1982k&kesslerpsychrevfromjournal

8/13/2019 1982K&KesslerPsychRevfromJournal

1/35

Psychological Review Copy right 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 T onal O rgan izationin aSpatial Representation ofMusical KeysCarol L. Krumhansl Edward J.KesslerCornell Un iversity Stanford University

The cognitive representation ofharmonic and tonal structure in Western musicis investigated using a tone-profile technique. In this method listeners rate howwell singletones(any one of the 12tonesof thechromaticscale)followam usicalelement such as a scale, chord, or cadence. Very stable rating profiles reflectingth e tonal hierarchies in major and minor keys are obtained, which, when inter-correlated and analyzed using mu ltidimensional scaling, produce a four-dimen -sional spatial map of the distances between keys. The keys ar e located on thesurface of a torus, in which the circle of fifths and the parallel and relativerelations betw een m ajor and m inor keys are represented. In add ition, singlechords (major, minor, diminished, and dominant seventh) are foundto becloselyassociated withthem ajor andm inor keysinw hich they play harm onic functions.The developing and changing sense of key during sequences of chords istracedby ob tainin g probe tone ratings following each chord in 10different sequences,8 of which contain m odulat ions (changes) between keys. Mod ulat ions betweenclosely related keys are found to be effected more immediately than are mod-ulationsbetween relatively distant keys. In all cases beyond the initial chord, thesenseof theprevailingkey is stronger than that producedby the last heard chordin isolation. Thus, listeners integrate harm on ic fun ction s over mu ltiple chords,developing a sense of key that m ay need to bereevaluated as additional chordsare sounded. It issuggestedthatthe perceived relations between chords andkeysan d between different keys are mediated through an internal representation ofthe hierarchy of tonal functions of single tones in music.

Musicconsistsof tones va ry ing in pitch, serveto highlightrhythmicpatterns,furtherduration, loudness,and timbre, but the per- emphasizephrase structure,anddistinguishception of music extends well beyond the betweentonesconstitutingthe primary me-registration of these physical attributes of lodiclineandtonesservingmore ornamentalthe musicalstimulus.Indeed,musiccontains or harmonic funct ions. Timbral character-considerablestructureeven in the relations istics may additionally provide importantthat obtainamongthe individualtones.For cues for theoverallstructureof themusicalexample,thedurationsare suchthatmetri- composition.calandrhythmicpatternsemerge from the lnWestern music pitchrelationshipsaresuccession of tones, and these patterns in probably the most developed and essentialcombinationwithotherfeatures define nat- to defining organization. Despite the tre-uralboundaries,orphrases, wi th in the mu- mendous variationfound in theselectionandsical composition. Variations in loudness orderingofpitchesin themusicalliterature,certain underlying regularities can be ob-This research w assupported inpart by agrant f r o m served an d described The perceptionof mu-th e National Science Foundation (BNS-81-03570)to gic lstructuredependson the processingofth efirstau thor .The au thorsaregratefu ltoDavidM. . , . ,. ... * 7Green for the use of the facilities in thePsychophysics Pltchinformation withreference to asystemLaboratory at Harvard Univer s i ty ,to Murray Spiegel ofknowledgeaboutthe conventional USCSofand David Wilson for technical adviceandassistance, pitches within the musical tradition. Thusand toRogerN.Shepardand ananonymous reviewer ognjtiveprocessesaresignificantlyinvolvedfW S5T 3iSd" nftfSro, L.perception and empirical work inKrumhansl, DepartmentofPsychology, U risHall,Cor- this area is directed at the description ofn e l l U n i v e r s i ty , I t h a c a ,NewY o r k 1 4 8 5 3 . these processes and the internal represen

334


2/35

T O N A L ORGANIZATIONI N M U S I C 335tation of pitch relations essential for the as-similation of musical organization.In our work w e rely heavilyonthe char-acterization of pitch structure provided bymusic theorists whose analyses of musicalscoresandintrospectiveprobings haveyieldedalarge bo dyoftheoretical w ritings. W hereasmuch of this work, such asPiston'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 studyo fmu sic perception is a matteryet to be decided, although preliminaryev-idence does suggest that listeners familiarwith Western music possess internal cogni-tive structures and processing strategies sim-liar to those presumed by music theorists.A t am inim um , m usic theory providesacon-venientterm inologyfo rdescribingth e struc-ture of music itself and is a rich source ofpredictionsand hypotheses fo rempirical in -vestigation.Music theorists suggest that pitch struc-ture in tradi t ional 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 th e tonic, functionsas a central reference point for the systemas awhole. Every other tone,and each chord,plays a unique function withrespecttothatabstracttonal center andthese functionsaresumm arized in a hierarchy of structural sta-bility. Moreover, different m usical keys havespecial associations to each other and aredescribed as more or less distantly related.This investigation provides a quanti tat ive,empirical measure of the perceived relat-edness between the ind ividu al pitches and aninstantiated tonal center, replicating and ex-tending anearlier study byK rumhansl andShepard (1979). These quantitative resultsare then used to obtain a spatial map ofmusical keys, reflecting th e distances be-tween the different keys. Next, w e charac-terizeth eharm onic rolesofindividual chordswith respect todifferent tonal centers in thissystem. Finally,wetracein the obtained spa-

tial map how the sense of keydevelops andchanges over time when the listener hearswell-structured sequences of chords.Music-Theoretic Descriptions of Tonali ty

Music-theoretic analyses are directed atspecifying th e tonal functions of the musicalelements'single tones and tones soundedsimul taneously as chordsin 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 hierarch y distinguishe s betwee ntones on the basis of their relative promi-nenceorstab ility.Thetonictoneisdescribedas the single most central and stable pitchwithin the set of musical pitches. This tonegivesthe nameto the key;for example, thenote C is the tonic of the k ey of C. T he tonicis also the first tone of the scale (sometimesreferred to as the first scale degree), is typ-ically sounded near th ebeginningo f a pieceofm usic, and is the tone that most freque ntlyoccurs at the end of major phrase bound-aries. Next in the tonal hierarchy are theother scale tones, usually conforming to theinterval patterns of the ma jo r or minor dia-tonic scale. A m ong 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 markero f the tonal center orkey.Final ly, the tones that are not containedwithin thescale, called nondiatonic, are theleaststableand appear infrequentlyandwithless rhythmic stress. This tonal hierarchy isat the core of music-theoretic accounts ofstructure in tonal music.In addi t ion, tones are frequ ently soundedsimul taneously in chords, and a similar hi-erarchy is presumed to apply to the set ofchords.The most frequently usedchordsarethose consisting of three different pitches,which are the refore called triads. T riads areconstructedfromaroottone,whichis a noteof the scale, the tone a major or minor thirdabove th e root, and a third tone a ma jo r orminor third above th e 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 thescaleisused todesignatethechord.For ex-


3/35

336 C A R O L L . K R U M H A N S L A N D E D W A R D J . K E S SL ERample, th e chord built on the first scale de-greeisindicatedby I, the chordbuilton thesecond by II, and so on. The chords builtoneachof thesevenscaledegrees constitutethebasic set of harm onies of the key. The ton ic,I, triad isdescribed as the most stable chordwithin th e system, followed by the V, ordom inan t chord, the IV, VI, and II chords,and final ly the III and VII chords. Chordsoutside this set are called nondiatonic andare infrequ ently sounded;whenp resent theyde ma nd are turnto themore stable elemen ts.These hierarchies among tones and chordsare used to provide varying degrees of ten-sionandresolutiona tdifferent points intimethroughout a composition, whichis thef u n -damental organizing principle of tonalmusic.Ton al s t ructure is even more complex thanjus t outl ined, for a hierarchy of levelsis be-lieved to exist in the instantiation of tonalregions. Chomsky (1965) has postulated asurface and deep structure forlanguage,andSchenker(1906/1954, 1935/1979) has sim-ilarly found tonal m usic to be an alyzable onforeground,middleground, and backgroundlevels. Three levels of key instantiation m aybe distinguished: the primary key, a tem-porary alternative key, and the key of a sin-gle chord. The pr imary key is the ma jo r orminorkeythat isgiven em phasis and usuallybeginsand ends th e piece. In between, mod-ulations (changes to different keys)m ay es-tablish new though usually related keys.Moreover, each individual chord m ay takeon some of the properties of the tonic of akey, a process called tonicization, whichdiffers from modulat ion by having a muc hbriefer influence. We will return to theseideas later, as they become relevant for theinterpretation of certa in experimental re -sults.

