasoning, problem solving, and decision making.
The cognitive psychologist is generally less concerned with the occasional exception or special case than with the more general rules govern- 6 Cognitive foundations of musical pitch ing human cognition. Furthermore, the focus is less on describing the experience of specially trained or exceptional individuals than with cognitive capacities exhibited more generally.
Finally, the motivation is not to provide a prescriptive account of human behavior—what individuals should do on logical or rational grounds—but to arrive at a descriptive account of what individuals actually do when engaged in cognitive activities. Investigations in various areas have led to the view that cognitive activity requires complex mental structures and processes. That is, observable behaviors can only be understood if we presuppose a mental system that encodes, transforms, and combines information.
Out of this work have emerged various characterizations or models of the cognitive system. Certainly, areas of cognitive activity are not yet fully understood, and important theoretical issues remain unresolved, particularly about the most appropriate format for describing or representing internal structures.
However, the field has now evolved to the point where quite precise questions have been formulated, laboratory techniques for investigating those questions have been devised, and various conceptual systems for summarizing and communicating the empirical observations have been developed. As a complex human activity, music perception is naturally an area of interest to cognitive psychology. However, introducing this domain of inquiry raises anew basic questions about how best to formulate the questions to be investigated, how to compile observations that bear on these questions, and finally how to devise a descriptive framework for communicating the results obtained.
In general terms, the aim is to describe the human capacity for internalizing the structured sound materials of music by characterizing the nature of internal processes and representations. In making choices about the most fruitful kind of analysis, a variety of considerations must be weighed. The musical materials employed in the experiments should not be so impoverished that only a small subset of the potentially relevant internal structures are evoked or engaged. For example, presenting tones in isolation or in random combinations would not reveal the nature of internal structures with specifically musical content.
The materials should, in some sense, be representative of actual music. On the other hand, complex musical materials evoke a multiplicity of responses that may be difficult, if not impossible, to disentangle. Moreover, detailed analyses of the response to particular pieces of music may inadvertently lead to overgeneralizations of the results obtained; the conclusions reached may depend strongly on the particular musical context and be largely uninformative with respect to broader questions.
Only to the extent that the materials are representative of a broad class of music can experimental results be generalized. In the end, the selection of materials depends to Objectives and methods 1 some extent on one's interests and intuitions about the most fruitful questions to be investigated experimentally. Significant questions also arise about the kinds of behaviors or responses that are to be measured. Various considerations come into play here as well.
The cognitive psychologist is interested in describing an internal system that cannot be examined directly. Consequently, its properties must be inferred from external, observable behaviors. For the most part, musical experience is not well suited to verbal description, which may be one reason why music has received less attention than other cognitive activities. Although one component of training in music is the acquisition of terminologies for characterizing musical structure and these often take a verbal form, although other symbolic devices are also used , not all listeners or even performers have sophisticated skills in music analysis.
If one is interested in understanding the musical experience of listeners with diverse levels of training, then the observed responses should not require special knowledge of established descriptive systems. A similar argument applies to responses requiring production, such as music transcription, playing an instrument, or even singing. Nonetheless, the observations made should be musically relevant, that is, they should be matched in some way to the experience itself.
The observations made are informative only to the extent that the mode of response mirrors essential components of the internal systems engaged in listening. Here again, a considerable degree of intuition is involved. Other important considerations in choosing behaviors to be recorded come from the experimental methodologies of cognitive psychology itself.
The researcher seeks a systematic body of observations that are well suited to treatment by various analytical techniques. The available techniques restrict the kinds of observations that can be treated. Moreover, to understand the relationship between the music and the listener's experience, it is essential to vary properties of the external musical event and trace how these changes affect the pattern of responding.
In this connection, we speak of the external musical event as the stimulus, and the changes made in the stimulus as experimental manipulations. For this strategy to be informative, considerable understanding of the relevant stimulus attributes or properties is required. It is essential to control irrelevant properties of the stimulus, and to change or manipulate only those of interest.
