Tracing the Dynamic Changes in Perceived Tonal ...

Psychological Review Copyright 1982 by the American Psychological Association, Inc.1982, Vol. 89, No. 4, 334-368 0033-295X/82/8904-0334$00.75

Tracing the Dynamic Changes in Perceived Tonal Organizationin a Spatial Representation of Musical Keys

Carol L. Krumhansl Edward J. KesslerCornell University Stanford University

The cognitive representation of harmonic and tonal structure in Western musicis investigated using a tone-profile technique. In this method listeners rate howwell single tones (any one of the 12 tones of the chromatic scale) follow a musicalelement such as a scale, chord, or cadence. Very stable rating profiles reflectingthe tonal hierarchies in major and minor keys are obtained, which, when inter-correlated and analyzed using multidimensional scaling, produce a four-dimen-sional spatial map of the distances between keys. The keys are located on thesurface of a torus, in which the circle of fifths and the parallel and relativerelations between major and minor keys are represented. In addition, singlechords (major, minor, diminished, and dominant seventh) are found to be closelyassociated with the major and minor keys in which they play harmonic functions.The developing and changing sense of key during sequences of chords is tracedby obtaining probe tone ratings following each chord in 10 different sequences,8 of which contain modulations (changes) between keys. Modulations betweenclosely related keys are found to be effected more immediately than are mod-ulations between relatively distant keys. In all cases beyond the initial chord, thesense of the prevailing key is stronger than that produced by the last heard chordin isolation. Thus, listeners integrate harmonic functions over multiple chords,developing a sense of key that may need to be reevaluated as additional chordsare sounded. It is suggested that the perceived relations between chords and keysand between different keys are mediated through an internal representation ofthe hierarchy of tonal functions of single tones in music.

Music consists of tones varying in pitch, serve to highlight rhythmic patterns, furtherduration, loudness, and timbre, but the per- emphasize phrase structure, and distinguishception of music extends well beyond the between tones constituting the primary me-registration of these physical attributes of lodic line and tones serving more ornamentalthe musical stimulus. Indeed, music contains or harmonic functions. Timbral character-considerable structure even in the relations istics may additionally provide importantthat obtain among the individual tones. For cues for the overall structure of the musicalexample, the durations are such that metri- composition.cal and rhythmic patterns emerge from the ln Western music pitch relationships aresuccession of tones, and these patterns in probably the most developed and essentialcombination with other features define nat- to defining organization. Despite the tre-ural boundaries, or phrases, within the mu- mendous variation found in the selection andsical composition. Variations in loudness ordering of pitches in the musical literature,

certain underlying regularities can be ob-This research was supported in part by a grant from served and described The perception of mu-

the National Science Foundation (BNS-81-03570) to gical structure depends on the processing ofthe first author. The authors are grateful to David M. . , . , ,. ... e * 7Green for the use of the facilities in the Psychophysics Pltch information with reference to a systemLaboratory at Harvard University, to Murray Spiegel of knowledge about the conventional USCS ofand David Wilson for technical advice and assistance, pitches within the musical tradition. Thus,and to Roger N. Shepard and an anonymous reviewer cognjtive processes are significantly involvedfW^S±5T!3iSd"^nftfSro, L. * ™ perception and empirical work inKrumhansl, Department of Psychology, Uris Hall, Cor- this area is directed at the description ofnell University, Ithaca, New York 14853. these processes and the internal represen-

334

TONAL ORGANIZATION IN MUSIC 335

tation of pitch relations essential for the as-similation of musical organization.

In our work we rely heavily on the char-acterization of pitch structure provided bymusic theorists whose analyses of musicalscores and introspective probings have yieldeda large body of theoretical writings. Whereasmuch of this work, such as Piston's (1962)Harmony, is limited to the description ofcompositional conventions in existing mu-sic, other treatments (e.g., Berry, 1976;Schenker, 1906/1954, 1935/1979; Schoen-berg, 1911/1978, 1969) point to the formthat a psychological model of music pro-cessing might take. Whether these intuitionswill in fact prove to elucidate the psycho-logical study of music perception is a matteryet to be decided, although preliminary ev-idence does suggest that listeners familiarwith Western music possess internal cogni-tive structures and processing strategies sim-liar to those presumed by music theorists.At a minimum, music theory provides a con-venient terminology for describing the struc-ture of music itself and is a rich source ofpredictions and hypotheses for empirical in-vestigation.

Music theorists suggest that pitch struc-ture in traditional Western music can bestbe understood in terms of the concept oftonality, or musical key, which will be de-scribed more fully later. In a tonal systemone single pitch, called the tonic, functionsas a central reference point for the systemas a whole. Every other tone, and each chord,plays a unique function with respect to thatabstract tonal center and these functions aresummarized in a hierarchy of structural sta-bility. Moreover, different musical keys havespecial associations to each other and aredescribed as more or less distantly related.This investigation provides a quantitative,empirical measure of the perceived relat-edness between the individual pitches and aninstantiated tonal center, replicating and ex-tending an earlier study by Krumhansl andShepard (1979). These quantitative resultsare then used to obtain a spatial map ofmusical keys, reflecting the distances be-tween the different keys. Next, we charac-terize the harmonic roles of individual chordswith respect to different tonal centers in thissystem. Finally, we trace in the obtained spa-

tial map how the sense of key develops andchanges over time when the listener hearswell-structured sequences of chords.

Music-Theoretic Descriptions of Tonality

Music-theoretic analyses are directed atspecifying the tonal functions of the musicalelements'—single tones and tones soundedsimultaneously as chords—in the prevailingtonality or key. Every musical key is asso-ciated with a hierarchy that applies to theset of musical pitches, 12 in each octaverange. This hierarchy distinguishes betweentones on the basis of their relative promi-nence or stability. The tonic tone is describedas the single most central and stable pitchwithin the set of musical pitches. This tonegives the name to the key; for example, thenote C is the tonic of the key of C. The tonicis also the first tone of the scale (sometimesreferred to as the first scale degree), is typ-ically sounded near the beginning of a pieceof music, and is the tone that most frequentlyoccurs at the end of major phrase bound-aries. Next in the tonal hierarchy are theother scale tones, usually conforming to theinterval patterns of the major or minor dia-tonic scale. Among these tones the fifth andthird scale degrees are relatively stable;these tones together with the tonic form themajor triad chord, which like the scale is adistinctive marker of the tonal center or key.Finally, the tones that are not containedwithin the scale, called nondiatonic, are theleast stable and appear infrequently and withless rhythmic stress. This tonal hierarchy isat the core of music-theoretic accounts ofstructure in tonal music.

In addition, tones are frequently soundedsimultaneously in chords, and a similar hi-erarchy is presumed to apply to the set ofchords. The most frequently used chords arethose consisting of three different pitches,which are therefore called triads. Triads areconstructed from a root tone, which is a noteof the scale, the tone a major or minor thirdabove the root, and a third tone a major orminor third above the second tone. The oc-tave equivalent of one or more of thesepitches is typically sounded simultaneously.The position of the root of the triad in thescale is used to designate the chord. For ex-

336 CAROL L. KRUMHANSL AND EDWARD J. KESSLER

ample, the chord built on the first scale de-gree is indicated by I, the chord built on thesecond by II, and so on. The chords built oneach of the seven scale degrees constitute thebasic set of harmonies of the key. The tonic,I, triad is described as the most stable chordwithin the system, followed by the V, ordominant chord, the IV, VI, and II chords,and finally the III and VII chords. Chordsoutside this set are called nondiatonic andare infrequently sounded; when present theydemand a return to the more stable elements.These hierarchies among tones and chordsare used to provide varying degrees of ten-sion and resolution at different points in timethroughout a composition, which is the fun-damental organizing principle of tonalmusic.

Tonal structure is even more complex thanjust outlined, for a hierarchy of levels is be-lieved to exist in the instantiation of tonalregions. Chomsky (1965) has postulated asurface and deep structure for language, andSchenker (1906/1954, 1935/1979) has sim-ilarly found tonal music to be analyzable onforeground, middle ground, and backgroundlevels. Three levels of key instantiation maybe distinguished: the primary key, a tem-porary alternative key, and the key of a sin-gle chord. The primary key is the major orminor key that is given emphasis and usuallybegins and ends the piece. In between, mod-ulations (changes to different keys) may es-tablish new though usually related keys.Moreover, each individual chord may takeon some of the properties of the tonic of akey, a process called "tonicization," whichdiffers from modulation by having a muchbriefer influence. We will return to theseideas later, as they become relevant for theinterpretation of certain experimental re-sults.

Psychological Models of Pitch Structure

Music theory suggests that musical struc-ture in tonal music can be accounted for bythe hierarchies that apply to the musical ele-ments and that each key is associated witha particular ordering of stability within theset of single tones and chords. A number ofpsychological investigations have obtainedquantitative empirical support for the inter-

nalization of these hierarchies by listenersfamiliar with tonal music. One study byKrumhansl and Shepard (1979) asked lis-teners to rate how well each different toneof the chromatic scale in an octave rangecompleted a seven-tone C major scale. Theratings given by listeners having the greatestfamiliarity with tonal music recovered thetonal hierarchy predicted by music theorists:The tonic tone received the highest ratings,followed by the third and fifth scale degrees,the remaining scale degrees, and finally thenondiatonic tones. This hierarchy was alsoobtained in a scaling study of tones presentedin a well-defined tonal context (Krumhansl,1979). In that study the tones most stablewithin the tonal system were perceived asmost closely interrelated, and those less sta-ble as more distantly related. Moreover, thislatter study revealed a pattern of temporalasymmetries in the judgments of pairs oftones that depended systematically on theirrelative stability in the instantiated key.Thus, in both these studies the pattern ofresults was highly specific to the hierarchythat applies to the tonality of the context.

One feature of these results should be em-phasized. In both studies it was found thatintervals of equal size were not perceived asequivalent but depended instead on the tonalfunctions of the individual tones within thecontext key. In the Krumhansl and Shepard(1979) scale-completion study, the tone acertain number of half steps above the im-plied tonic received a different rating thandid the tone an equal number of half stepsbelow the tonic. For instance, the note C#,which is one half step above the impliedtonic, C, was given a much lower rating thanwas the note B, which is a half step belowthe implied tonic. This may be understoodby the fact that the C# is nondiatonic withinthe context key, but the B plays the functionof the melodically significant leading tonewithin the key. Similarly, the scaling study(Krumhansl, 1979) found that the judgedmusical relatedness between two tones form-ing a fixed interval depended not simply oninterval size per se but on the tonal functionsof the individual tones. Again, for example,different ratings were given to intervals ofequal size, such as that formed by C and C*and that formed by C and B. These findings


indicate that some systematic transforma-tion must intervene between geometricallyregular representations of pitch structure,such as those proposed by Shepard (1982a;see also Shepard, 1981, 1982b), and thepitch relations as they are perceived in atonal context. Moreover, these modificationsmay be accounted for by the unique functionof each pitch within the instantiated key, asdescribed by music theorists.

There is also empirical support for themusic-theoretic description of a hierarchy ofstability that applies to chords. Two studies(Bharucha & Krumhansl, in press; Krum-hansl, Bharucha, & Kessler, 1982) have re-covered the predicted hierarchy in which thetonic, I, chord is the most stable, followedby the V chord, the IV, VI, and II chords,and finally the III and VII chords. This hi-erarchy has been obtained even in the ab-sence of a well-defined tonal context (Bhar-ucha & Krumhansl, in press). However, thejudged relations between chords were af-fected when a key context was provided(Bharucha & Krumhansl, in press; Krum-hansl, Bharucha, & Castellano, in press).These alterations in judgments, which aresummarized in three context-dependent prin-ciples, are a function of the distance betweenthe instantiated context key and the key inwhich the chords play harmonic roles. Aswith single tones, then, the perception ofchords depends significantly on their func-tions within the established key. This sug-gests that if invariant relationships are foundin music, they are neither at the level of sin-gle tones nor at the level of chords but in-stead may be found at the more abstractlevel of musical keys or tonal centers. Thefocus of the present investigation is the char-acterization of the psychological represen-tation of structure at the level of musicalkeys.

A number of psychological studies usingmemory performance provide further evi-dence for the importance of tonal organi-zation in music perception. Well-structuredtonal sequences have been found to be re-membered better than those not conformingto key structure. This result has been ob-tained using both melodic sequences (Cohen,1975; Dewar, 1974; Dewar, Cuddy, & Mew-hort, 1977; Frances, 1972) and harmonic

sequences, that is, chord sequences (Bhar-ucha & Krumhansl, in press; Krumhansl etal, in press). A number of results reflect thedifferential stability of diatonic and nondia-tonic elements. Tones and chords that arenondiatonic within an implied key have beenfound to be recognized correctly less oftenthan are diatonic tones and chords (Bharu-cha & Krumhansl, in press; Krumhansl,1979). Moreover, diatonic elements are morefrequently confused with other diatonic ele-ments than they are with nondiatonic ele-ments (Bharucha & Krumhansl, in press;Dewar, 1974; Dowling, 1978). Finally,asymmetries have been found such that non-diatonic tones and chords are more fre-quently confused with diatonic elementsthan diatonic elements are confused withthose that do not belong to the key (Bhar-ucha & Krumhansl, in press; Dowling &Bartlett, Note 1). The magnitude of each ofthese effects has been shown to depend sys-tematically on the choice of the context key(Krumhansl et a,!., in press). These memoryresults provide convergent evidence for thestructural relations obtained in the empiricalstudies described earlier.

Overview

In this article we investigate three closelyrelated aspects of the representation andprocessing of tonal structure, each of whichis presented in a separate section. In the firstwe obtain a quantitative measure of the de-gree to which each individual tone in an oc-tave range is related to an abstract tonalcenter. These quantitative measurements arethen used to derive a spatial map of themajor and minor keys, representing the dis-ftances between different tonal centers. In thesecond section we determine the relationsbetween chords of different types and thesetonal centers, representing this informationby points in the spatial representation oftonal regions. In the final section we assesshow the sense of key develops and changesover time during sequences of chords, someof which contain modulations (shifts) be-tween different keys. Our focus in the lasttwo sections is on the manner in whichchords and chord sequences relate to tonalregions as they are laid out in the spatial


representation of keys described in the firstsection. Each one of these sections beginswith a short description of the relevant musictheory, followed where possible by a briefsummary of previous empirical investiga-tions related to the issue considered in thatsection.

