Tonality Induction - A Statistical Approach Applied Cross-Culturally (Krumhansl 2000)

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Tonality Induction: A Statistical Approach Applied Cross-CulturallyAuthor(s): Carol L. KrumhanslSource: Music Perception: An Interdisciplinary Journal, Vol. 17, No. 4, Tonality Induction(Summer, 2000), pp. 461-479Published by: University of California PressStable URL: http://www.jstor.org/stable/40285829 .

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MusicPerception 2000 by he regents of the university f CaliforniaSummer000, Vol.17,No. 4, 461-479 all rights reserved.

Tonality Induction: A Statistical Approach AppliedCross-CulturallyCAROL L. KRUMHANSLCornellUniversity

Sensitivity to tone distributions has been proposed as a mechanism un-derlying tonality induction. This sensitivity is considered in a cross-cul-tural context using two styles of music, Finnish spiritual folk hymns andNorth Sami yoiks. Previous research on melodic continuation judgmentsshowed strong correlations with the statistics of the musical style, spe-cifically, the tone distributions and two- and three-tone transitions. Thisarticle develops models using these three kinds of statistics to categorizeshort initial segments as coming from one style or the other. The modelusing tone distributions was found to make numerous categorizationerrors,which can be understood because the tone distributions for thesestyles are similar.However, categorization was better for the models thatused two- and three-tone transitions. The major differences between thetransitional probabilities in the styles were analyzed, and these differ-ences were used to account for the cases that the models found difficult.These results point to listeners' sensitivity to higher order transition in-formation and its utility for style identification.

article explores in a cross-cultural setting basic psychological prin-ciples that may underlie tonality induction. In tonal-harmonic West-ernmusic, tonality induction is the process through which a listener identi-fies the key of a piece of music, that is, what the tonic is and whether thekey is major or minor. Identifying the tonality is an important first step inanalyzing a piece of music. Once the key is known, then the piece can beanalyzed harmonically, its grouping and phrase structures can be deter-mined, its formal organization can be specified, and so on. In other words,tonality (together with meter) usually precedes furthertheoretical analysisof the music. From a psychological point of view, identifying the tonality isimportant for encoding and rememberingthe music. Extensive experimen-tal research (summarized, e.g., in Dowling & Harwood, 1986; Handel,Address correspondence to Carol L. Krumhansl, Department of Psychology, Uris Hall,Cornell University,Ithaca, NY 14853. (e-mail: [email protected])ISSN: 0730-7829. Send requests for permission to reprint to Rights and Permissions,University of California Press, 2000 Center St., Ste. 303, Berkeley,CA 94704-1223.

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462 Carol L. Krumhansl1989; Krumhansl, 1990) has shown various effects of tonality. Among othereffects, tones in the key are betterremembered,are identified more rapidly,are preferredin final temporal positions, contribute to phrase ending, aremore expected than nonscale tones, and are less frequently confused withnonscale tones than the reverse. More generally,tonality induction is inter-esting psychologically because it appears to rely on two basic principles ofperception and cognition.The first principle is the existence of cognitive referencepoints (Rosch,1979). Cognitive reference points refer to instances within categories thatare used to encode, name, and remember other instances within the cat-egory. For example, in the category of fruit, some elements, such as apple,function as cognitive referencepoints. Other instances are described in ref-erence to these, for example, in the names given to pineapple and Apfelsine("apples from China" in German). Cognitive reference points are held tobe an efficientmechanism for categorization, requiringa minimum memoryload and generally enabling instances to be correctly categorized. In music,the tonality establishes certain tones as referencepoints, creating a hierar-chy of tones. The tonic heads the hierarchy,followed by the fifth and thirdscale degrees, followed by the other scale tones, and finally the nonscaletones. One method used to study this experimentallyis the probe-tone tech-nique (Krumhansl & Kessler, 1982; Krumhansl & Shepard, 1979). Musi-cal contexts (e.g., scales, tonic triads, chord cadences, or short excerptsfrom pieces) are followed by a single tone, called the probe tone. Instruc-tions variously ask how well the tone completes or fits with the context.The second principleis sensitivityto the frequencieswith which instancesoccur. Psychological research has shown quite accurate memory for howoften different instances have been presented during the learning phase ofmemory experiments. Various results also show that instances that occurmore frequently are more easily encoded, named, and remembered. Fre-quent instances are also considered to be better category members. Forexample, Rosch (1979) found that the word "apple" occurred more fre-quently in a corpus of written Englishthan did the word "pineapple."Thissuggestslinks between cognitivereferencepoints and sensitivityto frequency.Cognitive reference points may be established initially by their relativelyfrequent occurrence. In music, tones given higher ratings in the probe-tonetask are also sounded more frequentlyin Western music (Krumhansl,1990).This suggests that the reference points are established initially by frequentrepetition. Supportingevidence comes from a study that used North Indianclassical music (Castellano, Bharucha,& Krumhansl, 1984). In that study,Westernlisteners, who were unfamiliar with the style, gave responses simi-lar to the responses of Indian listeners. This similarity was attributed toWesternlisteners' basing their responses on the distributions of tones in thecontexts presented,that is, the frequencyand durationwith which the tones



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Tonality Induction: A Statistical Approach Applied Cross-Culturally 463were sounded.Frequencyeffects were also found by Oram and Cuddy(1995),who constructednovel tonesequencesaccording o predetermined(andnovel) requency istributions. heyshowed hatthesenovelfrequencydistributionsnduceda tonalhierarchy, s measured ntheprobe-tone ask.Frequencyensitivitymay playan especiallycentralrole in music,becausemost listenersdo not have absolutepitch and rely primarilyon relativepitch.Thefrequent epetitionof the reference onesprovidesa frameworkfor encodingandrememberinghe soundedtones.

