quantification of the hierarchy of tonal functions within a diatonic context€¦ · ·...

Journal of Experimental Psychology:Human Perception and Performance1979, Vol. S, No. 4, 579-594

Quantification of the Hierarchyof Tonal Functions Within

a Diatonic Context

Carol L. Krumhansl and Roger N. ShepardStanford University

Listeners rated test tones falling in the octave range from middle to high Caccording to how well each completed a diatonic C major scale played inan adjacent octave just before the final test tone. Ratings were well ex-plained in terms of three factors. The factors were distance in pitch heightfrom the context tones, octave equivalence, and the following hierarchy oftonal functions : tonic tone, other tones of the major triad chord, other tonesof the diatonic scale, and the nondiatonic tones. In these ratings, pitch heightwas more prominent for less musical listeners or with less musical (sinu-soidal) tones, whereas octave equivalence and the tonal hierarchy prevailedfor musical listeners, especially with harmonically richer tones. Ratings forquarter tones interpolated halfway between the halftone steps of the standardchromatic scale were approximately the averages of ratings for adjacentchromatic tones, suggesting failure to discriminate tones at this fine levelof division.

The study of perceived pitch and of theperceived relations between tones differingin pitch has generally been approached fromone of two quite different traditions: thepsychoacoustic and the musical. The psycho-acoustic approach has typically focused onsimple, physically specifiable properties oftones isolated from any musical context—properties of frequency, separation in logfrequency, or simplicity of integer ratios offrequencies. The results of such studies haveprovided some precise information about howthe ear responds to isolated tones or tones inrandom sequences. We believe that they havebeen less informative with regard to how thelistener perceives tones in organized musicalsequences. Music theory suggests that theperception of such sequences may rely on thelistener's sensitivity to different and struc-turally richer principles associated with tonaland diatonic organization. Such principles

are useful in explaining the cognitive phe-nomena of reference point, motion, tension,and resolution that underlie the dynamicforce of virtually all tonal music. They have,however, been subjected to relatively littlesystematic laboratory investigation or quan-titative formulation.

Background

Until recently, psychoacoustic studies weredominated by the assumption that pitch is asimple psychological counterpart of the singlephysical dimension of frequency, and so var-ies along a single psychological dimension ofwhat has come to be called pitch "height."The problem of pitch was in this way re-duced to the psychophysical one of determin-ing the functional form of the presumablymonotonic relationship between physical fre-

Copyright 1979 by the American Psychological Association, Inc. 0096-1523/79/0504-0579$00.75

579

580 CAROL L. KRUMHANSL AND ROGER N. SHEPARD

quency and psychological pitch (Stevens &Volkmann, 1940; Stevens, Volkmann, &Newman, 1937).

There is, however, evidence inconsistentwith such a simple, unidimensional, andmonotonic representation of pitch, namely,evidence that tones an octave apart (that is,tones standing in a frequency ratio of 2:1)are perceived as having something more incommon than tones somewhat less than anoctave apart (Allen, 1967; Blackwell &Schlosberg, 1943; Boring, 1942, pp. 376,380; Humphreys, 1939; Licklider, 1951, pp.1003-1004; Ruckmick, 1929). With increas-ing refinement, such psychoacoustic obser-vations have led to the view that pitch is amore complex attribute having at least twocomponents: pitch height (Balzano, 1977;Deutsch, 1972a; Dewar, 1974; Levelt, Vande Geer, & Plomp, 1966) and octave equiva-lence or tone "chroma" (Bachem, 1954; Bal-zano, 1977; Deutsch, 1973; Shepard, 1964).Certainly the precision (Ward, 1954) andcross-cultural consistency (Bowling, 1973a)

The work reported here was supported in partby National Science Foundation Grant BNS-7S-02806 to the second author and was carried outduring the first author's National Institute of Men-tal Health Traineeship (MH-10478-09) at Stan-ford University. We are greatly indebted to severalresearch assistants and colleagues: Shelley Hur-witz, who ran the subjects in Experiment 1 andcarried out much of the data analysis for bothexperiments; Zehra Peynircioglu, who ran the sub-jects in Experiment 2 (and served as our one sub-ject with absolute pitch); Kip Sheeline, who pre-pared the computer-generated tapes for Experiment2 using the facilities generously provided by theCenter for Computer Research on Music and Acous-tics at the Stanford University Artificial Intelli-gence Laboratory; and John Grey and Gerald Bal-zano, who provided additional help in the form ofequipment and suggestions.