Distances Between Keys

Determining the psychological distancebetween keys is important to an understand-ing of the perception of music in the Westerntonal tradition. In music the juxtapositionof keys can be used to highlight structuralelements, provide contrasts of varying de-grees, and add expressive qualities to a piece(Berry, 1976, p. 15; Rosen, 1972, pp. 26, 27;Salzer, 1962, p. 20). Some keys are so closelyrelated that a change from one to anothercan hardly be detected, whereas others areso distantly related that shifts between themseem abrupt and unpleasant. The measure-ment of the psychological distances betweentonal centers may elucidate the nature of theinternal representation of musical structurethat underlies these perceptual phenomena.In addition, as suggested earlier, invariantrelations may be contained in the structureat the level of abstract tonal centers.

Music-Theoretic Descriptions of KeyDistance

Music theory presumes that several fea-tures of keys may determine the accessibilityor ease of modulation between them, whichis considered to be a rough measure of dis-tance between the keys. For major keys threedifferent features all consistently predict thesame pattern of interkey relatedness: thenumber of tones shared by their scales, thedifference in the number of sharps or flatsin the key signature, and the distance aroundthe circle of fifths. For the major keys thedifference in the key signatures is inverselyrelated to the number of shared scale tones.For example, two major keys that differ onlyin terms of one sharp or flat have six sharedscale tones. Thus, G major (one sharp) andF major (one flat) both differ from C major(no sharps or flats) in terms of one scale tone,and the tonic of G is a fifth above and thetonic of F a fifth below that of C major. D

major (two sharps) and B6 major (two flats)each have two scale degrees not shared byC major, and their tonics are separated bytwo fifths above and below the tonic of Cmajor. This fifth connection continues to in-dicate distance between major keys untilafter six fifths above or below any given keythe key of the tritone, the most distantly re-lated major key, is reached in both direc-tions. Because this fifth relationship foldsback on itself, it is conveniently pictured asa circle, called the circle of fifths, first pro-posed by Heinichen (Forte, 1979, p. 27).Thus, the distances between major keys arefairly easy to specify unambiguously.

Major and minor keys are often usedwithin the same piece, making the distancebetween keys of different modes an impor-tant issue. However, introduction of the mi-nor keys complicates the picture considera-bly. To every major key there correspondsa minor key that has the same key signature,called the relative minor, whose tonic is aminor third below (or a major sixth above)that of the major key. For example, A minoris the relative minor of C major. Despite theshared key signatures, the relative major andminor keys do not overlap completely in theirscale tones. The scale of the relative minorkey has in fact three different forms: theharmonic minor form in which the seventhscale tone is raised by a half step, the as-cending melodic form in which the sixth andseventh scale tones are each raised by halfsteps, and the descending melodic form (ornatural form) in which the scale tones areexactly those of the relative major key. Thesedifferent forms make impossible the precisedefinition of major-minor key distance asthe number of shared scale tones. Nonethe-less, a major key and its relative minor keyare considered closely related by virtue oftheir shared key signatures.

Additionally, there is a close tie betweena major key and the minor key built on thesame tonic, which is called the parallel mi-nor. This is true even though their key sig-natures differ in terms of three sharps orflats. For example, C major (with no sharpsor flats) has C minor (with three flats) as itsparallel minor. Despite the differences in keysignatures, these keys share most of thestructurally stable tones—the tonic, fourth,


and fifth scale degrees—as well as the sev-enth, leading tone in the harmonic and as-cending melodic forms. Possibly even moreimportant is the fact that they also share theessential dominant, V, chord, which preparesthe tonic of either of the parallel keys(Schoenberg, 1969, p. 51).

Thus, every major key has two closely re-lated minor keys—the relative and parallelminors. Or, looked at another way, any sin-gle minor key is closely tied to two differentmajor keys that are relatively distant on thecircle of fifths. This fact makes it impossibleto accommodate the minor keys within thescheme of the circle of fifths and has ledSchoenberg (1969, pp. 20, 30) to abandonthe circle of fifths model and to propose analternative formulation, shown in Figure 1.In this model there is a single central ref-erence key, with other keys seen as relatedregions. The figure shows the key regionssurrounding the C major and C minor keys.For both, the representation consists of al-ternating columns of major and minor keyssuch that each major key is flanked by itsrelative minor key on one side and its parallelminor key on the other. The keys within eachcolumn are separated by fifths. The minor-key model is clearly a subset of the one formajor keys. According to Schoenberg (1969,p. 30) a minor tonic does not maintain asdirect control over its regions as does a majortonic, and consequently, the number of di-rectly related regions is smaller for minorthan for major keys.

To anticipate our results somewhat, weobtain a spatial representation of keys thatis strikingly similar to the one proposed bySchoenberg (1969, pp. 20, 30). There aretwo important differences, however. First,Schoenberg's key regions are centered arounda particular major or minor key, and thus,this scheme does not simultaneously accom-modate all major and minor key relations.Instead, a different representation is re-quired for each key. In contrast, we obtaina map of key regions that depicts all interkeyrelations as distances between points in ametric space. Second, Schoenberg does notattempt to make fine discriminations in tonaldistance as, for instance, he equates the dis-tance between a major key and its paralleland relative minors. Despite his statements

SCHOENBERG'S CHARTS OF KEY REGIONS

C MAJOR KEY REGIONS

F'f "

c1

G«

,D d F f

A [ Q 0 c

E e | G | g

A*

B* b>

f1

O

C MINOR KEY REGIONS

Figure I. Schoenberg's spatial map of the key regionssurrounding C major and C minor keys (adapted fromSchoenberg, 1969). (The circle of fifths relation is rep-resented as the vertical dimension, and the parallel andrelative relations between major keys — indicated bycapital letters — and minor keys — indicated by lowercaseletters — as the horizontal dimension.)

that in common practice a major key iscloser to its relative than its parallel minor(Schoenberg, 1911/1978, p. 153; 1969, p.68), this difference is not structurally evidentin the model. (There is some debate on thispoint. Piston [1962] finds that parallel ma-jors and minors are practically identical andthat classical composition in the 18th and19th centuries tended "to regard the twomodes as simply two aspects of one tonality"[p. 145].) In any case, our method producesinterkey distances that permit precise com-parisons.

Empirical Studies of Key Distance

Although the issue of key distance has notbeen extensively investigated in psychology,there are a number of studies demonstratingkey distance effects. Memory performancefor transposed sequences has been found tobe better when the sequence is shifted to aclosely related key than to a distantly relatedkey (Bartlett & Dowling, 1980; Cohen,1975; Cuddy, Cohen, & Miller, 1979). Inthese studies listeners are required to judgewhether two melodic sequences are the same,except for transposition to a different key(which maintains the relative sizes of all the


intervals). Accuracy in this task is higherwhen the melody is transposed to a closelyrelated key. More recently, Krumhansl et al.(in press) have identified three principlesgoverning harmonic relations that depend onthe distance of the context key from the keyin which the chords have basic harmonicfunctions. The probability that a chord iscorrectly recognized was found to be a de-creasing function of this distance. Moreover,the psychological proximity of two differentchords in the same key, measured both interms of direct relatedness judgments andmemory confusions, decreased as interkeydistance increased. Finally, this distance wasfound to affect the magnitude and directionof temporal asymmetries in relatedness judg-ments and memory confusions. Althoughthese studies have not provided a measureof distance between musical keys, they dosuggest that the music-theoretic descriptionof structure at the level of tonal centers isapplicable to the process of music percep-tion.

The Present Approach to Determining KeyDistance

In our empirical determination of key dis-tances, we chose not to employ recognitionmemory for transposed sequences, which hasyielded key distance effects in previous stud-ies (Bartlett & Dowling, 1980; Cohen, 1975;Cuddy et al,, 1979). Three reasons for thiswere, first, in order to get very precise re-sults, we would need sequences that are uni-formly tonally unambiguous, which wouldbe difficult if not impossible to construct.Second, we were concerned that transposi-tions between closely related keys might oc-casionally be rejected owing to confusionswith tonal answers (Dowling, 1978). Finally,if both major and minor sequences were em-ployed, transpositions between them wouldresult in changed intervals in melodic se-quences or changed chord types in harmonicsequences, making the results difficult to in-terpret solely in terms of key distance. Es-timating key distances from changes in thepatterns of confusion errors or relatednessjudgments as a function of the context key(such as those observed in Bharucha &Krumhansl, in press; Krumhansl et al., inpress) would be quite indirect as well as ne-

cessitate a very large number of observa-tions.

For these reasons we chose to employ themethod introduced in the earlier scale-com-pletion study (Krumhansl & Shepard, 1979),which we will refer to here as the tone-profiletechnique. The results of that study, in whichlisteners rated how well each musical pitchin an octave range (the pitches of the chro-matic scale) completed a seven-note majorscale, were described earlier. The rating pro-files for the listeners with a moderate to highlevel of musical training revealed consider-able structure, which accorded well withmusic-theoretic descriptions of the tonal hi-erarchy. This result suggests that rating pro-files of this sort may serve as a distinctivemarker of tonal organization that couldserve as a basis for determining distancesbetween keys.

In the first experiment we obtained ratingprofiles for the 12 different musical elementslisted in Table 1. Each of these musical ele-ments was presented with each of the 12tones of the chromatic scale, which will becalled probe tones. In other words, everypossible element-probe combination waspresented and for each the listeners ratedhow well the probe tone fit, in a musicalsense, with the element just heard. This pro-cedure yielded a rating profile for each ofthe 12 elements studied. The tones andchords employed were constructed of pitchessounded in five octaves using an amplitudeenvelope that tapered off at both low andhigh ends of the frequency range (afterShepard, 1964).

Four precautions were taken to maximizethe stability of the rating profiles. First, thesame musical element appeared throughouta block of trials. Second, to minimize carry-over effects between blocks, each new ele-ment was presented in a different key fromthat of the previous block, and each blockbegan with a few practice trials to acclimatethe listener to the new element in the newkey. Third, the experiment contained exactreplications and also replications in whichthe element was transposed to a differentreference pitch. Fourth, listeners selected forthe experiment formed a relatively homo-geneous group in terms of musical back-ground, with a moderate to high degree ofmusical sophistication.


Table 1Musical Elements Used in Experiment 1

Element Example

Ascending major scale

Ascending harmonic minor scale

Major chord

Minor chord

Diminished chord

Dominant seventh chord

IV-V-I cadence in major

II-V-I cadence in major

VI-V-I cadence in major

IV-V-I cadence in minor

II-V-I cadence in minor

VI-V-I cadence in minor

D major scale: D E F* G A B C» D

A minor scale: A B C D E F G ' A

E major chord: E G* B

D minor chord: D F A

F* diminished chord: F* A C

Dominant seventh chord on G: G B D F

Cadence in C major: F major chord, G major chord, C major chord

Cadence in F major: G minor chord, C major chord, F major chord

Cadence in D major: B minor chord, A major chord, D major chord

Cadence in E minor: A minor chord, B major chord, E minor chord

Cadence in F* minor: G* diminished chord, C* major chord,F* minor chord

Cadence in B minor: G major chord, F' major chord, B minor chord

Experiment 1

Method

Subjects. Ten Harvard University undergraduatesserved as subjects and were paid at the rate of $3 perhour. Each subject participated in two experimental ses-sions lasting a total of approximately 2.5 hours. All sub-jects had a minimum of 5 years of formal musical in-struction. Responses to a questionnaire on musicalbackground indicated that the subjects had received in-struction on musical instruments and voice for an av-erage of 10.9 years. In terms of performing experience,the participants in the experiment had performed foran average of 12.0 years in instrumental groups and 2.1years in choral groups. Currently, subjects were playingmusic an average of 3.1 hours per week and listeningto music for 23.3 hours per week. One subject reportedhaving taken a "very basic" introductory course in musictheory, but the remaining subjects had no formal musictheory background. All subjects reported having normalhearing and no one reported having absolute pitch.

Apparatus. The stimulus tapes (Maxell UD 35-90)were recorded at 7.5 inches per second (19 cm per sec-ond) on a Sony TC-540 tape recorder. During the ex-perimental sessions the tapes were played back usingthe same tape recorder and Telephonies TDH-50 head-phones. All the stimuli were prepared digitally on aPDF-15 computer in the Psychophysics Laboratory atHarvard University and were played through a digital-to-analog converter under computer control. The digitalrepresentations were sampled at a rate of 5,000 pointsper second. A temporal amplitude envelope with a 10msec rise and decay reduced onset and offset clicks in-troduced by the hardware. Finally, a low-pass filter withcutoff set at 2450 Hz eliminated high-frequency noise.

Two kinds of musical units were produced: chords

and single tones. The major, minor, and diminishedchords each consisted of 15 sine wave components (threein each of five octaves), the dominant seventh chordsconsisted of 20 sine waves (four in each octave), and thesingle tones consisted of 5 sine waves (one in each oc-tave). The components were selected from the 12 notesof the chromatic scale, whose frequencies were spacedequally in log frequency (equal tempered tuning) andwere based on a 440 Hz A. The amplitudes of the se-lected components were determined by a loudness en-velope over the five-octave range (from 77.8 Hz to 2349Hz), which consisted of three parts: a gradually increas-ing section over the first octave and a half, a constantsection over the middle two octaves, and a symmetricallydecreasing section over the last octave and a half. Thecorresponding amplitude of the sine wave componentsat each frequency was determined using the amplitude-loudness curves of Fletcher and Munson (1933). Thismethod, originally introduced by Shepard (1964) anddescribed in detail elsewhere (Krumhansl et al., 1982),produces tones and chords with an organlike sound,without any clearly defined lowest or highest componentfrequencies. The composite chords and single tones wereplayed at 71 db (A) that, because of the difference innumber of sine wave components, necessitated a slightupward shift of the loudness envelope for the single tonesand a slight downward shift for the dominant seventhchords.

