perceived harmonic structure of chords in three related...

Journal of Experimental Psychology:Human Perception and Performance1982, Vol. 8, No. 1, 24-36

Copyright 1982 by the American Psychological Association, Inc.0096-1523/82/080l-0024$00.75

Perceived Harmonic Structure ofChords in Three Related Musical Keys

Carol L. KrumhanslCornell University

Jamshed J. Bharucha andEdward J. Kessler

Harvard University

This study investigated the perceived harmonic relationships between the chordsthat belong to three closely related musical keys: a major key, the major keybuilt on its dominant, and the relative minor key. Multidimensional scaling andhierarchical clustering techniques applied to judgments of two-chord progressionsshowed a central core consisting of those chords that play primary harmonicfunctions in the three keys. Th& separation of chords unique to the keys and themultiple functions of chords shared by the different keys were simultaneouslyrepresented. A regular pattern of asymmetries was also found that suggests ahierarchy among different types of chords. In addition, there was a preferencefor sequences ending on chords central to the prevailing tonality. Comparisonwith earlier results on single tones points to differences between melodic andharmonic organization.

The most striking characteristic of West-ern music is its harmonic structure, that is,the prevalent use of simultaneously soundedtones in chords. Indeed, music theory de-scribes in detail the construction of chordsequences, the function of chords in estab-lishing tonal organization, and the intimateconnection between harmonic and melodicorganization. There is no objective empiricalbasis, however, for this extensive literatureon harmonic structures. Instead, music the-orists attempt to characterize the commonpractice of composers in using chords inmusic. Nor have psychological studies onmusic addressed the question of how theseharmonic properties are perceived. Virtuallyall earlier studies employing organized mu-sical stimuli have used melodic sequences,that is, sequences of single tones (cf. Cuddy& Cohen, 1976; Dowling, 1978; Dowling

This research was supported by grants from the Mil-ton Fund and from the National Science Foundation(BNS-81-03570) to the first author.

We are grateful to David Green for generously ex^tending the use of the Psychophysics Laboratory atHarvard University for stimulus preparation, to MurraySpiegel and David Wilson for technical assistance, andto those who reviewed ,an earlier version of this manu-script for their helpful comments.

Requests for reprints should be sent to Carol L.Krumhansl, Department of Psychology, Uris Hall, Cor-nell University, Ithaca, New York 14853.

& Fujitani, 1971), and thus have been con-cerned only with melodic properties, thoseprinciples that relate sequentially soundedtones to one another. This study, which ex-plores the perceived similarities betweenmusical chords, is a first step toward under-standing how harmonic organization is rep-resented by the listener.

The chords used in this experiment werethose chords that belong to three closely re-lated keys. The primary purpose was to char-acterize the pattern of interrelationships be-tween these chords, with particular concernfor understanding how the internal repre-sentation of musical harmonies is compatiblewith the common musical practice of mod-ulation between different keys. It was ex-pected that those chords that play strongharmonic roles in the three keys studiedshould be judged as closely related, since apattern of associations of this sort would fa-cilitate smooth modulation between keys.The results are also used to determine howthe seven chords in any single key are per-ceived in relation to one another and, sinceboth major and minor keys were used,whether this depends on mode.

Multidimensional scaling and hierarchicalclustering techniques are applied to the data,which take the form of judgments of twochord sequences. Multidimensional scaling

24

PERCEIVED HARMONIC STRUCTURE 25

has been applied to musical tones in a num-ber of earlier studies. Levelt, VandeGeer,and Plomp (1966) obtained a two-dimen-sional representation of musical intervals. Inthat study each point in the spatial config-uration corresponded to a harmonic interval,that is, two simultaneous tones. The solutiontook a U-shaped form, with the musical in-tervals ordered along this contour accordingto the pitch separation of the two tones ofthe interval. For intervals constructed usingcomplex tones (tones with harmonics), a di-mension that correlated with the simplicityof the ratio of the fundamental frequenciesalso emerged. In a more recent study, Krum-hansl (1979) scaled single tones, rather thanintervals. This study also differed in that thesubjects were asked to judge how related thetones were to one another in the context ofa particular musical scale that they heardat the beginning of each trial. This procedureyielded a three-dimensional conical shape,with the tones ordered around the cone ac-cording to pitch height. In this configurationthe tones most central to the key of the scale(in particular those tones forming the majortriad on the tonic tone) were clustered to-gether near the vertex of the cone, with thetones less related to the key losated fartherfrom the vertex. That this representationreflected the tonal organization of the keyof the context scale was taken as evidencefor the influence of knowledge about howtones are related to one another in the tra-ditions of Western music and was used toargue against the position that perceivedmusical organization is derived directly frompsychoacoustic properties of tones.

In the present study musical chords madeup of three different simultaneously soundedtones (and their octave multiples) were usedas the elements of comparison. The choiceof triadsu (defined as those chords formed bythe superposition of two intervals of a third)was motivated by the fundamental role thattriads play in our harmonic system (see Pis-ton, 1962). The chord stimuli were con-structed using a technique originated byShepard (1964). Each chord contained 15sinusoidal components, corresponding to thethree different notes of the triad in five oc-taves. The amplitude of the various com-ponents varied as shown in Figure 1, so com-

ponents in the center of the frequency rangewere all of equal apparent loudness, and theloudness level of the components at the highand low ends of the frequency range taperedoff to threshold levels. The chords producedin this way have an organlike quality withno well-defined lowest or highest pitch.