Psychological M odels of Pitch Stru ctureMusic theory suggeststhat musical struc-ture in tonal music can be accounted for by

the hierarchiesthatapplyto the musicalele-me n t s and that each key is associated withaparticular orderingofstability within theset ofsingle tones an d chords. A n u m b e ro fpsychological investigations have obtainedquant i tat ive empirical support for the inter-

nalization of these hierarchies by l istenersfamiliar with tonal music. O ne study byK r u m h a n s l and Shepard (1979) asked lis-teners to rate howwell each different toneof the chromatic scale in an octave rangecompleted a seven-tone C m a jo rscale. T hera t ingsgivenby listeners h av ing the greatestfamiliarity with tonal music recovered thetonal hierarchy predicted by music theorists:T he tonic tone received th e highest ratings,followedb y the third and fifth scale degrees,th e remaining scale degrees, and finally thenondiatonic tones. Thishierarchy wasalsoobtained in a scaling stud y of tones p resentedin a well-defined tonal context (Krumhans l ,1979). In that s tudy th e tones most stablewithin th e tonal system were perceived asmost closely interrelated, and those less sta-ble as more distantly related. Moreover, thisla t ter s tudy revealed a pat tern of temporalasymmetries in the j u d g m e n t s of pairs oftones that depended systematically on theirrelative stability in the instant ia ted key.Thus, in both these studies th e pattern ofresults was highly specific to the hierarchythatapplies to the tonality of the context.On e featur e of these results should be em -phasized.Inboth studiesit wasfound thatintervals of equal size were not perceived asequivalent but depended instead on the tonalfunctions of the individual tones wi thin th econtext key. In the Krumhansl and Shepard(1979) scale-com pletion stud y, th e tone acertain number of half steps above the im-plied tonic received a different rating thandid the tone an equal number of half s tepsbelow the tonic. For instance, the note C#,which is one half step above th e impliedtonic,C, wasgivenamuch lowerrating thanwas th e note B, which is a half step belowth e implied tonic. This may be understoodby the fact that the C#is n ondiatonicw ithinthe con text key , but the B plays the functionof th e melodically significant leading tonewithin th e key. Similarly, th e scaling study(Krumhansl , 1979) found that the judgedm usical relatedness between tw o tones form-in ga fixedinterval dependednot simply oninterval size per se but on the ton al function sof the individualtones. Again, fo rexample,different ra t ings were given to intervalsofequal size, such as that formed by C and C*and that formed by C and B. These f indings


4/35

TONAL ORGANIZATION IN MUSIC 337indicate that some systematic t ransforma-tion must intervene between geometricallyregular representations of pitch structure,such as those proposed by Shepard (1982a;se e also Shepard, 1981, 1982b), and thepitch relations as they are perceived in atonal context. M oreove r, these mod ificationsmay beaccounted for by the unique functionof each pitch within th e instantiated key, asdescribed by music theorists.There is also empirical support for them usic-theoretic d escription of ahierarchyofstability that applies tochords. T w o studies(Bharucha & Krumhansl , in press; Krum-hansl, Bharucha, & Kessler, 1982) have re-coveredthepredictedhierarchy inw hichthetonic, I, chord is the most stable, followedby the V chord, the IV, V I, and II chords,and finally the III and VII chords. This hi-erarchy has been obtained even in the ab-sence of awell-defined tonal context (Bhar-ucha & K r u m h a n s l ,inpress) . How ever, th ejudged relations between chords were af-fected when a key context w as provided(Bharucha & Krumhansl , in press; K r u m -hansl, Bharucha , & Castel lano, in press).These al terations in judgments , which aresummarized inthree con text-depen dent prin-ciples, are afunction of the distance betweenth e 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-geststhatif inv arian t relationships are foundin music, they are n eithe r at the level of sin-gle tones nor at the level of chords but in-s tead may be found at the more abstractlevel of musical keys or tonalcenters. Thefocusof the present investigation is the char-acterization of the psychological represen-tation of s tructure at the level of musicalkeys.A n u m b e r of psychological studies usingmemory performance provide further evi-dence for the impor tance of tonal organi-zation in music perception. Well-structuredtonal sequences have been found to be re-m embered better than those not conform ingto key structure. This result has been ob-tainedusing b oth m elodicsequences(Cohen,1975;D ewar , 1974; De war , Cudd y,&M e w -hort, 1977; Frances, 1972) and harmonic

sequences, that is , chord sequences (Bhar-ucha & K rum han s l , in press ; Krum hans l e tal,inpress).An u m b e ro fresults reflect th edifferential stability ofdiatonic and nondia-tonic elements. Tones and chords that arenondiatonic w ithinanimplied keyhave beenfound to be recognized correctly less oftenthan are diatonic tones and chords (Bharu-cha & K r u m h a n s l , in press; Krumhansl ,1979). Moreover, diatonic elementsare morefrequently confused with other diatonic ele-ments than they are with nondiatonic ele-ments (Bharucha & Krumhansl , in press;Dewar, 1974; Dowling, 1978). Finally,asymm etries ha ve been found suchthatnon-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 & K r u m h a n s l , in press; Dowling &Bartlett, Note 1).The m agn itude of each ofthese effects has been shown to depend sys-tematically on the choice of the context key(Krumhans let a, ., in press). T hese m em oryresults provide convergent evidence for thestructu ral relations o btained in the empiricalstudies described earlier.

OverviewIn this article w e investigate three closelyrelated aspects of the representation andprocessing of tonal structure, each ofwhichis presented in a separate section. In the firstw e obtain a quanti tative measure of the de-gree to w hich each ind ividual tone in an oc-

tave range is related to an abstract tonalcenter . These quanti tative m easurem entsarethen used to derive a spatial map of themajor and minor keys,representing th e dis-ftances between different tonal centers. In thesecond section w e determine th e relationsbetween chords of different types and thesetonal centers, representing this informationby points in the spatial representation oftonal regions. In the final section w e assesshow the sense of key develops an d changesover t ime during sequences of chords, someof which contain modulations (shif ts) be-tween different keys. O ur focus in the lasttw o sections is on the manner in whichchords and chord sequences relate to tonalregions as they are laid out in the spatial


5/35

338 C A R O L L .K R U M H A N S LA N D E D W A R D J. K E S S LE Rrepresentationof keysdescribedin the firstsection. Each one of these sections beginswitha short description of the relevan t mu sictheory, followed where possible by a briefsumma ry of previous empirical investiga-tions related to the issue considered in tha tsection.

Distances Between K eysDetermining th e psychological distancebetween keys is im portan t to an un derstand -in gof the perception of music in the W esterntonal tradition. In music the juxtapositionof keys can be used to highlight structural

elements, provide contrasts of vary ing de-grees,and addexpressive qua litiesto apiece(Berry,1976,p. 15;Rosen, 1972,pp. 26, 27;Salzer, 1962,p. 20). S ome keys are so closelyrelated that a change from one to anothercan hardly bedetected, whereas othersaresodistantly related th at shifts betwee n themseem abrupt and unpleasant . T he measure-men to f the psychological distances betweentonal centersm ayelucidate th ena tureof theinternal representation of musical structurethat underl ies these perceptual phenomena.In addition, as suggested earlier, invariantrelationsmay be contained in the structureat the levelof abstract tonal centers.Music-Theoretic Descriptions of KeyDistance

Music theory presumes that several fea-tureso fkeysm aydetermineth e accessibilityor ease of modulation between them, whichisconsidered to be a rough measure of dis-tance between the keys. For m ajor keys threedifferent features all consistently predict thesame pattern of interkey relatedness: th enumber of tones shared by their scales, th edifference in the numb er of sharps or flatsin the keysignature, and thed istance arou ndth e circle of fifths. For the major keys th edifference in the keysignatures is inverselyrelated to the n u m b e rof shared scale tones.Forexample,tw om ajor keys thatdifferonlyin t e rm sof onesharp or flat haves ixsharedscale tones. Thus, G m ajor (one sharp) andF major (one flat) both differ from Cm a jor(nosharps or flats) in term s of one scale tone,and the tonic of G is a fifth above and thetonicof F a fifthbelowthatof Cmajor. D

major ( twosharps)a nd B6major ( two flats)each have tw o scale degrees not shared byC major , and their tonics are separated bytw o fifths above and below the tonicof Cmajor. This fifthconnection continuesto in-dicate distance between major keys unti lafter six fifthsaboveor belowany givenkeythe key of the tri tone, the most distantly re-lated major key, is reached in both direc-tions. Because this fifth relationship foldsback on itself, i t is conven iently pictured asacircle,calledthecircleof fifths, first pro-posed by Heinichen (Forte, 1979, p. 27).Thus, the distances betw een m ajor 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-tan t issue. H owev er , introduction of the m i-nor keyscomplicatesthe picture considera-bly. T o every major key there correspondsam i n orke ytha th as thesamek eysignature,called the relative minor, whose tonic is aminor third below (or a major sixth above)tha tof them ajor key .F orexample,Am inoristhe relative minorof Cm ajor. D espite th eshared key signatures, the relative m ajor andminorkeysdo notov erlap completelyintheirscale tones. The scale of the relative minorkey has in fact three different forms: th eharmonic minor form in which the seventhscale tone is raised by a half step, the as-cendingm elodicform inwhichthe sixthandseventh scale tonesareeach raisedby halfsteps, and the descending melodic form (o rna tu ra l form) in which the scale tones areexactly those of the relative m ajor key. Thesedifferent forms make impossible th e precisedefinition of ma jo r -mino r key distance asthe n um be r of shared scale tones. No nethe-less, a major key and its relative minor keyare considered closely related by virtueofthei r shared keysignatures.Additionally, there is a close tie betweena major key and the minor keybuilt on thesame tonic, which is called the parallel mi-nor. This is true even though their key sig-natures differ in te rms of three sharps orflats. For exam ple, C major (wi th no sharpsorflats) has C m inor (with three flats) as i tsparallel m inor. Despitethe differences in keysignatures, these keys share most of thestructurallystabletonesthetonic, fourth,