Tone Frequenciesand Tonal Hierarchies in Tonality InductionThe research ust summarized hows a connection betweencognitivereference ointsinmusic and distributions f tones. Thisapparent onnec-tionled to thedevelopment f a key-finding lgorithm by myselfand MarkSchmuckler nd described n Krumhansl,1990). The algorithmwas pro-posedas a possiblemodel of how a listenermightidentify he key initiallyand trackmodulations rom onekeyto another.Onepremiseof the modelis that listenershave internalized one frequenciesn piecesin majorandminorkeys through prior experience.The second premiseis that whenlistenersheara section of a pieceof music,theymatch the distributionof

soundedtones to the internalizedonal hierarchies.The algorithmbeginswitha sampleof tones,forexample,each measureof a musicalpiece.Thetotal duration(or numberof occurrences) f each of the chromaticscaletonesis thendetermined.This tone distribution s then correlatedwith thetonalhierarchies f allmajorandminorkeys(using he probe-tonevaluesfrom Krumhansl& Kessler,1982). The highestcorrelation s taken to bethekeyidentifiedby the algorithm.The firstapplicationof the algorithmwas to the initial four-toneseg-mentsof the pianopreludesof J. S. Bach,Shostakovich,and Chopin.Forthe 48 Bachpreludes, he algorithmdentified hecorrect(composer's es-ignated)key44 times with justthe first four tones.When an incorrectkeywasfound, t wascloselyrelated o the correctkey.Forthe 24 Shostakovichpreludes, he algorithmwas correcton 17 preludes,and a closelyrelatedkeywas identified orthe otherpreludes.Thealgorithmwas far less accu-rateon theChopinpreludes, dentifyingonly 1 1 keys correctlyon the basisof the firstfour tones. Of the remaining13 preludes,8 Preludeexcerptsproducedcloselyrelatedkeys. In threemorecases,a diffuse collection ofkeysnearthe correctone was identified,with no probe-toneprofilecorre-latingstronglywiththe tonedistributions.nthelast two cases,manyotherkeyshadhighercorrelations han the correctkey had. Both thesepreludesegmentscontained a chromaticalterationof one of the scale tones (thetonic in one case, and the third scaledegree n the other).Thus, it is not



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464 Carol L. Krumhanslsurprisinghat the algorithm ailedin these casesto identifythe keyfromthe four-tonesegments.

The second applicationwas to the fugue subjectsof J. S. Bach andShostakovich.Here,thequestionwashow manytoneswouldbe needed ofind the correctkey.In the case of the Bachpreludes, he correctkey wasfound 44 times out of 48, with an averageof 5.1 tones neededfor thecorrect dentification. n thecaseof the Shostakovichpreludes, he correctkeywas found22 out of 24 times,with an averageof 4.3 tones needed orthe correct identification.Both these applicationssuggestthat the shortinitialportionsof thesepiecestend to emphasize,by repetitionanddura-tion, those tones that arerelativelyhigh in the tonal hierarchy.Althougherrorsoccurred,the correct identificationswere far more frequentthanone wouldexpect by chance(1 out of 24). Most of the errorswere identi-fying closelyrelatedkeys, allowinga processsuch as thatmodeledby thealgorithmo orient he listener o the correct onalregion, f not the correctkey.The thirdapplicationwas to a singleprelude C Minor,BookII)of J. S.Bach. We examinedthe entirepiece on a measure-by-measureasis.Thepreludecontainsan interestingpatternof modulations,and the questionwas whether he algorithm ould trackthesemodulations. nthiscase,wealso had experts'judgments; heywere askedto indicatethe primarykeyfor each measureand alsoany keysof lesserstrength.Theexpertsappearedunableto focusexclusivelyon singlemeasures, howinginfluencesof pre-cedingand followingmeasures.Thus, the algorithm'snputconsistedofthe tone durations n the currentmeasure withfullweights)and the tonedurations n the precedingmeasureand the followingmeasure eachwithone-halfweights).Thealgorithm howedgoodgeneralagreementwith theexperts,although t lackedsome of theprecision oundintheexperts' udg-ments.However, hedegree o which thealgorithmwas successful uggeststhat it is ableto identifysectionscontaining onalambiguityandmodula-tions in generalagreementwith the experts.The Krumhansl-Schmucklerlgorithm s simple,usingonly priorpsy-chologicaldata and the durationsof tones in shortexcerpts.It does notcarryoutanymusic-theoreticalrocedures,uch asharmonicanalysis see,e.g.,Rowe, 1993, 2000; Vos, 2000; Vos& VanGeenen,1996).Nor doesittake meter ntoaccount,orphrasing.The mostimportantmissingelementforpresentpurposes,however, s thatthealgorithmdoesnot take thetem-poralorderof the tones into account.Initialexplorations,describedbrieflyin Krumhansl1990), suggested hatthe algorithmcouldbe sharpenedbythe additionof informationabout tone order. nthe extensionof the algo-rithm,the numberof two-tone transitions(how often one tone was fol-lowed by another)was comparedwith listeners' udgmentsof melodic n-tervals n tonal contexts(Krumhansl, 979, 1990). Thismethod,however,could beappliedreadilyonlyto monophoniemusic becauseorder s coded



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Tonality Induction: A Statistical Approach Applied Cross-Culturally 465easily onlyforsingle-voicedmelodies.Thus,we did not develop t further,althoughextensivepsychologicaldatashow the importanceof tone orderas is discussednext.