There were two reasons for constructingthe chord stimuli in this fashion. The firstwas that we were interested in harmonicrather than melodic effects. Using conven-tionally constructed chords, the topmosttones of a chord progression tend to be heardas a melody. This method, then, which pro-duces chords without a clearly defined high-est component, minimizes melodic effects.Similarly, of course, the method also reduceseffects that have to do with the movementof the bass (the lowest) tones. The secondreason was to eliminate pitch height differ-ences between the chords. In the originalapplication Shepard (1964) demonstratedcircularity in judgments of relative pitchusing complex tones with components inmultiple octaves. These. results were inter-preted in terms of a circular dimension,called pitch chroma, that is orthogonal tothe rectilinear pitch height dimension pro-posed by Stevens (Stevens & Volkmann,1940; Stevens, Volkmann, & Newman,1937). An alternative way of eliminatingpitch height differences in the present studymight have been to select the three tones ofthe triads from different octaves, yieldingdifferent "inversions" so that there would beno large overall pitch height differences be-tween chords. This method of controlling forpitch height, however, would be complex andonly approximate. In addition, different in-versions of the same triad sound slightly dif-ferent and do not play identical roles in mu-sical composition.

The selection of the 13 chords used in theexperiment (listed in Table 1) was based oncertain music theoretic considerations, whichare outlined briefly. The chords are thosethat belong to three keys: C major, G major,and A minor. Both G major and A minorare closely related to C major, as measuredby the number of scale tones in common andthe frequency of modulation from one keyto the other in musical compositions. As thekey of the dominant (the fifth scale degree

26 C. KRUMHANSL, J. BHARUCHA, AND E. KESSLER

of C major), G major is closely related toC major, with the scales of the two keys over-lapping in all but one tone; C major containsthe note F, whereas G major contains F#.A minor is called the relative minor of Cmajor, and the natural minor scale in A con-tains exactly the same tones as the C majorscale, except that it begins on the note Ainstead of the note C. The chords in minorkeys, however, use a raised seventh scalenote. Thus a G# substitutes for G in thechords in A minor. The Roman numerals I-VII in Table 1, referring to the scale positionof the root of the triad, denote the functionalpositions of the triads within a given key.For example, I denotes the triad built on thefirst scale degree (C in C major, G in Gmajor, and A in A minor). The term ana-logue will be used to indicate the set of triadsthat have the same functional positions intheir respective keys.

The chords used are of four differenttypes: major, minor, diminished, and aug-mented, which differ in terms of the intervalsbetween the tones making up the triads. Ina major triad the middle tone is a major third(four half steps) above the root, and thehighest tone is a minor third (three halfsteps) above the middle tone. A minor triad

Table 1Thirteen Chords Used in Scaling Experiment

Chord

C major (C)D minor (d)E minor (e)F major (F)G major (G)A minor (a)B diminished (b°)D major (D)F# diminished (f#°)B minor (b)E major (E)G# diminished (gf)C augmented (C+)

Com-po-

nents

CEGDFAEGBFACGBDACEBDFDF#AF#ACBDF#EG#BG#BDCEG#

Cmajor

IIIIIIIVVVIVII——

————

Gmajor

IV

—VI—III—V

VIIIII———

Aminor

_IV—VI—III———V

VIIIII

Note. Roman numerals refer to scale position of rootof triad.

also contains a major and a minor third, ex-cept that the lower interval is a minor thirdand the upper interval is a major third. Adiminished triad consists of two superposedminor triads, and an augmented triad is builtup of two major thirds. These chords, whensounded, have different effects. Ratner(1962, p. 19) described the major chord as

60 -

oXQ.

lil>

THRESHOLD

C E

LOG

1

\

G C E

FREQUENCY

E G C2349 Hz

Figure 1. Loudness envelope used in constructing the chords as a function of log frequency. (Shown arethe 15 frequencies comprising the C-major chord. The three triad tones are sounded with varying loudnesslevels in each of five octaves.)


giving one of the most satisfying impressionsin music, a firm and pleasant sense of a tonalcenter, an absolute effect of stability. Minortriads are characterized as sounding lessbright and assertive than major triads butalso capable of representing a tonal center.In contrast, diminished triads, because of thedissonant interval of a diminished fifthformed by the lowest and highest tones, aredescribed as having an unstable, compactsound that creates a strong demand for res-olution (pp. 26-27). Finally, augmentedtriads also contain a dissonant interval be-tween the lowest and highest tones, an aug-mented fifth, and are consequently unstable,although more diffuse in sound than dimin-ished triads (p. 187).

It should be clear that the analogoustriads, I-VII, are of the same type for allmajor keys, and the same is true for all minorkeys. Analogous triads differ in type betweenmajor and minor keys, however, giving thesedifferent modes their distinctive flavors. Forexample, whereas the I chord is a major triadin a major key, it is minor in a minor key.In any single key the set of seven triads differfrom each other in.type and also in theirharmonic functions with respect to the key.In a major key the I, IV, and V chords areall major, and a progression involving thesechords gives what may be the strongest pos-sible instantiation of a major key. Possiblyrelated is the fact that these three triads con-tain all of the notes of the scale. The II andVI chords are minor and play more second-ary harmonic roles, although they often en-ter into chord progressions. In contrast, theIII chord, which is also minor, and the VIIchord, which is diminished, are relativelyrare in harmonic progressions. In minor keysthe V chord is also major, but the I and IVchords are minor, possibly making the minorharmonic system as a whole somewhat lessstable than major harmony. Again, however,the I, IV, and V triads take the most centralroles in establishing the key, followed by theII, which is diminished, and the VI, whichis major, and finally the augmented III andthe diminished VII triads.