6/35

TONALORGANIZATION IN MUSIC 339an d fifth scale degreesas well as the sev-enth, leading tone in the harmonic and as-cending melodic forms. Possibly even moreimpor tan tis thefact that they also share th eessential dom inant, V, chord, whichpreparesthe tonic of either of the parallel keys(Schoenberg, 1969, p. 51).Thus, every major key has two closely re-lated minor keysthe relative and parallelminors. Or, looked at another way, any sin-gle m inorkey is closely tied to two differentmajor keys that are relatively distant on thecircle of fifths. This fact makes it impossibleto accommodate th e 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 Cmajor and Cm inor keys.For both, the representation consists of al-ternat ing columnso f m a jo r and minor keyssuch that each major key is flanked by itsrelative m inor key on one side and its p arallelminorkey on theother.Thekeys withineachcolumn are separated by fifths. The minor-key model isclearly a subset of the one formajorkeys. Acc ording to Schoe nberg(1969,p. 30) a minor tonic does not maintain asdirect con trol overitsregionsasdoesamajortonic, and consequently, th e n u m b e r of di-rectly related regions is smaller fo r minorthan fo r 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 aretw o impor tan t differences, however. First,Schoenberg's keyregionsarecentered aroun da 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 , w e obtaina m ap of key regionsthatdepicts all interkeyrelations as dis tances between points in ametric space. Second,Schoenbergdoes notattem pt to m ake fine discriminations in tonaldistance as, for instance, he equates the dis-tance between a major key and i ts paralleland relative minors. Despite his statements

SCHOENBERG'S C H A R T S O F K E Y REGIONSC MAJOR KEY REGIONS

F'fc1G

,D d F fA [ Q 0 cE e | G | g

A *

B * b >

f1O

C MINOR K EY REGIONS

Figure I.Schoenberg's spatial map of the keyregionssurroundingC m a jo r and C m inor keys (adapted fromSchoenberg, 1969). (The circle of fifths relation is rep-resented as the ver t ical dimension,and the parallel andrelative relations between major keysindicated bycapitallettersan dm inorkeysindicatedbylowercaselettersasthe hor izontal dimen sion. )

that in common practice a major key iscloser to its relative than it s parallel minor(Schoenberg, 1911/1978, p. 153; 1969, p.68), thisdifference is nots tru cturally evidentin the model. (There is some debate on thispoint. Piston [1962] f inds that parallel ma-jors andminorsarepractically identical andthat classical composition in the 18th and19th centuries tended "to regard the twomodes as s im ply twoaspectsof onetonality[p. 145].) In anycase, our method producesinterkey distances that permit precise com-parisons.Empirical Studies of Key Distance

A lthoug h the issue of key distance has notbeen extensively investigated inpsychology,there are a num be r of s tudies dem onstratingkey distance effects. Memory per fo rmancefor transposed sequences has been found tobe better when the sequence is shifted to aclosely related keythanto adis tan tly relatedkey (Bartlett & Dowling, 1980; Cohen,1975; Cuddy, Cohen, & Miller, 1979). Inthese studies listeners are required to judgewhethertwo m elodic sequences are the same,except fo r transposition to a different key(whichm ain ta insth e relative sizesof all the


7/35

340 C A R O L L. K R U M H A N S L A N DEDWARDJ.KESSLERintervals) . Accuracy in this task is higherw h e n the melodyistransposed to a closelyrelated key.More recently, Krumhansl et al.(in press) have identified three principlesgoverningharmonic relationsthatdependonth e distanceof thecontext keyfrom the keyin which the chords have basic harmonicfunctions. The probability that a chord iscorrectly recognized w as found to be a de-creasingfunction ofthis distance. Moreover,th epsychological proximityof two differentchords in the same key, measured both interms of direct relatedness judgments andm em ory confusions, decreased as interkeydistance increased. Finally, this distance wasfound to affect themagnitude anddirectionoftemporal asymmetriesinrelatedness judg-me nts and memory confusions. Althoughthese studies have not provided a measureof distance between musical keys, they dosuggest thatthemusic-theoretic descriptionof structure at the level oftonal centers isapplicable to the process of music percep-tion.The Present Approach toDetermining K eyDistance

In ourempirical determinationof keydis-tances, w echose not toemploy recognitionm e m o r yfortransposed sequences, whichhasyieldedkeydistanceeffectsinprevious stud-ie s(Bartlett &Dowling, 1980; Cohen, 1975;C u d d y e t al,, 1979). Three reasons fo r thiswere , first, in order to get very precise re-sults,wewould need sequences thatareuni-formly tonally unambiguous, which wouldbe difficult if not impossible to construct.Second, w ewere concerned that transposi-tions betweencloselyrelatedkeys mightoc-casionally be rejected owing to confusionswithtonalanswers(Dowling, 1978). Finally,ifbothmajor andminor sequences wereem -ployed, transpositions between them wouldresult in changed intervals in melodic se -quences orchanged chord typesinharmonicsequences,makingtheresults difficult to in-terpret solely in termsof keydistance. Es-t imat ing keydistances from changes in thepa t te rns of confusion errors or relatednessj udgme nt s as a function of the context key(such as those observed in Bharucha &K r u m h a n s l , in press; Krumhansle t al., inpress)would bequite indirect as well as ne-

cessitate a very large number of observa-tions.Forthese reasonswechosetoemploythemethod introduced in the earlier scale-com-pletionstudy (Krumhansl&Shepard, 1979),which w ewill refertohereas thetone-profiletechnique.T heresultso fthat study,inwhichlisteners ratedhowwelleachmusical pitchin anoctave range (the pitchesof the chro-maticscale)completed a seven-note majorscale, were described earlier. The rating pro-filesforthelistenerswithamoderatetohighlevel ofmusical training revealed consider-able structure, which accorded well withmusic-theoretic descriptionsof thetonalhi-erarchy. Thisresult suggeststhatratingpro-filesof this sort m ay serve as a distinctivema rke r o f tonal organization that couldserve as a basis fo r determining distancesbetween keys.In the firstexperimentw eobtained ratingprofiles for the 12different musical elementslisted inTable 1.Each ofthese 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 th e probe tone fit, in a musicalsense,with the element just heard. This pro-cedure yielded a rating profile fo r each ofthe 12 elements studied. The tones andchords employed were constructed ofpitchessounded in fiveoctaves 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 betweenblocks, each new ele-m e n t w aspresented in a different key fromthat of the previous block, and each blockbeganwitha fewpractice trialstoacclimateth e listener to the newelement in the newkey.Third, theexperiment contained exactreplications and also replications in whichthe element was transposed to a differentreferencepitch. Fourth, listeners selected fo rthe experiment formed a relatively homo-geneous group in terms of musical back-ground, with a moderate to high degree ofmusicalsophistication.


8/35

TONAL ORGAN IZATION IN MUSIC 341Table1MusicalElements Used inExperiment 1

Element ExampleAscending major scaleAscending harmonicminor scaleMajor chordMinor chordDiminished chordDom inant seventh chordIV-V-Icadence in majorII-V-I cadence inmajorVI-V-I cadence in majorIV-V-I cadence inminorII-V-I cadence inminorVI-V-I cadence inminor

D major scale: D E F* G A B C DA minor scale: A B C D E F G ' AE major chord: E G * BD minor chord: D F AF* diminished chord: F* A CDominant seventh chord on G: G B D FCadence in C major: F major chord, G major chord, C major chordCadence in F major: G minor chord, C major chord, F major chordCadence in D major: Bminor chord, A major chord, D major chordCadence in E minor:A minor chord, Bmajor chord, E minor chordCadence in F*minor:G *diminished chord, C*major chord,F* minor chordCadence in Bminor:G major chord,F'major chord, Bminor chord

