The representation of harmonic structure in music: hierarchies of stability as a function of context.

The ability to appreciate most Western music presupposes cognitive structures capable of abstracting an underlying harmonic structure from a complex string of musical events. In this paper we provide a description of the listener’s knowledge of hierarchies of harmonic stability. The organization of harmonic information may be summarized by Osix principles. Three of these ;vrinciples--Key Membership, Intrakey Distance and Intrakey Asymmetry-govern harmonic organization Independent of context. Three principles-Contextual Identity, Contextual Distance and Contextual Asymmetry-govern harmonic organization in the presence of a tonal context. Chords that are members of the same’ key are represented in a hierarchy of stability that is independent of context. chords from different keys are represented in a hierarchy of stability that is dependent upon the prevailing context. Two different experimental tasks were used to provide convergent evidence for these principles: 1.) multidimensional scaling of chords in the absence of any context or in the presence of different tonal contexts, and 2) recognition memory for chords in random and tonal contexts. It is suggested that harmonically stable chords function as cognitive reference points for the system as a whole. The importance of representations of hierarachies of harmonic stability is discussed with respect to generative accounts of musical competence. *This research was supported in part by a grant from the National Science Foundation (BNS-8103570) to Carol L. Krumhansl. The authors are grateful to David M. Green for the use of the Psychophysics Laboratory at Harvard and to William K. Estes for helpfful suggestions. Murray Spiegel, David Wilson, Michael Hacker, Robert Nosofsky and an anonymous reviewer gave valuable advice. Mary Castellano assisted in running subjects and in data analysis. Reprint requests may be addressed to Jamshed Bharucha (who is now at Cor;lell) or Carol L. Krumhansl at Department of Psychology, Uris Hall, Cornell University, Ithaca, NY 14853, U.S.A. OOlO-0277/83/010063-40/$12.30 0 Elsevier Sequoia/Printed in The Netherlands 64 J. Bharuchn and C L. Krumhansl The perceptual processing of music involves the abstraction of underlying invariants in auditory events of considerable surface complexity. A piece of music can undergo substantial surface alteration (in terms of instrumentation, embellishment, transposition, or variation) and yet be recognized as the same piece. Only a level of description more abstract than that of the acoustic signal can capture the structural properties of music that enable the listener ‘to perceive musical sequences with marked surface differences as similar. This ability suggests that the mental representation of music is governed by a system of abstract structural principles. Substantial work has been done on the cognitive systems that enable us to process melodies, that is, sequences of tones (for example, Bartlett and Dowling, 1980; Cuddy ef al., 1979; Deutsch, 1969, 1980; Deutsch and Feroe, 1981; Dewar et al, 1977; Dowling, 1978; Idson and Massaro, 1978; Krumhansl, 1979; Krumhansl and Shepard, 1979; Massaro et al., 1980). Only recently has experimental research been done on the cognitive systems that enable us to process harmonic organization in sequences of chords (Krumhansl, Bharucha and Kessler, 1982). In this paper, we argue that the extraction of harmonic structure is fundamental to the processing of Western music, and propose six basic structural principles governing the representation of harmony. Music theoretic descriptions of musical structure Music theorists and composers have been concerned with the formulation of theories of the abstract structure of music (Schenker, 193511979, 1906/ 1954; Schoenberg, 1969). The prevailing theory that has emerged offers a description of music in terms of two fundamental concepts, which are constitutive of musical structure referred to as tonal. The first is the hierarchy of stability that applies to the twelve tones of the chromatic scale within an octave range. The concept of stability refers to the fact that certain tones are perceived as more final and serve as better completions of melodic phrases than do others, which demand ‘resolution’ to more stable tones in the system. The more stable tones are generally those that appear more frequently, in prominent positions, and with rhythmic stress. In the hierarchy of stabili9y, one single tone, called the tonic, is the most stable tone. It is also the first tone of the scales of Western tonal music. Following the tonic, the other tones of the scale are the next most stable, particularly the fifth and the third tones of the scale. The tones not contained in the scale are the least stable wiihin the system. In Western music, the major and minor diatonic scales are most frequently employed. In Eastern music, The representation of harmonic stnrctut e in music 6 5 other scales are often used, but the hierarchies are often analogous. Since any tone can function as the tonic, in other words, melodies are transposable, the placement of the tonal hierarchy is dependent upon the context and is determined by the use of tones from a particular scale and their temporal ordering. In its most general form, this account seems to capture the basic structural organization of most music, Western and Eastern alike. Additional structure over and above this may or may not emerge cross-culturally, but this question is beyond the scope of this paper. In Western music, a second fundamental organizing concept is the harmonic functioning of simultaneously sounded tones in chords. By far the majority of the work of music theorists has dealt with harmonic organization, which provides a description of the abstract structure of most Western music over the last four centuries. It is noteworthy that although many surface aspects of music have changed drastically over this time, the underlying harmonic structure has remained constant for the most part. Popular forms of music still cling to the traditional emphasis on triads built on the tonic (I), the dominant (V), and the subdominant (IV) chords. The units of harmonic analysis, as suggested by music theory, are chords (or more precisely, root functions). The chords into which a piece of music is analyzed may not actually be present in the music as simultaneously sounded tones. They may, instead, be implied by a sequence of successive tones (i.e., a melody). The analysis of tonal melodies in music theory thus invariably makes