Experiment 1 was first reported at the sym-posium "Cognitive Structure of Musical Pitch" atthe 1978 meeting of the Western Psychological As-sociation in San Francisco (Krumhansl, Note 1).

Requests for reprints should be sent to eitherCarol L. Krumhansl, who is now at Dept. of Psy-chology and Social Relations, Harvard University,Cambridge, Massachusetts 02138, or to Roger N.Shepard, Department of Psychology, Stanford Uni-versity, Stanford, California 94305.

with which listeners adjust a variable toneso that it is an octave away from a fixed toneappear difficult to explain in terms of anypurely one-dimensional psychophysical scaleof pitch. In addition, some suggestive indica-tions have been reported for augmented simi-larities of tones differing by simple ratios offrequencies other than the simplest two-to-one ratio that presumably underlies octaveequivalence (Balzano, 1977; Levelt et al.,1966).

There is evidence, moreover, that the ef-fects of even these physically specified fac-tors depend on whether the tones are inter-preted by the listener as musical or nonmusi-cal. On the basis of data from a nonmusicaltask of fractionation, Stevens et al. (1937)originally constructed, and Stevens andVolkmann (1940) refined, the mel scale ofpitch, which is a markedly nonlinear, thoughmonotonic, function of both frequency andlog frequency. In a more musical task inwhich subjects produced transpositions of afamiliar sequence of pitches, Attneave andOlson (1971) found, by contrast, that thetranspositions maintained the interval rela-tions of the original melody with respect tolog frequency, rather than with respect tothe psychoacoustic mel scale. The extent towhich the presented tones are interpreted asmusical also appears to influence the degreeto which tones separated by an octave areperceived as similar. For example, Allen(1967) found that octave equivalence wasstrong in music students but largely absentin nonmusical listeners. Similarly, Thurlowand Erchul (1977) found that only somesubjects were able to recognize musical in-tervals (defined, in effect, in terms of tonechroma) when the higher tone of the intervalappeared randomly within any of severaldifferent octaves.

Recognizing the powerful role that musicaland cognitive factors may play in the per-ception of tones in a musical context, a num-ber of experimental psychologists interestedin the perception of music have recently col-lected data on tones contained in more orless well-structured musical sequences. Suchstudies have emphasized the importance oflarger structural, cognitive, and Gestalt-like

TONES IN A DIATONIC CONTEXT 581

properties such as melodic contour, proxim-ity grouping, and, especially, musical scaleand tonality.

Bowling and his co-workers, particularly,have established the importance of contour.Dowling and Fujitani (1971), for example,found that listeners easily distinguished trans-positions that preserved melodic contourfrom those that did not; but, for transposi-tions that preserved qualitative contour, theyfound that listeners had difficulty in dis-criminating transpositions that also preservedthe quantitative sizes of the pitch intervalsfrom those that did not. Moreover, althoughDeutsch (1972b) found that subjects haddifficulty identifying familiar melodies whenthe tones were dispersed in different octaves,Dowling and Hollombe (1977) subsequentlyshowed that this disruptive effect of octavedispersal is reduced if the dispersed versionis constructed so as to preserve the quali-tative contour of the original melody.

The residual difficulty of octave-dispersedmelodies, even after contour has been pre-served, undoubtedly stems in large part fromthe resulting violation of the Gestalt prin-ciple, well-known in vision, of perceptualgrouping and coherence on the basis of prox-imity. More direct evidence for the operationof such a proximity principle in the domainof auditory pitch comes from studies of audi-tory "stream segregation" or "melodic fission"(e.g., see Bregman & Campbell, 1971; Dowl-ing, 1973b; Shepard, in press). Still furtherevidence for the importance of proximitycomes from experiments by Deutsch (1978)in which listeners attempted to recognizethe repetition of a given tone following theexperimental interpolation of a sequence ofother tones. Deutsch reported that accuracyof recognition varied inversely with the aver-age size of the intervals in the interpolatedsequence.