Exper iment 1Method

Subjects. Ten Harvard University undergraduatesserved as subjects and were paid at the rateof $3 perhour. Each subject participated in two experimental ses-sions lasting atotalofapproximately 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 averageof 12.0 yearsin instrumental groupsand2.1years inchoral groups. C urrently, subjects were playingmusic an average of 3.1 hours per week and listeningtomusic for 23.3 hourspe rweek.O ne subject reportedhavingtakena"verybasic"introdu ctory courseinmusictheory, but the remaining subjects had no formal musictheory background. All subjects reported having normalhearing and no onereported having absolutepitch.Apparatus. The stimulus tapes (Maxell UD 35-90)were recorded at 7.5inches 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 an d Telephonies TDH-50 head-phones. All the stimuli were prepared digitally on aPDF-15 computer in thePsychophysics Laboratory atHarvard University and were played through a digital-to-analog converter under computer control.Thedigitalrepresentations were sampled at a rateof5,000 pointsper second. A temporal amplitude envelope with a 10msec rise and decay reduced onset and offset clicks in-troduced b y the hard war e. Finally, a low-pass filterwithcutoff 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 15sine wave components (threei n each of five octaves), the dominant seventh chordsconsistedof 20sine waves (fourineach 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 overthe five-octaverange (from 77.8Hz to 2349Hz ), which consisted ofthree parts: agradually increas-ing section over the first octave and a half, a constantsection overthemiddletwooctaves,and a symmetricallydecreasing section over the lastoctave and ahalf. Thecorresponding amplitude of the sine wave componentsateachfrequency was determined using the amplitude-loudness curves of Fletcher and Munson(1933). Thismethod, originally introduced by Shepard (1964) an ddescribed in detail elsewhere (Krumhansl et al., 1982),produces tones and chords with an organlike sound,withoutanyclearlydefinedlowestorhighest componentfrequencies. Thecom posite chordsandsingle tones wereplayed at 71 db (A ) tha t, because of the difference innumber of sine wave components, necessitated a slightupward shift of the loudne ss envelope for the single tonesand a slight downward s h i f t 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,minorchord, diminished chord, dom inant seventh chord,and three cadences (IV-V-I,II-V-I, an dVI-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. T he first two trials in each block were forpractice.


9/35

342 CAROL L. K R U M H A N S L AND E D W A R D J. KESSLERO n th e other 12 trialsof theblock, each possible probetone from th e chromatic scale w aspaired with th e ele-men t in a randomorder.Astimulus tape contained 12elementblocks,one foreach element.Areplication tapecontained th e identicalelementsas the firsttapebut ina different random order an d with a different order ofprobe tones.Atranspo sit ion tape usedth esame e lementtypes but each w as soundedin a different pitch rangethanon theoriginal tapes. A gain, therew as arep licationtape withthetransposed elements randomizedasabove.T he subjects heard th e four different tapes in differentorders such that during each of the two experimentalsessions, a subject heard onecom pleterep licationei-ther both original or both transposit ion tapes.O n each t r ia l th e e lement w as heard first, followedby a 1-sec pause, an d then th e probe tone sounded fo r.5sec. The t imingof the different elementswas asfol-lows: The tonic tones of the two scales (major andminor) were sounded for .5sec, and the remainingscaletones for .25 sec, with a .19-sec pause between scaletones.T he single-chord elem ents were played for .5sec,as were the chords in the three-chord cadences. In thecadences there was approximately .25 sec betweenchords. A4-sec in terval betweentrials allowed subjectsto record their responses.Procedure. At thestartof the first session, subjec tswere instructed to rate on a 7-point scale how well, ina m usical sense, each prob e tone fit into or w e n t withth e musical element just heard. O n this scale, 1 wasdesignated "fits poorly" and 7 was designated "fitswell," and subjects w ere encourag ed to u se the fullrangeof the response scale. Subjects responded to three com-plete blocks ofpractice trials before beginning the ex-per imentaltapes. After al lexperim ental tapeshadbeenpresented, th e subjects completed th e short question-naire abo ut their m usical backgro unds.Results and Discussion

The 12types of musical elements listedin Table 1 were each probed by the 12 tonesof the chromatic scale.The set ofjudgmentsfor each of the musical elements will be re-ferred to as the profile fo r that element.These profiles will be analyzed in a numberof ways. First, th e stabilities of the profilesare evaluated using intersubject correlationsand correlations between th e profiles ob-tained fo r transposed elements. Second, th eprofiles themselves are intercorrelated andseparatecomposite 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 aspatialmap of the 24major and minor keys.The discussion ofchord-key relations thatderive from th eobtained profiles willbe de-ferred unt i l the next section.Intersubject differences. No large indi-

vidualdifferences were expected because thesubjects were selected to be comparable interms of their m usical backgrou nds. In orderto evaluate consistency across listeners, in -tersubject correlations on the response pro-files(averaged over the twoexact replica-t ions) we re com puted. These correlationsaveraged .585, ranging from .412 to .740,and werea ll significant(p < .01). (The cor-relations between the two replications fo reach subject ranged from .241 to .658, av-eraging .492, suggesting thaterror variabil-ity may be substant ia l ly a t tenuating the in-tersubject correlations that were computedon th e responses averaged over the two rep-lications.) Multidimensional scaling (Krus-kal, 1964; Shepard, 1962a, 1962b; th e com-puter program used w as KYSTKruska l ,Young, & Seery, Note 2) and hierarchicalclustering (Johnson, 1967;the p rogram usedw as AGCLUSOlivier, Note 3) techniqueswere applied to the matrix of intersubjectcorrelations. Neither of these analyses pro-duced solutions that grouped listeners to-gether in a waythat related to anyaspectoftheir m usical backgrounds. Co nsequently,individual differences werenotexploredfur-ther, and the remaining analyses are basedon theratingprofilesaveragedacrossthe 10subjects.Transpositions. Each element was rep-resented in twod ifferent 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 iscom m only believedthattranspositions are perceived as equivalent,except perhaps by listeners with absolutepitch. Because not one of our subjects hadabsolute pitch, high correlations were ex-pected between th e ra t ings whentheprofileswere shiftedappropriatelytocompensate fo rthe interval of transposition. These t rans-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. O ne cen-tral purpose of this experiment was to es-tablishreliablepro files fo r major and minorkeysso that these could be used in later anal-yses. In addition, the major key profile from


10/35

T O N A L ORGANIZATION INMUSIC 343this experiment can be compared with thatof Krumhans l and Shepard (1979). Corre-lations between the 12 different profilesshowed av ery s im ilar pattern of probe tonejudgments for the major chord element andthe three cadencesinmajor ,IV-V-I,II-V-I,andVI-V-I.The average correlation be-tween these(.896,p


11/35

344 C A R O L L.K R U M H A N S L AND E D W A R DJ. KESSLER-C M A J O R PROFILE-A MINOR PROFILE

C C' 0 D 1 E F F ' G G a A1 BP R O B E T O N E

Figure 3.The C major keyprofile and the A m inor keyprofile are superimposed to i l lustrate the derivation ofth e interkey distances. (The A minor profile was ob-tained by shifting the C m inorprofile down th reestepsof 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 supposedby music theorists to de-te rmine key distance. (Other measures ofkey distance, such as the summed squareddistances between th e rating profiles, werealso explored and produced almost identicalresults). The regularities in these measuresofkeydistance arebroughtoutm ore clearlyin the following analyses, however,so a fulldiscussion willbe postponed untillater.Multidimensional scaling of major an dminor profile correlations. T he interkeycorrelation matrix w as then analyzed us-in g the multidimensional scaling programMDSCAL (Kruskal , Note 4). A satisfactorysolution w as obtained only in four d imen-sions, w ith a stress v alue (Fo rm ula 1) of.017. The standard, principal-axis orienta-tion option was employed. This solution isshown in its 2 two-dimensional projectionsinFigure4. In the first twodimen sions, themajor and m inor keys are ar ranged arounda circle sothat any major key is flanked oneither sideby itsdominan tandsubdominan tmajor keys (whose tonics are a fifth aboveand belowthat of the first major key) . T hesame holds true for the minor keys. Thus,in the first two dimensions w e obtain th ecircle of fifths for major and minor keys.How ever, the placement of the minork eycircle with respect to the major key circleposes problems fo r interpretation. In the so-lutiontheclosestminorkey to anymajorkey

is neither its relative nor its parallel minorbut insteadth eminorkeybuilton thesecondscale degree (the relative minor of the sub-dominant) . This placement undoubtedly re-flects acomprom ise betw eenth e close tie ofa major key to both its relative and parallelm inor keys, which are separated by threesteps around the circle of fifths. (For ex-ample, the C majorprofile correlates stronglywith both the Aminor and C minor profilesbut less stro ngly w iththat of Dm inor ).Be-causeo fthis tension producedby the relativeand parallel relations, th e two-dimensionalsolution (which was almost identical to theleft p a n e lofFigure 4) was rejected in favorof th e four-dim ensio nal solution picturedhere.T he projection in the third and four thd i-mensions represe nts the parallel and relativerelations between m ajor and minor keysd i-rectly.The keys again form acircular con-figuration, butwi thaslightly sm aller radius.O nth is circlean ygiven m ajorkey is flankedby it sp aralle l m inoron one side and by itsrelativeminoron theother,andlikewise,anyminor key is located next to its parallel andrelative m ajor keys. In order to accomplishthis, the solution identified as a single pointallm ajor keys whose tonicsdiffer by amajorthird and also all minor keysdiffering 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 clearlyth e s tructure of the interkey relations.