The Importance of Temporal Order in MusicIn tonal contexts,the orderof tones is highlyimportant.This kind oforderdependencyeflects hetemporalnatureof music,withauditory ventsoccurringalong the asymmetricdimensionof time going forward n onedirection.Temporal-orderffectsin music have been treatedexperimen-tallyat leastfrom thebeginningof the 20th century seeKrumhansl, 990,p. 122 forcitations).Theseearlystudiesdemonstrated ifferencesn judg-ments(suchas preference ndfinality)as a functionof the order n whichthe toneswereheard.A scalingstudy(Krumhansl, 979) studiedtempo-ral-orderasymmetriesof all pairsof tones in an octave range.Listenerswereaskedto makesimilarity udgments n sequentially resentedpairsoftonesin a cleartonalcontext.Specifically,ollowingthe tonalcontext,twotoneswereheardandthe listenerswere askedto judgehow closelyrelatedthe first tone is to the secondtone in the tonal system suggestedby thecontext.Theratings orpairsof toneswere oftenasymmetric,hatis, they

dependedon the order n whichthe two toneswerepresented.Theasym-metriesfollowed a regularpattern.Highersimilarityratingswere givenwhen the firsttone was lower in the tonal hierarchy han the second.Inotherwords,two toneswerejudgedas moresimilarwhenthe second tonewasrelativelyhighin the tonalhierarchy,uchas thetonic,dominant,andthird.Two otherexperimentsKrumhansl,979) foundtemporal-orderffectsin memoryconfusions.A to-be-rememberedone was soundedbefore amelodicsequence.After the sequence,anothertone was soundedand thelistenershadto judgethattone as the same as or different rom the to-be-rememberedone. In both experiments, istenersconfused the previouslyheardnonscale onesmorefrequentlywith scale tonesthantheyconfusedthepreviouslyheardscaletoneswithnonscale ones.Thus,bothsimilarityandmemory udgmentsontainedaregularpatternof temporal-ordersym-metries, eadingto the conclusion,"Theseresultssuggestthat, in the psy-chologicalrepresentation,hosetoneslesscloselyrelated o thetonalityarelessstable thantonescloselyrelated o the tonality,and that the represen-tationincorporateshetendency or less stabletonesto move towardmorestabletonesin time.Thus,thesetemporalasymmetrieseflect hedynamiccharacterof musical onesin time"(Krumhansl, 979, p. 372).Similaremporal-orderffectswerefoundinvariousstudiesdoneincol-laborationwithJamshedBharucha summarizedn Krumhansl, 990) forchordsequences.The effectswere largeand appeared n both similarity



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466 Carol L. Krumhanslandmemoryjudgments.Theycould be manipulatedn a predictableandinterprtableway by varying he tonal context. Thetemporal-orderffectsfor both tones and chords could be related o tonal andharmonichierar-chies.And,most importantn thepresentcontext,thepsychologicaludg-mentscorrelatewith the frequencyof chordsand chordprogressionsnmusic.Insum,it appears hatfrequencyof successiveelements n musicisincorporatedn listeners'cognitiverepresentationsnd that this informa-tionmaybe used to orientto thetonalityof a pieceof music.Thispossibil-ity will now be considered n an algorithm hat uses tone-order nforma-tion. The algorithmwill be appliedto two styles of vocal monophoniemusic,Finnishspiritual olk hymnsand North Samiyoiks, thatwere usedin two recentstudies.

Two Cross-Cultural Studies:Finnish SpiritualFolk Hymns andNorth Sami YoiksThe two studies investigatedmelodic expectancy.Expectancyhas animportant unctionin a wide varietyof behaviors, ncludingperception,speechunderstanding nd production,and skilledperformance.Diversemethodshave beendeveloped o studymusicalexpectancy,ncludingpro-

duction,memory,detection,priming,and structural udgments.The twostudiesgrewoutof a collaborationbetweenmyself,JukkaLouhivuori,PetriToiviainen,TopiJarvinen,PekkaToivanen,TuomasEerola,all fromtheUniversityof Jyvskyl,andAnnukkaHirvasvuopio romUtsjokiandtheUniversityof Tampere Krumhansl,Louhivuori,Toiviainen,Jarvinen,&Eerola, 1999; Krumhansl,Toivanen, Eerola, Toiviainen,Jarvinen,&Louhivuori, 000). Ourapproachwasto compareresults rom behavioralexperiments,a statisticalanalysisof the musicalstyles,and a neuralnet-work model of the self-organizing ype (Kohonen,1997).In both behavioralexperiments,eight test excerptswere chosen fromeach of the two stylesof music,Finnishspiritual olk hymnsand NorthSamiyoiks. They were chosen so that they endedwhen expectations orcontinuationwould be expectedto be strong,and the actual musiccon-tained an unexpectedcontinuation hat mightbe known to the more ex-pertlistenersbut not to the others.The excerptsweretransposed o thattheyfell within the rangefromG3(theG below middleC) to E5(theE anoctave and a thirdabove middleC)and hadC as a reference one (tonic nWesternmusic).The methodused nthebehavioral xperimentwassimilarto that used in earlierresearchon musicalexpectancy (e.g., Krumhansl,1995; Schmuckler, 989). Eachtrialbeganwith the excerptfollowedby asingletone,calledthecontinuation one.Allcontinuation ones withintherange(from heG3 o thehighE5)wereused on different rials n a random



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Tonality nduction:A StatisticalApproachAppliedCross-Culturally 467order.The listenerswere askedto judgehow well the continuation one fitswith theirexpectationsabout what mightfollow in the melody.