Stimulus materials. The 12 elements used in theexperiment, illustrated in Table 1, are ascending majorscale, ascending harmonic minor scale, major chord,minor chord, diminished chord, dominant seventh chord,and three cadences (IV-V-I, II-V-I, and VI-V-I) inboth major and minor keys. Each trial consisted of anelement followed by a probe tone. The same elementwas used in 14 consecutive trials to form a block oftrials. The first two trials in each block were for practice.


On the other 12 trials of the block, each possible probetone from the chromatic scale was paired with the ele-ment in a random order. A stimulus tape contained 12element blocks, one for each element. A replication tapecontained the identical elements as the first tape but ina different random order and with a different order ofprobe tones. A transposition tape used the same elementtypes but each was sounded in a different pitch rangethan on the original tapes. Again, there was a replicationtape with the transposed elements randomized as above.The subjects heard the four different tapes in differentorders such that during each of the two experimentalsessions, a subject heard one complete replication—ei-ther both original or both transposition tapes.

On each trial the element was heard first, followedby a 1-sec pause, and then the probe tone sounded for.5 sec. The timing of the different elements was as fol-lows: The tonic tones of the two scales (major andminor) were sounded for .5 sec, and the remaining scaletones for .25 sec, with a .19-sec pause between scaletones. The single-chord elements were played for .5 sec,as were the chords in the three-chord cadences. In thecadences there was approximately .25 sec betweenchords. A 4-sec interval between trials allowed subjectsto record their responses.

Procedure. At the start of the first session, subjectswere instructed to rate on a 7-point scale how well, ina musical sense, each probe tone fit into or went withthe musical element just heard. On this scale, 1 wasdesignated "fits poorly" and 7 was designated "fitswell," and subjects were encouraged to use the full rangeof the response scale. Subjects responded to three com-plete blocks of practice trials before beginning the ex-perimental tapes. After all experimental tapes had beenpresented, the subjects completed the short question-naire about their musical backgrounds.

Results and Discussion

The 12 types of musical elements listedin Table 1 were each probed by the 12 tonesof the chromatic scale. The set of judgmentsfor each of the musical elements will be re-ferred to as the profile for that element.These profiles will be analyzed in a numberof ways. First, the stabilities of the profilesare evaluated using intersubject correlationsand correlations between the profiles ob-tained for transposed elements. Second, theprofiles themselves are intercorrelated andseparate composite profiles are constructedfor major and minor keys. Finally, correla-tions between these appropriately shiftedmajor and minor profiles, providing a mea-sure of interkey relatedness, are analyzedusing multidimensional scaling, yielding aspatial map of the 24 major and minor keys.The discussion of chord-key relations thatderive from the obtained profiles will be de-ferred until the next section.

Intersubject differences. No large indi-

vidual differences were expected because thesubjects were selected to be comparable interms of their musical backgrounds. In orderto evaluate consistency across listeners, in-tersubject correlations on the response pro-files (averaged over the two exact replica-tions) were computed. These correlationsaveraged .585, ranging from .412 to .740,and were all significant (p < .01). (The cor-relations between the two replications foreach subject ranged from .241 to .658, av-eraging .492, suggesting that error variabil-ity may be substantially attenuating the in-tersubject correlations that were computedon the responses averaged over the two rep-lications.) Multidimensional scaling (Krus-kal, 1964; Shepard, 1962a, 1962b; the com-puter program used was KYST—Kruskal,Young, & Seery, Note 2) and hierarchicalclustering (Johnson, 1967; the program usedwas AGCLUS—Olivier, Note 3) techniqueswere applied to the matrix of intersubjectcorrelations. Neither of these analyses pro-duced solutions that grouped listeners to-gether in a way that related to any aspectof their musical backgrounds. Consequently,individual differences were not explored fur-ther, and the remaining analyses are basedon the rating profiles averaged across the 10subjects.

Transpositions. Each element was rep-resented in two different pitch ranges in theexperiment; these are called transpositionreplications. Although subjects, particularlythose with less musical training, have beenfound to have some difficulty producingtranspositions of tone sequences (Attneave& Olson, 1971), it is commonly believed thattranspositions are perceived as equivalent,except perhaps by listeners with absolutepitch. Because not one of our subjects hadabsolute pitch, high correlations were ex-pected between the ratings when the profileswere shifted appropriately to compensate forthe interval of transposition. These trans-position correlations were high, averaging.839, so the profiles used in the remaininganalyses were those obtained by averagingthe transposed ratings.

Major and minor key profiles. One cen-tral purpose of this experiment was to es-tablish reliable profiles for major and minorkeys so that these could be used in later anal-yses. In addition, the major key profile from


this experiment can be compared with thatof Krumhansl and Shepard (1979). Corre-lations between the 12 different profilesshowed a very similar pattern of probe tonejudgments for the major chord element andthe three cadences in major, IV-V-I, II-V-I, and VI-V-I. The average correlation be-tween these (.896, p < .01) indicates a con-sistent pattern of ratings for these elements.The profile for the major scale was somewhatless similar, having an average correlationof .796 with the other major profiles. Con-sequently, we will take as the major key pro-file the average ratings given the 12 probetones for the major chord and the three ca-dences in major. This profile is shown at thetop of Figure 2, where the average probetone judgments are plotted with respect toC major. Of course, the profile for any othermajor key would be identical, except shiftedthe appropriate number of semitones up ordown. Similarly, the profiles for the minorchord and the cadences in minor were verymuch alike, with an average correlation of.910 (p < .01). Again, the minor scale profilewas less similar, with an average correlationof .727 with the others. Thus, the minor keyprofile, also shown in Figure 2, is the averageof the profiles for the minor chord and theminor cadences.

The major and minor key profiles sharea number of features with each other andalso with the scale-completion judgments ofKrumhansl and Shepard (1979). In both keyprofiles the tonic, C, received higher ratingsthan all the other tones: t(9) = 16.84 formajor, and t(9) = 13.42 for minor, p < .001for both. In addition, all nontonic scale tones(using the harmonic form for minor) hadhigher average ratings than did nondiatonictones: *(9) = 6.05 and 9.23, /?<.001, formajor and minor, respectively. Within theset of diatonic tones, the components of thetonic chord, C, E, and G in major and C,E* (D#), and G in minor, were judged asfitting more closely with the major and mi-nor elements than were the other diatonictones: *(9) = 16.28 and 9.77, p < .001, formajor and minor, respectively. Thus, the rat-ings in this study strongly confirmed the hi-erarchy obtained by Krumhansl and Shep-ard (1979). This hierarchy was also evidentin the multidimensional scaling solution oftones judged in a major scale context

MAJOR KEY PROFILE

I I I I I I I IC C" D D« E F F« G G1 A A( B

PROBE TONE

Figure 2. The obtained major and minor key profilesfrom the first experiment. (The profile for the major key[upper graph] is the average rating given each of the12 tones of the chromatic scale following a major chordand the three cadences [IV-V-I, II-V-I, and VI-V-I]in a major key. The minor key profile [lower graph] isaveraged over the minor chord and the three cadencesin minor. The profiles are shown with respect to C majorand minor, respectively.)

(Krumhansl, 1979) and is entirely consistentwith the qualitative predictions of musictheory.

Interkey distance. We next used the ma-jor and minor key profiles, which have beenshown to be extremely reliable and inter-pretable, as an indirect measure of interkeydistance. The correlations between the pro-files, shifted to the appropriate tonics, aretaken as a measure of this distance. To il-lustrate this process, Figure 3 shows the pro-file for C major superimposed on the profilefor A minor (the C minor profile shifteddown three half steps or equivalently up ninehalf steps). The ratings of the two profileswere then correlated. These two particularprofiles were quite similar, giving a high cor-relation, as would be expected for a majorkey and its relative minor. This same pro-cedure was applied to all major-major, ma-


-C MAJOR PROFILE-A MINOR PROFILE

C C' 0 D1 E F F' G G" a A1 B

PROBE TONE

Figure 3. The C major key profile and the A minor keyprofile are superimposed to illustrate the derivation ofthe interkey distances. (The A minor profile was ob-tained by shifting the C minor profile down three stepsof the chromatic scale. The values of the two profileswere then correlated to give a measure of the distancebetween C major and A minor keys.)

jor-minor, and minor-minor key pairs. Theobtained pattern of correlations was ex-tremely regular and reflected to a strikingdegree the several features discussed earlierthat are supposed by music theorists to de-termine key distance. (Other measures ofkey distance, such as the summed squareddistances between the rating profiles, werealso explored and produced almost identicalresults). The regularities in these measuresof key distance are brought out more clearlyin the following analyses, however, so a fulldiscussion will be postponed until later.

Multidimensional scaling of major andminor profile correlations. The interkeycorrelation matrix was then analyzed us-ing the multidimensional scaling programMDSCAL (Kruskal, Note 4). A satisfactorysolution was obtained only in four dimen-sions, with a stress value (Formula 1) of.017. The standard, principal-axis orienta-tion option was employed. This solution isshown in its 2 two-dimensional projectionsin Figure 4. In the first two dimensions, themajor and minor keys are arranged arounda circle so that any major key is flanked oneither side by its dominant and subdominantmajor keys (whose tonics are a fifth aboveand below that of the first major key). Thesame holds true for the minor keys. Thus,in the first two dimensions we obtain thecircle of fifths for major and minor keys.

However, the placement of the minor keycircle with respect to the major key circleposes problems for interpretation. In the so-lution the closest minor key to any major key

is neither its relative nor its parallel minorbut instead the minor key built on the secondscale degree (the relative minor of the sub-dominant). This placement undoubtedly re-flects a compromise between the close tie ofa major key to both its relative and parallelminor keys, which are separated by threesteps around the circle of fifths. (For ex-ample, the C major profile correlates stronglywith both the A minor and C minor profilesbut less strongly with that of D minor). Be-cause of this tension produced by the relativeand parallel relations, the two-dimensionalsolution (which was almost identical to theleft panel of Figure 4) was rejected in favorof the four-dimensional solution picturedhere.

The projection in the third and fourth di-mensions represents the parallel and relativerelations between major and minor keys di-rectly. The keys again form a circular con-figuration, but with a slightly smaller radius.On this circle any given major key is flankedby its parallel minor on one side and by itsrelative minor on the other, and likewise, anyminor key is located next to its parallel andrelative major keys. In order to accomplishthis, the solution identified as a single pointall major keys whose tonics differ by a majorthird and also all minor keys differing by amajor third. The musical interpretation ofthis identification is not immediately ob-vious; however, viewing this four-dimen-sional solution in a slightly different butequivalent way reveals much more clearlythe structure of the interkey relations.

Toroidal representation of key distance.The low stress value of the four-dimensionalsolution is evidence that this representationprecisely accounts for the intercorrelationsbetween profiles. However, the 2 two-di-mensional projections posed problems forinterpretation. In each of the two-dimen-sional projections, circular configurationswere obtained. Thus, the points fall on a sur-face that can be described as a circle in twodimensions crossed with a circle in two moredimensions. But this is exactly the topolog-ical definition of a torus (S1 XS1). Eachpoint in four dimensions can be uniquelydescribed by the angular distance, 9, aroundone circle and the angular distance, 0,around the second circle. The (6, 4>) valuesfor each key can then be plotted in two di-

TONAL ORGANIZATION IN MUSIC

MULTIDIMENSIONAL SCALING SOLUTION OF KEYS

345

ft ««Db ^g» f Ek

Bc' c gi,

E»i

D e G°

g»Al>E c e

f c8 BrEkf a G

bOk ~ g d*

F A

fc DBk

d b* p»

DIMENSIONS I AND 2 DIMENSIONS 3 AND 4

Figure 4, The four-dimensional multidimensional scaling solution of the intercorrelations between the24 major and minor key profiles (stress = .017). (The projection of the solution onto the first twodimensions is shown on the left. In this projection the circle of fifths for major and minor keys wasobtained. The projection onto the last two dimensions is shown on the right. Major and minor keysseparated by an interval of a major third were represented as single points in the solution. Each majorkey was located next to its parallel minor key on one side and its relative minor key on the other.Similarly, each minor key was flanked by its parallel and relative major keys.)

mensions as shown in Figure 5, where it isunderstood that the opposite edges of therectangle are identified.

To see the similarity between this repre-sentation and the more familiar (but mis-leading) description of a torus as an inner-tube, first visualize folding the rectangle overto identify the two horizontal edges, makinga hollow tube, followed by wrapping itaround to line up the two open ends. Notonly does this introduce distortions of dis-tances (the inner radius is smaller than theouter radius) but it also leads to false intu-itions about the interdependence of coordi-nates in the different dimensions. For thesereasons the flattened out representationshown in Figure 5 is preferable.

The (B, <j>) coordinates of the 24 keys werecomputed from the four-dimensional scalingsolution and are shown in the torus repre-sentation in Figure 5. Because of small de-viations from perfect circularity in the ob-tained scaling solution, ideal coordinateswere computed for each key such that thepoints were constrained to fall on the surfaceof the torus. The program CMH2 (Cliff,1966), which shifts, rotates, and expands orcontracts one configuration to best match asecond configuration, was used to comparethe ideal configuration with that actuallyobtained in the scaling solution. This pro-

gram gave a correlation of .998, indicatingthat the torus is in fact a very accurate rep-resentation of the four-dimensional scalingsolution of the profile correlations.

More importantly, when viewed as (6,<j>) coordinates on a torus, the pattern of in-terkey distances becomes entirely interpret-able. First of all, the keys separated by fifthsfall on a path that wraps three times aroundthe torus before joining up with itself again;the major keys fall on one such path, andthe minor keys on another, parallel path.These are lined up so that any major key isflanked by its relative minor on one side andits parallel minor on the other. These par-allel, relative, and fifth relations are madeexplicit for C major in Figure 5.