Finally, and most importantly from ourstandpoint, there are indications that tonalityand the organized pattern of pitch intervalscontained within a musical scale influence theperception of musical sequences. Tonality isdescribed by music theorists (e.g., Piston,1941; Ratner, 1962) as the single most criti-cal factor in musical organization. Briefly,

tonality is synonymous with key and desig-nates a set of tones as the musical scale tobe used in a particular context. The mostcommon scale used in traditional Westernmusic is the major diatonic scale, which con-sists of seven of the 12 musical tones con-tained within each octave (namely, the seventones called do, re, mi, fa, sol, la, ti, gener-ally completed by the eighth tone, do', anoctave above the first). These tones conformto a fixed pattern of intervals that repeats inevery octave. In the key of C, they are repre-sented by the white keys of the piano key-board. The remaining five tones in each oc-tave (in C major, the black keys of thepiano) are nonscale or nondiatonic tones.

Recent archeological evidence suggeststhat the use of the diatonic scale goes backwell over 3,000 yr. (Kilmer, Crocker, &Brown, 1976). And, although scales do dif-fer considerably from one culture to another,they generally share certain basic structuralfeatures with the diatonic scale (Dowling,in press; Dowling, Note 2), which seem toreflect universal structural constraints thatmay have both a cognitive (Dowling, Note2) and a group-theoretic (Balzano, Note 3)basis.

In addition to defining the musical scale,tonality designates one single tone in thescale, called the tonic, as the most structur-ally stable tone in the system. This tonefunctions as a kind of reference point towhich each of the other tones has a well-defined relationship, called its tonal function.In accordance with a hierarchy describedin music theory (Meyer, 1956, pp. 214-215),other tones differ with respect to how closelyrelated they are to the tonic and, conse-quently, how structurally stable they arewithin the system. For example, in the keyof C major the tonic, C, is the most stabletone, but other tones, particularly the thirdand fifth tones of the scale, E and G (whichtogether with C constitute the so-called ma-jor triad chord), are also relatively stable,whereas other tones of the scale are lessstable, "tending" (as music theorists say)toward more stable tones—particularly thetonic itself. The tones outside the scale, ornondiatonic tones, are still less stable and


tend even more strongly toward the morestable tones within the system.

Empirical evidence for the importance oftonality and the diatonic structure comesfrom a number of recent studies. Cohen(Note 4) has shown that musical subjects,after hearing a short excerpt from a musicalcomposition, are able to generate the asso-ciated diatonic scale with some accuracy.Other investigators (Dewar, 1974; Dewar,Cuddy, & Mewhort, 1977) have found thatmemory is better for tones contained in se-quences conforming to diatonic structurethan for tones in sequences not conformingto this structure. Dewar (1974) found thatchanges in musical sequences that took suchsequences outside the diatonic scale weremore easily detected than changes that leftthem within the diatonic scale, and Dowling's(1978) subjects found it difficult to distin-guish exact transpositions (a constant shiftin log frequencies, but necessarily departingsomewhat from the diatonic scale) from con-tour-preserving shifts along the diatonicscale (changing the intervals somewhat, butkeeping within the scale). Cohen (1975)and Attneave and Olson (1971) have found,further, that subjects are better able to recog-nize and to produce transpositions of tonalor familiar sequences than other less well-structured sequences. Finally, in a series ofexperiments undertaken subsequently to theexperiments to be reported here, Krumhansl(1979; Krumhansl, Note 5) demonstratedthat in an explicitly tonal context, the pat-tern of perceived similarities between toneswas highly specific to the tonality of thatcontext and recognition memory was betterfor diatonic than for nondiatonic tones.

These various recent findings strongly re-inforce the notion that the representation ofmusical pitch must be more highly structuredthan had been supposed on the basis of ear-lier psychoacoustic studies in which experi-menters assiduously avoided any semblanceof a musical context. Apparently, in addi-tion to the relations of the frequencies ofthe physically presented test tones them-selves, which underlie pitch height and oc-tave equivalence, structurally richer factorsinduced by a particular tonal context will

have to be taken into account in attemptinga complete description of the interval repre-sentation of musical pitch.

Our Approach

The experiments that we now report wereinitially motivated by what, despite the re-cent advances just reviewed, we still per-ceived as a noticeable gap between the rich-ness and power of the total structure im-plied by music theory, on the one hand, anddetailed experimental demonstrations andquantifications of that structure, on theother. Studies of the perception of musicalintervals had clearly demonstrated system-atic effects of simple separation in log fre-quency and of octave equivalence, but thehierarchy of tonal functions or stabilities thatare to be expected on the basis of the music-theoretic notions of tonality and diatonicismhad curiously failed to assert itself in thelaboratory.