One purpose of the present experimentwas to determine in what way this variationin the stability and harmonic functions ofthe triads might be reflected in judgments

of relatedness between chords. In an earlierscaling study on musical tones (Krumhansl,1979), a regular pattern of asymmetries wasfound. Higher similarity ratings were givenwhen the first tone was less related to thekey of the context scale, hence less stable,and the second tone was more related thanwhen the opposite held. The result was in-terpreted in light of Rosch's (1975) notionof cognitive reference points, with thosetones central to the tonality functioning asreference points in relation to which othertones might be perceived and remembered.Given the above characterizations of the dif-ferent triads, similar asymmetries might alsobe found for chords in this experiment.Therefore, subjects were asked to judge foreach pair of chords how well the secondchord followed the first, a task focusing ontemporal order, which would allow for thepossibility of such temporal asymmetries.

The trials in the experiment were con-structed in the following way. Each trialbegan with an ascending scale (C major, Gmajor, or a harmonic minor scale in A). Thiswas followed by the two comparison chordsplayed in succession. Thus, the trials werestructured in the same way as the earlierscaling study on single tones (Krumhansl,1979). This will then permit a comparisonbetween the results of the two studies thatmay point to differences between harmonicand melodic organization. Although thereis a strong functional similarity between anote and the triad built on that note, it neednot be the case that the scaling of the sevenscale tones of a key will be identical to thescaling of the seven chords built on thosetones,

Method

SubjectsTen subjects (5 male and 5 female), all from the

Harvard community, participated in two sessions thatlasted a total of approximately 4 hours. Seven were paidat the rate of $3/hr, and 3, including the first author,were volunteers. On the average the subjects had playedmusical instruments or sung for 9.4 years, participatedin group instrumental performing for 3.4 years andchoral singing for 1.5 years, currently played or sang2 hours per week, and listened to music for 15.4 hoursper week. Only two subjects had taken a course in musictheory. Normal hearing was reported by all and perfectpitch by none.


Apparatus

The stimuli were generated under the control of aPDF-15 computer in the Psychophysics Laboratory atHarvard University. The computer was programmed toproduce the single tones of the scale contexts by meansof a Wavetek signal oscillator and the chords by creatingdigital representations and playing them out through adigital-to-analog converter. Recordings were made onMaxell UD35-90 tapes at 7.5 in./sec (19 cm/sec) ona Sony TC-540 tape recorder. Subjects listened to thetapes on the same tape recorder through Superex(Swinger) headphones.

Stimuli

Each trial consisted of an ascending scale followedby two chords in succession. The scale tones and thetones comprising the chords were all drawn from the setof equally spaced semitones on a logarithmic scale with12 semitones per octave, as would be obtained by equaltempered tuning based on 440 Hz (A). Thus, the fre-quency of the «th semitone above (« > 0) or below(n < 0) 440 Hz was 440 X 2"/l2.

The tones of the musical scale context were generatedusing a triangular waveform. The choice of the trian-gular waveform was based on evidence that the har-monics produced in this way contribute to the perceivedmusicality of the tones. For example, Levelt et al. (1966)found in a scaling study that the qualities of consonanceand dissonance were more salient for intervals made upof complex tones than for intervals made up of sinewaves. In a different task requiring subjects to rate howwell different tones completed a scale sequence, Krum-hansl and Shepard (1979) found that the ratings re-flected the musical functions of the individual tonesmore for complex tones than for sine wave tones. In theexperiment the tones of the scale were played back atapproximately 67 dB (SPL).

Three different ascending scale contexts were usedin the experiment: C major, G major, and A minor, eachplayed over an octave range, from C4 (261.6 Hz) to C5,from G3 (196 Hz) to G4, and from A3 (220 Hz) to A4,for the three scales, respectively. The first and last tonesof each scale, both tonics, sounded for .5 sec each andthe six intervening tones for .2 sec each. There were nopauses between scale tones. Following the last tonic ofthe scale were a 1.5-sec pause, the first chord soundingfor .5 sec, another 1.5-sec pause, and finally, the secondchord for .5 sec. The two tonics in the scale were length-ened to provide a possible rhythmic interpretation thatplaced the two chords that followed the scale both onstrong beats, avoiding differential rhythmic stress on thetwo chords. The intertrial interval was 6 sec, except afterevery 10 trials, when an additional 2 sec was used toannounce the trial number.

The chords were constructed to obscure the topmostand bottommost tones of each triad. The tones of eachtriad were sounded over a five-octave range (from 77.8Hz to 2349 Hz), using the loudness envelope shown inFigure 1. The loudness envelope was made up in threesections. Over the first octave and a half, the loudnessenvelope increased from threshold to 60 phons, wherethe shape of this increasing function was a cumulative

normal distribution from ^2.125 a to 2.125 a, followedby a constant loudness level of 60 phons over the nexttwo octaves and a symmetric decline over the last octaveand a half. Equal-loudness curves (Fletcher & Munson,1933) were used to determine the correct intensity level(in dB) for each frequency. Following this, the corre-sponding amplitude was calculated for each frequencyusing the following equation, derived from Roederer(1975, p. 80):

£ _ J Q(//20 + log/(Ol

where A is the amplitude of the desired frequency, / isthe sound intensity level (in dB), and A(> is the amplitudeof a 1000-Hz tone at loudness threshold (i.e., at 0 dBfor 1000 Hz).

For each of the 13 chords in the experiment (see Table1), a digital representation consisting of 15 sinusoidalcomponents (corresponding to the three triad tones ineach of the five octaves) was constructed with a samplingrate of 5000 Hz. During production a low-pass filterwith a cutoff value of 2500 Hz eliminated high-fre-quency noise. Onset clicks were eliminated automati-cally by the system with a 5-msec ramped onset.