Potentially separable properties must not be experimentally covaried or confounded if the pattern of response is to be interpreted unambiguously. Finally, the variables manipulated should be musically relevant. This requires that the manipulated stimulus attributes correspond to appropriate conceptualizations about the way music is constructed. If this general strategy is carried out successfully, it leads to a scientific account of particular aspects of the musical experience. Admittedly, one could still argue about the choice of stimulus materials, the 8 Cognitive foundations of musical pitch nature of the response required, the interpretation of the data, and generality of the conclusions.
However, as a minimum, these arguments take place in the context of a controlled and systematic body of observations about the musical experience. Contemporary cognitive psychology is closely aligned with other subfields of the cognitive sciences, particularly philosophy of mind, linguistics, and artificial intelligence. It should be clear, however, that cognitive psychology differs in certain fundamental ways from these other fields, despite their overlapping concerns.
Cognitive psychology is, at the core, an empirical approach to understanding human thought and experience. This means that theories are developed from and constrained by observations about human behavior made largely in laboratory situations. The conclusions, or generalizations, reached are intended as summary descriptions of the external behaviors observed in response to controlled external situations.
In its reliance on laboratory methods, cognitive psychology is patterned closely after the physical sciences. With this alliance comes a concern for quantification. Advances in these other scientific disciplines have come largely through successful measurement of physical attributes and the discovery of lawful relationships among them.
Special problems arise, however, in quantifying psychological properties. At the most fundamental level, these problems arise because the observations are of some external correlate of the internal system, and not of the internal system itself. The measurements are thus indirect, making it necessary to employ a variety of approaches.
To the extent that different approaches converge on the same results, it can be argued that we are describing the internal system and not the effects of some particular method of measurement. Moreover, observable behaviors are notably complex and variable, and so it is necessary to constrain the mode of responding and record aspects that can be coded precisely. Mathematics provides numerical structures well suited to this, and therefore we tend to record quantifiable aspects of responses. In involving mathematics, however, we must be careful not to make unwarranted assumptions about the nature of the measured quantities.
The kind of conclusions that are justified on the basis of particular kinds of data is the subject of considerable theoretical analysis and debate. Despite the difficulties, quantification is unavoidable. The variable nature of human behavior impels us in that direction. Under what are seemingly identical external circumstances, different individuals will respond differently, and the same individual will respond differently on different occasions. Even so, we intuitively believe that there are underlying regularities and lawful principles operating.
To uncover these, we routinely employ statistical and other analytical methods. Although a technical understanding of these methods will not generally be necessary for what follows, the empirical work presented should be understood with an ap- Objectives and methods 9 preciation for the difficulties involved in quantifying psychological attributes and achieving general statements in the face of considerable variability. The final issue concerns the choice of listeners to participate in the experiments.
This choice interacts with a variety of other considerations, including the objective of the experiment, the nature of the materials employed, and the kind of response that is to be measured. Because cognitive psychology is directed at describing mental capacities exhibited quite generally, the majority of studies do not employ participants with extraordinary talents or severe deficits.
However, even the general population includes large variations in the extent of musical talent, training, and performing and listening experience. If we are especially interested in how musical knowledge is acquired, then a variety of strategies can be followed. One strategy is to take a developmental approach, using listeners of different ages. Another strategy is to select listeners so that they have different levels of musical training or prior experience with the kinds of materials used in the experiment.
Alternatively, the experiment can provide extended experience with a particular set of materials and trace how this experience changes the pattern of responding. If, however, we are less concerned with the process of acquisition, and more concerned with describing a fully developed cognitive system for processing musical materials, we tend to employ listeners with considerable training and experience.
In any case, because music background may affect the results, care must be taken to assess participants' music backgrounds and treat the data in a way that allows for the possibility of individual differences. The plan of the research Having described in general terms some of the considerations involved in studying music from the perspective of cognitive psychology, I turn now to more specific methodological and theoretical issues surrounding the work described in the following chapters.