TheFinnishspiritual olk hymnscombineelementsof Lutheranhymnsand Finnish olksongs.Thestyleis generallyassociatedwith the Beseecherreligious prayermovement,which createdthe texts and melodiesin themiddleof the 18thcentury n southwestFinland.Twoexamplesare shownin Figure1 Forthe studyon Finnishspiritual olk hymns,two groupsoflistenersparticipated.The firstgroupconsisted of 12 subjectswho weremembersof the YouthChoirof the Beseechers.Theexperimentwas runattheir rehearsal n the parishhall of Eura,Finland.They had heard andlearned he spiritual olk hymnsat an early age and were highlyfamiliarwith the style,and most subjectsknew most of the hymnsin the experi-ment.Thesecondgroupconsistedof 14 undergraduatemusicologymajorsat the Universityof Jyvskyl.Theywere not familiarwith the particularhymnsanddid not recognizeanyof the melodies.However, heyreportedbeingfamiliarwith thestylethat is to beexpectedbecause hespiritual olkhymnscombineelements romLutheranhymnsand Finnish olk music.The Samiyoiks, fromLapland,have a style quitedistinctfrom Westernmusic.Yoiksare almostexclusivelyvocal andtypicallyusea few nonsensesyllablesrather han words.Yoiksmostlydeal with topicsof everydayife,

Fig.1. Two Finnish piritual olkhymns ransposedo C major.



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468 CarolL. Krumhanslsuchas people,animals,places,and nature.Theyhavepersonalmeaningfor the yoikerand are sung eitherin solitude or with people who shareknowledgeof theyoik'stopic.Yoiksarehighly mprovisatory nd have notbeenextensivelyanalyzed roma theoreticalpointof view.Theyare char-acterizedby short,repeatedmotives that oftencompriseonlya few notes,in most cases fromthe majorpentatonicscale.Theytendto includerela-tively arge ntervaleaps.Examplesare showninFigure2. Threegroupsoflistenersparticipatedn thestudyon North Samiyoiks.The firstconsistedof seven native Samiwho had extensiveexperience earningandperform-ing yoiks. They reported hat theywerehighlyfamiliarwith the style,al-though theywereveryfamiliarwith only two of the yoiks selectedas ex-perimentalmaterials.The secondgroupconsistedof 11undergraduate usiceducationmajorsat the Universityof Jyvskyl.Theirprimary rainingwas in Westernmusic.However,they had learnedto play a number ofyoiksduringa 1-yearperiod, ncluding wo of theyoiksusedin theexperi-ment. Thethirdgroupconsistedof Westernmusiciansunfamiliarwith theyoiksandtheyoik style.This allowedus to separate wo factors: 1) musi-cal cultureand (2) experiencewith the particular oiks.Thefollowingsummaryocusesexclusivelyon theconvergencebetweenthe melodiccontinuation udgmentsand the statisticalstyle analysis.Thestatisticalstyle analysisconsidereda corpusof 18 Finnishspiritualfolkhymns(onlythe 9 hymnsin majorwill be consideredhere)and 18 yoiks.For eachcorpus,theanalysis ound the tonedistributionshowfrequently

Fig.2. Two North Samiyoiks transposed o that Cand G arefrequentones.



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Tonality Induction: A Statistical Approach Applied Cross-Culturally 469eachchromatic caletoneappeared) nd thefrequencyof allpossibletwo-and three-tone ransitions.

The correlationswith the behavioralresultsfromthe Finnishspiritualfolkhymnstudy(Krumhanslt al., 1999) are shown at the top of Table1.Asseeninthefirst ine,the melodiccontinuationudgments f bothgroupsof listenerscorrelated tronglywith the distributionsof tones in the cor-pus. (Thedegreesof freedomof these and all the other correlations n thetableis 174.) The next line shows the correlationsbetweenthe judgmentsand the two-tonetransitions, hat is, the relativeprobabilitiesof the con-tinuationtone conditionalon the last tone of the experimental ontexts.Again,these correlationswerehighlysignificant.The next line shows thecorrelationbetweenthe judgmentsand the three-tone ransitions, hat is,the relativeprobabilitiesof the continuation one conditional on the lasttwo tonesof theexperimentalontexts.Again, hesecorrelationswerehighlysignificant,and herea differencebetweenthe groupswas found such thatthe correlations or the expertswere reliablyhigherthan the correlationsforthenonexperts.Finally, he correlationswith a variablecodingthe cor-Table1Correlations Between Statistics of Music and Behavioral Judgments ofMelodic Continuations

Spiritual olkHymnsSubjectGroup

Finnish BeseecherChoirTone distributions .64*** .72***Two-tone transitions .50* * * .56***Three-tone transitions .41 * * * .59* * *Correctnext tone .15 .52 * * *

North SamiYoiksSubjectGroup

Western Finnish SamiAllYoiksTone distributions .63*** .74*** .66***Two-tone transitions .67* * * .61 ** * .57* * *Three-tone transitions .41*** .38*** .43***Correctnext tone .11 .21** .32***YoiksKnownOnlyto SamiTone distributions .65*** .78*** .67***Two-tone transitions .69* * * .60* ** .54* **