reference to the underlying-or implied-harmonic structure. The music theoretic analysis of music into abstract chord functions suggests the possibility that listeners have an internal representation of these chord functions which may be activated not only by direct sounding of these chords, but also by certain melodic patte-rns. The rhythmic analysis of a piece of music may also rely upon harmonic structure. It is often the case that the ‘strong beat’ may be found by the listener with no difficulty even though it may be cued neither by loudness nor temporal parameters. In such cases, the strong beat will be perceived at the position of chords that are harmonically stable in the prevailing context. The stress is therefore implied by harmonic functions even though an actual acoustic stress may occur in a different temporal position in the sequence or may not occur at all. It is of course true that just as melodic and rhythmic analyses rely on harmonic structure, the converse holds as well to some extent. For instance, harmonic analysis often involves melodic considerations such as voiceleading (which governs the expectations of melodic progression), and the expectations for harmonic progression may be strongly influenced by metrical or rhythmic factors. However, it is harmonic structure that music 65 J. Bhamcha and C L. kwnhansl theorists have been predominantly concerned with, and with which they have been remarkably successful in constructing sophisticated analytic theories. Although there is by no means unanimous agreement on the fine details of a standard theory of harmony, the principles are codified in rigorous textbooks (e.g., Piston, 1962) and are routinely taught in courses on harmony. In sharp contrast, a comprehensive theory of melodic structure has eluded music theorists. Rarely does one read a textbook or take a course on melody. This is not because melodic structure is necessarily simpler than harmonic structure. Rather, it is because, with the exception of principles of voice-leading, only vague principles of good melodic form have been fox%coming. Furthermore, as stated earlier, attempts at rigorous analyses of melodic structure invariably invoke the harmonic structure that is implied by the melody. The analysis of root functions in music theory is the identification of the stable tones of the composition as components of chords that function within the prevailing tonality or key. Seven chords constitute the basic set of chords drawn from any given key, which are those built on the seven steps of the diatonic scale. Often the Roman numerals I-VII are used to indicate the scale step on which a chord is built (i.e., the root tone). For the most part, chords of three tones (triads) are used, with a second tone sounded as either a minor or major third (three or four chromatic steps, respectively) above the root tone, and a third tone sounded as either a major or minor third above the second tone. Depending on the particular combination of major or minor thirds, the resulting chord is major, minor, diminished, or augmented. The chord type is determined by the position of the root of the chord within the scale, since all component tones are drawn from the diatonic scale, which contains different intervals between scale steps. Chord type also depends on whether the key is major or minor, giving the distinctive qualities of the major and minor modes. Often a fourth tone is also sounded which repeats one of the three triad components in another octave. Music theorists have also described a hierarchy of stability for the basic set of chords. The tonic (I) chord is the most stable, followed by the chords built on the fifth (V), fourth (IV), sixth (VI) and second (II) scale tones. The chords built on the third (III) and seventh (VII) scale tones, although within the basic set of chords from a key, are less essential for establishing the key, and tend to appear less frequently and with less rhythmic or met&G stress. In addition chords outside the prevailing key (cahed nondiatonic) are sometimes employed, but these produce very unstable effects, demanding resolution to the more stable harmonies of the system. The use The representation of harmonic structure in music 67 of chords, then, is highly constrained by membership in the key, and the sequencing of chords tends to follow very regular patterns. Particular chord sequences, called cadences, are routinely used as formulae for establishing keys and signaling phrase endings in music. The empirical investigation of harmonic structure The relative success of comprehensive theories of harmonic structure and the fact that they quantify over abstract entities (chord functions) suggest that the internal representation of harmony is a fruitful focus cf study. We argue that the experimental study of this structure as perceived by listeners familiar with the musical tradition is a necessary step toward characterizing the process of music perception. Music theorists and researchers in non-experimental branches of cognitive science consult their intuitions in developing theories about the psIychologically salient characteristics of musical organization. However, in the absence of careful empirical investigation we cannot be certain that the average listener is extracting all or any part of this structure. This point is even more crucial for music than it is for language for the following reason. Everybody produces novel sentences in a language. Since linguistic competence is a necessary condition for the production of novel sentences, we can infer such competence in every speaker. We cannot infer musical competence in the majority of people, who do not produce novel musical sequences in composition or improvisation, Most people do listen to music, however, and there is strong reason to suppose that they ‘comprehend’ it, that is, extract some unifying underlying structure. In support of this is the fact that the acceptance of unstructured or atonally structured music has been limited, even though various composers have attempted to introduce music based on other than tonal structures in the present century. It would seem, then, that most listeners do have considerable knowledge of structure in music. The present investigation focuses on the knowledge that listeners familiar with Western music have about the harmonic functions of chords. In the first study reported here, half the listeners had some formal training in music theory, in addition to training on an instrument or voice. These listeners had explicit knowledge of music theory. The remaining listeners had no music theory background, although