The experiments summarized here are directed at characterizing the listener's internal system for processing pitch structures, focusing primarily on those found in traditional Western music. I will refer to this music as tonal-harmonic to indicate, first, that it is tonal in the sense of being organized around a central reference pitch tonic and, second, that harmony is important for establishing the tonal framework. The aim is to describe what the listener knows about pitch relationships in this style, how this knowledge affects the processing of sounded sequences, and how this system arises from stylistic regularities identifiable in the music.
In tonal-harmonic music, three distinct types of basic elements can be identified: single tones, chords, and keys. Although pitch is continuously variable, Western tonal music selects from the potentially infinite set of pitches a relatively small set of discrete pitch categories. This property, 10 Cognitive foundations of musical pitch in fact, holds true for most if not all musical cultures.
In Western music, there are 12 pitch categories in each octave range, with a total of about eight octaves in musical usage. Both psychologically and compositionally, however, a natural identification exists between corresponding pitches in different octaves that is, pitches one or more octaves apart , and so the set of musical tones can be described as 12 pitch categories or classes. Chords in tonal-harmonic music consist of three or more tones sounded simultaneously, although the same harmony can also be expressed as successively sounded tones.
Diatonic triads, three-tone chords built on the seven tones of the diatonic scale, form the core of harmonies in eighteenth- and nineteenth-century music. A musical key is defined by its tonic and its scale or mode, which is usually major or minor. The tonic is one of the 12 pitches in an octave range, and the scale is a specific pattern of intervals defined relative to the tonic.
The major and minor scales and twelve tonics give rise to a total set of 24 keys. The construction of these elements will be treated in considerably more detail later. For the present purposes, it is important to note only that tonal-harmonic music is constructed from small sets of basic elements.
Consequently, it is possible to measure experimentally the degree to which they are perceived as related to one another. The experiments constituting the basic line of inquiry follow this approach quite directly. The studies assess the perceived relationships between elements, either of the same type e. For reasons of practicality, the sets are restricted in certain ways.
The tones employed in the studies all have simple harmonic spectrums, employ equal-tempered tuning, and are generally restricted to a one-octave range. The experiments do not consider various borrowed or altered chords, and do not employ modes other than major and minor. These restrictions, although they ignore characteristics of potential interest, permit interdependencies between perceptual relations for elements of different types to be evaluated in a straightforward way. In the basic experiments, the elements of interest tones or chords are presented singly or in pairs following a musical unit, such as a scale or chord cadence, that would be expected to establish a particular key quite unambiguously.
The listeners are required to judge, using a numerical scale, how related the presented elements are, either to each other, or to the established key. Some justification for this procedure is needed. The rationale for presenting tones in key-defining contexts is that key, or tonality, has important implications for the way in which tones and chords are used in simultaneous and successive combinations.
Moreover, the key establishes hierarchies of structural significance for tones and chords. Certain of these elements are more central or important within the tonality, with other elements functioning in relation to these central elements. It will be argued that this kind of hierarchical ordering is an essential point of contact between the way in which tonal-harmonic music is constructed Objectives and methods 11 and the psychological system that organizes, interprets, and remembers music of this style. The rating task required of the listener is simply to make a judgment on a numerical scale about the degree to which different elements are related.
The instructions to listeners do not specify the particular features or attributes to be considered in arriving at their decision. The task is openended and unconstrained, except by the particular musical materials presented and the mode of responding—the numerical scale. In defense of the method, it turns out that there is considerable consistency in the results obtained.
Different listeners exhibit similar patterns of responses, and the same individual tends to produce the same or nearly the same response on different occasions when the identical musical stimulus is presented. This internal consistency lends some support to the method. Moreover, it is possible to interpret the patterns obtained in terms of various music-theoretical concepts, serving as a source of external validation for the experimental task.