There is a striking similarity between themap of tonal regions given by Schoenberg(1969) for major and minor keys (Figure 1)and local regions of the toroidal represen-tation obtained here (Figure 5). The torus,however, has the advantage of simulta-neously depicting all interkey relations, notjust those immediately surrounding a singlemajor or minor key. Moreover, precise quan-titative comparisons between interkey dis-tances can be made from the torus repre-sentation. For example, a major key is infact found to be more closely related to itsrelative than to its parallel minor as seen,

346

ek/d

CAROL L. KRUMHANSL AND EDWARD J. KESSLER

f » d b k -

qt

a ' fx /

Relative^ /

V«. 1 Parallel A"

/ Circle of fifths

eFigure 5. An equivalent two-dimensional map of the multidimensional scaling solution of the 24 majorand minor keys. (The vertical [dashed] edges are identified and the horizontal [solid] edges are identified,giving a torus. The circle of fifths and parallel and relative relations for the C major key are noted.)

for instance, as the closer distance betweenC major and A minor than between C majorand C minor. Another finding is that a majorkey is closer to the minor key built on itsthird scale degree (the relative minor of thedominant key) than it is to the minor keybuilt on its second scale degree (the relativeminor of the subdominant key). For in-stance, C major and E minor are closer thanare C major and D minor. In fact, this wasanticipated by Schoenberg (1969, p. 68) andmay reflect the greater number of chordsshared by the major key and the relativeminor of the dominant key. Other compar-isons of this sort can easily be made usingthis spatial map of key regions. Moreover,this representation provides a framework forapproaching the problem taken up later ofhow chords relate to different tonal centersand how the sense of key develops as thelistener hears sequences of chords.

Other spatial representations. Other pre-viously proposed spatial representations havebeen described in detail by Shepard (1982a;see also Shepard, 1981, 1982b) and will bementioned only briefly here. The first suchrepresentation in the literature is the helicalstructure of single tones (Drobisch, 1846,cited in Ruckmick, 1929; Pickler, 1966; Re-vesz, 1954; Shepard, 1964). In this three-dimensional configuration, the single tonesare spaced along a helical path in order of

increasing pitch height such that tones sep-arated by an octave interval are relativelyclose as the helix winds back over itself onsuccessive turns. The projection of the pointson the plane perpendicular to the verticalaxis of the helix is often referred to as "tonechroma," and the projection on the axis as"tone height." This representation, then, si-multaneously specifies the close relation be-tween tones separated by small intervals andthat between tones at octave intervals. Thishelical representation is preferable for mu-sical pitches to the unidimensional psycho-physical scale of pitch that combines bothfrequency and log frequency as proposed byStevens (Stevens & Volkmann, 1940; Ste-vens, Volkmann, & Newman, 1937), be-cause there is a strong identification betweentones differing by octaves. This octave effectis seen in their interchangeable use in musictheory and composition and in judgments ofintertone relatedness (Allen, 1967; Boring,1942, p. 376; Krumhansl, 1979; Licklider,1951, pp. 1003-1004; and numerous othertreatments). Shepard (1982a) argues thatthe tones should be equally spaced in termsof log frequency around the helix becauseboth the selection of musical tones and per-formance in transposition tasks (Attneave& Olson, 1971) indicate that the log fre-quency scale applies to musical pitches.

Shepard (1981, 1982a, 1982b) has re-


cently proposed more complex representa-tions of single tones. In the representationreferred to as the "melodic map" (Shepard,1982a, Figure 4), the set of 12 tones in anoctave range are placed on the surface of thetorus in four dimensions. As in the config-uration obtained here, the first two dimen-sions correspond to the circle of fifths. Thelast two dimensions constitute the chromacircle of the helical structure proposed ear-lier. However, introducing the circle of fifthssplits apart the chroma circle into two whole-tone scales forming a double helix that wrapsaround the surface of the torus. Shepardprovided evidence that an additional fifthdimension might be needed to account fortone height.

Empirical support for Shepard's melodicmap comes from the earlier scale-completionstudy (Krumhansl & Shepard, 1979). Thereare a number of reasons why the presentanalysis, which is based on similar data, pro-duces different results. The first reason isthat the method of obtaining the matrix ofinterelement distances was different in thetwo cases. In the Shepard analysis the valuefor tones separated by each number of halfsteps was the average of the rating given tothe tone that number of half steps above theimplied tonic and the rating given to the tonethat number of half steps below the tonic.As noted earlier, however, intervals of equalsize often received significantly different rat-ings, and these differences could be under-stood in terms of their functional interpre-tations within the tonal hierarchy of theinstantiated key. Consequently, Shepard'sanalysis loses considerable structure con-tained in the rating profiles. The presentanalysis used instead the correlation betweenthe shifted profiles as the distance measure.This latter measure is most properly inter-preted as a measure of interkey distance be-cause it is based on the similarity of tonalfunctions of the individual musical tones inthe different keys. A second, related reasonfor differences between the two representa-tions is that tone height does not apply toan abstract tonal center, because it is inde-pendent of the octave in which the tonic issounded. Moreover, the probe tones and con-text elements used in this study were con-structed of frequencies sounded in five oc-taves simultaneously, thus minimizing tone

height effects. Nonetheless, it is significantthat the chroma circle did not emerge fromthe present analysis because Shepard (1964)demonstrated that this circular dimension isindependent of tone height.

Shepard's (1982a, Figure 8) "harmonicmap," which is a transformation of the me-lodic map, is identical to the spatial repre-sentation obtained here except for scale fac-tors and the inclusion of points for major andminor keys in the present results. Here, how-ever, we give this configuration a differentinterpretation. Shepard's proposed represen-tation depicts the relations between singletones, with certain chord, scale, and key re-lations reflected as patterned subsets withinthe spatial map. Thus, this repesentationbrings out structures at all three levels: singlepitches, chords, and keys. In contrast, weinterpret the spatial configuration obtainedhere as depicting structure at only the levelof abstract tonal centers and presume thatadditional or other structural features applyto single tones and chords, particularly thosethat reflect the context-specific hierarchy ofstability described by music theorists. Em-pirical evidence supporting this view hasbeen obtained in a number of studies (Bhar-ucha & Krumhansl, in press; Krumhansl,1979; Krumhansl et al, 1982, in press;Krumhansl & Shepard, 1979) and was sum-marized earlier in this article.

That Shepard's (1982a) harmonic mapappears as a subset of the spatial represen-tation of musical keys obtained here, how-ever, may be accounted for by the centralfunction of the tonic tone in the structuralhierarchy that applies to each maj6r andminor key. In other words, the convergencebetween Shepard's harmonic map and thetoroidal map of musical keys may be under-stood in terms of the close association be-tween an abstact tonal center and the tonethat is its tonic. To test this hypothesis anadditional analysis of the profiles obtainedin the present experiment was performed. Inthis analysis a measure of the psychologicalproximity between single tones was esti-mated by correlating the ratings given to onetone in the 24 major and minor keys withthe ratings given to the other tone in the 24keys. This measure reflects how similar thetwo tones are in their tonal functions acrossall possible musical keys. Multidimensional


scaling applied to these correlations re-covered the spatial configuration of Shep-ard's harmonic map (stress = .007), The firsttwo dimensions contained the circle of fifths,and the second two dimensions contained acircular configuration in which tones differ-ing by a major third were identified as asingle point; the ratio of the radii was 4:3.Thus, Shepard's harmonic map may be ob-tained from the probe tone ratings and maybe understood as being derivative of struc-ture at the level of abstract tonal centers andmediated through the central function of thetonic tone in the hierarchy of stability.

Relations Between Chords and Tones

In much of tonal music, groups of tonesare sounded simultaneously in chords. Al-though it is a straightforward matter to spec-ify the chords that constitute the basic setof harmonies of a particular key, as de-scribed earlier, the converse problem of de-termining the prevailing key from a sequenceof chords is much more difficult. The key ofa piece of music is not explicit in the music,except as the key signature of the writtenscore, and even this does not distinguish be-tween major and minor modes. The listenermust extract key information from the se-quence of chords itself, using knowledgeabout the harmonic functions of chords inthe different musical keys. In this section weturn to the problem of characterizing thelistener's knowledge of these harmonic func-tions.

Music-Theoretic Descriptions of ChordFunctions

Seven chords form the basic set of har-monies of a particular key and are oftendenoted by roman numerals to designate theposition of the root of the chord in the scaleof the key. The tones of the chords are usu-ally selected from the major scale for majorkeys and from the harmonic minor scale forminor keys. Because of differences betweenthe intervals of the majqr and minor scales,different chord types appear in major andminor keys. In a major key, I, IV, and V aremajor chords; II, III, and VI are minor; andthe VII chord is diminished. A minor keyhas major chords as V and VI, minor chordsas I and IV, diminished chords as II and VII,and an augmented chord as III, which, how-

ever, is often used in its major form. (Forthis reason we do not consider the aug-mented chord further here.) Despite thesedifferences in chord types, the roman nu-meral notation used in music theory impliesthere is a similarity in the effect of chordsbuilt on the same scale degrees in major andminor keys. In practice, usually a fourth tonerepeating one of the other three in a differentoctave is also simultaneously sounded. Inaddition, the other chord tones may beplaced in any different octave, yielding dif-ferent inversions. These inversions differ interms of the lowest bass tone and in the spe-cific intervals contained in the chords.Whether these inversions produce signifi-cantly distinctive effects or simply serve tointroduce greater variety is a matter of somedebate (see, for example, Rameau, 1722/1971; Schoenberg, 1969, p. 6). The issue ofchord inversions will not be considered fur-ther here because our experiments employedchords with each component sounded in fiveoctaves, using an amplitude envelope thattapered off at both low and high ends of thefrequency range.

Not only are there restrictions on the par-ticular chords used in a key but there arealso restrictions on the orders in which theycan appear. Moreover, certain chords withinthe set are described as playing more struc-turally significant roles than others, andthese are typically sounded more frequentlyand in more prominent temporal positions.In Western music the key sense is primarilya product of chords used in this highly con-strained fashion. Despite these conventionsthere usually remains some residual ambi-guity of key^ At the most basic level, thisambiguity results because any given chord(and sometimes even longer sequences ofchords) can simultaneously function in morethan one key. In fact, major and minorchords each play roles in at least five differ-ent keys, and a diminished chord in threekeys. For instance, a C major chord canfunction as I in C major, V in F major, IVin G major, V in F minor, or as VI in Eminor. For Berry (1976) the dual or multiplemeanings of harmonies are fundamental totonal music:

The designation by roman numeral symbols of a har-mony in two or three lights . . . is not an inconsistency;it is a realistic representation of the range of functional


significances attending a harmony of both primary andsecondary contexts of tonal reference. The indication ofsuch multiple functions is a reflection of the depth oftonal-harmonic meaning and of the way harmony isheard in tonal music, (p. 67)

Thus, any single chord may be interpretedas simultaneously playing various differentharmonic functions, but there may be a ten-dency to assign the chord one single mostlikely role. Schenker (1906/1954) notes thetendency to "ascribe to any major or minortriad, first of all, the meaning of the tonic"(p. 252). This interpretation may, however,need to be reevaluated as subsequent chordsare heard that are incompatible with thepreviously assigned harmonic function. InWestern music, then, there is a complicatedset of circumstances that leads to the des-ignation of the key of a piece of music.

Empirical Studies of Chord Functions

Previous investigations of chord functionshave focused on the relations between dif-ferent chords rather than those betweenchords and abstract tonal centers. Nonethe-less, the results of these studies on chordshave shown clear influences of organizationat the level of tonal centers. One scalingstudy (Krumhansl et al., 1982) used the setof chords from three closely related keys: Cmajor, G major, and A minor. In that studythe chords of the three keys were shown tobe organized in an overlapping fashion thatpreserved the internal chord hierarchy withineach key, despite the multiple functioningof chords in several keys. The chords of Gmajor and A minor that are shared with Cmajor were centrally located, whereas thoseunique to G major were separated from thoseunique to A minor, which is relatively remotefrom G major. In a complementary studyBharucha and Krumhansl (in press) inves-tigated the chords of maximally distant ma-jor keys, finding a clear separation betweenthe chords of distant keys. This separationwas also reflected in memory confusions.Finally, Krumhansl et al. (in press) observedsystematic alterations of perceived interre-lations between chords as the context keywas varied. These effects were a function ofthe distance between the key in which thechords function and the key of the context,as measured by the number of steps aroundthe circle of fifths. (Only major key contexts,

whose distances can be described by the cir-cle of fifths, were used in that study.) Thus,there is considerable evidence from theseempirical investigations that chords andtheir interrelations are perceived with re-spect to a system of related tonal centers.

The Present Approach to RepresentingChord Functions

The following analysis addresses the prob-lem of how the perceived relation betweena chord and its abstract tonal center, thatis, its harmonic function, may be representedusing the spatial map of musical keys de-scribed in the last section. We test the spe-cific hypothesis that a chord is perceived asclosely related to the abstract tonal centersin which it plays its most harmonically sig-nificant roles. If this is the case, then inter-key distances in the spatial map of the majorand minor keys must reflect the multiple in-terpretations that are possible for each singlechord. In other words, keys sharing chordsmust be located in proximal positions on thesurface of the torus. The profiles obtainedfor major, minor, diminished, and dominantseventh chords using the method describedearlier are used to test these predictions.


Chord-key profile correlations. Each ofthe four chord profiles (for major, minor,diminished, and dominant seventh chords)was correlated with each of the compositemajor and minor key profiles (Figure 2). Thekey profiles were shifted to all possible tonicsso that the relation between each of the 4chords and each of the 24 major and minorkeys could be determined. Not surprisingly,the major chord profile correlated highlywith the key in which it is the I chord, asdid the minor chord with the correspondingminor key. This would be expected becausethe chord profiles and the cadence profileswere averaged to determine the key profiles,and the tonic interpretation is assumed mostlikely for major or minor chords. Other pat-terns in the correlations are brought outmore clearly in the next analysis.

Chord positions on the torus. Earlier, theargument was made that keys sharing dhordsshould be perceived as closely related be-


cause this would enable listeners to entertainmultiple functional roles in different keys.The torus representation provides a conve-nient framework for testing this predictionas well as for representing the relativestrengths with which a single chord is inter-preted in its different possible roles.