Toroidal representation of key distance.The low stress valu eof thefour-dime nsionalsolution is evidencethat this representationprecisely accounts for the intercorrelationsbetween profiles. However , the 2 two-di-mensional projections posed problems fo rinterpretation. In each of the two-dimen-sional projections, circular con figurationswereobtained. Th us,the pointsfallon asur-facethatcan bedescribedas acirclein twodimensionscrossed withacirclein twomoredimensions. But this is exactly th e topolog-ical definition of a torus (S1 X S 1 ) . Eachpoint in four dimensions can be uniquelydescribed by the ang ular dis tance, 9,aroundone circle and the angular dis tance, 0,around the second circle. The (6 , 4 > valuesfor each key can then beplotted in two di-


12/35

TONAL ORGANIZATION IN MUSICM ULTID IM ENSIONAL SCALING SOLUTION OF K EY S

345

D b ^g f E kB c' c gi,E

D e G

g A l> E c e

f c8 BrEk a G bOk ~ g d*

F Afc DBkd b* p

DIM ENSIONS I AND 2 DIM ENSIONS 3 AND 4Figure 4, T he four-d ime nsional mu l t id ime nsional sca l ing so lut ion of the intercorrelations between th e24 major and minor key profiles (stress = .017). (T he projection of the s olutio n onto the first twodimensions is shown on the left. In this projection th e circle of fifths for major an d minor keys w asobtained. The projection onto the last two dim ension s is shown on the r igh t . Major and minor keysseparated by an in terva lof a major third were represented as single points in the solution. Each majorkey was located next to i ts paral lel minor key on one side and i ts relative minor key on the other.Similarly, each minor key was f lanked by its parallel and re la t ive major keys.)

mensions as shown in Figure 5, where it isunderstood that the opposite edges of therectangle areidentified.To see the similarity between this repre-sentation and the more famil iar (but mis-leading) description of a torus as an inner-tube, first visualize folding the rectangle overto identify the twohorizontal edges, mak inga hollow tube, followed by wrapping itaround to line up the two open ends. N otonly does this introduce distortions of dis-tances (the inner radius is smaller th an theouter radius) but i t 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 ispreferable.The (B, < j > coordinates of the 24 keys werecomputed from th e four-dimensional scalingsolution and are shown in the torus repre-sentation in Figure 5. Because of smal l de-viations from perfect circularity in the ob-tained scaling solution, ideal coordinateswere computed fo r each key such that th epoints were constrained tofallon the surfaceof th e torus. T he program CMH2 (Cliff,1966),w hichshifts, rotates, and expandsorcontracts oneconfiguration to best match asecond configuration,w as used to comparethe ideal configuration with that actuallyobtained in the scaling solution. This pro-

gram gave a correlation of .998, indicatingthatthe torusis infactaveryaccurate rep-resentation of the four-dimensional scalingsolution of the profile correlations.More impor tan t ly , when viewed as (6 ,< j > coordinates on a torus, the pattern of in-terkey distances becomes entirely interpret-able.First ofall,the keys separated by fifthsfallon apaththatwrapsthree times aroundth e torus before joiningup with itself again;th e major keys fall on one such path, andth e minor keys on another, parallel path.These are lined up sothat any major key isflanked by its relat ive m inoron oneside andi ts parallel minor on the other. These par-allel, relative, and fifth relations are madeexplicit for C ma jo r in Figure 5.Thereis a striking similaritybetween themap of tonal regions given by Schoenberg(1969) fo rm a jo r and m inor 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 surroundinga singlemajor orm inor key. Moreover , precise qu an-titative comparisons between interkey dis-tances can be made from the torus repre-sentation. For example , a ma jo r key is infact found to be more closely related to itsrelative than to its parallel minor as seen,


13/35

346

ek/d

C A R O L L . K R U M H A N S L A N D E D W A R DJ . KESS LERf d b k -

qt

a fx /Relative^ /V. 1 Paral le l A

/ Circle o f f i f t h s

Figure 5.A n equivalent two-dimensional map of the multidimensionalscaling solution of the 24 majoran d minorkeys.(Thevertical[dashe d] edgesareidentified and thehorizontal[solid] edgesare identified,giving a torus.T he circleof fifths and paralleland relativerelations for the C major key arenoted.)

for instance, as the closer distance betweenC m ajor and A m inor than between C m ajorandC m inor. A nother finding isthata m ajorkey is closer to the m i n o r key bui l t on i tsthird scaledegree (the relative minorof thedom inan t key) than it is to the mino rkeybuilton its secondscale degree (the relativeminor of the subdominant key) . For in-stance,Cm a jo rand Em inora recloserthanare C majorand D minor. In fact, thiswasanticipated by Schoenberg (1969, p . 68) andm ay reflect the greater number of chordsshared by the ma jo r key and the relativeminor of the dominant key. Other compar-isons ofthissortcan easilybe made usingthis spatial map of key regions. Moreover,this representation providesaf ram eworkforapproaching the problem taken up laterofhow chords relate to different tonal centersan d how the sense of key develops as thelistener hears sequences of chords.Otherspatial rep resentations. Other pre-viouslyproposed spatial representations ha vebeen described in detail by Shepard (1982a;see also Shepard, 1981,1982b) and willbementioned onlybriefly here. The first suchrepresentation in theliteratureis thehelicalstructure of single tones (Drobisch, 1846,citedin Ruckmick, 1929;Pickler, 1966; Re-vesz, 1954; Shepard, 1964). In this three-dimensional configuration,th e single tonesare spaced along ahelical path in order of

increasing pitch height suchthat tones sep-arated by an octave interval are relativelyclose as the helix w ind s back over itself onsuccessive turns.Theprojectionof the pointson the plane perpendicular to the verticalaxisof the helixisoften referred to as tonechroma, and the projection on the axis as toneheight. This representation, then,si-multaneously specifiesth e close relation be-tween tonesseparatedbysmallintervalsandthat 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 logfrequency asproposedbyStevens (Stevens & Volkmann, 1940; Ste-vens, Volkmann, & Newman, 1937), be-cause thereis astrong identification betwee ntonesd iffering by octaves. This octave effectisseenintheir interchangeableuse in musictheory an dcomposition and injudgmen tsofintertone relatedness (Allen, 1967; Boring,1942, p. 376; Krumhansl, 1979; Licklider,1951, pp. 1003-1004; and numerous o thertreatments). 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 appliesto musical pitches.Shepard (1981, 1982a, 1982b) has re-


14/35

TONAL ORGANIZATION IN MUSIC 347cently proposed more complex representa-tions of single tones. In the representationreferred to as the melodicm ap" (Shepard,1982a, Figure 4), the set of 12 tones in anoctave rangeareplaced on thesurfaceof thetorus in four dimensions. As in the config-uration obtained here, the f irs t two dimen-sions correspond to the circle of fifths. Thelast tw o dimensions constitute th e chromacircle of the helical structure proposed ear-lier. How ever, introdu cing th e circleof fifthssplits a pa rtth echrom a circle intotw owhole-tone scalesformingadouble helix that w rapsaround the surface of the torus. Shepardprovided evidence that an additional fifthdimension m i g h t be needed to account fo rtone height.Empirical suppor t fo r Shepard's melodicm apcomesfrom th e earlier scale-completionstudy( K r u m h a n sl &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 m ethod of obtaining the m atr ix ofinterelement distances w as different in thetw ocases. In the Shepard analysisthevaluefor tones separated by each numbero f halfstepswas the average of the rating given tothe tone tha t n um be r of half s teps above theimplied tonicand therating givento thetonethat number of half s teps below the tonic.A snoted earl ier , how ever, interva lsofequalsize often receivedsignificantly 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 th ecorrelation betw eenth e shifted profilesas the distance measure.This latter measure is most properly inter-preted as am easureof interk ey dis tance be-cause it is based on the similarity of tonalfunctions of the individual musical tones inth e different keys. A second, related reasonfo r differences between the two representa-tions isthat tone height does not apply toan abstracttonalcenter,because it is inde-pendent of the octave in whichth e tonic issounded.M oreover,thep robe tonesa ndcon-text elements used in this s tudy were con-structed of frequencies sounded in five oc-taves s imultaneously, thus minimizing tone