all had instruction on an instrument or voice. Listeners in this second group are comparable to native speakers who have implicit but not explicit knowledge of the language. The pattern of responses of these two groups of listeners provides an 68 A Bharucha and C L. Krumhansl indication of the degree of correspondence between the intuitions of listeners with and without explicit knowledge of music theory. The second study employed almost exclusively listeners with musical experience but with no music theory training. The choice of these subject populations was based on a concern for assessing the effect of explicit knowledge of music theory in the first study, and for investigating rather precise characteristics of implicit knowledge about harmonic structure in both studies. Beyond the scope of the present paper is the question of whether the harmonic principles identified here would be evident for listeners with less or even no experience with the Western musical tradition. The experimental study of harmonic structure has been difficult in the past because of confounding with melodic factors. This is because the outermost voices in a sequence of chords tend to have salient melodic properties. For instance, the topmost tones typically outline a melody or theme. An additional problem with studying chords is the existence of inversions, which are variants of a basic chord function that depend on which component tone of the chord is in the bass; the root tone of a chord need not be in the bass. In order to minimize these effects, Krumhansl et al. (1982) used chords with component tones sounded over a five octave range. The amplitudes of the components were determined by an amplitude envelope, such that the loudness gradually increased over the first octave and a half, was constant over the center two octaves, and decreased symmetrically over the highest octave and a half. Thus, at both the high and low ends of the frequency range, the loudness tapered off to threshold (after Shepard, 1964), producing chords without clearly defined lowest and highest pitches. The results of that study will be described in detail !ater. This method, which minimizes melodic factors and obscures inversions, will also be used in the present study. Principles governing relationships between chords with and without tonal contexts In this section we outline a number of principles that we presume to govern the perceived relationships between chords. Two basic kinds of principles, formulated in terms of psychological distance, will be stated: those that govern relationships between chords either in no context or in a random context (context independent principles), and those that depend on the establishment of a tonal context (context dependent principles). The context independent principles are strongly suggested by data from. Krumhansl et al (1982), and also embody some very basic ideas in music theory. The representation of harmonic structure in music 69 In a multidimensional scaling study of the perceived relatedness between chords from three closely related keys, Krumhansl et al. (1982) found that 1) chords from a given key cluster together, 2) the I, IV and V chords form a cluster within the group of chords from each key, and 3) a chord pair in which the second chord is either I, IV or V and the first chord is not is judged more closely related than the same pair in the opposite order. In the present paper WE’I have formulated these three principles as Key Membership, Intrakey Distance and Intrakey Asymmetry, respectively. Although none of these principles is stated in quite this manner in music theory, they nevertheless describe very fundamental music theoretic concepts. Our purpose is not, however, to simply formalize music theoretic principles. Principles of music theory describe the structure of musical camposjtions and reflect the intuitions of composers and theorists. We wish to formulate and test psychological principles governing the percepticn of music by listeners with substantial exposure to music but with little explicit knowledge of musical structure. Toward this end, music theory can provide hypotheses, since it is hoped that psychological and music theoret& principles would show considerable convergence. The context dependent principles are proposed primarily on the basis of data concerning the perception of tones rather than chords. The perceived organization of tones is largely contextual. Krumhansl (1979) found that the tones that form the tonic triad of the tonal context are perceived as most closely related to each other and the nondiatonic tones least closely related, and that a pair of tones is perceived as more closely related when the first tone is nondiatonic and the second diatonic than vice versa. We hypothesize similar principles for the domain of chords: Contextual Distance and Contextual Asymmetry. Finally, the principle of Contextual Identity below is based mainly r I1 the finding that a tonal context increases the recognizability of a diatonic tone in that context (Krumhand, 1979). We will refer to the chords I-VII as the chords rhat are rirdmbers of a key, chords that function in a key, or simply as chords from a key. Strictly speaking, all chords have a harmonic function in any given key. However, we will simply deal with those triads built upon the seven notes of the diatonic scale, with all the triad tones drawn from the scale. We will refer to the prevailing context as either tonal or random. In a tonal context, a given key has been established as the prevailing tonality. Chords from this key will be referred to as being in the context key, chords not from this key as being out of the context key. In a random context, or with no context at all, no single key has been established as the prevailing tonality. The set consisting of the I, IV and V chords will be referred to as the harmonic core. 70 J. Bhamcha and C L. K~rumhansl in order to precisely state the harmonic principles, let the psychological distance between two chords, Ci and C2, in temporal succession be denoted by d(C,, C,). Let K denote the basic set of seven chords that are members of a key, and let C E K mean that the chord C is a member of K, and C $ K mean that it is not from K. The set, K, contains the seven triads built upon the seven degrees of the diatonic scale; the set, S, contains the three chords (I, IV, and V) of the harmonic core. There are three principles governing context independent relationships. The first principle, Key Membership, states that chords from the same key are perceived as more closely related than chords that are not from the same