Experimental Design and Procedure

Given the set of 13 chords, there are a total of 156ordered pairs of nonrepeating chords. In the experimentas a whole, each possible pair of chords was presentedtwice with each of the three different scale contexts,giving two complete replications. For each scale contextsix blocks of 55 randomly ordered trials with the samescale context were taped. The first 3 trials in each blockwere included to allow the subject to adjust to the keyof the scale, but the results from these trials were notincluded in the analyses. Subjects heard the tapes indifferent random orders, following the constraint thatno two consecutive blocks of trials had the same contextscale. At the start of the first session, a practice tapewas played, with five randomly selected trials for eachof the three scales. Subjects were asked to rate, usinga ranking from 1 to 7, how well the second chord fol-lowed the first in the context of the preceding scale: 1was designated "follows poorly" and 7 was designated"follows well." The directionality of the relationshipbetween the two chords was stressed in the instructions,as was the need to attend to the context scale.

Results

Individual Subject Differences

To determine whether there were differ-ences between the subject's responses thatwere systematically related to musical back-ground, intersubject correlations on the re-sponse matrices were computed. These cor-relations averaged..323, and all but 3 of the45 correlations were significant. Moreover,the hierarchical clustering method (Johnson,1967) applied to the intersubject correlations


did not yield clusters that related system-atically to musical background. Conse-quently, the remaining analyses were per-formed on the data averaged over subjects.

Differences Between Contexts

The data for each scale context (C major,G major, or harmonic minor in A) consistedof a matrix of relatedness judgments on the156 pairs of distinct chords. The correlationbetween the C major and G major contextmatrices was .89, that for C major and Aminor contexts was .81, and that for the Gmajor and A minor contexts was .78, allhighly significant (/x.OOl). Thus, similarresults were obtained for the three scale con-texts, indicating that the pattern of resultsmay reflect characteristics of the chords,such as chord type, that are independent ofthe context scales. Alternatively, or in ad-dition, the similarity of the results for Gmajor and A minor may be mediated throughthe close relationship of each of these tonal-

ities to C major. Analyses evaluating thesetwo kinds of explanations are presentedlater.

A t test for differences between nonin-dependent correlations determined, however,that the correlation between the results forC and G major contexts was significantlygreater than the other two correlations,t(\53) = 3.30 and 4.88, ps < ,01, for the twocomparisons, respectively. The pattern ofcorrelations may reflect the strong relation-ship between a major scale and the majorscale built on its dominant, the slightlyweaker relationship between a major scaleand its relative minor, and finally, the rel-atively remote harmonic relationship be-tween G major and A minor.

Analyses on the Set of 13 Chords

Multidimensional scaling and clusteringof the 13 chords. Multidimensional scaling(Kruskal, 1964; Shepard, 1962) and hier-archical clustering (Johnson, 1967) methods

(a) 13 CHORDS

00o

(c) G MAJOR

0

o(b) C MAJOR

o

O

o(d) A MINOR

Figure 2. Multidimensional scaling of the results for the 13 chords in C major, G major, and A minor.(Panels a, b, c, and d all show the identical configuration. In Panel a the points are labeled accordingto the name of the corresponding chords. The major chords are indicated by upper case letters, C, D,E, F, and G. The minor chords are indicated by lower case letters, a, b, d, and e. The diminished chordsare indicated as b°, f#°, and g#°, and the augmented chord as C-K In Panels b, c, and d the points arelabeled according to the scale position of the root of the triad in the keys of C major, G major, and Aminor, respectively. An open circle indicates that the corresponding chord does not belong to the key.)


1.0-

z 2.0UJ

g= 3.0'

<f>

z 4.0'LJ

<J 5.0 •or

6.0

r~

nb fS D G

G Major' 16 Major' 3U Stt S IA Minor

C (

1

L~1

1

b° F d a E 98 C*

I m szn E ii sim 21 E

n 2t is i 2 2E n

5. Hierarchical clustering solution using maxi-mum (complete link) method on results for the 13chords. (The position of each triad within C major, Gmajor, and A minor is indicated below the chord name.See caption to Figure 2 for names and explanation ofchord symbols.)

were applied to the matrix of relatednessjudgments averaged over the context types.(These methods applied to the three contextmatrices separately did not give solutionsvery different from that obtained from theaveraged matrix.) The actual program usedto accomplish the scaling was KYST (Krus-kal, Young, & Seery, Note 1), and AGCLUS(Olivier, Note 2) was used for the clustering.Figure 2 shows the two-dimensional solution,which had a stress value of .181 (Stress For-mula 1).

In Figure 2a the configuration is labeledwith the names of the 13 chords. The centerof the configuration contains a cluster ofmajor and minor chords, with the 3 dimin-ished chords and the 1 augmented chordoccupying peripheral positions. The chordsin central positions are also those chords thatplay important harmonic roles in the threekeys used in the experiment. This aspect ofthe scaling solution is brought out moreclearly in Figures 2b, 2c, and 2d, in whichthe same configuration is relabeled to showthe functions of the chords in the keys of Cmajor, G major, and A minor, respectively.In these panels an unlabeled circle indicatesthat the corresponding chord does not ap-pear in the key. In Figure 2b the I, IV, andV chords of C major are clustered together

in the center, with the II, VI, and III chordsalso fairly close to that cluster. The VIIchord and the chords unique to the otherkeys are generally distant from the cluster,the only exceptions being the D and E majorchords, which play the role of V in G majorand A minor, respectively. In Figure 2c itcan be seen that the chords belonging to Gmajor are generally located in the left halfof the configuration, the only exceptionbeing the II (the A minor chord), which alsobelongs to C major and A minor. Again, theI, IV, and V chords are clustered together,and the II and VI chords are also fairly close.The III and VII chords are more distant, asare the chords not belonging to G major. InFigure 2d it can be seen that the chords be-longing to A minor are located on the righthalf of the configuration. Here again, the I,IV, and V chords cluster together, and theVI chord is also fairly close. The II (whichis a diminished chord in a minor key), III,and VII chords are generally farther away,as are the chords not belonging to A minor.