Tasks of this sort have found useful application in a wide variety of domains. Measures of relatedness or proximity take different forms, including the frequency with which two things co-occur, the probability that two things are confused or misidentified as each other, and numerical ratings about the similarity between objects in the domain of interest.
Because data of this general sort arise in various ways, analytical techniques have been devised for their treatment. These techniques require that a measure of relatedness has been obtained for every pair, or at least most pairs, of objects within the set under consideration. The techniques then reduce this set of numbers, which is usually quite large, to a form, or representation, that expresses the underlying patterns contained in the complex set of data. Usually the form of the representation is that of a spatial configuration or map, a graph or tree structure, making the relationships among the elements relatively easy to understand.
For this reason, these techniques have proved fruitful as exploratory methods for discovering patterns in complex data that may otherwise be difficult to uncover. These methods are used frequently in the experiments reported here. As a matter of subsidiary interest, however, it turns out that characteristics of the data obtained in the experiments on music perception make them not entirely suited to analysis by existing methods, suggesting desirable extensions and modifications.
These basic experiments describe the perceived relationships between the three types of elements of tonal-harmonic music in an abstract sense. The measurements are presumed to reflect part of a system of knowledge summarizing how the elements are generally employed within the style. Any particular piece exhibits its own unique features: melodic figures, rhythmic and metrical groupings, patterns of repetitions, variations and developments, harmonic progressions, and modulations between keys.
These features are apprehended by the listener, in part, by reference to 12 Cognitive foundations of musical pitch more abstract knowledge about pitch structures. For example, encoding and remembering a melodic figure is influenced by appreciating how its component tones function within the scale of the key.
Or, to take another example, a shift between keys is traced by the listener through understanding the multiple functions of the sounded chords within the different keys. Thus, the experience of longer and more complex sequences is assumed to engage the abstract knowledge about pitch structures and be influenced by it.
To establish this, various experiments were conducted in conjunction with the rating studies. Some of the experiments are concerned with how tonality affects the listener's ability to remember component elements in longer melodic and harmonic sequences. Certain variables manipulated in these memory studies are those that emerge as important in the rating studies, so the memory data provide a source of convergent evidence.
Other experiments employ the rating methods in conjunction with relatively extended and complex sequences, permitting comparisons with the simpler and more abstract materials. The choice to focus on tonal-harmonic music was based on two primary considerations. First, I was concerned with investigating the knowledge of pitch structures resulting from relatively extensive experience with a musical style.
It seemed that information about the final form this knowledge takes could suggest processes through which it is acquired. The availability of listeners to participate in the experiments who were familiar with this style was important, then, in choosing this focus. The second consideration was the extensive body of music theory and analysis treating this idiom. This literature identifies general stylistic principles of organization that could aid in constructing the stimulus materials.
Inasmuch as the stimulus materials incorporate features typical of the style, the results obtained have a reasonably wide range of generality. Moreover, the terminology of music theory provides a useful means of interpreting and communicating some of the empirical results. Finally, it was hoped that the perceptual studies might provide a psychological explanation for some of the principles articulated in more theoretical treatments of music.
It should be emphasized, however, that the experimental methods do not depend on a particular idiom; they can be extended to other styles when appropriate modifications are made in the stimulus materials. This approach was taken in a number of studies that explore the perceptual effects of music outside the tonal-harmonic tradition. These provide information about the perception of music in less familiar styles that can be compared with the results for tonal-harmonic music. Outline of the following chapters The basic empirical results, outlined in Figure 1.
The results show that a context defining a major or minor key imposes on the set of tones a well-defined ordering of structural significance or stability. This ordering is called a tonal hierarchy. The tonal hierarchies of different keys can be used to produce a quantitative measure of the degree to which the keys are related to one another. The argument is based on the idea that keys are related to the extent that their tonal hierarchies are similar. The analysis that produces a spatial representation of key distances from the tonal hierarchies completes Chapter 2.