Even the reported effects of the simplicityof the ratio of physical frequencies have notbeen as robust as expected. Thus, in thestudy by Levelt et al. (1966), such effectsappeared only in the case of already har-monically rich tones as opposed to pure sinu-soidal tones; and in the study by Balzano(1977), such effects emerged only for har-monic intervals (simultaneous tones), notfor melodic intervals (successive tones).What effects had been reported might there-fore be attributed to relatively sensory in-teractions between related frequencies orharmonics in the ear more than to cognitivefactors having to do with the tonal functionsindicated by music theory. And, even so,some of the results suggestive of structurebeyond the simple acoustic one of separa-tion in log frequency—such as the virtuallyidentical multidimensional scaling solutionsindependently obtained by Levelt et al.(1966) and Wickens (reported in Shepard,1974)—may have been partly artifactualconsequences of attempts to extract more di-mensions than the nonmetric analysis wouldsupport (see Shepard, 1974, pp. 387-388).

Our experiments were based on the con-jecture that previous failures to bring out


sharply the tonal functions that are expectedwithin the diatonic system might stem from afailure to meet three conditions simultane-ously : (a) The tones must be presentedwithin a musical context that strongly andunambiguously establishes the tonal frame-work with respect to which all test tonesare to be interpreted, (b) Detailed quantita-tive information' must be obtained for eachof the alternative tones of interest relativeto this tonal framework, (c) Data must beanalyzed separately for individual listenerswho may differ widely in musical experience,training, or aptitude, and so vary markedlyin their responsiveness to the experimentallyintroduced musical context.

In both of the following experiments, weestablished the tonal context by simply play-ing the seven tones of an ascending or de-scending major scale (omitting the eighthtone, an octave from the first, that normallycompletes the sequence). We then played asingle tone from somewhere in the octaverange extending from the omitted tonic tonethat would normally have completed the con-text series to the tone just one octave be-yond that note. We asked listeners, for eachsuch final tone, to rate how well that tone fitin with or completed the preceding contextsequence. We hoped that the judgments ob-tained for different tones within the test oc-tave would provide a direct, quantitativeindication of the stability and tonal functionof each tone relative to the tonal frameworkinduced by the preceding context.

Experiment 1

In Experiment 1, subjects judged howwell each of the 13 (equally tempered chro-matic) tones within the octave range be-tween middle C and the tone an octave above(C') completed ascending and descendingC major scales. The two alternative scalesequences and the 13 final tones used inExperiment 1 are shown in Figure 1. Sincethe ascending C major scale began on theC an octave below middle C and the descend-ing C major scale began on the C two oc-taves above middle C, the octave range offinal tones was bound by the two alternativecontext sequences.

DESCENDING CONTEXT

C B A G F E D

ASCENDING CONTEXT

FINAL NOTES

» fr> » H» °—°—»" 'C CB D 01 E F FJ G GJ A AJ B C1

Figure 1. The descending and ascending contextsequences of seven tones of the C major scale andthe 13 chromatic alternative test tones from theintervening octave that might immediately followthe context sequence.

The choice of the key of C (and conse-quently of the octave range from C to C')instead of any of the 11 other possible majorkeys and ranges should not, of course, re-strict the generality of our results. For inaccordance with music theory, previous psy-chological results (e.g., Attneave & Olson,1971; Dowling, 1978; Dowling & Fujitani,1971) confirm that the structure of the per-ceived relations between the various scaledegrees is invariant under transpositionalong a log frequency scale. (We disregardhere the exceptions that might conceivablyarise in the case of those rare individualspossessing true absolute pitch.)

Earlier studies suggest that a number offactors may influence such judgments ofscale completion. First, if pitch height is animportant dimension, then tones close in fre-quency to the context tones should be pre-ferred to tones farther from the contexttones. Second, if the tones separated by anoctave are equivalent or closely related, thenthe Cs at both the high and the low ends ofthe octave range of test tones should bejudged as approximately equally good com-pletions. Third, if tones of the diatonic scaleare more closely related to the tonal centerimplied by the scale context, then these dia-tonic tones should be preferred to the non-diatonic tones. Fourth, if the more detailedmusic-theoretic notions concerning a hier-


archy of tonal functions within the diatonicsystem have psychological correlates, thenthe most preferred tones, following the tonic,should be the other tones of the major triadchord (the third and fifth degrees of thescale), followed by the other diatonic andfinally the nondiatonic tones.