Three-tone transitions .49* * * .46* * .65***Correct next tone .07 .18 .50*****p


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470 Carol L. Krumhanslrect next tone of the hymn was much stronger for the experts than for thenonexperts. In terms of Bharucha's (1987) distinction between schematicand veridical expectations, the experts showed stronger veridical expecta-tions than the nonexperts.The correlations with the behavioral results from the Sami yoik study(Krumhansl et al., 2000) are shown in the rest of Table 1. Again highlysignificant correlations were found between the continuation tones andtone distributions, the two-tone transitions, and the three-tone transitions.The correlation with the correctnext tone (veridicalexpectations)was stron-gest for the Sami, followed by the Finnish, and finally the Westernsubjects.Recall that only some of the yoiks were familiar to the Sami subjects. Theresults for this subset of yoiks are shown at the bottom of Table 1. Here theeffect of veridical expectations is more pronounced. In addition, for theseyoiks, the correlation with the higher order three-tone transitions was no-ticeably higher for the Sami than for the other two groups of listeners. Thecentral point to take from both studies is that listeners, even those withlittle prior familiarity with the style, appear to be sensitive to statisticaldistributions of tones. This result is similar to those of the North Indianmusic study (Castellano et al., 1984). The present results show that listen-ers are not only sensitive to the distributions of tones, but also to higherorder statistics, such as the two- and three-tone transitions.

Statistical Distributions and Musical StyleThese findings stimulated furtheranalysis of the problem of tonality in-duction. They suggested that it might be interesting to develop an exten-sion of the Krumhansl-Schmuckleralgorithm (Krumhansl,1990) that usednot only the tonal hierarchies(which correlatewith the frequenciesof singletones) as was done in the earlier algorithm, but also the frequencies ofsuccessive tone combinations. In addition to evaluating the utility of thisextension of the earlier algorithm, this approach may be especially usefulin a cross-cultural context where the music has not been treated theoreti-cally. Using the statistics of the music may yield insights into how a theo-retical analysis of the style might be developed. The two styles of musicused in these studies, Finnish spiritual folk hymns and North Sami yoiks,lend themselves to this analysis because music in both traditions is mono-phonic, thus eliminating the problem of how to code relationships betweensuccessive tones. The particular focus in what follows is on whether thealgorithm can correctly distinguish between hymns and yoiks given theinitial seven tones.Figure 3 shows the tone distributions from the corpus of 9 hymns inmajor and 18 yoiks. Both sets of values were normalized before being



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Tonality Induction: A Statistical Approach Applied Cross-Culturally 471

Fig. 3. Tone distributions for a corpus of 9 Finnish spiritual folk hymns transposed to Cmajor and a corpus of 18 North Sami yoiks transposed so that C and G are frequent tones.

graphed. Two points should be noted. First, the tone distributions for themajor hymns and yoiks were very similar (with a correlation of r (10) =.87, p < .0005). This correlation is higher than the correlation between thetonal hierarchies of any major or minor keys (Krumhansl, 1990, p. 38).Thus, the problem of distinguishing between these styles on the basis of thetone distributions would be harder than distinguishing between any twomajor and minor keys on the basis of their tonal hierarchies. Second, al-though the hymn and yoik tone distributions are very similar,they differ inminor ways. They differ in as much as the F and B (and to some extent A)appearrelatively infrequently in the yoiks compared with the hymns. Also,the G appears more frequently than C in the yoiks, whereas this differencedoes not appear in the hymns. For this reason, the yoik hierarchy might besaid to be pentatonic in orientation and to have both C and G as referencepoints.Three models were tested with the materials listed in Table 2. All eightFinnish spiritual folk hymns used to test the model were transposed to Cmajor and began with the tones shown in the table. The eight yoiks werethose used in the behavioral experiment, which were transposed so that Cand G were relatively frequent. Their initial seven tones also are listed inTable 2. Seventones were chosen for the tests because the previous applica-tion (Krumhansl, 1990) found this was approximately the number of tones



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472 Carol L. KrumhanslTable 2Materials to Test the Models

Finnish SpiritualFolk HymnsHymn No. Halullisten Sielujen Hengelliset Laulut (HSHL) First Seven Tones1 HSHL a CCBABCD2 HSHL 4a CCEGGAB3 HSHL 2a GGECEGG4 HSHL 0 EGCCBAD5 HSHLia CDEFDGA6 HSHL 5a CDEFDGG7 HSHL 7 GCCCDED8 HSHL 2 CBCDEDC

North Sami YoiksYoik No. Name of Yoik First Seven Tones1 Anden Inga CCCDDCD2 Elle Sunna GCDDCGD3 tfappaMagdalena CDGCDCE4 Hald EEEDCDD5 Bierra Bierra GCCCDDC6 Val'gon guoi'ka GEDEGAD7 ilen Niga Elle GCGFFFG8 Lemmon Elle GGCCAAG

neededto find the key in Westernmusic. Seven tones also gavea reason-ablenumberof two- and three-tone ransitions sixandfive,respectively).Thetone durationswerenot taken into account.The first modelcorrelated he distributionof the first seventoneswiththe tone distributionsn the hymnand yoik corpora.The second modelcorrelated he distributionof two-tone transitions n the initial segmentwiththe two-tone transitionsn thehymnandyoik corpora.Similarly,hethirdmodel correlated he distributionof three-tone ransitionswith thethree-tone ransitions n the two corpora.If the highercorrelation s withthecorrectcorpusstatistics,we will saythat the model hasmade a correctstyleclassification. f not, it has made a classification rror.Consequently,we will concentrateon the differencesbetween the correlationswith thetwo stylestatistics.Figure4 shows,for the threemodels,the differencebetween hecorrela-tionwith thehymnstatisticsandtheyoikstatistics.As canbeseen,the firstmodelmakesnumerous lassification rrors.Correlationsorcorrectclas-sificationsdid not reliablyexceedthose for errorclassifications,(1,15)=.32, ns. This resultwould be expectedgiventheverysimilar one distribu-tions shown in Figure3. The secondmodel, using two-tone transitions,