The next two chapters are also concerned with tonal hierarchies. Chapter 3 asks what objective properties of music are correlated with the tonal hierarchies measured in the experiments. Two possibilities are examined in detail: tonal consonance and the statistical distribution of tones in tonal-harmonic music. Strong correlations are found between the tonal hierarchies and the durations and number of occurrences of tones in various compositions, suggesting a mechanism through which the tonal hierarchy is internalized.
These findings suggested the key-finding algorithm described in Chapter 4. This simple algorithm takes as input the durations of each of the 12 chromatic scale tones in a musical segment; these values are then matched to the tonal hierarchies of all possible major and minor keys. The algorithm returns a set of values indicating the relative strength of each possible key. The algorithm is applied to initial segments of a number of compositions, and to segments throughout a single composition to trace key changes within this piece.
Chapters 5 and 6 turn to the question of how tones are heard in relation to one another. The experiment that is central to Chapter 5 presents all possible pairs of successive tones following a context that defines either a major or minor key. Listeners judge how closely related the first tone of the pair is heard as being to the second.
These data are then analyzed to determine the variables influencing the listeners' judgments: distance on the chroma circle, distance on the circle of fifths, and the position of the two tones in the tonal hierarchy of the context key. The judgments are compared with statistical summaries of the frequency with which melodic intervals appear in tonal-harmonic music.
The judgments depend on temporal order and key context, which limits the utility of spatial models for representing perceived pitch relations. This problem is treated more theoretically in Chapter 6, which proposes three principles of contextual dependency.
These principles, which account for various general patterns found in the experiment of Chapter 5, are also supported by a number of pitch memory studies described in Chapter 6. Other principles governing perceptual organization of pitch sequences are also summarized briefly. The next two chapters examine the perceptual relationships that obtain for chords. Chapter 7 begins with two experiments that measure the relative structural significance of all possible major, minor, and diminished chords in major and minor key contexts.
A well-defined harmonic hier- 14 Cognitive foundations of musical pitch Fig. Schematic outline of the organization of the chapters that describe the basic experiments. These explore the perceived relationships in tonal-harmonic music between elements of three types: tones, chords, and keys. Other chapters present material related to these basic experiments.
Chapter 7 closes by demonstrating that the harmonic hierarchy gives rise to a measure of key distance virtually identical to that obtained from the tonal hierarchy. Chapter 8 summarizes a number of studies measuring the degree to which different chords are perceived as related to one another. These perceptual judgments depend strongly on the tonal context according to the principles of contextual dependency proposed in Chapter 6.
Supporting results are obtained from studies of memory for chord sequences, also described in Chapter 8. The chapter concludes with a summary of the commonalities found for tonal and harmonic structures. Chapters 9 and 10 summarize four detailed case studies of the psychological effects of pitch structures in more complex musical sequences. The four experiments all use the same methodology introduced in Chapter 2.
Chapter 9 begins with an investigation of how the sense of key develops and changes as a harmonic sequence unfolds in time. The next experiment tests the possibility that two keys can be perceived simultaneously when the musical context employs materials of two different keys in parallel. Chapter 10 considers two musical styles outside the Western tonal-harmonic tradition. One study examines the perception of pitch structures in materials drawn from two compositions in the style of 12tone serialism.
The other study takes a cross-cultural approach, using rags from North Indian music. Each of these case studies reveals special psychological capacities and limits for perceiving complex musical sequences. The final chapter steps back from the empirical results to consider what Objectives and methods 15 they reveal about the psychological basis of musical pitch structures. Certain properties of pitch systems are identified that may be important for accurate encoding and memory.
A number of traditional and novel pitch systems are analyzed for the presence or absence of these properties. The chapter concludes with a summary of the empirical findings, and a discussion of what they indicate about the nature of the perceptual and cognitive abilities underlying our musical experience.