The hierarchical clustering solution, shownin Figure 3, is also included, since it bringsout additional structure not contained in themultidimensional scaling solution. Kruskal(1977) has stressed the complementary na-ture of these two techniques. In the treestructure shown, obtained using the maxi-mum (complete link) method, the centralcluster consists of those chords belonging toC major together with the V chords of Gmajor and A minor. This main cluster is sub-divided into two subclusters: one with the I,IV, V, and VI of G major and another withthe I, IV, V, and VI of A minor. Of thechords moderately or strongly associatedwith these keys, only the II chords are miss-ing from these subclusters. That the II of Gmajor (the A minor chord) is not containedin the G major subcluster is undoubtedly dueto its multiple function as the I of the keyof A minor, which draws it into the A minorsubcluster instead. And although the II ofA minor (the B diminished chord) is not in-cluded in the A minor subcluster, possiblyreflecting the unstable nature of diminishedchords, it does join the A minor subclusterat the next level in the tree. Thus, the 'hi-erarchical clustering solution shows a fairly


clear separation of the group of C majorchords into the chords that also function inG major and those that function in A minor,a pattern that is not brought out clearly inthe two-dimensional spatial configuration.

Asymmetries in relatedness judgments.Substantial asymmetries were found in thedata matrix, with the response for a pair ofchords depending on the temporal order ofthe two chords. In other words, for a pairof chords, the judgment of how well onechord followed another depended on whichchord was first and which chord was second.These asymmetries depended on chord typeand, to some extent, on the context scale thatpreceded the two to-be-judged chords.

Generally when the first chord was majorand the second chord was not, the rating waslower than for the reverse order. This wastrue whether the other chord was minor,*(59) = 5.02, p < .001, diminished, t(44) =5.02, p < .001, or augmented, t( 14) = 12.54,p < .001. In addition, there was a preferencefor moving away from, rather than toward,the augmented chord whether the otherchord was diminished, f(8) = 6.48, p < .001,minor, r ( l l ) = 5.12, p < .001, or as above,major, f(14)= 12.54, p<.001. There wasno consistent pattern, however, for trials con-taining one diminished and one minor chord;the slight preference for diminished-to-mi-nor sequences over the reverse order was notsignificant, t(35) = .645.

There was a preference for sequences endring, rather than beginning, on a chord in thekey of the scale context for C major contexts,f(41) = 3.55, p < .001, and for G major con-texts, f(41) = 2.729, p < .01. Post hoc anal-yses showed, however, that this result for theG major scale trials was entirely attributableto those chords shared with C major. For Aminor the asymmetries were less regular.Some slight preference was shown for se-quences ending on chords unique to A minor,but 'no other.asymmetries related to the Aminor context key were observed.

Additional analyses on the 13 chords.On any given trial neither of the chords, oneof the chords, or both of the chords mightbelong to the context key. This variable hada significant correlation with the averageresponse to the chord pairs (r = .288, p <

.001), with the highest relatedness judg-ments found when both chords belong to thekey. Thus, it would seem that the contextscale enhances perceived harmonic strengthbetween chords in its key. This point needsto be qualified, however, since differenceswere found when the three context keys wereconsidered separately. The correlation wasstrongest for the C major scale (r = .490),followed by the G major scale (r = .312),both of which are highly significant (ps <.001). A nonsignificant correlation was found,however, for A minor (r = .035). This andother results described earlier indicate thatthe strongest harmonic center established inthe experiment may have been that of Cmajor.

Chords sharing more tones actually re-ceived lower relatedness judgments. The cor-relation between the number of tones sharedby the two chords and the judged relatednesswas -.170 (p < .001). The negative sign ofthe correlation may reflect the fact thatmany of the chords sharing two componentsdiffer because a single component tone israised or lowered a half step in one triadwhen compared to the other. This alterationmeans that the chords do not belong to thesame key and, thus, would not be expectedto be judged as harmonically related.

Pitch height distance between the com-ponents of the two chords did not correlatesignificantly with rated relatedness (r =.029). The distance between chords was de-fined as the average of the following threenumbers: the number of semitones betweenthe roots of the triads, the number betweenthe middle tones of the triads, and the num-ber between the highest tones of the triads.To avoid any ambiguity due to the multipleoctave range in which the triad componentswere sounded, these distances were mea-sured as the minimum number of half stepsbetween the corresponding components.Thus, the loudness envelope effectively elim-inated pitch height effects.

Analyses on the Seven Analogous Chords(l-Vll) in the Three Keys

Comparisons across scale contexts Theset of 13 chords used in the experiment con-tained the chords that function as the I-VII


chords in each of the three keys of C major,G major, and A minor, with some chordsplaying multiple functions in the differentkeys. Thus, for each scale context, a subsetof the data could be viewed as judgments onall pairs of the chords from the set I-VII.The correlations of the analogous chordpairs in the different contexts were .594 forC and G major contexts, .584 for C majorand A minor, and .547 for G major and Aminor (ps < .001). These indicate that theresponses in the experiment strongly reflectthe particular harmonic functions that theanalogous chords play in the different tonal-ities. That the correlations do not simplyreflect triad type (major, minor, diminished,or augmented) is indicated by the fact thatthe correlation between the major keys (inwhich analogous chords are of the sametype) is not much different from the corre-lations between major and minor keys (inwhich analogous chords are of differenttypes).