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Fig. 4. Categorizations made by the three models. The values plotted are the correlationbetween the test material and the hymn corpus minus the correlation between the test ma-terial and the yoik corpus. Positive values mean that the model classified the test excerpt asa hymn, and negative values mean that the model classified the test excerpt as a yoik. Model1 correlated the test excerpts with the tone distributions. Model 2 correlated the test ex-cerpts with the two-tone transitions. Model 3 correlated the test excerpts with the three-tone transitions. Errors are indicated by x's.

made fewer classification errors. With some notable exceptions (Hymns 2,3, and 7), the second model discriminates better between the styles, and thecorrelations for correct classifications exceeded those for error classifica-tions, (1,15) = 2.1, p = .03, one-tailed. Finally, the third model shows thebest discriminationbetween the styles.Only Hymn 7 is stronglymisclassified.Overall, the three-tone model performs quite well, with the correlationsfor the correct classifications higher on average than those for the incorrectclassifications, (1,15) = 4.4, p = .0003). In sum, the results of these analy-ses are that the algorithm's ability to discriminate style increases from thetone distribution model to the two-tone model to the three-tone model. Wenow consider the differences in the sequential order of tones that distin-guish between the hymns and the yoiks.



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474 Carol L. KrumhanslSequential Order in the Hymns and Yoiks

The following analysis was done to compare the two-tone and three-tone transitions in the hymns and yoiks. We consider the two-tone transi-tions first. The frequency of all two-tone transitions in the yoiks was sub-tracted from the frequency of the corresponding two-tone transitions inthe hymns (after both sets of values were normalized). The 10 biggest dif-ferences on both ends of the scale were selected for presentation. Figure 5shows on the left the 10 two-tone transitions that appeared in the hymnsfar more frequently than in the yoiks. The dashed lines represent two-tonetransitions that are scalar movement. As can be seen, the hymns are char-acterized by scalar movement, with the exception of the BD sequence only.Figure 5 shows on the right the 10 two-tone transitions that appeared inthe yoiks far more frequently than in the hymns. The circles here representrepeatedtones, for example, CC. Here, tone repetition is far more frequentthan in the hymns, and only two of the sequences, CD and DC, are scalarmovement. The pentatonic orientation of the yoiks is apparentin that noneof the frequent two-tone transitions includes F or B. Nor does A appearfrequently in two-tone transitions. The interpretation of the yoiks as hav-ing two reference pitches is supported by the frequent succession of CG.These two reference pitches appear to be reinforced by movement to andfrom D and E, respectively.Figure 6 shows the same analysis done for the three-tone transitions.Scalar movement in the hymns is even more clear here than for the two-

Fig. 5. Two-tone transitions that were more common in the hymns (left), and two-tonetransitions that were more common in the yoiks (right).



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Fig. 6. Three-tone transitions that were more common in the hymns (left), and three- onetransitions that were more common in the yoiks (right).tone transitions. Many of the patterns in the hymns consist of successivesteps along the scale, for example, DEF,EFG, CBA, and so on. Two othercases involve alternations with E, CEC and FEE Finally, two cases involverepetition followed by a scalar step, CCBand FFE. The yoik results, shownon the right, contain no movement of three tones along the scale. The rela-tively frequent sequences involving C and G are again apparent, as are thesequences involving C and D and the sequences involving G and E. Again,the prevalence of tone repetitions is seen, especially in the GGG sequencerepresentedby the double circle.With these results in mind, we can turn to the cases that were problem-atic for the algorithm. The first seven tones of Hymn 2 are CCEGGAB.That this was classified as a yoik can be understood by the following fac-tors: the repeated C and the repeated G, which are relatively frequent in theyoiks, and the fact that the only scalar movement, GAB, is not one of themost common scalarsequencesin the hymns. The second problematichymnwas Hymn 3, in which the first seven tones are GGECEGG. Note here thefrequent repetition of the G and the absence of scalar movement, both ofwhich are typical of the yoiks. Also the excerpt does not contain any of thetones F, A, or B, a feature that is also typical of the yoiks. The next prob-lematic case was Hymn 7 with the first seven tones GCCCDED. Here therelevant factors seem to be the repeated C, the GC sequence, the absence ofF,A, and B, and the fact that the only scalar movement is C D E (which isnot one of the three-tone transitions that strongly distinguishes betweenhymns and yoiks). Finally,Yoik 7 was not strongly classified as eitherhymn



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476 Carol L. Krumhanslor yoik. Its firstseventones are GCGFFFG.Yoik 7 has some features ncommonwith the hymns: he appearance f the F,the repetitionof the F,andthe alternationbetweenF andG. However, t also has important ea-tures n commonwith theyoiks, namely, he sequencesCG andGC.Thus,the algorithm'sdifficultywith thisyoik can be understood.