The correlations between the chord andkey profiles were analyzed using the programPREFMAP (Chang & Carroll, Note 5; seeCarroll, 1972, for a more complete descrip-tion of the method). This program deter-mines a point or vector in multidimensionalspace so as to best reflect the psychologicaldistances (or preferences) between an objectand a set of other objects. The set of otherobjects is also represented as points in thespace in an a priori configuration that isoften a previously obtained multidimen-sional scaling solution. The a priori config-uration used here was the idealized coordi-nates of the musical keys on the torus asdescribed earlier. We elected to use thePhase 4 option to find a vector to representthe correlations between each obtained pro-file and that of each of the 24 major andminor keys. This vector was determined inthe first two dimensions separately from thelast two dimensions because this ensures thatthe combined vector in four-dimensionalspace would in fact intersect the torus. (Fit-ting the correlations simultaneously in allfour dimensions, using the Phase 4 optionand using the ideal point option of Phase 3,was also tried in various cases, yielding verysimilar results.)

The results for the C major, A minor, andB diminished chords are shown in Figure 6.These chords were selected for visual pre-sentation because they all function in the Cmajor key, which is a convenient referencekey. Analogous results would hold for anyother chords. The point shown for the Cmajor chord at the top of Figure 6 is theintersection with the torus of the vector ob-tained in the first two dimensions and thevector obtained in the last two dimensions.That this is an accurate representation of thecorrelations between the chord and the keysis indicated by the measure of fit for the firsttwo and last two dimensions (r = .975 and.585, respectively). As expected, the C majorchord is very close to the C major key inwhich it functions as the I chord. It is, how-

ever, drawn slightly toward the F major andF minor keys in which it plays the strongdominant, V, function. The C major chordis also the IV of G major and VI of E minor,which are its slightly weaker harmonic roles,and these keys are more distant from the Cmajor chord. Thus, we see that the positionof the C major chord is a compromise of itsroles in the different keys, but it is locatedrelatively close to all keys in which it func-tions.

The center of Figure 6 shows the positionof the A minor chord in the map of key re-gions. The goodness-of-fit measure wasslightly lower in the first two dimensions,r = .923, and slightly higher in the last twodimensions, r = .615. Again, the strongestinterpretation of the chord is as the tonic,I, of A minor. In addition, its roles in F major(as III), C major (as VI), G major (as II),and E minor (as IV) are also reflected bythe close distance between the A minorchord and these keys. That these functionsdo not draw the chord point closer to thesekeys may be a result of the strong tendencyfor the parallel major key, A, to dominateits parallel minor. Schenker (1906/1954)notes the tendency for "every passage inminor to be resolved into major" (p. 54).Given this, and the strong parallel major-minor relation, the position of the minorchord can be understood as reflecting thetension caused by the opposite forces pullingit toward its relative and parallel major keys.

The position of the B diminished chord isshown at the bottom of Figure 6. For thischord the measure of fit was .963 in the firsttwo dimensions and .636 in the last two. Thisdiminished chord is located near C major,in which it is the VII chord; C minor, inwhich it is also the VII chord; and A minor,in which it is the II chord. That it is drawnslightly toward G major may reflect the over-lap between the B diminished chord and thetonic, I, chord of G major. The position onthe torus of the dominant seventh chord wasalso determined. The vectors obtained forthe dominant seventh chord built on G(which is the V7 of C major) intersected thetorus at a point between C and G major,almost identical to the position of the B di-minished chord. This confirms the interpre-tation of the leading tone triad, VII, as thedominant seventh chord with a missing root.


Thus, this result supports the music-theo-retic description of these two chords as func-tionally similar.

In sum, this analysis of chord functionsshowed that major, minor, diminished, anddominant seventh chords are perceived asclosely related to the tonal centers in whichthey play significant harmonic roles. Theprecise position obtained for chords thatfunction in multiple keys represents a com-promise of the combined influences of theirvarious roles in these keys. Moreover, thisanalysis made clear the fact that keys shar-ing chords are located in proximal positionson the surface of the torus. In this is seenthe intimate connection between chords andkeys in tonal music. That an extremely reg-ular representation of interkey distances wasobtained may be understood as resulting inpart from the complex interlocking patternof multiple chord functions. The relative po-sitions of the keys on the torus, then, dependto some extent on the chords that simulta-neously function in different tonal centers.Also, the perceptual coherence achieved dur-ing well-structured chord sequences wouldseem to result because each of the chords inthe key and in closely related keys is locatedproximally in this representation, a propertyalso obtained in the scaling studies of chords(Bharucha & Krumhansl, in press; Krum-hansl et al, 1982, in press). The pattern ofchord-key relations, then, substantiates ear-lier characterizations of harmonic relationsbetween chords, partially explains the struc-ture found at the level of abstract tonal cen-ters, and indicates a means through whicha sense of key might evolve during the per-ception of harmonic sequences. We turn nowto an investigation that traces how the keysense develops and changes as a well-struc-tured sequence of chords unfolds in time.

Chord Sequences: The Developing andChanging Sense of Key

Neisser (1976) has described perceivingas an ongoing process in which the internalschema is continuously modified as externalstimulus information becomes available:

The listener develops more or less specific readinesses(anticipations) for what will come next, based on in-formation he has already picked up. These anticipa-tions—which themselves must be formulated in termsof temporal patterns, not of isolated moments—govern

Figure 6. The relations between the C major, A minor,and B diminished chords and the 24 keys are illustratedin the top, middle, and bottom panels, respectively. (Thepoint for each chord is the intersection of the vectorsobtained by PREFMAP and the torus representation ofthe keys. For each chord its harmonic functions in thekeys are indicated by roman numerals. The dominantseventh chord on G [not shown] was located in approx-imately the same position as the B diminished chord.)

what he will pick up next, and in turn are modified byit. (p. 27)

In music perception this continually inter-active process between the internal schemaand external stimulus may perhaps be seenmost clearly as the developing and changingkey sense when a harmonic sequence is beingheard over time. We have already describedthe process of perceiving tonal organizationas one in which chords are interpreted intheir multiple functions, giving rise to cer-tain hypotheses about a likely key region andinducing expectations that its closely asso-ciated tones and chords will be sounded.However, the hypothesized tonal center mayneed to be reevaluated as the sequence pro-gresses. Given the spatial map of key re-gions, and the inherently temporal structure


of music, we are in a unique position to in-vestigate the dynamic changes that occur inthe internal representation during the per-ception of harmonic sequences. In particu-lar, we are concerned with the question ofhow the sense of key is initially extractedfrom the sequence and how modulations(shifts) between tonal regions are assimi-lated.

Music-Theoretic Descriptions of TonalOrganization

In Western music a composition is typi-cally organized around a primary key, andthe chords employed are for the most partcontained within the basic set of harmoniesof that key. However, shifts to other tem-porary keys are often employed and are usedto provide organization to the piece as awhole. Generally, this shift, or modulation,is to a closely related key and is effectedthrough the use of a chord that is shared bythe two keys. A chord that functions in thisway is called a pivot chord and is usuallyfollowed fairly immediately by a cadence,that is, a strong key-defining sequence ofchords that most frequently contains the Vand I chords of the new key. Schenker(1906/1954) notes, "In general, a cadencein the new key has proved to be the mostsuitable means to fortify the new key andthus to make the modulations real and com-plete" (p. 322).

In addition, Schenker (1906/1954, 1935/1979) has suggested a hierarchy of instan-tiated tonal regions, which was mentionedearlier in this article. Schenker's influenceis clearly evident in recent theoretical writ-ings by Lerdahl and Jackendoff (1977, inpress) and Deutsch and Feroe (1981). Hiswork also has direct relevance to the presentresults. Schenker proposed a system of mu-sical analysis that identifies the surface ele-ments of the music in terms of their relationsto fixed underlying musical structures. Thatis, at the very deepest level a few structuralelements are identified, and the musicaltones are described as they relate to thesestructural elements in a hierarchy of increas-ing abstraction. This system characterizesthe way in which the momentary influencesof the surface level are transformed into acoherent whole.

According to Schenker any piece of musiccan be analyzed at its most basic level, calledbackground or Ursatz, in terms of a singletonal center. This key is most often explicitlyinstantiated at the beginning and end andcontinues to exert its influence throughoutthe piece, despite possible modulations. In-deed, Schenker (1906/1954, p. xxii) rejectsthe concept that a complete modulation evertakes place on the background level; a newkey may achieve some independence but willfunction only on the middle ground or fore-ground levels, giving preeminence to the pri-mary key. A surprising number of music the-orists agree on the importance of the primarykey. For example, Rosen (1972) states, "Apassage in a tonal work that is outside thetonic is dissonant in relation to the wholepiece, and demands resolution if the formis to be completely closed" (p. 26). Essen-tially the same position is taken by Schoen-berg (1969):Every digression from the tonic is considered to be stillwithin the tonality, whether directly or indirectly,closely or remotely related. In other words, there is onlyone tonality in a piece, and every segment formerly con-sidered as another tonality is only a region, a harmoniccontrast within that tonality, (p. 19)

However, the position that there is a uniqueunderlying tonality has been criticized asbeing too inflexible, and to date there is noevidence that most listeners actually main-tain a fixed sense of key throughout an ex-tended musical passage.

The middle ground in Schenker's hierar-chy represents the piece of music in termsof the more structurally significant events(chords and tones). Thus, at this level themusic is specified in a more simplified formthan in the complete score. Modulations be-tween keys are reflected at this level of anal-ysis as well as the foreground level. The fore-ground level is the actual music itself. Eachindividual chord has a tendency to be inter-preted as the tonic of a key, a process calledtonicization. Schenker (1906/1954) says,"Not only at the beginning of a compositionbut also in the midst of it, each [chord]manifests an irresistible urge to attain thevalue of the tonic for itself (p. 256). Ton-icization is expected to be most pronouncedwhen the chord is preceded by its own dom-inant (V) and during sections of greatest keyambiguity and instability. Thus, Schenker

TONAL ORGANIZATION IN MUSIC

CHORD: I 2 3 - 4 5 6 7

353

TWELVEPROBETONES

\

t

J

w^

\

J

-VA* t

i

J

\A-«' \

\

J

y\'

i

j

w\/

3

j

v^1

j

J

hf\i >

J

J

/Wv

^

J

\AWVPROFILE; i 8

Figure 7. The design of Experiment 2, using chord sequences. (The first chord of each sequence wasfollowed by each of the 12 probe tones within the chromatic scale, giving the first rating profile. Next,the first two chords of each sequence were paired with the 12 probe tones, giving the second profile. Thisprocess was continued until profiles were obtained for sequences of Lengths 1 through 9.)

proposes that pitch structure may be ana-lyzed in terms of a hierarchy of increasingabstraction, with the least abstract levelbeing the actual tone, intermediate levelscorresponding to the more structurally sig-nificant elements within the key prevailingin that section of the piece, and the deepestlevel specifying a single key that is presumedto underlie the entire composition.

The Present Approach to AssessingPerceived Tonal Organization

Various aspects of Schenker's (1906/1954,1935/1979) description of a tonal hierarchywill be investigated in the next experiment,which again used the tone-profile techniqueemployed earlier. The design of the experi-ment is shown in Figure 7. Each of the se-quences consisted of nine chords. Ten dif-ferent sequences were used; their structurewill be described later. In the experiment thelistener first heard the very first chord ofeach sequence, followed by each of the 12possible probe tones of the chromatic scale,as in the first experiment. This gave a profileof ratings for each of the first chords of the10 sequences. Following this, the listenerhears the first two chords of each sequence,again paired with all possible probe tones,giving a profile for two-chord sequences.This procedure was continued until profileswere obtained for sequences of Lengths 1through 9. It is important to note that at notime had the listener heard the sequencebeyond the point at which it was beingprobed. That is, the rating profiles werebased only on the chords presented up to thatpoint of the sequence. These profiles were

then correlated with the major and minorkey profiles obtained earlier to trace the un-folding process through which a sense of keyis achieved.

The 10 harmonic sequences are shown inTable 2. The first two sequences were con-structed from the chords of a single key,those of C major for the first sequence andthose of C minor for the second; these willbe referred to as the intended keys. Thechord functions within each key are shownin the table as roman numerals, as in a stan-dard harmonic analysis. These sequenceswere constructed so as to be relativelytonally unambiguous, although as noted ear-lier the multiple harmonic functions of in-dividual chords necessarily entails a certainresidual ambiguity. Nonetheless, the selec-tion of chords from a single key tends toreduce this ambiguity. In addition, the tem-poral order of the chords was guided bymusic-theoretic descriptions of permissible(or probable) chord sequences in music. Forthis we relied on the table of chord progres-sions contained in Piston (1962, p. 18). Fur-thermore, these two sequences began withthe harmonically significant IV-V progres-sion and ended with the relatively strong II-VI-I cadence, both of which would be ex-pected to be effective instantiators of key.