Multidimensional scaling and clusteringon the analogous chords (I-V11). Figure 4shows the two-dimensional scaling solution(Stress = .083) and the tree structure fromthe hierarchical clustering technique (max-imum method) on the analogous chord pairscollapsed across contexts. The spatial con-figuration contains the core of I, IV, and Vchords, with the remaining chords spacedaround this cluster. Comparing this config-uration with the scaling solution on the 13chords, we see the same general pattern ap-pearing for the I-VII chords in each of the

three context keys, with the core of I, IV,and V chords surrounded by the less closelyrelated chords of the key. This was true des-pite the fact that the representation of all13 chords incorporated the complex patternof multiple functions that the individualchords play in the different keys.

The clustering method applied to the samedata shows a definite hierarchy within theset of chords I-VII that is not apparent inthe spatial solution. This hierarchy beginswith the clustering of I and V, after whichthe IV, VI, II, III, and VII join the clusterin turn. This kind of chaining pattern is moretypical of results using the minimum methodand is somewhat unusual for the maximummethod. Given the bias of the maximummethod against patterns of this sort, we areconfident that the data strongly indicate thishierarchical organization. The tree structurefirst joins the chords most closely associatedwith the tonality and step-by-step adds thosechords less related to the tonality. As thefairly high correlations across contexts wouldindicate, similar results were also obtainedfor the data from each of the three contextkeys separately.

Discussion

In summary, the multidimensional scalingrepresentation of the relatedness judgmentsof the 13 chords revealed a core of eightchords. This core included the major andminor chords that play important harmonicroles in the three keys used in the experi-

Figure 4. The left panel shows the multidimensional scaling solution for the seven analogous chords inthe three context keys, C major, G major, and A minor (solution shown for data averaged across thethree keys); the right panel shows the hierarchical clustering solution using the maximum (completelink) method for the same data.


ment. This core contained the I, II, III, IV,V, and VI of C major; the I, II, IV, V, andVI of G major; and the I, IV, V, and VI ofA minor. Particularly close, together werethe chords that play the roles of I, IV, andV in each of the three keys. In addition, thehierarchical clustering solution of these datarevealed a split in the core between thosechords that function in G major and thosethat function in A minor. Moreover, both

' methods represent the remote relationshipbetween the chords unique to G major andthose unique to A minor. Thus despite thefact that a single chord may play multiplefunctions in different related keys, thesetechniques were able to accommodate thesemultiple functions and, at the same time,represent the separation of the differentkeys.

In addition, asymmetries in the related-ness judgments were found. There was astrong pattern of asymmetries in the ratingsthat depended on triad type, which is con-sistent with a hierarchy of triad types inwhich the major triads are the most stablefinal chords, minor and diminished triadsintermediate, and augmented triads the leaststable. The effect of the key of the scale con-text on asymmetries was somewhat less reg-ular. There was a strong preference for se-quences ending on chords in the C major keywhen a C major scale preceded the chords.When the scale was in G major, the resultswere somewhat less clear, although consis-tently higher ratings were given to sequencesending on chords in C major. For A minorno particularly regular pattern emerged. ForC major and to some extent for G majorcontexts, then, there was a preference forsequences ending on chords contained in thecompact core of C major chords in the scal-ing solution. This is the same -pattern as thatfound in an earlier scaling study on singletones (Krumhansl, 1979), in which only aC major context was used. Asymmetries re-lated to the spatial density of points in ageometric representation have been noted inother domains (see Krumhansl, 1978) andare entirely consistent with Rosch's (1975)notion that central, reference stimuli serveas anchors in perception and memory. '

Other evidence that the C major key wasperceived the most strongly of the three keys

used as contexts in the experiment comesfrom considering the effect of whether thechords were in the key of the scale playedat the beginning of each trial. There was astrong positive relationship between thenumber of chords in the context key (whethernone, one, or two) and the judged relatednessof the two chords when the context was Cmajor, a weaker relationship when the con-text was G major, and virtually none whenthe context was A minor. Thus, both thisanalysis and the pattern of asymmetriespoint to the C major key as the predominanttonality in the experiment, with G majorperceived as having a closer relationship tothe tonal center than A minor.

The finding that the scale context had rel-atively weak effects in this experiment standsin contrast to the earlier study on tbnes(Krumhansl, 1979), which found definitecontext-specific effects. This difference maystem from a number of factors. First, in theearlier study the scale context exerted itsinfluence on individual tones, whereas chordswere used in the present experiment. Chordsare richer and more complex musical entitiesand as such may be used to establish tonalcenters in their own right; this is particularlythe case for major and, to some extent, minorchords. Thus, chords may be less susceptiblethan tones to influences from scale contexts.

Second, here we used three, rather than. one, different scale contexts. The particular

keys were chosen so that two of the keys areclosely related harmonically to a third butless closely related to each other. One scale(G major) is the major scale built on thedominant of C major, and the other (A mi-nor) is the relative minor of C major. Cor-relations between the results for the differentcontexts showed the highest agreement be-tween judgments for the C and G major con-texts, followed by C major and A minor con-texts. The largest differences were foundbetween G major and A minor contexts, aswould be predicted. Since G major and Aminor take these special roles in relation toC major, it would seem plausible that theprevailing tonal center would be that of Cmajor.