General DiscussionTo summarize irst the resultsof the algorithmdescribedhere, it wasbasednot only on the relativefrequenciesof tones (as was done in theearlierKrumhansl-Schmucklerlgorithm;Krumhansl,1990), but also onthefrequencies f two- and three-tone ransitionsn the initialsegmentsofthe music.The algorithmwas tested by askingit to distinguishbetweentwo musicalstylesusingonlythe first seventones of eightFinnish piritualfolk hymnsand eightNorth Samiyoiks. The relativefrequenciesof thetonesarequitesimilar n thesestyles.As would thus beexpected, healgo-rithmbasedonlyon therelative requencies f tones was largelyunabletoclassify heexcerptscorrectly.However,greateraccuracyn distinguishingbetweenthe styleswas foundwhen two- and three-tone ransitionswereconsidered.Comparinghefrequent wo- andthree-tone ransitionsn the

stylesyielded nformationabouthow the stylesdiffer.Thehymnsare dis-tinguishedbya diatonic onalityandfrequent calarmovement.Theyoiks,in contrast,appearto have a pentatonicorientation(in which the A isrelativelyweak), and two referencepitches,C and G, that arefrequentlysoundedin succession.Scalarmovement s far less common than in thehymns.These characteristicsould be used to understandwhichexcerptspresenteddifficulties o the algorithm.Theyalso aresuggestiveof furthertheoreticaldevelopments,especiallyof the yoik style, which has not re-ceived extensive heoreticalanalysis.At a more general evel, this articlewas motivatedby examininghowtwo basicpsychologicalprinciplesare used and elaborated n music. Thefirst is the existenceof cognitivereferencepoints.These have been estab-lishedby a varietyof experimental esultssummarized arlier.Cognitivereferencepoints also have been shown to have statisticalsupportin thefrequencyof reference ones in music. The present analysesof two- andthree-tone ransitionsextendsthese observationsby consideringpatternsof tone movement o and from thereferencepitches,andin the case of theyoiks, betweenthe two apparentlyequallystrongreferencepitches.Thesecondprinciples sensitivityo frequencynformationand use of that sen-sitivity in learning and classification. Previous experimental results(Krumhanslt al., 1999;Krumhansl t al., 2000) showedthatmusicalex-pertisewas associatedwith melodicexpectationsbased on sensitivity olonger sequences.In a complementaryway, the algorithm hat took into



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Tonality Induction: A Statistical Approach Applied Cross-Culturally 477accountsequentialorderperformedmoreaccuratelyn distinguishingbe-tween the two musicalstyles.Inclosing,let us consider wo recentexperi-mentalresultsdemonstratinghe importanceof sequential nformationnother communication ystems.A recentpaper(Aslin,Saffran,& Newport, 1998) is concernedprima-rilywithlanguageacquisitionbut hasintriguingparallelswith music.Aslinet al. reported hat 8-month-old nfantscan segmenta continuousstreamof speechsyllables nto wordlikeunits afteronly 3 min of exposure.Thisabilityappears o derive rom theirsensitivity o the successiveorderingofsyllables.Aslin et al. createdan artificial anguageof trisyllabicnonsensewords;onelanguage onsistedof thewords:pabiku, ibudo,golatu,daropl.The familiarizationperiod consisted of a continuous stream of thesetrisyllabicnonsensewords. The words wereheardin random order withno breaks between them, such as "pabikugolatudaropitibudodaropi-golatu...." Afterthe familiarizationperiod,which lasted 3 min, infantswerepresentedwith words andpart-words trisyllabic equences panningwordboundaries).Theirbehavioral esultsshowedthattheydiscriminatedbetween risyllabicwordsandpart-words.The words andpart-wordsdif-feredin the transitionprobabilities.The transitionprobabilitiesbetweenthe syllableswere higherin the words than in the part-words.In otherwords,infantsappear o besensitive o the successiveorderingof syllablesin the artificialanguage.Thus,the research uggests hatin bothlanguageand music, listenersmay be sensitive to order informationand use thissensitivity o formlargerunitsin perceptionandmemory.An interestingapplicationof sequentialprobabilitiesalso appears n arecentstudyof birdsongs (Gentner& Hulse, 1998). Birdsongs serve atleastthree functions(whichare different or differentbirdspeciesand arenot mutuallyexclusive).First,songscan be usedbymalesto attractmates.Second,songscan be usedto defend he nestsite.Third,songscan be usedto recognize ndividualbirds,such as a femalerecognizingts mate. Theirstudyfocusedon the last function,individualrecognition,and examinedwhat about an individualbird'ssong mightcarrythe informationaboutthe identityof the singer.Theyusedthe following terminology o describesongs.A songconsistsof a numberof sequentiallypatternednoteclusters,called "song types."Thesearegenerally ess than 1 s long and are oftenrepeatedbefore the next song type is sung. Sequencesof song types arestrung ogether n timeto producewhat is called a "songbout."Apparently,different ndividualbirdsstring togetherthe song types indifferentways. Gentnerand Hulse (1998) did a statisticalanalysisof thefrequencies f successive ong typesin European tarlings.Thefirstanaly-sis describes he frequencydistributionof song types, that is, how oftendifferent ong typesoccur.The secondanalysisdescribes he frequencyofsuccessivepairsof song types(liketwo-tonetransitionsnmusic).The thirdanalysisdescribes he frequencyof successivetriplesof song types (like