height effects. Nonetheless, it is significantthat th e chroma circle did notemerge fromth epresent an alysis because Sh epard(1964)demonstrated that this circular dimension isindependent of tone height.Shepard's (1982a , Figure 8) "harmonicmap," which is a t ransformationof the m e-lodic map, is identical to the spatial repre-sentation obtainedhere except forscalefac-torsand theinclusionofpointsfo rmajorandminorkeys in the presen t results. He re, how-ever, w e give this configurationa differentinterpretation.Shepard'sproposed represen-tation depicts th e relations between singletones, with certain chord, scale, and key re-lations reflected as patterned subsets withinthe spatial map. Thus, this repesentationbringsouts tructuresat allthree levels: singlepitches, chords, and keys. In contrast, weinterpret the spatial configuration obtainedhere asdep icting s tructure at onlyth e levelof abstract tonal centers and p resumethatadditional or other s tructural fea tures applyto single tonesandchords, particu larly thosethat reflect th e context-specific hierarchyofstability described by music theorists. Em-pirical evidence supporting this view hasbeen obtained in an u m b e rofstudies(Bhar-ucha & Krumhansl , in press; Krumhansl ,1979; K r u m h a n s l et al, 1982, in press;K r u m h a n s l &Shepard, 1979) and wassum-marized earl ier in thisarticle.That Shepard's (1982a) harmonic m apappears as a subset of the spatial represen-tation of musical keys obtained here, how-ever, may be accounted for by the centra lfunction of the tonic tone in the s tructuralhierarchy that applies to each maj6r andminor key. In other words, th e convergencebetween Shepard's harmonic map and thetoroidal map ofm usical keysmay beunder-stood in te rms of the close association be-tween anabstact tonal center and the tonethat is its tonic. Totest this hypothesis anadditional analysis of the profiles obtainedin the present experimen t was performed .I nthis analysis a measure of the psychologicalproximity between single tones was esti-matedby correlating theratingsgivento onetone in the 24 major and minor keys wi ththe ratingsgiven to the other tone in the 24keys. This measure reflects how similar thetw o tones are in their tonal functions acrossall possible musical keys. Multidimensional


15/35

348 C A R O L L.K R U M H A N S L A N D E D W A R D J. KESSLERscaling applied to these correlations re-covered the spatial configuration of Shep-ard'sharmonicmap(stress=.007),The firsttwo dimensionscontainedthecircleof fifths,and th e second tw o dimensions contained acircular configuration in which tones differ-in g by a major third were identif ied as asingle point; th e ratio of the radii w as 4:3.Thus ,Shepard's harmonic map m ay be ob-tained from the probe tone ratings and maybe understood as being derivative of struc-tureat thelevelofabstract tonal centers andmediated throu gh the cen tral function of thetonic tone in the hierarchy of stability.

Relations Between Chords and TonesIn m u c h of tonal music, groups of tonesare sounded simultaneously in chords. Al-though it is as traightforward m atter tospec-ify the chords that constitute the basic setof harmonies of a particular key, as de-scribed earlier, th e converse problem of de-te rminingthe prev ailing keyfroma 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. T he listenerm u s t extract key information from the se-quence of chords itself, using knowledgeabout the harmonic functions of chords inth edifferent musical keys. In this sectionw eturn to the problem of characterizing thelistener's know ledgeoftheseharmonic func-tions.Music-Theoretic Descriptions of ChordFunctionsSeven chords form the basic set of har-monies of a particular key and are oftendenoted byroman num eralstodesignate theposition of the root of the chord in the scaleof th e key. The tones of the chords are usu-allyselected from th e major scale fo r m ajorkeys and from th e harmonic minorscalefo rmi nor keys. Because of differences betweenthe intervals of the majqr and minorscales,different chord types appear in major andminorkeys. In am ajor key, I , IV, and V aremajor chords; II, III, and VI arem inor;andthe VII chord is diminished. A minor keyhasm ajor chordsas V and VI ,m inor chordsasI and IV,dim inished chordsas II and VII ,an dan augmentedchordas 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, th e roman nu-meral notation used in music theory impliesthere is a similarity in the effect of chordsbuilton the same scaledegrees inmajor andmi norkeys.Inpractice, usu allyafourth 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 inversionsdiffer int e rmsof the lowest bass toneand in the spe-cific intervals contained in the chords.W hether these inversions produce signifi-cantly distinctive effects or simply serve tointroducegreater varietyis am at te r ofsomedebate (see, for example, Rameau, 1722/1971; Schoenberg, 1969, p. 6). The issue ofchord inversionswill not be considered fur-ther here because our experim ents employedchordsw itheach component sounded in fiveoctaves, using an ampli tude envelope thattapered off at both low and high ends of thefrequency range.

Not only are there restr ictions on the p ar-ticular chords used in a key but there arealsorestrictionson theordersin which theycan appear. M oreover, certain chords w ithinthe set are described asplaying more s truc-turally significant roles than others, andthese are typically sounded more frequentlyand in more prominent temporal positions.In W estern m usic the key sense is prim arilya 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 functioninmorethan one key. In fact, major and minorchords each play roles in at least five differ-ent keys, and a diminished chord in threekeys. F or 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 dua l or multiplemeanings of harmonies are f u n d a m e n t a l totonal m usic:The designation by roman nu mera l symbolsof a har-monyin tw o or three l ights. . .is no t an inconsistency;it is arealisticrepresentationof the rangeoffunctional


16/35

TONAL ORGANIZ AT IONIN MUSIC 349significances a t tending a harm ony of bo th pr imary an dsecondarycontextsof tonal reference. Theindicationofsuch m ul t ip le funct ions is a reflection of the depth oftonal-harmonic meaning and of the way harmony isheard in tonal music, (p. 67)Thus , any single chord ma y be interpretedas simultaneously playing various differentharmonicfunctions, but therem a y be aten-dency to assign the chord one single mostlikely role. Schenker(1906/1954)notes th etendency to ascribe to any major or minortriad,first of all, the meaningof thetonic(p. 252). Th is interpretation m ay, how ever,need to be reevaluated as subsequentchordsare heard that are incompatible with thepreviously assigned harmonic function. InWestern music, then,thereis acomplicatedset of circumstances that leads to the des-ignation of the key of a piece of music.Empirical Studies of Chord Functions

Previousinvestigations ofchord functionshave focused on the relations between dif-ferent chords rather than those betweenchordsandabstract tonalcenters. Nonethe-less, th e results of these studies on chordshave show n clear influence sof organizationat the level of tonal centers. One scalingstudy ( K r u m h a n s let al., 1982) used the setofchordsfromthreeclosely relatedkeys:Cmajor, G m a jor ,and A mino r .In that studythe chordsof thethree keys were showntobe organized in an overlapping fashion thatpreserved the internal chord hierarchy w ithineach key, despite the multiple functioningof chords in several keys. The chords of Gmajorand A minor thatare shared withCmajor were centrally located, whereas thoseuniqueto G m ajor w ere separated from thoseuniqueto Aminor, w hichisrelativelyremotefrom G major . In a complementary studyBharucha and Krumhansl (inpress) inves-tigated th e chordsofm axim al ly d is tan t m a-jo r keys,finding a clear separation betweenthe chords of distant keys. This separationw as also reflected in m e m o r y confusions.Finally, Krumhanslet al. (in press) observedsystematic alterations of perceived interre-lations between chords as the context keyw asvaried. These effects werea functionofth e distance between the key in which th echords functionand the key of the context,asm easured by the n u m b e rof steps aroundthe circle of fifths. (O nly m ajor key contexts,

whose distances can be described by the cir-cle of fifths, we re used inthatstudy.) 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

T he followinganalysis addresses the prob-lem of how the perceived relation betweena chord and its abstract tonal center,thatis ,its harm onicfunction,m ay be representedusing the spatial map of musical keys de-scribed in the last section. W e test th e spe-cific hypothesis that a chord isperceived asclosely relatedto theabstract tonal centersin which it plays it s most harmonically sig-nificant roles.Ifthisis thecase, then inter-keydistances in the spatialm ap of themajorand m inor keys mu st reflect th e multiplein -terpre tations th at are possible for each singlechord. In other words, keys sharing chordsmustbelocatedinproximal positionson thesurface of the torus. The profiles obtainedfo r m ajor , m inor,diminished, anddominantseventh chords usingth e method describedearlier are used to test these predictions.Results and Discussion

Chord-key profile correlations. Eachofth e four chord profiles (for major , minor ,diminished, and dominant seventh chords)was correlated with each of the compositemajorandm inorkeyprofiles(Figure2). Thekeyp rofiles w ere shiftedto allpossible tonicsso that the relation between each of the 4chords andeachof the 24m ajor 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 withth e correspondingminor key.This wouldbeexpected becauseth e chord profiles and the cadence profileswere averaged todetermine the keyprofiles,an d th e tonic interpretation is assumed m ostlikely for m ajor or m inor chords.Other pat-terns in the correlations are brought outmore clearly in the next analysis.Chordpositionson thetorus. Earlier,theargumentw asmad e that keys sharingdhordsshould be perceived as closely related be-


17/35


18/35

TONAL ORGANIZATION IN MUSIC 351Thus, this result supports the music-theo-retic description of these two chords as func-t ional ly similar.In sum, this analysis of chord functionsshowed that major, minor, diminished, anddominant seventh chords are perceived asclosely relatedto the tonal centers inwhichthey play significant harmonic roles. Theprecise position obtained for chords thatfunction inmultiple keys represents a com-promise of the combined influenceso f theirvarious roles in these keys. Moreover, thisanalysis made clear the fact that keys shar-in gchords are located in proximal positionson th e surface of the torus. In this is seentheintimateconnection between chordsandkeysintonal music.Thatan extremely reg-ular representation of interkey distances wasobtained may be understood as resulting inpart from the complex interlocking patternofmultiple chord functions. The relative po-sitions of the keys on the torus, then, dependto some extent on the chords that simulta-neously function in dif ferent tonal centers.Also, the perceptual coherence achieveddur-in g well-structured chord sequences wouldseem to result because each of the chords inthe key and in closely related keysislocatedproximal ly inthis representation, apropertyalso obtained in the scaling studies of chords(B ha ruc ha & Krumha ns l , i n press; Krum-hansletal, 1982,inpress). Thepatternofchord-key relations, then, substantiates ear-lier characterizations ofharmonic relationsbetween chords, partially explainsthestruc-ture foundat the levelofabstracttonal cen-ters, and indicates a means through whicha sense of key might evolve during the per-ceptionofharmonic sequences.Weturnnowto an investigation that traceshow the keysense developsandchanges as a well-struc-tured sequence ofchords unfolds in time.