Method

Subjects

Twenty-four Stanford University undergradu-ates participated. Twenty-three of the participantsreceived credit toward an introductory course re-quirement. The 24th, a research assistant who wasas yet naive about the purposes of the experiment,was later added because of her extensive musicaltraining and her possession of absolute pitch. Eachlistener participated in a single 1-hr, experimentalsession, and filled in a short questionnaire describ-ing his or her musical training and experience.

Apparatus

We prepared the stimulus tapes by recording at7.5 inches per second (ips) directly from a Farfisa(Chicago Musical Instrument Co.) electronic or-gan onto tape using a Revox A77 Stereo Tape-recorder. We used the flute stop on the organ asthe best available approximation to a pure sinu-soidal tone generator. During the experimentalsessions, we played the prerecorded tapes at acomfortable loundness level via the Revox Tape-recorder, a Dynaco SCA-80Q amplifier, and twoBang and Olafsen Beovox S4S speakers.

Stimulus Sequences

Each trial consisted of the presentation of aseven-tone sequence of either an ascending or adescending C major scale followed immediately bya final tone, which was any one of the 13 equallytempered notes within the octave range bounded bythe two possible scale contexts. The duration ofeach tone was approximately .75 sec, and the in-tertrial interval during which listeners made theirresponses was approximately 6 sec. Each of the13 final notes appeared with each of the contexttypes (ascending and descending scales) once ineach of eight blocks of trials. Trials were ran-domly ordered within blocks, and the blocks werepresented to the subjects in different random orders.

Procedure

Listeners were instructed to judge on each trialhow well the final tone completed the presentedsequence. We asked them to express their judg-

ments by means of numerical ratings of goodnessfrom 1 to 7, where 7 was designated "very good,"4 was designated "moderate," and 1 was desig-nated "very bad." We encouraged them to usethe full range of these ratings.

Results

Individual Differences

Each listener's response profile consistedof the average ratings for each of the 13notes for the two types of context. Of theoriginal 23 listeners, 22 of these responseprofiles showed one of the three general pat-terns in Figures 3, 4, and 5. Seeking moreobjective support for our informal divisionof the listeners into these three groups, wecomputed correlations between the data foreach pair of listeners and then analyzed theresulting matrix of correlations by means ofthe hierarchical clustering method HICLUS(Johnson, 1967). The HICLUS solution us-ing the compactness option consisted of threegroups of listeners and a single unclassifiedlistener, as shown in Figure 2. These moreobjective results, as well as those (notshown) obtained by the connectedness op-tion, were in essentially complete agreementwith our original subjective classification.

All eight listeners in Group 1 played aninstrument or had vocal instruction (meanyears of instruction, 7.4; mean years per-forming, 5.6). Of the eight listeners in Group2, all but one played an instrument or sang(mean years of instruction, 5.5; mean yearsperforming, 3.3). Of the six listeners in Group3, only two played an instrument and onesang (mean years of instruction, .7; meanyears performing, .0). Owing to rather largeintragroup variation, only Groups 1 and 3differed significantly from each other interms of years of instruction and performingexperience. The t values comparing Groups1 and 3 were t(\2) = 3.00, p < .05, andt(\2) = 4.62, p < .01, for years of instruc-tion and years performing, respectively. Thecomparisons involving Groups 2 and 3 ap-proached significance, t(\2) = 2.03 and^(12) =2.07, . 0 5 < / > < . 1 0 for both, foryears of instruction and years performing,respectively. The single unclassified listenerhad 1 yr. of musical instruction and 11 yr.


.00

.20

.40

.60

.80

t.oo1-

HICLUS ON

INTERSUBJECTCORRELATIONS

GROUP 1 GROUP 2 GROUP 3 *

Figure 2. Hierarchical clustering (compactness or complete link method) of the first 23 listenersin Experiment 1, based on the matrix of correlations between rating profiles for all pairs oflisteners.

of performing experience. None of the origi-nal 23 listeners responded affirmatively to aquestion asking whether they have absolutepitch. The subsequently added 24th listener,however, reported having absolute pitch inaddition to exceptionally extensive musicaltraining—17 yr. of formal instruction and10 yr. of performing experience.