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478 Carol L. Krumhanslthree-tone transitions in music). They then synthesized song bouts that con-formed to these different statistics. That is, they took the statisticsfrom dif-ferent birds and creatednew song bouts in accordancewith the relativefre-quenciesof song types,two-song type sequences,andthree-song ypesequences.Three groups of birds were first trained. The first group consisted ofmales who learned (through reinforcement) to identify their own songsfrom those of four familiar males. The second group was familiar with thesongs, but none of the songs was their own; they were trained to identifythe songs of a particular bird. The third group consisted of females whowere previously unfamiliar with all the songs; they were also trained toidentify the songs of a particular bird. The question, then, was whether thetrained birds could generalize this training to the novel synthesized songbouts, correctly identifying the individual. The training successfully trans-ferredto the novel song bouts. Moreover, the transferwas more successfulfor the synthesized song bouts that matchedthe relativefrequenciesof longersequences of song types. These resultssuggestthat birds,as well as humans,are sensitiveto the successiveorderingof sound events.Inbirdsong, it appearsto be used to identifythe individualbirds,at least by Europeanstarlings.Thus, sensitivity to the order of events appears to play a role in a varietyof communicative contexts. It appearsto be a basic psychological principlein both humans and other species. As a consequence of this generality,wewould expect to find its influence in music cross-culturally.This has beenshown not only in the behavioral experiments in previous research, butalso in the statistical style analysis summarizedhere. The algorithmshowedthe utility of sequential information in classifying musical styles. This gen-eral principleof sensitivity to sequential order is elaborated in each of thesesystems in differentways. In music, it is coordinated with referencepitches,scale structure,and harmony, and it is elaborated by rhythm and phrasing.In language, it appears to yield information about the sequences of soundsthat form words of the language and possibly also syntactic categories. Inanimal communication, there is at least one example of sequential orderingof sound types yielding identification of individuals. Despite these differ-ences, sensitivity to sequential ordering appears to be a core principle inencoding and interpreting sequences of sounds.1

ReferencesAslin, R. N., Saffran,J. R., & Newport, E. L. (1998). Computation of conditional probabil-ity statistics by 8-month-old infants. Psychological Science, 9, 321-324.Bharucha,J. J. (1987). Music cognition and perceptual facilitation: A connectionist frame-work. Music Perception, 5, 1-30.

1 The researchin Finland was supported by a Fulbright Fellowship awarded to CarolKrumhansl and a grant from the Finnish Academy awarded to the Cognitive Musicologygroup at the University of Jyvskyl.



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Tonality Induction: A Statistical Approach Applied Cross-Culturally 479Castellano, M. A., Bharucha, J. J., & Krumhansl, C. L. (1984). Tonal hierarchies in themusic of North India. Tournaiof Experimental Psychology: General. 113. 394-412.Dowling, W.J., & Harwood, D. L. (1986). Music cognition. Orlando, FL: Academic Press.Gentner,T. Q., & Hulse, S. W. (1998). Perceptualmechanisms for individual vocal recogni-tion in European starlings, Sturnus vulgaris. Animal Behavior, 56, 579-594.Handel, S. (1989). Listening: An introduction to the perception of auditory events. Cam-bridge, MA: MIT Press.Kohonen, T. (1997). Self-organizingmaps (2nd d.). Berlin:Spring-Verlag.Krumhansl,C. L. (1979). The psychological representationof musical pitch in a tonal con-text. Cognitive Psychology, 11, 346-374.Krumhansl,C. L. (1990). Cognitive foundations of musical pitch. New York:Oxford Uni-versity Press.Krumhansl, C. L. (1995). Music psychology and music theory: Problems and prospects.Music Theory Spectrum, 17, 53-80.Krumhansl,C. L., & Kessler,E. J. (1982). Tracingthe dynamic changes in perceived tonal

organization in a spatial representationof musical keys. Psychological Review, 89, 334-368.Krumhansl,C. L., & Shepard,R. N. (1979). Quantification of the hierarchyof tonal func-tions within a diatonic context. Journal of Experimental Psychology: Human Percep-tion and Performance, 5, 579-594.Krumhansl,C. L., Louhivuori,J., Toiviainen, P.,Jrvinen,T., & Eerola, T. (1999). Melodicexpectancy in Finnish folk hymns: Convergence of statistical, behavioral, and computa-tional approaches. Music Perception, 17, 151-195.Krumhansl, C. L., Toivanen, P., Eerola, T., Toiviainen, P., Jarvinen, T. 6c Louhivuori, J.(2000). Cross-culturalmusic cognition: Cognitive methodology applied to North Samiyoiks. Cognition, 75, 1-46.Oram, N., & Cuddy,L. L. (1995). Responsiveness of Western adults to pitch-distributionalinformation in melodic sequences. Psychological Research, 57, 103-118.Rosch, E. (1979). On the internal structure of perceptual and semantic categories. In T. E.Moore (Ed.), Cognitive development and the acquisition of language. New York: Aca-demic Press.Rowe, R. (1993). Interactivemusic systems: Machine listening and composing. Cambridge,MA: MIT Press.Rowe, R. (2000). Key induction in the context of interactive performance. Music Percep-tion, 17, 511-530.Schmuckler,M. A. (1989). Expectation in music: Investigations of melodic and harmonicprocesses. Music Perception, 7, 109-150.Vos, P. G. (2000). Tonality induction: Theoretical problems and dilemmas. Music Percep-tion, 17, 403-416.Vos, P. G., & Van Geenen, E. W. (1996). A parallel-processing key-finding music. MusicPerception, 14, 185-223.

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