Chord Sequences: The DevelopingandChang ingSense of KeyNeisser (1976) has described perceiving

as anongoing process inwhicht he internalschema iscontinuouslym odified asexternals t imulus information becomes available:The listener develops moreor less specific readinesses(anticipations) for what willcome next, based on in-formation he has already picked up. These anticipa-t ionswhich themselvesmust be formulated intermsoftemporal patterns, not ofisolated momentsgovern

Figure 6.T he relationsbetweenthe C m ajor, Aminor,and Bdiminishedchordsand the 24 keys areillustratedinth etop,m iddle ,andbottom panels, respectively. (Thepoint fo r each chord is the intersection of the vectorsobtained by PREFMAP and thetorus representationofth ekeys.F oreach chord it s h a rmo n i c functions in thekeys are indicatedb y ro ma n numerals.T he dom inan tseventhchordo n G[notshown] w aslocatedinapprox-imately th esameposition as the Bdiminishedchord.)

what he will pick upnext, and in tu rn are modified byit. (p. 27)In music perception this continually inter-active process between the internal schemaandexternal stimulus may perhaps be seenmost clearlyas thedevelopingandchangingkeysense whenaharmonic sequencei sbeingheard over time.Wehave 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 andinduc ing expectations that its closely asso-ciated tones and chords will be sounded.However,the hypothesized tonalcentermayneedto bereevaluatedas the sequence pro-gresses. Given the spatial map of key re-gions,and the inherently temporal structure


19/35

352 CAROL L.KRUMHANSLAND E D W A R DJ. KESSLERof music, we are in a unique position to in-vestigate the dynamic changes that occur inth e internal representation during the per-ception of harmonic sequences. In particu-lar, we are concerned wi th th e question ofhow the sense of key is initially extractedfrom th e sequence and how modulat ions(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 key s areoften employed and are usedto provide organization to the piece as awhole. G enerally, this shift, or modulat ion,is to a closely related key and is effectedth rough 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,tha t 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 m odulations 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 alsohas direct relevance to the presentresults. Schenker proposed a system of mu-sical analysis that identifies the surface ele-m ents of themusicintermsof their relationsto f ixed un derlying m usical s tructures .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 inf luencesof the surface level are transformed into acoherent whole.

According toSchenker an ypieceo fm usiccanbeanalyzedat itsmost b asic level,calledbackground or Ursatz, in terms of a singletonal center. This key is mostoften explicitlyinstantiated at the beginning and end andcontinues to exert it s influence throughoutth e piece, despite possible modulations. In-deed, Schenker (1906/1954, p. xxii) rejectsth e concept that a complete modulation evertakes place on the background level; a newkeymay achieve some indep ende nce but willfunction onlyon the middle ground or fore-groundlevels, giving preem inence to the pri-m ary key. A surpris ing num ber of music the-orists agree on the imp ortance of the prim arykey. F or 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 th e same position istaken by Schoen-berg (1969):Everydigression from the tonic is considered to be stillwithin the tonality, whether directly or indirectly,closely or remotely related. Inother w ords, there isonlyonetonality in apiece,and every segment form erly con-sidered as another tonali ty is only a region, a harmoniccontrast within that tonali ty, (p. 19)However, the position that there is a uniqueunderlying tonality has been criticized asbeing to o inflexible, and todate there is noevidence that most listeners actually main-tain a fixed sense of key throughout an ex-tended musical passage.T he middle ground inSchenker's hierar-chy represents the piece of music in termsof the more structurally significant events(chords and tones). Thus, at this level th emusic isspecified in a more simplified formthan in the complete score. Modulations be-tweenkeys are reflected atthis level of an al-ysis as well as the fo reg rou nd level. The fore-ground levelis the actual music itself. Eachindividual chord has a tendency to be inter-preted as the tonic of akey,a process calledtonicization. Schenker (1906/1954) says,"Not onlyat the beginningof a compositionbut also in the midst of it, each [chord]manifests an irresistible urge to attain th evalue of the tonic fo ritself (p. 256). Ton-icization is expected to be most pronouncedwhen the chord is preceded by its own dom-inant (V) and during sections ofgreatestk eyambigui ty and instability. Thus, Schenker


20/35

T O N A L O R G A N I Z A T I O NIN M U S I CC H O R D : I 2 3 - 4 5 6 7

353

T W E L V EPRO BET O N E S

\

t

J

w^

\J

-VA t

i

J\ A - ' \

Jy\

jw \

3j

v 1

jJhf\

J

J

/Wv

^

J

\ W VP R O F I L E ; i 8

Figure 7. T he design of Exper iment 2 , using chord sequences. (The first chord of each sequencew asfollowed byeach of the 12probe toneswithin th e chromatic scale, giving the first ra t ing profile. Next ,the first two chordsofeach sequence were pa ired w ithth e 12pro be tones,giving th e secondprofile. Thisprocess w asc on t inuedun t i l profiles were obtained fo r sequencesof Lengths 1t h rough 9.)proposes that pi tch s tructure may be ana-lyzed in t e r m s of a hie ra rchy of increasingabstraction, with the least abstract levelbeing the actual tone, intermediate levelscorresponding to the more s tructural ly s ig-nificant elements within the key prevai l ingin that sectionof the piece,and the deepestlevelspecifyingasinglekeythat is presumedto under l i ethe ent i re composi t ion.The Present Approach to AssessingPerceived Tonal Organization

Variousaspects ofSchenk er's (1906/1954,1935/1979)description of a tonal hierarchywill be investigated in the next exper iment ,which again used th e tone-profi le techniqueemployed earlier. The design of the experi-m e n t is shown in Figure 7. Each of the se-quences consisted of nine chords. Ten dif-ferent sequences were used; their structurewillbedescribedlater.In theexper im ent thel istener first heard the very first chord ofeach sequence, followed by each of the 12possible probe tonesof the chromatic scale,as in the first experiment . This gaveaprofileof ratings for each of the first chords of the10 sequences. Following this, th e listenerhearsthe first twochordsof each sequence,again paired with all possible probe tones,giving a profile for two-chord sequences.This proced ure w as continued un ti l profi leswere obtained fo r sequences of Lengths 1through9. It is im por t a n tto notethatat not ime had the l istener heard the sequencebeyond th e point at w hi c h i t was beingprobed. That is, the ra t ing profiles w e rebased only on thechords presented up to tha tpoint of the sequence. These profiles w e re

then correla ted with the major and minorkeyprofiles obtained earlier totracethe un-foldingprocess throug h w hich a sense of keyis achieved.The 10harm onic sequences ar e shown inTable 2. The first two sequences were con-structed from th e 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 wi thin each key are shownin the table asroman numera l s , as in a stan-dard harmonic analysis . These sequenceswere constructed so as to be relativelytona lly unam biguou s , a l thoughasnoted ear-l ier the mul t iple harmonic functions of in-div idual chords necessarily entails acertainresidual ambigui ty. Nonetheless , the selec-tion of chords from a single key tends toreduce this ambigui ty.In addi t ion,the tem-poral order of the chords was guided bymusic-theoretic descriptions of permissible(or probable) chord sequences inm usic .Forthisw erelied on the table of chord progres-sions con tainedinPiston (1962,p. 18). Fu r -thermore, these tw o sequences began withthe harmonical ly s ignif icant IV-V progres-sionand ended withthe 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-ulat ions (changes) between keys; these areshown as Sequences 3 through 10 inTable2.Boththechords themselvesand the roma nnumeral analysis in the two keys are showninthe table . T hese m odu lat ing sequences allhad the same general s t ructure . Each ofthem began wi tha cadence, IV-V-I or II-V-I, inthef irs t key, which was either m ajor