Quantifying tonal hierarchies and key distances The experiments reported in this chapter assess the degree to which single musical tones are perceived as being related to tonal centers, or keys. The experiments provide a quantitative measure of the hierarchical ordering imposed on the individual tones in tonal contexts. In music-theoretical terms, the ratings might be identified with the relative stability or structural significance of tones as they function within tonal contexts. It will be argued that this hierarchy is, in some sense, basic to the structuring of music itself and also to the psychological response to music.
This identification of a music-theoretical construct and a pattern of psychological data, then, represents a point of contact between the structure contained within the music and described by music theory, and the listener's response to that structure. One very general feature of music is that one particular pitch is established as a central reference pitch. This pitch is called the tonic, or tonal center. The means of emphasizing the tonic and organizing the other elements around it vary considerably across musical styles.
In most cases, the tonic is emphasized both melodically and rhythmically; it is sounded with relative frequency and with longer duration; and it tends to appear near the beginning and end of major phrase boundaries and at points of rhythmic stress. Other mechanisms for establishing the tonal center may also be used.
For example, in classical Indian music the tonic appears in the form of a drone that is sounded continuously throughout the rag, often accompanied by another tone, called the secondary drone. In Indonesian music, the gong tone the most important pitch is sounded at the end of major melodic sections together with the gong. In traditional Western music, tonality is associated with a system of harmony, with certain chords and chord sequences acting as important indicators of the tonal center. In twentieth-century Western music, compositional techniques have evolved that represent significant steps away from the basic principles of tonal harmony.
Examples are extreme chromaticism, the construction of chords not assignable to traditional harmonic functions, the simultaneous employment of materials from more than one key, and serial tone techniques. In some of these cases, the music is structured so as to deny or prevent reference to a single, stable tonal center. The effect of these techniques on the listener is a topic of considerable psychological interest, and certain cases will be considered later.
Despite these develop16 Quantifying tonal hierarchies and key distances 17 ments in the West, much contemporary music, as well as music in other cultures, adheres to the basic principle of tonality that, defined in its most general sense, is the centering of the pitch materials around one particular tone. A number of considerations suggested that the most appropriate place to start the exposition was with the tonal hierarchy imposed on tones by key contexts in Western music.
First, the experimental method is quite simple and illustrates certain issues in experimental design. Second, there are relatively clear intuitions and theoretical predictions for this case; other cases with less clear predictions will be presented later. Third, the techniques of data analysis introduced here, correlation and multidimensional scaling, are used again in numerous other studies. Finally, the results obtained here parallel those found in other areas of human cognition and perception, suggesting that a general psychological principle is operating in the particular musical case considered.
Briefly, this general psychological principle is that particular perceptual and conceptual objects have special psychological status. This notion appears in a variety of contexts in the psychological literature, notably in the work of the Gestalt tradition and more recent work by Garner , Rosch , Rosch and Mervis , and Goldmeier The basic idea is that within categories certain members are normative, unique, selfconsistent, simple, typical, or the best exemplars of the domain sometimes called "prototypes".
They are reference points to which other category members are compared. To illustrate with some examples that have been investigated experimentally, colors are often described with respect to "focal" colors, such as red, green, blue, and yellow. A color may be described as off-red or brownish-red, with implicit reference to a "focal" red. Similarly, numbers are rounded off to other numbers with special cognitive status, such as multiples of tens and hundreds.
One says that 9 is almost 10, or that 95 is almost Certain visual forms seem somehow "better" than others, because they are simpler, more regular, or more symmetric. The other, less regular forms are described as variants of these "good" forms. Thus, for example, a line may be described as almost vertical, and a quadrilateral figure as almost a square.
These are all examples in which the elements are described in reference to certain other elements having special status within the category. From a psychological point of view, the existence of singular, central, or prototypical elements within categories is thought to reflect a drive toward maximizing the efficiency of coding or minimizing the complexity of cognitive objects Goldmeier, , p. This is, in Rosen's terms , the principle of "cognitive economy"; one seeks a system of internal coding that is best suited for making distinctions that are relevant to the domain in question, at the same time conserving finite cognitive resources.