Judgments of Scale Completion

Group 1. The average scale completionjudgments for the 13 final tones for Group1 listeners are plotted for both ascendingand descending contexts in Figure 3. Theseeight listeners manifested a clear preferencefor the low and high Cs over all other notest(7) =21.59, p < .001. In addition, therewas a preference for scale notes other thanthe tonic (D, E, F, G, A, B) over the non-scale notes (C#, D#, F#, G#, A#), t(7) =12.59, p < .001. Among the scale tones, Dand B, which are the scale tones adjacent tothe tonic C, were judged to be relativelygood scale completions, as was the tone G,the fifth degree of the scale. The profiles for

ascending and descending scale contextswere similar, the ratings for the notes show-ing very little dependence on whether thescale context was ascending or descending.In particular, both high and low Cs werejudged to be equally good completions of as-cending and descending scales.

Group 2. The average responses of Group2 listeners, presented in Figure 4, showed astrong preference for the two Cs over allother 11 notes in the octave range, t(7) =13.66, p < .001, giving a deep U-shapedpattern to the profile. In addition, there wasa small but significant difference between thescale notes other than the tonic and the non-scale notes, *(7) =4.63, p < .01. Further-more, the low tones were judged to be some-what better completions for the ascendingcontext, and the high tones were judged tobe better completions for the descendingcontext. In particular, every subject showeda preference for the low C on ascending trialsand the high C on descending trials.

Group 3. In the ratings of Group 3 listen-ers, shown in Figure 5, again there was apreference for the two Cs over other tones


zins(3Q3

GROUP—— Ascending Context— Descending Coi

Dtt Fit Gtt A« B C'

NOTE

Figure 3. Mean rating profiles for Group 1 listeners in Experiment 1. (The mean rated good-ness of each test tone as a completion of the preceding context sequence is plotted separatelyfor ascending and descending contexts.)

in the octave range, t(5) = 6.45, p < .01,although the magnitude of the effect was notas large as in Group 1 and Group 2. Also,there was a small but significant preferencefor the other scale tones over nonscale tones,£(5) — 2.78, p < .05. However, distancefrom the tones of the preceding scale con-text accounted for most of the variance inratings of these listeners. Low tones werejudged to be better completions of ascendingscales, and high tones were judged to bebetter completions of descending scales. In

particular, the low C was judged to be amuch better completion of ascending scalesthan was the high C, and the opposite heldfor descending contexts. While the Group 2subjects showed this distance effect pri-marily at the two Cs, this effect was foundthroughout the octave range for Group 3.Indeed one of the listeners in Group 3 ex-hibited a pure distance effect with no indi-cation of octave equivalence. For this listener,each of the two crossing curves like thoseplotted in Figure 5 fell off monotonically

aa

GROUP 2Ascending ContextDescending Context

• Scale Degreeo Nonscale Note

c* Fit Git

NOTE

Figure 4. Mean rating profiles for Group 2 listeners in Experiment 1.


5oQ

GROUP 3—— Ascending Context

Descending Context• Scale Degreeo Nonscale Note

y

/

fit Git A," B

NOTE

Figure 5. Mean rating profiles for Group 3 listeners in Experiment 1.

from the C nearest to the preceding contextbut without any upswing at the C at theother end of the test range.

Unclassified subject. The single listenerin this original set of 23 who could not beclassified as falling into any one of the threegroups showed a pattern similar to Group 1listeners for descending contexts, but a vir-tually flat response profile for ascendingcontexts.

Subject 24. Figure 6 shows the responseprofile for the subsequently added listenerwho, in addition to exceptionally extensivemusical training, reported having absolute

pitch. As with Group 1, there was little dif-ference between ascending and descendingcontexts, and again there was a general pref-erence for scale over nonscale tones. Withinthe set of scale tones there was, in additionto the high ratings for the two Cs, a verymarked preference for the other members ofthe major triad, namely, the third and fifthdegrees of the scale (E and G), over theother scale tones.