21/35

Table 2Chord SequencesSequencenumber

123

4

5

6

7

8

9

10

FirstModulation keyN o C majorNo C minorYes: Direct C majorand closeYes: Indirect C majoran d remoteYes: Direct C majorand closeYes: Indirect C majoran d remoteYes: Close C minor

Yes: Distant C minor

Yes: Close C minor

Yes: Indirect Cminorbu t close

Secondkey Chord 1C major F majorC: IVC minor F minorc: IVG 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 Ddiminished

'c:IIC:A* major D diminishedc:I IA*:

Chord 2G majorVG majorVG majorV

(DG majorVG majorVG majorVG ma jorVG m a j o rVG major

V(V )G majorV

Chord3A minorV IA* majorV IC majorI( I V )C majorIC majorIC majorIC minorIC minorIC minor

IC minorI( I I I )

Chord4F majorIVF minorIVA minorV I(I DA minorV IF majorIV( V I )F majorIVA* majorV IG m a j o rVA* major

V IA* majorV I

(I)

Chord 5C majorIC minorIE minorIIIV IF majorIVVD minorIIIVD minorIIIF minorIVIA* majorV IVG major

VVF minorIVVI

Chord6A minorV IA* majorV IB minor

II IG minor

V IE major

VB* major

V ID* major

VIA major

VIA minor

V IE* major

V

Chord 7D minorIID diminishedIIE minor( H I )V IE* major

IVB diminished( V I I )IIE diminished

IIB* minor

IVF* minor

IVF major

IVB* minor

II

Chord 8G majorVG majorVD major

VF major( I V )VE major

VA major

VC major

VG * major( V I )VG major

(V )VE* ma jor

V

Chord 9C majorIC minorIG major(V )IB* major

IA minor(VI)ID minor(IDIF minor( I V )IC* minor

IC major

IA* major(VI)I

O78Ofr7 3GXZCOr>aD;>.78D~ cVIfa3C


22/35

TON AL O R G A N I Z A T I O NINM U S I C 355or minor , and ended with one of these tw ocadences in the second key, which again waseither majororm inor . That is, the firstthreeand last three chords constituted a cadencein the first and second keys, respectively. Ineach sequence the fourth chord belonged tothe first key. T he modulat ionw as effectedthrough the use of a pivot chorda chordthat functions harmonical lyi n both keysin the fifth position. Again w e were guidedin this choice by music theory, which de-scribes the use of pivot chords as the mostcommon m e a n s of modulat ion. The sixthchord belonged to the new key but not theinitial key, thus reinforcing the sense of thenew key, whichwasexpectedto becomeso-lidified by the cadence in the last three po-sitions.A s notedearlier, keys sharing chords arelocated inproximal regionson the torus,andthus,allmodulations employing pivotchordsare necessarily between relatively close re-gions. However, some of the pairs of keysrepresented in the m od ulating sequences aremore closely related than others. Of them odulating sequences beginning in a majorkey, the relations between C major and Gmajor (Sequence 3) and between C majorand A m inor (Sequence 5) are classifiedbySchoenberg (1969,p. 68) asdirect and close,that between C major and B* m ajor (Se-quence 4) and C major and D minor (Se-quence6) asindirect and remote. These arealso more distant on the toroidal represen-tation of the key regions. Of the sequencesbeginningin Cm inor,th erelation toFm inor(Sequence 7) and to C major (Sequence 9 )are described by Schoenberg (1969, p. 75)as close, the relation to A* m ajor (Sequence10) as indirect but close, and the relation toC* minor (Sequence 8) as distant. Again,these classificationsa re substantiated by theobtained interkey distances. Consequently,ofthem odulating sequences,five(Sequences3, 5, 7, 9, and 10)w illbeconsideredasmov-ing between closely related keyregions, andthree (Sequences 4, 6, and 8) between rel-atively distant regions.Certain l imitations of the roman numeralanalysesof the chord sequences provided inTable 2,whicharethose yieldedb y the stan-dard theory of harm onic analysis , should beem phasized. Such analyses do not m ake ex-plicit th e full rangeoftonal amb iguityof 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 th echord functions in terms of a wider varietyof keysand indicatehow earlier chord func-tions may be reinterpreted in light of har-mon ic inform ation following later in the se-quence. However,the tone-profile techn iqueemployed here enables us toassess directlyhow the listener's senseof keydevelops andchanges, providing a quanti tative measureof th e relative strengths of tonal interpre-tations at each point in time. In this way weare able tom onitorthe perceptualprocessesgiving rise to the sense of key.In addition, a n u m b e r of specific issuesraised by Schenker's (1906/1954, 1935/1979) description of tonal hierarchies willbeaddressed. Hisclaim thatat the deepestbackground level the l is tener maintains afixed und er lying sense of key, will be eval-uated by d eterm ining the extent to which theprofile ratings reflectth e tonali tyof the firstkey after a modulation has occurred. Changesat the middle ground in Schenker 's hierar-chy are investigated by describing the timecourse of the developmento f these alterna-tive, contrasting tonal interpretations in themodulating sequences. In this connectiondifferences between modulations to moreclosely and more dis tantly related tonal re-gionsare examined. Finally, the m om entaryinfluence of the individua l chords, whichm ay be considered to be the psychologicalcorrelate ofSchenker's foregroun d level,willbe assessed by determining th e influence ofthe last heard chord on the rating profiles.

Exper iment 2Method

Subjects. Fourteen adultsubjects fromthe HarvardUniversi ty and Cornell Univers i ty communi t ies , w howere recruited through posted noticesin 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-backgroundquestionnaireshowedthatsubjects had anaverage of-8.6 years instruction on musical instrumentsand voice and 6.4years experience performing in in-strumental groupsand 2.3years inchoral groups. Cur-rently,the subjects were playingan ins t rumentor sing-in g an average of 7.7 hours per week and were listening


23/35

356 C A R O L L.KRUMHANSLAND E D W A R D J. KESSLERto music for 14.6 hours per week. Four of the subjectshad taken one or more courses in music theory; th eremaining subjects had no background in music theory.Again, al l reported norm al hearing, and no one reportedhaving absolute pitch.

Apparatus andstimulus materials. The appara tusw as identical to that used in Experim ent 1 , and thechords of the sequences and probe tones were producedin the sam e way. The 10 sequences shown in Table 2were described in detail earlier. The durations of thechords and probe tones were asfollows: Each chord hadadura t iono f .5sec,with.12 5secbetweenchords. Therew as a pause of .5 sec between the last chord and theprobe tone, which sounded fo r.5sec, and a4-sec pausebetween trials.Procedure. Before beginning, the design of the ex-per iment wasexplained to each subject. That is, th eywere told that they w ould hear chord sequences tha tvaried from one to nine chords, each followedb y a singleprobe tone. For eachtrialthey weretorateon ascalefrom 1 to 7 howwel l the final probe tone fit in wi th th esequenceof chordson the trial.The firstsession beganwith a practice tape containing trials of SequenceLengths 1, 3, 5, 7, and 9, with a modu la t ingsequencetha t wasdifferent from any actually heard in the ex-periment. They then heard the first chords of each ofthe 10sequences (duplications el im inated ) with al l pos-sible probe tones. These were followed by all of thedifferent two-chord sequences, and the sequence lengthw as increased untilfinally the complete nine-chordse-quences were presented. Although th e sequences in Ta-ble 2 are described withrespectto C major or minor ,in the experiment the reference key was shifted ran-domly from tr ial totrial, and all chord sequences of agivenlengthwere randomlyintermixed.Lastly, aques-t ionnaire about musical background w as completed byeach subject.Results and Discussion

Individual differences. I'ntersubject cor-relations of the responses were generallyhigh, with an average correlation of .669between any in div idua l 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. T hefirst main results are pictured in Figure 8.T he solid line in the graph on the top leftshows th e average correlation between th eobtained profiles for the nonmodulat ingS e-quences 1 and 2 and the profile for the in-tended Cm ajor and Cminor keys. In otherwords, th e profilefound after each chord inthe sequence w as correlated w ith the profilefor the key from which the chords weredrawn.Thesecorrelationswereall quitehigh

and tendedtoincreaseas thesequences pro-gressed. Thus, it appears tha t the chord se-quences do in fact instantiate, at least tosomedegree,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 w ithinthe prevailing keys.In addition,there is a somewhat lower peak after th esecond chord, where th e harmonically s ig-nificant IV-V progression has occurred.Th us, even before the ton ic has been h eard ,th e obtained profile correlated stronglywith

that of the intended key. At each of the nineserial positions, it is possible to determineth e extent of the local key effect of the lastheardchordon the prev ail ing tonali tyof thesequence. To do this the chord-key corre-lations described earl ier were com pared w iththe correlations obtained in this experim ent.These correlations be tween the indiv idua lchordsand the keypredicthowstrong ly eachchord in isolationwouldb eexpected torelateto the intended key of the sequence and,thus, ser

1982k&kesslerpsychrevfromjournal

Documents