The other eight sequences contained mod-ulations (changes) between keys; these areshown as Sequences 3 through 10 in Table2. Both the chords themselves and the romannumeral analysis in the two keys are shownin the table. These modulating sequences allhad the same general structure. Each ofthem began with a cadence, IV-V-I or II-V-I, in the first key, which was either major

Table 2Chord Sequences

Sequencenumber

1

2

3

4

5

6

7

8

9

10

FirstModulation key

No C major

No C minor

Yes: Direct C majorand close

Yes: Indirect C majorand remote

Yes: Direct C majorand close

Yes: Indirect C majorand remote

Yes: Close C minor

Yes: Distant C minor

Yes: Close C minor

Yes: Indirect C minorbut close

Secondkey Chord 1

C major F majorC: IV

C minor F minorc: IV

G major F majorC: IVG:

B6 major F majorC: IVB*: (V)

A minor F majorC: IVa: (VI)

D minor F majorCIVd:

F minor D diminishedc: IIf:

C* minor D diminishedc: IIc*:

C major D diminished'c: IIC:

A* major D diminishedc: IIA*:

Chord 2

G majorV

G majorV

G majorV(D

G majorV

G majorV

G majorV

G majorV

G majorV

G majorV

(V)

G majorV

Chord 3

A minorVI

A* majorVI

C majorI

(IV)

C majorI

C majorI

C majorI

C minorI

C minorI

C minorI

C minorI

(III)

Chord 4

F majorIV

F minorIV

A minorVI(ID

A minorVI

F majorIV

(VI)

F majorIV

A* majorVI

G majorV

A* majorVI

A* majorVI(I)

Chord 5

C majorI

C minorI

E minorIIIVI

F majorIVV

D minorIIIV

D minorIII

F minorIVI

A* majorVIV

G majorVV

F minorIVVI

Chord 6

A minorVI

A* majorVI

B minor

III

G minor

VI

E major

V

B* major

VI

D* major

VI

A major

VI

A minor

VI

E* major

V

Chord 7

D minorII

D diminishedII

E minor(HI)VI

E* major

IV

B diminished(VII)

II

E diminished

II

B* minor

IV

F* minor

IV

F major

IV

B* minor

II

Chord 8

G majorV

G majorV

D major

V

F major(IV)

V

E major

V

A major

V

C major

V

G* major(VI)

V

G major(V)V

E* major

V

Chord 9

C majorI

C minorI

G major(V)

I

B* major

I

A minor(VI)

I

D minor(ID

I

F minor(IV)

I

C* minor

I

C major

I

A* major(VI)

I

O

78Ofr

73G

X

ZCOr>araD

;>.78D!~c

raVI

fa3C


or minor, and ended with one of these twocadences in the second key, which again waseither major or minor. That is, the first threeand last three chords constituted a cadencein the first and second keys, respectively. Ineach sequence the fourth chord belonged tothe first key. The modulation was effectedthrough the use of a pivot chord—a chordthat functions harmonically in both keys—in the fifth position. Again we were guidedin this choice by music theory, which de-scribes the use of pivot chords as the mostcommon means of modulation. The sixthchord belonged to the new key but not theinitial key, thus reinforcing the sense of thenew key, which was expected to become so-lidified by the cadence in the last three po-sitions.

As noted earlier, keys sharing chords arelocated in proximal regions on the torus, andthus, all modulations employing pivot chordsare necessarily between relatively close re-gions. However, some of the pairs of keysrepresented in the modulating sequences aremore closely related than others. Of themodulating sequences beginning in a majorkey, the relations between C major and Gmajor (Sequence 3) and between C majorand A minor (Sequence 5) are classified bySchoenberg (1969, p. 68) as direct and close,that between C major and B* major (Se-quence 4) and C major and D minor (Se-quence 6) as indirect and remote. These arealso more distant on the toroidal represen-tation of the key regions. Of the sequencesbeginning in C minor, the relation to F minor(Sequence 7) and to C major (Sequence 9)are described by Schoenberg (1969, p. 75)as close, the relation to A* major (Sequence10) as indirect but close, and the relation toC* minor (Sequence 8) as distant. Again,these classifications are substantiated by theobtained interkey distances. Consequently,of the modulating sequences, five (Sequences3, 5, 7, 9, and 10) will be considered as mov-ing between closely related key regions, andthree (Sequences 4, 6, and 8) between rel-atively distant regions.

Certain limitations of the roman numeralanalyses of the chord sequences provided inTable 2, which are those yielded by the stan-dard theory of harmonic analysis, should beemphasized. Such analyses do not make ex-plicit the full range of tonal ambiguity of the

individual chords, their varying degrees ofstability or the possibility that the tonalfunctions of the chords may need to be re-evaluated as the sequence progresses. Lessconservative approaches might interpret thechord functions in terms of a wider varietyof keys and indicate how earlier chord func-tions may be reinterpreted in light of har-monic information following later in the se-quence. However, the tone-profile techniqueemployed here enables us to assess directlyhow the listener's sense of key develops andchanges, providing a quantitative measureof the relative strengths of tonal interpre-tations at each point in time. In this way weare able to monitor the perceptual processesgiving rise to the sense of key.

In addition, a number of specific issuesraised by Schenker's (1906/1954, 1935/1979) description of tonal hierarchies willbe addressed. His claim that at the deepestbackground level the listener maintains afixed underlying sense of key, will be eval-uated by determining the extent to which theprofile ratings reflect the tonality of the firstkey after a modulation has occurred. Changesat the middle ground in Schenker's hierar-chy are investigated by describing the timecourse of the development of these alterna-tive, contrasting tonal interpretations in themodulating sequences. In this connectiondifferences between modulations to moreclosely and more distantly related tonal re-gions are examined. Finally, the momentaryinfluence of the individual chords, whichmay be considered to be the psychologicalcorrelate of Schenker's foreground level, willbe assessed by determining the influence ofthe last heard chord on the rating profiles.

Experiment 2

MethodSubjects. Fourteen adult subjects from the Harvard

University and Cornell University communities, whowere recruited through posted notices in the psychologydepartments, participated in the experiment. They werepaid at the rate of $12 for two experimental sessionslasting a total of approximately 3 !/i hours. A musical-background questionnaire showed that subjects had anaverage of-8.6 years instruction on musical instrumentsand voice and 6.4 years experience performing in in-strumental groups and 2.3 years in choral groups. Cur-rently, the subjects were playing an instrument or sing-ing an average of 7.7 hours per week and were listening


to music for 14.6 hours per week. Four of the subjectshad taken one or more courses in music theory; theremaining subjects had no background in music theory.Again, all reported normal hearing, and no one reportedhaving absolute pitch.

Apparatus and stimulus materials. The apparatuswas identical to that used in Experiment 1, and thechords of the sequences and probe tones were producedin the same way. The 10 sequences shown in Table 2were described in detail earlier. The durations of thechords and probe tones were as follows: Each chord hada duration of. 5 sec, with. 125 sec between chords. Therewas a pause of .5 sec between the last chord and theprobe tone, which sounded for .5 sec, and a 4-sec pausebetween trials.

Procedure. Before beginning, the design of the ex-periment was explained to each subject. That is, theywere told that they would hear chord sequences thatvaried from one to nine chords, each followed by a singleprobe tone. For each trial they were to rate on a scalefrom 1 to 7 how well the final probe tone fit in with thesequence of chords on the trial. The first session beganwith a practice tape containing trials of SequenceLengths 1, 3, 5, 7, and 9, with a modulating sequencethat was different from any actually heard in the ex-periment. They then heard the first chords of each ofthe 10 sequences (duplications eliminated) with all pos-sible probe tones. These were followed by all of thedifferent two-chord sequences, and the sequence lengthwas increased until finally the complete nine-chord se-quences were presented. Although the sequences in Ta-ble 2 are described with respect to C major or minor,in the experiment the reference key was shifted ran-domly from trial to trial, and all chord sequences of agiven length were randomly intermixed. Lastly, a ques-tionnaire about musical background was completed byeach subject.


Individual differences. I'ntersubject cor-relations of the responses were generallyhigh, with an average correlation of .669between any individual subject's responsesand the average responses for the group.Again, no systematic differences betweensubjects appeared that related to musicalbackground. All further analyses are basedon the average responses across subjects.

Correlations between sequence and keyprofiles: Nonmodulating sequences. Thefirst main results are pictured in Figure 8.The solid line in the graph on the top leftshows the average correlation between theobtained profiles for the nonmodulating Se-quences 1 and 2 and the profile for the in-tended C major and C minor keys. In otherwords, the profile found after each chord inthe sequence was correlated with the profilefor the key from which the chords weredrawn. These correlations were all quite high

and tended to increase as the sequences pro-gressed. Thus, it appears that the chord se-quences do in fact instantiate, at least tosome degree, the intended key.

There are peaks in the correlations at thefifth and ninth chord positions, where thetonic, C major or C minor chord, wassounded. These can be interpreted as the lo-cal effects of the chords in their tonic func-tions within the prevailing keys. In addition,there is a somewhat lower peak after thesecond chord, where the harmonically sig-nificant IV-V progression has occurred.Thus, even before the tonic has been heard,the obtained profile correlated strongly withthat of the intended key. At each of the nineserial positions, it is possible to determinethe extent of the local key effect of the lastheard chord on the prevailing tonality of thesequence. To do this the chord-key corre-lations described earlier were compared withthe correlations obtained in this experiment.These correlations between the individualchords and the key predict how strongly eachchord in isolation would be expected to relateto the intended key of the sequence and,thus, serve as a baseline against which thestrength of the intended key can be com-pared. To the extent that the obtained cor-relations exceed the local chord effects, theprofiles reflect integration over multiplechords, leading to a sense of the overall key.These chord-key correlations averaged overSequences 1 and 2 are plotted as the dottedline in Figure 8.

If complete tonicization is occurring, theobtained correlations will be expected to ap-proximate these chord-key correlations.Complete tonicization is seen when the tonic,I, chord of the intended key is sounded inPositions 5 and 9. At all the nontonic posi-tions, however, we see that the correlationswith the prevailing key were stronger thanwould be predicted on the basis of the singlechords alone. The difference between thegraphs, indicated by the shaded area, is ameasure of the extent to which the listenerperceives the prevailing tonal organizationas opposed to the tonal organization of thelast heard chord. (However, this differencein the first position undoubtedly simply re-flects some instability of the ratings that wascharacteristic of the early positions of thesequences.) These results indicate that dur-


ing a harmonic sequence, listeners do de-velop a sense of key that is at least partiallyindependent of the individual chords. Thatthe two functions tend to move in paralleldoes suggest, however, that local effects oftonicization are occurring to some degree.Moreover, the fact that the correlations be-tween the obtained profile and the prevailingkey profile are less than unity means that theintended key is less than completely instan-tiated.

To summarize the results for the non-modulating sequences, a sense of the in-tended key is evidenced by the relatively highcorrelations with that key's profile. The cor-relations tended to increase as the sequenceprogressed. Some effects of tonicization oc-curred, as indicated by the covariation be-tween the obtained correlations and the cor-relations between the last chord profile andthat of the prevailing key. For each nontonicchord in the sequence, however, the corre-lation with the prevailing key was strongerthan would be predicted on the basis of thelast heard chord. This suggests that listenersdo in fact process the multiple harmonicfunctions of the chords and integrate thisinformation as additional chords are heard,giving rise to a key sense that is partiallyindependent of the individual chords. Thecorrelations with the intended key, though,were less than perfect, reflecting some am-biguity and some local effects of the chordsthemselves. Thus, the profile measure is si-multaneously reflecting both the underlyingprevailing key (at the background and mid-dle ground levels) and the effect of the singlechords (at the foreground level).

Sequences modulating between closelyrelated keys. The modulating sequencescan be used to separate effects at backgroundand middle ground levels. The results for thefive modulations between closely relatedkeys are shown at the center of Figure 8. Onthe left the correlations between the profileobtained at each position of the sequence andthe profile for the intended first key aregraphed as the solid line; the correlationsfrom the first experiment between the lastheard chord and the first key are indicatedby the dashed line. On the right are the anal-ogous correlations with the profile of the in-tended second key.

The graph on the left shows that the

— Obtained correlationwith key.

Correlation of lastchord with key

CLOSE MODULATIONS: FIRST KEY SECOND KEY

1.0

0.6

S.4

DISTANT MODULATIONS FIRST KEY SECOND KEY

1.0

.8

,6

.4

.2

0

-.2

-.4

-.6

3 4 5 6 7 8 9 1 2 3 4 5 6POSITION IN SEQUENCE

7 8 9

Figure 8. The correlations with the intended key or keysin Experiment 2. (The solid line on the top left figureshows the correlation between the obtained profile ateach point along the sequence and the profile of theintended key averaged over the two nonmodulating se-quences. The dashed line indicates the correlation be-tween the last heard chord and the intended key derivedfrom the first experiment. The extent to which the pre-vailing key was stronger than the local chord effects isindicated by the shaded area. The graphs in the centerof the figure show the average results for the five se-quences that modulated between closely related keys.The correlations between the profile for the first key andboth the obtained and single chord profiles are plottedas the solid and dashed lines, respectively. On the rightare shown the analogous correlations with the secondkey at each point along the sequence. The graphs at thebottom of the figure show the results for the three se-quences that modulated between relatively distant keys.)

strength of the first key grows during thefirst three chords, which constitute a cadencein the first key, and then gradually declinesthroughout the remainder of the sequence.A dip occurs at the sixth chord, which is amember of the basic set of harmonies of thesecond, but not of the first, key. At eachnontonic position along the sequence afterthe initial chord, the correlation with theprofile of the first key is stronger than wouldbe predicted on the basis of the last heardchord alone, as indicated by the shaded re-gion. This demonstrates the continued influ-ence of the first key even after the modu-


lation has occurred. Again the effect oftonicization of each chord is apparent as thecovariation between the obtained correlationwith the first key profiles and the correlationbetween the last heard chord and first keyprofiles. The difference between these func-tions, however, tends to decrease toward theend of the sequence. Although listeners doachieve a sense of the initial key and tendto maintain it throughout the sequence,there is some evidence here that the strengthof the first key, beyond the local chord ef-fects, declines as the sequence moves into thenew key.

On the right, the strength of the second,closely related key is seen to increasethroughout the sequence. Initially, the ob-tained correlations matched almost perfectlythose that would be expected on the basisof the last heard chord alone. As soon as achord belonging to the second but not thefirst key was heard in the sixth position, how-ever, the obtained and single-chord corre-lations deviate, with a stronger second-keyeffect than would be expected on the basisof the remaining chords in isolation. Thus,although local chord effects account for theresponses throughout the first five chords,the listeners were apparently prepared toshift to the new key as soon as a chord uniqueto it was heard.

These results for modulations betweenclosely related keys showed that listenersgradually shifted their key sense from theregion of the first key toward the region ofthe second key. Some residual effect of thefirst key, however, was maintained through-out the entire sequence. This supports thenotion that, at least for modulations betweenclosely related keys, once a key has beenestablished its influence continues to be ex-erted despite the modulation to a differentkey. Thus, effects of the predominant key atthe background level are separated from theevents at middle ground and foreground lev-els. However, the predominant key appearsto prepare the listener for modulations toclosely related regions, for as soon as a chordunique to the new key is heard, that key isperceived more strongly than would be ex-pected on the basis of that single chord inisolation. In this we see that even when theinitial key sense has been achieved, listenersentertain alternative, closely related key re-

gions and are prepared to shift toward themas soon as chords unique to them are sounded.The strength of the new key continues toincrease and achieves a relatively high leveleven before the cadence in the new key hasbeen heard.