Discussion

The results of Experiment 1 demonstratethe influence of a number of factors on the

oQ

SUBJECT 24 Ascending ContextDescendingContext

• Scale Degree° Nonscale Note

Ctt Fit Git Alt C1

NOTE

Figure 6. Mean rating profiles for the listener in Experiment 1 who was subsequently addedbecause of her exceptionally extensive musical training and possssion of absolute pitch.


perceived suitability of alternative tones ascompletions of a scale sequence. Further,the relative contributions of these differentfactors evidently vary markedly with themusical backgrounds of the listeners. Thepreviously well-established factor of pitchheight emerged once again, with tones closein frequency to the context tones preferredover tones farther from the context. How-ever, the influence of pitch height was strongonly for the least musical listeners in Group3. For these listeners, the context apparentlyserved primarily to define a location on thecontinuum of pitch height, so that there werenearly opposite effects when the contexttones were presented above and below therange of test tones. In marked contrast, themusical Group 1 listeners gave strikinglysimilar ratings following ascending and de-scending scale sequences. Thus, the contextevidently defined for these listeners a tonalstructure that, under octave equivalence, ap-plied to tones in any octave according totheir individual functions within this invari-ant structure regardless of their location inpitch height. In particular, the high and lowCs were judged by Group 1 to be equallygood completions of both ascending and de-scending scales. This octave equivalence wasweaker for less musical subjects, with the lowC preferred in ascending scale contexts andand the high C preferred in descending scalecontexts, in agreement with earlier findings(Allen, 1967).

The orderly relationship found betweenthe results for individual listeners and theirreported levels of musical interest and train-ing is consistent with similar relationshipsreported in earlier studies, including thoseby Allen (1967), Attneave and Olson(1971), Cohen (197S), Cuddy (1970), De-war et al. (1977), and Shepard (1964).Apparently, the kinds of structural relationsthat are extracted from tonal sequences varywith the extent of the listener's musicalsophistication, whether innate or acquired.

Despite these individual differences, therating profiles of all three groups of listenersrevealed a general preference for scale overnonscale tones, although this preference fordiatonic tones was strongest for the most

musical listeners. Within the set of diatonictones, moreover, the tonic was the moststrongly preferred by all listeners. Particu-larly for musical listeners, the fifth and, tosome extent, the third degrees of the scalewere also judged to be relatively good scalecompletions, as were the second and seventhdegrees. However, proximity to the tonicin pitch height probably contributed to thepreferences for the second and seventh inmany subjects; certainly the rated prefer-ences for these two tones were lower in thelistener with the most extensive musicalbackground (Subject 24; see Figure 6).

Our quantitative results are consonant,then, with expectations based on musictheory that the full musical interpretation ofa set of tones entails the internal assignmentof those tones to a hierarchy of tonal func-tions. Indeed, in the case of musical sub-jects, this hierarchy, together with virtuallycomplete octave equivalence, essentially re-places the effect of distance or separation inpitch height that has been the dominant re-sult of previous psychoacoustic studies in theabsence of a tonal context. Further evidencefor the reality of such an underlying hier-archy comes from a subsequent study(Krumhansl, 1979; Krumhansl, Note 5) inwhich musically trained listeners judged thesimilarity between pairs of tones in an es-tablished tonal context. Multidimensionalscaling of the results of that study yielded ahierarchy in essential agreement with thehierarchy found here. A tightly organizedcluster corresponding to the tones of themajor triad chord was surrounded by amore widely dispersed set corresponding tothe remaining diatonic tones, which set inturn was surrounded by a much more widelydispersed set corresponding to the nondia-tonic tones.

Experiment 2

Two limitations of Experiment 1 led usto undertake a second experiment. First,since in Experiment 1 we had divided theoctave test range only into the standardhalftone (or semitone) steps of the 13-tonechromatic scale, we precluded the possibility


of discovering any still lower levels of thehierarchy of tonal functions. The possibilityoccurred to us (and to Balzano, Note 6)that, just as the tones directly implied by aparticular diatonic context dominate all othertones from the chromatic scale of halftonesteps, those other tones are themselves in-directly implied by such a context undertransposition (modulation of key) ; and sothey might in turn dominate other tones noteven indirectly implied under transpositionsuch as, for example, tones interpolated byquarter-tone steps, halfway between thestandard tones of the chromatic scale.

Second, the flute-stop tones presented inExperiment 1 were only approximations topure sinusoidal tones. So, even though thetones were presented sequentially ratherthan simultaneously, the listeners' judgmentsof goodness of sequence completion may havebeen partly influenced by some perceptionof overlap between the harmonics present inthe test tone and the frequencies of the pre-ceding context tones. As we have alreadynoted, there is evidence that judgments ofharmonic relations reflect a richer structurewhen the tones themselves are harmonicallyricher (e.g., Levelt et al, 1966).