To determine possible differences betweenharmonic and melodic organization the re-sults for the I-VII chords in this experiment


can be compared to the results for the sevencorresponding root tones in the earlier ex-periment (Krumhansl, 1979). Two majordifferences appear. First, whereas Chords I,IV, and V formed a closely related clusternear the center of the scaling solution in thepresent experiment, Scale Degrees I, III, andV were clustered near the center of the so-lution for individual tones. In other words,the musically important harmonic relationsbetween the tonic, subdominant, and domi-nant triads are reflected in judgments ofchord pairs. In contrast, when single tonesare being compared, the most closely asso-ciated subset of tones consisted of the first,third, and fifth scale degrees. This patternof associations may be mediated through themembership of these tones in the tonic triad,which takes a primary role in establishingthe key. The second main difference was thatthe solution for tones was organized arounda circular dimension of pitch chroma, similarto that found by Shepard (1964), which wasabsent from the configuration for chords.

Emerging from this and the earlier studyon tones (Krumhansl, 1979) is a pattern ofmultiple levels of musical organization, withsimilar structural principles obtaining ateach different level. Further, the pattern ofinterelement associations that holds withinone level seems to reflect in part the func-tions of the elements as components of mu-sical entities at other levels of the generalstructure. This structure can be considereda hierarchy, with single tones comprising thelowest level, chords an intermediate level,and the different tonal systems or keys thehighest level. Our conception of this struc-ture is one in which links between elements,which represent the varying strengths of in-terelement associations, travel both horizon-tally and vertically. In the vertical dimensionascending and descending links are assumed.These would allow for a series of incomingtones to be interpreted in terms of their har-monic and tonal functions and for an estab-lished tonal system to create expectanciesabout which tones and chords are likely tooccur in the context.

The occurrence of such interlevel effectshas been documented in a number of psy-chological studies. For example, Cohen (Note3) has shown that after hearing a short ex-

cerpt from a musical composition, musicallytrained listeners are able to identify the keyof the composition with some accuracy. Al-though the growth of this tonal sense overtime has not been studied systematically,numerous studies point to the effect thattonal organization has on memory for se-quences of tones. Better memory for tonesin sequences that are well structured withrespect to a tonal system has been found re-peatedly (Cuddy & Cohen, 1976; Dewar,Cuddy, & Mewhort, 1977). Further, Krum-hansl (1979) has shown that nondiatonictones (tones not belonging to the key) aremore difficult to remember than tones thatfit into the key of the context; Dowling(1978) has demonstrated that changes pro-duced by replacing a tone by another tonealso in the key are difficult to detect. Thesestudies all point to the strong effect that aninstantiated tonal system has on the way inwhich sequences of tones are encoded andremembered. Finally, a recent study byCuddy, Cohen, and Miller (1979) has shownan effect of the relatedness between differentkeys. They found that it was easier to rec-ognize a melody when it is transposed to thekey of the dominant, a closely related key,than when it is transposed to the key of thetritone, a distantly related key. Bartlett andDowling (1980) obtained similar key dis-tance effects in both children and adults.These studies force the consideration of linksbetween levels in the kind of hierarchy pro-posed here.

Although the focus of this and the earlierscaling study of musical tones (Krumhansl,1979) has been on the horizontal intralevelorganization, interlevel influences are clearlyevident. In the context of a tonal system, thesingle tones that were judged most similarare those that comprise the tonic triad (theI chord). The strongest interchord associa-tions are those that are most essential for theestablishment of the key (the I, IV, and Vchords). Furthermore, these three chordsfrom the three closely related keys studiedwere all judged to be strongly related. Thiswas true despite the multiple functions thatthe chords play in the different keys, indi-cating that the psychological representationsupports the interpretation of a given chordin a number of related tonalities. This prop-


erty of the representation would seem tocomplement the practice in musical compo-sition of modulation between keys, sincemodulations between closely related keyswould not necessitate a radical rearrange-ment of the pattern of interchord associa-tions.

These interlevel influences do not neces-sarily imply that the structure at any givenlevel is entirely derivable from structure athigher levels. In support of such partial in-dependence, a circular dimension of pitchchroma has been found for single tones(Krumhansl, 1979; Shepard, 1964). No cor-responding circular dimension was apparent,however, in the spatial representation forchords in this study. Moreover, new factorsemerged at the level of chords that have todo with special properties of chords. In par-ticular it was found that the major and minortriads tended to be relatively closely related,with the diminished and augmented chordsless so. Asymmetries were found, with apreference for sequences ending on more sta-ble chords, particularly major triads, oversequences ending on less stable chords. Al-though asymmetries were also found in thejudgments on individual tones, the patterndid not mirror in any direct way those foundfor chords.

Despite these differences between tonesand chords, similar horizontal organizingprinciples apply at both these levels of thehierarchy. In both domains a central com-pact cluster of musical elements closely re-lated to the instantiated key was surroundedby less closely related elements. Further,strong temporal asymmetries in judgmentsof pairs of elements indicated a preferencefor moving toward, rather than away from,the elements in the core. In both cases mem-bership in the core could be accounted forby influences from the next higher level: themajor triad chord components formed thecentral cluster for single tones, those triadsmost essential for defining the key formedthe central cluster for chords. Thus multi-dimensional scaling and clustering tech-niques applied to judgments of sequences oftones and chords, complemented by analysesof temporal asymmetries, has revealed acomplex multilevel internal representationof musical pitch with similar organizational

principles at different levels and strong de-pendencies between levels.