Sequences modulating between distantlyrelated keys. A somewhat different pictureemerges, however, for the sequences thatcontain modulations between distantly re-lated keys. These results are shown at thebottom of Figure 8. On the left, the corre-lations with the first key are seen to risesteadily throughout the cadence in the firstthree positions and continue to maintain arelatively high level until they drop dramat-ically as soon as chords of the new key areintroduced in Positions 6 through 9. Despitethis rapid decline the correlations with thefirst key continue to be stronger than thosepredicted by the most recently heard chorduntil the last two positions, in which the sin-gle-chord correlations with the first key be-come stronger. That is, the effect of the firstkey is maintained for some time after themodulation but is greatly weakened, andeventually the association between the in-dividual chords and the first key appears tobe suppressed by the introduction of the newkey. The extent of chord tonicization, as be-fore, is evident in the parallel movement ofthe two functions.

The suppression effect between distantlyrelated keys is even more apparent in theobtained correlations with the second key,shown at the bottom right of Figure 8. Afterthe first chord, where local chord effects ac-count for the results, there is an increasingtendency for the strength of the second keyto be lower than predicted by its associationwith each individual chord. This differencecontinues to increase through the fifth po-sition, despite that fact that the fifth chord—the pivot chord—is contained in the secondkey. Even after the sixth chord of the se-quence, which is unique to the new key, thestrength of the new key is somewhat weakerthan the local chord effects would predict.Only following the seventh chord does thestrength of the new key exceed that whichwould be predicted solely on the basis of themost recently heard chord.

These results contrast with those obtainedfor the modulations between closely related


keys. For those sequences the sense of thefirst key was maintained throughout the en-tire sequence despite the modulation to thenew key. For the distant modulations, al-though the first key continued to exert itsinfluence after the modulation, its effect wasvery much weakened, and the relations be-tween the individual chords and the first keywere suppressed even before the completionof a cadence in the new key. These listenersappeared unable to maintain the sense of theprimary key at the background level whenthe sequence modulated to a relatively dis-tant key. Moreover, whereas there was areadiness to move to new, closely relatedkeys, there was a certain measure of resis-tance to shift to distantly related keys, asseen in the later development of the senseof the new key in the sequences modulatingbetween distant keys.

Although three of the sequences are de-scribed here as distant modulations, this isonly relative to the close modulations. Infact, only one of these three modulations(that between C minor and C* minor) wasdescribed by Schoenberg (1969, pp. 68 and75) as truly distant; the others were classi-fied as indirect and remote. By electing tostructure the modulating sequences using apivot chord shared by the two keys, modu-lations between more distantly related re-gions could not be included. This is a con-sequence of the fact, discussed earlier, thatkeys sharing chords are at least relativelyclose. Perhaps even more striking suppres-sion effects might have been obtained formodulations that moved between less closelyrelated keys constructed in some way otherthan through a single pivot chord. However,the pivot-chord method is most representa-tive of traditional harmonic practice, andeven within the range of key distances em-ployed, regular and interpretable differenceswere found between the close and relativelydistant modulations.

Traces of the developing and changingkey sense on the torus. The last analysispointed to a sense of key that developed andchanged as the chord sequences unfolded intime. In that analysis we grouped togetherthe two nonmodulating sequences, the fiveclose modulations, and the three relativelydistant modulations. In order to explore inmore detail the perceived tonal structure of

each of these sequences, we again used theprogram PREFMAP (Chang & Carroll, Note5). The input to this program was the matrixof correlations between the rating profileafter each successive chord of each of thesequences and the 24 major and minor keyprofiles of the first experiment. Again, thea priori configuration was the set of idealizedcoordinates of the keys on the torus, and thePhase 4 option was used to find vectors inthe first two and last two dimensions sepa-rately, as described earlier. The results ofthese analyses can be represented as the in-tersection with the torus of the two resultingvectors for each of the nine positions alongthe sequence. The measure of fit between theresulting vectors and the correlation datawill not be given for each case separately.However, in all cases the fit in the first twodimensions was better (average r = .940)than in the last two dimensions (averager = .589). There was a tendency, however,for the first two dimensions to account formore of the variance for the sections ana-lyzed in major (/• = .970) than for the sec-tions in minor (r = .906). The last two di-mensions, in contrast, accounted for morevariance for sections in minor (r = .632) thanin major (/• = .553).

Figure 9 shows the results for the sequencein C major (Sequence 1) on the left and forthe sequence in C minor (Sequence 2) on theright. The points are labeled by the positionin the sequence of the last heard chord. Onthe left the first point shows the tonicizationof the first chord, F major. When the Gmajor chord follows, the point moves downtoward C major. Following this the pointstend to cluster in the region around the Cmajor key, although the positions vary some-what. This movement reflects local influ-ences of the individual chords, as can be seenby comparing the positions of the points withthe individual chords of the sequence in Ta-ble 2. For example, movement of the thirdpoint toward A minor is accounted for bythe A minor chord as the third chord in thesequence, the movement of the fourth pointtoward F major by the F major chord inposition four, and so on. Thus, in these traceswe again see the joint influences of the pre-vailing C major region and the local influ-ences of chords as they function in theirmultiple harmonic roles.


SEQUENCE^ C MAJOR. f« d -

Db/C"

1 F'

e

efc/d'

-bk—I

SEQUENCE: c MINOR-,f«

Figure 9. The developing sense of key is traced on the torus for the nonmodulating sequences in the Cmajor key (on the left) and the C minor key (on the right). (The points, which are the intersection ofthe vectors obtained by PREFMAP and the torus, are labeled by the position of the last heard chord inthe sequence.)

On the right the position of the first pointindicates tonici/ation of the F minor chordin the first position of the sequence in Cminor. After the second chord, G major, isheard, the C minor key appears to be fullyestablished. However, following this, stronglocal chord effects are evident, as the thirdpoint moves toward A* major, and the fourthpoint toward F minor. After this, the tonic,C minor chord, is sounded, and the remain-ing points cluster more closely around theC minor key. For this sequence, then, it ap-pears that the prevailing key of C minorbecomes established later in the sequence

than C major did in the first sequence, per-haps due to the greater instability of theminor mode in its attraction to both relativeand parallel major keys. In spite of the useof chords with the same roman numerals(chord functions) in these two sequences,this analysis points to significant differencesin the functioning of the analogous chordsacross modes and in the ways in which minorand major keys are processed and organized.

Figure 10 shows the results of the analyseson the modulating sequences that began inC major. On the left are the two close mod-ulations, to G major and A minor, and on

SEQUENCE^ c MAJOR - G MAJOR SEQUENCE c MAJOR - & MAJOR

O'/C" A $> Dl/C" Dk/C» . A d® Ol/C»ii

SEQUENCE'• C MAJOR -A MINOR. f» d

Figure 10. The key sense is traced on the torus for the four modulating sequences that began in the Cmajor key. (The two plots on the left show the results for those sequences that modulated betweenclosely related keys, and the two plots on the right show the results for the more distant modulations.)


SEQUENCE' C MINOR-F MINOR

D'ye"

SEQUENCE' C MINOR - C« MINORd

SEQUENCE: C MINOR - C MAJOR SEQUENCE' C MINOR- At MAJOR, f« d bk-. • f dI I I

Dl/C* DI/C"

Figure 11. The results for the four sequences that begin in the C minor key are shown in the four plots.(The two on the left and the one on the lower right are for sequences that modulated between closelyrelated keys; the one on the upper right shows the trace obtained for a modulation to a distantly relatedkey.)

the right are the more distant modulations,to B* major and D minor. In all of thesecases, the key of C major is firmly estab-lished by the time the third chord has beenheard, that is, after the IV-V-I cadence inC major. Following this, the points tend tomove toward the new key center, showingsome local effects of the individual chords,until by the eighth and ninth chords the newkey has been quite firmly established. Forboth close modulations the points have allmoved to the vicinity of the second key fromthe sixth chord on. For the more distantmodulations, the continued pull of the firstkey is evident later in the sequence.

Similar results were found for the mod-ulating sequences that began in C minor(shown in Figure 11). The two modulationson the left were close modulations, to F mi-nor and C major. The one on the lower rightto A* major was called by Schoenberg (1969,p. 75) indirect but close. For these sequencesthe points have all shifted toward the newkey from the sixth chord on. The figure inthe upper right is the only sequence includedin this experiment that was classified bySchoenberg as truly distant. In that plot we

see a dramatic shift occurring between thefirst and second keys. One special feature ofthis last result should be noted. As the toruswraps around, the C minor and Cf minorkeys are not in fact maximally distantly re-lated, as is necessitated by the shared A*(G*) major chord. However, the trajectorythat is followed between the two keys doesnot follow the shortest path between the keysbut rather takes the longer path around thetorus in the other direction.

This last result, that the path taken be-tween two keys need not be the shortest pathon the torus, indicates that the perceiveddistance between keys during modulationsdepends on more than just absolute key dis-tance. Instead, it depends on how the mod-ulation is achieved, that is, on the specificchords that appear in the sequence. In par-ticular, the number and nature of pivotchords, the duration of the transitional re-gion, whatever intermediate regions are em-phasized, and also the direction of the mod-ulation (i.e., which key is first and which issecond) are all factors that may influencewhether the modulation is heard as smoothor abrupt and whether the keys are perceived


as closely or distantly related. We suggest,then, that trajectories of the sort presentedhere, which trace the key sense as it developsand changes over time, may more fully cap-ture the perceptual phenomena of shiftingtonal regions than simple schemes that clas-sify pairs of keys in terms of their relativedistances.

General Discussion

The experiments reported here were di-rected at describing the internal represen-tation of pitch relations in listeners familiarwith Western music. Music theorists char-acterize pitch structure in tonal music as thefunctioning of tones and chords within anabstract tonal center, or musical key. Dif-ferent tone and chord functions are asso-ciated with distinctive properties, certainelements taking more structurally signifi-cant roles than others. Additional structureis presumed to exist in the varying degreesof relatedness between different musicalkeys. Here we investigated the question ofwhether listeners do, as suggested, processindividual tones and chords with referenceto a system of abstract tonal centers, and ifso, how the perceived tonal functions of thetones and chords and the relations betweenkeys might be represented quantitatively:

We first obtained a measure of the strengthof association between each musical pitch ofthe chromatic scale and an abstract tonalcenter. The abstract tonal center was in-stantiated by a musical element, such as ascale, chord, or cadence, that would be ex-pected to be a relatively unambiguous in-dicator of the major or minor key. Each ofthe 12 notes of the chromatic scale was pre-sented following each of these musical ele-ments. The listeners' ratings of how well theindividual tones fit with these musical ele-ments contained considerable structure, thehighest rating being given to the tonic tone,the first degree of the scale. This was fol-lowed by the third and fifth scale degrees,which together with the first scale degreeform the tonic, I, chord. For major key ele-ments the rating for the fifth scale degreewas higher than that for the third, whereasthe opposite order was found for minor keys.The fourth scale degree for major and thesixth scale degree for minor followed next.

The remaining scale degrees had interme-diate ratings and, finally, the nondiatonictones had the lowest ratings. These ratingprofiles agree with the qualitative accountof tonal functions provided by music theo-rists, and the profile for the elements defin-ing major keys replicated the results of anearlier study (Krumhansl & Shepard, 1979),which introduced the basic technique usedhere.

After reliable rating profiles were ob-tained for major and minor keys, which serveas distinctive markers of the instantiatedkey, the profiles were shifted and intercor-related. This provided a measure of distancesbetween any pair of keys (major or minor)based on the similarity of the tonal hierarchyin the two keys. These correlations showedthat each major key was closely related toits dominant and subdominant major keysand to its relative and parallel minor keys.Similarly, each minor key was closely tiedto its dominant and subdominant minor keysand to its relative and parallel major keys.The multidimensional scaling technique,which transforms a set of measured inter-object proximities into a spatial configura-tion of points so as to best account for theproximity data, was applied to the matrixof correlations. That analysis produced avery regular configuration of points in fourdimensions that accounted almost perfectlyfor the correlations computed between therating profiles. The 24 major and minor keyswere situated on the surface of a torus so asto directly reflect the circle of fifths relationsas well as the parallel and relative relationsbetween major and minor keys.

When the obtained torus representationwas viewed as a rectangle in two dimensionswith opposite edges identified, the resultswere strikingly similar to the map of keyregions provided by Schoenberg (1969, pp.20, 30), arrived at through analysis of struc-ture in musical compositions. Thus, this em-pirically derived spatial representation of the24 keys strongly confirms the music-theo-retic description of structure at the level ofabstract tonal centers. However, measuresof key distance proposed by music theorists,such as overlapping scale tones, differencesin key signature, or distances around thecircle of fifths, do not provide the kind ofprecise quantitative measure obtained here


through comparing the relative strengths ofthe individual tones within the different keys.Although all of these factors are reflected,directly or indirectly, in the profiles, neithersingly nor in combination do they yield thekind of unambiguous measure of key dis-tance produced by the present method.

What is more interesting from the psy-chological point of view is that listeners, atleast those with fairly extensive experiencewith tonal music, have apparently internal-ized musical structure at the more abstractlevel of tonal centers. In the present contextthis representation was derived in a some-what indirect way from the profile ratings.However, other studies have demonstratedkey distance effects on recognition of trans-posed melodic sequences (Bartlett & Dowl-ing, 1980; Cohen, 1975; Cuddy et al., 1979).Confusions between chords and judgmentsof harmonic relations have similarly beenfound to depend on key distance (Krumhanslet al., in press). In combination with thesestudies, the findings obtained here providequite strong evidence for an internal repre-sentation of interkey structure. Moreover,the present approach suggests a mechanismthrough which the relations between keysmight become internalized.

The method used here to recover the dis-tances between keys was based on the rel-ative stability of the single tones within thekey instantiated by the context element. Thisraises the possibility that the internalizationof interkey structure might likewise be de-rivative of the perceptual assignment oftones to positions within the hierarchy oftonal functions. The tonal hierarchy of singlepitches is, according to this view, psycholog-ically primary, and perceived key distancesdepend on the similarity of the hierarchiesassociated with the keys. The primacy of thelevel of tones makes logical sense because,after all, music is constructed from the si-multaneous and successive combination ofindividual pitches.