In her work, this is codified in a structural theory, sometimes called cue validity, in which the best members of categories are those 18 Cognitive foundations of musical pitch having the greatest number of features in common with other elements of the category and the fewest features in common with members of other categories. It should be noted, however, that psychologically special category members may exist even when the objects cannot be decomposed into discrete features.
The example of color, mentioned earlier, is a case in which cue validity theory based on discrete features does not apply. Empirical support for psychological reference points consists of two related kinds of findings. The first establishes that elements can be rated reliably in terms of "goodness" Garner, or typicality e.
That is, observers agree with one another when asked to make ratings about how good an exemplar each member is of the category. This establishes a kind of hierarchical ordering on the elements in the category, often expressed in quantitative terms. The second kind of finding shows that the hierarchical ordering influences various measures of perceptual or cognitive processing. For example, "good" figures are remembered better, and are subject to fewer distortions than are less "good" figures e.
The hierarchical ordering is also reflected in the speed with which observers classify category members, the time required and the number of errors made when learning novel classifications, the order of acquisition of concepts in children, the order and probability of item output, judgments of the degree of relatedness between elements, and various linguistic forms for describing category membership. See Rosch, , for a summary of these empirical findings. In summary, considerable empirical work supports the general notion that human cognitive and perceptual systems invest certain elements with special status: these elements are given priority in processing, are most stable in memory, and are important for linguistic descriptions.
The tonal hierarchy This description of a hierarchical ordering of category members would seem readily applicable to tones in tonal contexts. A tonal context designates one particular tone as most central. The other tones all have functions specified with respect to this tone, in terms of their relatedness to the tonic and secondary reference points established by the tonic. This suggests the operation of a system of reference points in music similar to those emerging from studies in other domains.
One important difference should be noted at the outset, however. Whereas other perceptual and cognitive reference points are fixed, the tonic depends on the particular context. No tone is inherently more "tonic" than others, in contrast to, for example, certain colors that appear to be inherently more "focal" than others. Quantifying tonal hierarchies and key distances 19 The relative stability of a tone may depend to some degree on its treatment within a particular compositional context as, for example, by the degree to which it is stressed metrically or rhythmically, its place in the contour of a melodic line, its pitch range, and the timbre and dynamics with which it is sounded.
However, it is presumed that there is a more abstract, invariant hierarchy of stability that is typical of a musical style more generally, and that this more abstract hierarchy is an important characteristic contributing to the perceived stability of each tone within a complex musical sequence. In this connection, Bharucha b, p. The tone C may occur many times in a musical piece; each occurrence is a distinct musical event.
But all the occurrences are instances of a class of tones tokens of a type denoted by "C. It is with the latter kind of hierarchy that the present experiments are concerned. The notion of an abstract tonal hierarchy is suggested in theoretical descriptions of musical structure, in which it is expressed using a variety of terms. Stability will be taken here to be the basic term and to be loosely identified with other terms used, such as relative structural significance, priority, resolution versus tension , and rest versus activity.
It is taken to refer to the dimension along which musical tones differ, with some tones producing an unstable effect and requiring resolution, and other tones producing a stable effect and giving a sense of completion. Central to the work of L. Meyer , this concept of stability is intimately connected to his notions of emotion and meaning in music.
Tonal stability plays a central role in numerous other treatments of musical structure, but in Meyer's presentation it takes on specifically psychological content in which he generalizes Gestalt theory to relate principles of tonal organization to the listener's response. In addition to the psychological functions ascribed by Meyer to the concept of musical stability, his treatment is notable in two other respects. First, it is striking how similar his language is to that used in connection with perceptual and cognitive reference points in the psychological literature summarized earlier.
Second, his treatment gives very specific predictions about the ordering expected for Western tonal music, which can be compared to the empirical results presented later. For these 20 Cognitive foundations of musical pitch reasons, the following passage from L. Meyer , pp. They tend to move toward the more stable points in the system—the structural or substantive tones.