Moreover, many aspects of the resultshave direct correlates in music theory. Tothe extent that the internal representationof musical relations, as revealed in experi-mental studies of this sort, mirrors variousmusic theoretic structures, we can make atleast four points. First, there is empiricaljustification for the intuitively conceivedclaims of music theory. Second, the listenersin our studies, who were not trained in musictheory, have a highly structured internal rep-resentation of pitch relationships, one thatcoincides with that of music theorists andcomposers. This is not a trivial claim becausethere is no a priori reason to assume thatlisteners without such formal training canabstract the underlying structures from com-positions, most of which have considerablesurface complexity. Third, the data can beused to construct psychological models of thesort outlined here, which can be further elab-orated and tested in subsequent experiments,guided in part by music theoretic consider-ations. Finally, studies of this sort may pointto basic psychological principles that governthe organization of pitch relationships un-derlying both musical composition and mu-sic theory.

Reference Notes

1. 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 Telephone Laboratories, 1973.

2. Olivier, D. C. Aggregative hierarchical clusteringprogram: Program write up. Computation CenterDocumentation, Harvard University, 1973.

3. Cohen, A. Inferred sets of pitches in melodic per-ception. Paper presented at a symposium on Cog-nitive Structure of Musical Pitch at the annual meet-ing of the Western Psychological Association, SanFrancisco, Calif., April 1978.

References

Bartlett, J. C., & Dowling, W. J. Recognition of trans-posed melodies: A key-distance effect in developmen-tal perspective. Journal of Experimental Psychology:Human Perception and Performance, 1980, 6, 501-515.

Cuddy, L. L., & Cohen, A. J. Recognition of transposedmelodic sequences. Quarterly Journal of Experimen-tal Psychology, 1976, 28, 255-270.

Cuddy, L. L., Cohen. A. J., & Miller, J. Melody rec-


ognition: The experimental application of musicalrules. Canadian Journal of Psychology, 1979, 33,148-157.

Dewar, K. M., Cuddy, L. L., & Mewhort, D. J. K.Recognition memory for single tones with and withoutcontext. Journal of Experimental Psychology: Hu-man Learning and Memory, 1977, 3, 60-67.

Dowling, W. J. Scale and contour: Two compounds ofa theory of memory for melodies. Psychological Re-view, 1978, 85, 341-354.

Dowling, W. J., & Fujitani, P. S. Contour, interval, andpitch recognition for melodies. Journal of the Acous-tical Society of America, 1971, 49, 524-531.

Fletcher, H., & Munson, W. A. Loudness, its definition,measurement and calculation. Journal of the Acous-tical Society of America, 1933, S, 82-108.

Johnson, S. C. Hierarchical clustering schemes. Psy-chometrika, 1967, 32, 241-254.

Krumhansl, C. L. Concerning the applicability of geo-metric models to similarity data: The interrelationshipbetween similarity and spatial density. PsychologicalReview, 1978, 85, 445-463.

Krumhansl, C. L. The psychological representation ofmusical pitch in a tonal context. Cognitive Psychol-ogy, 1979, //, 346-374.

Krumhansl, C. L., & Shepard, R. N. Quantification ofthe hierarchy of tonal functions within a diatonic con-text. Journal of Experimental Psychology: HumanPerception and Performance, 1979, 5, 579-594.

Kruskal, J. B. Nonmetric multidimensional scaling: Anumerical method. Psychometrika, 1964, 29, 28-42.

Kruskal, J. B. The relationship between multidimen-sional scaling and clustering. Classification and clus-tering. New York: Academic Press, 1977.

Levelt, W. J. M., VandeGeer, J. P., & Plomp, R. Triadiccomparisons of musical intervals. British Journal ofMathematical and Statistical Psychology, 1966, 19,163-179.

Piston, W. Harmony (3rd ed). New York: Norton, 1962.Ratner, L. G. Harmony: Structure and style. New

York: McGraw-Hill, 1962.Roederer, J. G. Introduction to the physics and psy-

chophysics of music (2nd ed.). New York: Springer-Verlag, 1975.

Rosch, E. Cognitive reference points. Cognitive Psy-chology, 1975, 7, 532-547.

Shepard, R. N. The analysis of proximities: Multidi-mensional scaling with an unknown distance function.I & II. Psychometrika, 1962, 27, 125-140; 219-246.

Shepard, R. N. Circularity in judgments of relativepitch. Journal of the Acoustical Society of America,1964, 36, 2346-2353.

Stevens, S. S., & Volkmann, J. The relation of pitch tofrequency: Revised scale. American Journal of Psy-chology, 1940, 53, 329-353.

Stevens, S. S., Volkmann, J., & Newman, E. B. A scalefor the measurement of the psychological magnitudeof pitch. Journal of the Acoustical Society of Amer-ica, 1937, 8, 185-190.

Received January 26, 1981

Epstein Appointed Editor, 1983-1988

The Publications and Communications Board of the AmericanPsychological Association announces the appointment of WilliamEpstein, University of Wisconsin, as Editor of the Journal of Ex-perimental Psychology: Human Perception and Performance fora 6-year term beginning in 1983. As of January 1, 1982, manu-scripts should be directed to:

William EpsteinDepartment of PsychologyW. J. Brogden Psychology BuildingUniversity of Wisconsin1202 West Johnson StreetMadison, Wisconsin 53706

Manuscript submission patterns on JEP: Human Perception andPerformance make the precise date of completion of the 1982volume uncertain. Therefore, authors should note that althoughthe current editor, Margaret Jean Intons-Peterson, will receive andconsider manuscripts until December 31, 1981, should the 1982volume be completed before that date, Intons-Peterson will redirectmanuscripts to Epstein for consideration for the 1983 volume.

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