However, the suggestion that the tonalhierarchy is fundamental raises the questionof how the tonal hierarchy itself may be ac-quired. In music certain tones are empha-sized by their more frequent occurrence, par-ticularly at phrase beginnings and endings,and these tones typically have longer dura-tions and are given greater rhythmic stress.

Moreover, the selection of these pitchesmaintains certain fixed intervals, conformingin Western music to the interval patterns ofthe major and minor diatonic scales. Thediatonic scales are constructed of a distinc-tive pattern of whole and half steps betweensuccessive scale degrees, and this intervalstructure, which is maintained in all tonalmusic, may further facilitate the internal-ization of pitch relations (Balzano, 1980).Thus, music itself contains numerous fea-tures that would promote the assignment ofpitches to various positions within a hierar-chy. In this connection it should be notedthat despite the fact that other musical cul-tures employ different sets of pitches thanthose employed in Western music, there area number of other traditions in which aroughly analogous hierarchy exists withinthe set of musical tones. For example, Indianmusic is typically constructed around a fewsingle pitches that are used more frequentlyand appear in more prominent positionswithin the sequence; other tones appear lessfrequently, are given less rhythmic emphasis,and tend to be followed by the more focaltones (Jairazbhoy, 1971; Meyer, 1956).

Both the construction and perception ofpitch sequences around one or a few tonesmay reflect a general cognitive phenomenonthat has been shown to apply to a wide va-riety of perceptual and semantic domains(Rosch, 1975; see Smith & Medin, 1981, fora recent summary of related work). Accord-ing to this view there are certain membersof natural categories that function as cog-nitive reference points, or prototypes, for thecategory as a whole. These elements are de-scribed as the most representative of the cat-egory, in relation to which all other categorymembers are seen. Krumhansl (1979) notedthe applicability of this description to themost structurally stable pitches in the tonalsystem in music.

Finally, psychophysical accounts (Helm-holtz, 1863/1954) of musical structure havefocused on the coincidence between the in-tervals formed by the tones most stable inthe tonal hierarchy, such as fifths and majorthirds, and the overtone structure of natu-rally produced tones. The frequency ratiosof tones forming these intervals can be rep-resented as ratios of small integers, whichmay be related to the phenomenal descrip-


tion of these intervals as sounding more con-sonant than others. Despite Krumhansl's(1979) demonstration that judgments of therelations between tones forming fixed inter-vals depend on the broader context in whichthey are embedded, and the cross-culturaldiversity found in the selection of intervals,the role played by the factors of overtonestructure and frequency ratios in the initialconstruction of the tonal hierarchy, at leastthat found in Western music, cannot beruled out.

In any case, we suppose that the hierarchyof tonal stability is acquired through expe-rience with the structural relations that ob-tain in the music itself. In support of thishypothesis, Krumhansl and Shepard (1979)found that listeners with more extensivemusical backgrounds produced much morefinely differentiated judgments of how wellsingle tones completed a scale sequence.Whereas the responses of these listeners re-flected the tonal hierarchy, listeners with lesstraining emphasized tone-height differences.In a recent developmental study, Krumhansland Keil (in press) found a developmentaltrend in judgments of pairs of tones embed-ded within short tonal melodies. The youn-gest children, who were in the first twogrades of elementary school, distinguishedonly between scale and nonscale tones. Theadditional distinction between the tonic,third, and fifth scale degrees and the otherscale tones emerged only later in the devel-opmental sequence, although it was firmlyestablished by Grades 5 and 6. Only the re-sponses of adult listeners in that study re-flected the functional equivalence of thetonic tones separated by an octave. A tone-height effect was found for all groups of lis-teners, and the magnitude of this effect didnot change with the age of the listeners. Thedevelopmental trend observed may be char-acterized as a pattern of increasing differ-entiation between the tonal roles of the in-dividual pitches within the context key. Thisdescription also applies to differences ob-served as a function of training in the Krum-hansl and Shepard (1979) study, althoughthe ordering of the components differed inthe two cases.

Dowling (1978) has described the diatonicscale structure as a well-learned perceptualschema and has stressed its importance for

the perceptual organization of tone se-quences. That this schema appears quiteearly developmentally was suggested to ac-count for Bartlett and Bowling's (1980)finding that recognition of transposed se-quences was affected by the distance of thetransposition for children of elementaryschool age. Arbitrary pitches cannot be ac-commodated within the interval schema ofmusical pitches. In support of this, Krum-hansl and Shepard (1979) found that pitcheshalfway between chromatic scale steps(quarter tones) did not appear to be discrim-inated from their neighboring musical tones,and Siegel and Siegel (1977a, 1977b) haveshown categorical perception of pitch amongmusicians. Moreover, the tones must con-form to diatonic scale structure as evidencedby poorer memory for random sequences(Cohen, 1975; Dewar, 1974; Dewar et al.,1977; Frances, 1972). Finally, even if thepitches conform to some scale, the sequencewill be most easily assimilated and remem-bered if it is constructed so as to emphasizethose pitches most important within the dia-tonic heirarchy (Cohen, 1975). Thus, thereis considerable support for the position thatthe tonal hierarchy described by music the-orists is internalized by listeners and signif-icantly affects the perceptual processesthrough which muscial tones are encodedand remembered. In the present article wehave suggested that this tonal hierarchy maybe psychologically fundamental and haveshown that the perceived relationships be-tween different abstract tonal centers maybe derived from it.

Although we have emphasized the tonalinterpretation of single tones here, musictheorists have focused on the description ofthe harmonic functions of chords rather thansingle tones. Indeed, much of traditionalWestern music is constructed from tonessounded simultaneously in chords. The ques-tion addressed in the second section of thisarticle, then, was how chord functions mightbe represented within the obtained spatialrepresentation of tonal regions. In order toassess the relations between chords and keys,the ratings profiles for chords were comparedwith those for the major and minor keys.Each of the four chords—major, minor, di-minished, and dominant seventh—was foundto be closely associated with the keys in


which it appears, and the degree of associ-ation reflected the relative strengths of thedifferent functional roles in the hierarchydescribed by music theorists and obtainedempirically in the studies of Krumhansl etal. (1982) and Bharucha and Krumhansl(in press). In the present investigation boththe major and minor chords were found tobe closest to the key in which they functionas the tonic, I, chord. In the case of the majorchord, it was next closest to the major keyin which it is the dominant, V, chord. Forthe minor chord, after the tonic interpreta-tion, the VI function in a major key was thenext strongest. The diminished chord wasclosely tied to the major chord in which itplays the VII function, and the dominantseventh chord was close to the major key inwhich it is the V7.

A subsequent analysis of these chord-keyrelations represented the chords as points inthe map of musical keys. This analysisbrought out more clearly the feature thateach chord is closely associated with themusical keys in which it functions. Thus, theobtained chord positions reflected the inher-ent ambiguity of chord functions that comesabout because each chord may simulta-neously function within multiple tonal cen-ters. As noted earlier, it is a simple matterto specify the chords of any single key butimpossible to assign a single key to any in-dividual chord or, indeed, a set of chords. Ifa sense of key is ever to be achieved, thelistener must be able to entertain the possibleharmonic functions of each chord and inte-grate this information as the musical se-quence progresses. This process will be fa-cilitated if the internal representation is suchthat each chord is closely tied to the tonalcenters in which it functions, and this prop-erty was found in the present results. Thus,the chord-key distances indicate that theinternal representation supports the simul-taneous interpretation of chords in their var-ious harmonic roles.

Given the finding that chords are posi-tioned close to the keys in which they func-tion, it must necessarily be the case that keyssharing chords are located close to one an-other in the map of key regions. This resultmight suggest that chord overlap can be usedas a measure of key distance. There are prob-lems, however, with this approach. Most no-

tably, although parallel major and minorkeys were found to be closely related, theyin fact share only two chords. Thus, althoughshared chords belong to closely associatedtonal centers, not all closely positioned keyshave a substantial number of chords in com-mon. The measure of key distance derivedfrom the tonal hierarchies of single tonesprovided a more satisfactory measure of keydistance, which nonetheless gave rise to aspatial map that was able to accommodatethe complex interlocking pattern of chordfunctions.

In the final section we investigated theway in which the sense of key develops andchanges during extended chord sequences.In discussing the results it was useful to em-ploy the notion of different levels of tonalorganization at foreground, middle ground,and background levels (Schenker, 1906/1954, 1935/1979). The first level corre-sponds to the local effects of the individualchords, particularly as major and minorchords tend to suggest the key in which theyare the tonic chord. For all of the sequences,this effect of tonicization was apparent asthe covariation between the observed strengthof the prevailing key and what would be pre-dicted if each chord of the sequences hadbeen heard in isolation. In addition, whenvectors representing the relative strengths ofthe different keys at each point in time wereprojected onto the torus, the resulting tracesclearly exhibited variations attributable tothe effect of the individual chords.

In all cases, however, a sense of the pre-vailing key above and beyond these single-chord effects was clearly evident at all stepsbeyond the initial position in the sequences.To separate middle ground from backgroundlevels, some of the sequences contained mod-ulations between keys. Schenker (1906/1954,1935/1979) holds that at the very deepestbackground level, a single key once estab-lished remains invariant despite modulationsthat may affect the other levels. For themodulations between closely related keys,the sense of the first key was maintainedthroughout the sequence despite modula-tions to the new key. The strength of the firstkey, however, tended to decrease comparedwith the new key as the sequence progressedand to become more dependent on localchord effects. In contrast, for the modula-


tions between relatively distant keys, thesense of the first key was entirely absent bythe end of the sequences. In fact, the secondkey appeared to suppress the first key belowthe level of local chord effects, indicatingthat listeners may not maintain the key atthe background level when relatively distantmodulations occur. This suppression effectbetween distantly related keys was also ev-ident at the beginning of the sequences con-taining distant modulations. For these thestrength of the second key (which was to beestablished later) was again lower thanwould be expected based on the individualchords early in the sequence, and this dif-ference tended to increase as the first keybecame more firmly established.

One other difference was found betweenclose and relatively distant modulations. Forthe close modulations, as soon as a chordunique to the new key was heard, a sense ofthe new key was apparent. In contrast, whenthe modulation was to a less closely relatedkey, it took longer to achieve the new keysense. This effect for distant modulationswas seen in both the correlations with thenew key and in the traces on the torus, wherethere was a greater tendency for the pointsto remain in the vicinity of the first key afterthe modulation had occurred. Thus, it ap-pears that an established sense of key tendsto prepare the listener for modulations tonew closely related keys but induces a cer-tain measure of resistance to move to remote.tonal centers.

The obtained differences between closeand distant modulations indicate that keydistance does indeed affect the perception ofmodulations between keys. However, keydistance alone does not entirely account forthe ease or difficulty of modulation. Instead,difficulty depends to a great degree on theway in which the modulating sequence isconstructed, that is, on the particular chordsof the sequence. In some cases the path takenbetween keys may be relatively direct, inother cases less so. In one of the modulatingsequences studied, the path taken betweenthe first and second keys followed the longer,rather than the shorter, path between thetwo keys on the surface of the torus. Hadthe sequence used different intermediarychords, a shorter path might have been tra-versed instead and the modulation perceived

as being more direct. Thus, key distance perse is only one factor that affects the easewith which a modulation is assimilated.Tracing the changing key sense in a spatialmap of the sort obtained here provides moreinformation about the shifting tonal sensethan does a simple classification of key re-lations.

In these results we find that listeners in-tegrate information from successive chords,extracting a sense of key that grows overtime, which in turn induces expectanciesabout possible alternative tonal centers thatmay later be introduced. The key sense, then,comes about through a process of perceivingthe individual chords in their relations to oneanother and in their multiple functional roleswithin different tonal centers. This processis facilitated by an internal representationof interkey distances that supports thesemultiple interpretations, one in which keyssharing chords are closely positioned and thechords are closely tied to the keys in whichthey function.

At the most basic level, however, we be-lieve that relations between keys and be-tween individual chords and keys are allmediated through the internalized hierarchyof tonal functions at the level of individualtones. Indeed, the present results are allbased on the analysis of tonal hierarchies,as reflected in the rating profiles of singletones. In Western music this hierarchy givesrise to harmonic properties of simulta-neously sounded tones in chords that func-tion within different, interrelated tonal cen-ters in a way that appears to be unparalleledin other musical cultures. The present studyprovides empirical evidence that listenersfamiliar with Western music do in fact per-ceive musical organization at these moreabstract levels and provides a quantitativedescription of the internal representation ofthese musical structures.

Reference Notes

1. Dowling, W. J., & Bartlett, J. C. Assimilation ofbrief atonal melodies to tonal prototypes: Asym-metrical effects of judgment. Paper presented at themeeting of the Psychonomic Society, Philadelphia,November 1981.

2. Kruskal, J. B., Young, F. W., & Seery, J. B. Howto use KYST, a very flexible program to do mul-tidimensional scaling and unfolding. Murray Hill,N.J.: Bell Laboratories, 1973.


3. Olivier, D. C. Aggregative hierarchical clusteringprogram: Program write up (Computation CenterDocumentation). Cambridge, Mass.: Harvard Uni-versity, 1973.

4. Kruskal, J. B. How to use MDSCAL, a program todo multidimensional scaling and multidimensionalunfolding. Murray Hill, N.J.: Bell Laboratories.

5. Chang, J.-J., & Carroll, J. D. How to use PREFMAPand PREFMAP2: Programs which relate preferencedata to multidimensional scaling solutions. MurrayHill, N.J.: Bell Laboratories.

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Attneave, F., & Olson, R. K. Pitch as medium: A newapproach to psychological scaling. American Journalof Psychology. 1971,54, 147-166.

Balzano, G. J. The group-theoretic description of 12-fold and microtonal pitch systems. Computer MusicJournal, 1980, 4, 66-84.

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Received July 2, 1981Revision received February 8, 1982 •

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