Accordingly, although Experiment 2 wascarried out in essentially the same way asExperiment 1, we modified the set of tonesin two ways: To the 13 chromatic tonesseparated by halftone steps within the octavetest range, we added the 12 quarter toneshalfway between these to obtain a morefinely graded series of 25 test tones. And,instead of generating the tones by means ofan organ flute stop, we used a computer togenerate all tones as pure sinusoids, free ofupper harmonics.

Method

Subjects

We recruited seven Stanford undergraduateswho served as paid listeners by posting noticesin the music department. The eighth listener wasthe same highly musically trained research as-sistant who participated in Experiment 1. As be-fore, all listeners completed a questionnaire de-scribing their musical training and experience.

Apparatus

We recorded the stimulus tapes at 7.5 ips on aRevox A77 Taperecorder from a digital signalprocessor controlled by the PDP-10 computer atStanford's Center for Computer Research on Musicand Acoustics. During the experimental sessions,we played these tapes by a Revox A77 Tapere-corder through earphones at a comfortable loud-ness level.

Stimulus Sequences and Procedure

The trials were similar in construction to thoseof Experiment 1. The context, consisting of anascending or descending diatonic scale, was fol-lowed by a final tone, which was any one of the25 tones in the set containing the 13 standard chro-matic tones and the 12 interpolated quarter tonesin the octave range falling between middle C(261.6 Hz) and the C an octave above (S23.2 Hz).The tones were pure sinusoids with a duration of .5sec each and a linear amplitude envelope havinga .05-sec rise time, a .40-sec steady state, and a.OS-sec fall time. Each of the 25 final tones ap-peared with each of the two context types once ineach of the 10 blocks of trials. We randomly or-dered trials within blocks and presented the blocksto the listeners in different random orders. Therewere five practice trials at the start of theexperiment.

The procedure was identical to that of Ex-periment 1.

Results

Individual Differences

Again there was considerable variation inthe rating profiles. Consequently, the inter-subject correlations were analyzed by meansof the program HICLUS. Using the compact-ness method, Subjects 3, 4, 6, and 8 wereclustered together; Subjects 2, 5, and 7formed another cluster; and Subject 1 wasseparated from the other listeners in thesolution. Subject 1 was again the only sub-ject claiming to have absolute pitch. Sub-jects 3, 4, 6, and 8 had on the average 5.2 yr.of formal instruction and 4.5 yr. of perform-ing experience. Subjects 2, 5, and 7 had anaverage of 6.5 yr. of instruction and 3 yr. ofperforming experience. Not -surprisingly,since these listeners were recruited from themusic department, all subjects had at least amoderate level of instruction and perform-ing experience and did not differ markedlyfrom each other in this respect.


Judgments of Scale Completion

Subject 1. The rating profiles of thishighly trained musical listener, shown inFigure 7, exhibit similarities to the profilesshe produced (as Subject 24) in Experiment1. Her profiles were very similar for as-cending and descending contexts. There wasa strong preference for the tonic, C, over allother tones, and relatively high ratings werealso given for the third and fifth degrees ofthe scale, E and G. In addition, there was ahint of a preference for the other scale tonesover nonscale musical tones. However,there was no overall difference between thestandard chromatic tones and the interpo-lated quarter tones. Rather, the ratings ofthe quarter tones seemed to approximateaverages of the ratings of the neighboringchromatic tones.

Subjects, 3, 4, 6, and 8. These listenersalso showed a general preference for thetonic, C, over the other tones. In addition,there were higher ratings for the scale tonesthan for nonscale tones. In particular, therewas a rise in the ratings for tones near thethird and fifth scale degrees, E and G. Aswith Subject 1, the ratings for the quartertones approximated averages of the ratingsfor the neighboring chromatic tones. Theprofiles for this group reflected an increasein the distance effect at the expense of oc-tave equivalence. Especially in the case ofthe lower tones in the range, tones nearerthe context tones in pitch height were pre-ferred to tones farther from the contexttones. Thus the low C was judged as abetter completion of an ascending contextthan a descending context.

SUBJECT I

ASCENDING CONTEXT • MUSICAL TONEDESCENDING CONTEXT O QUARTER TONE

SUBJECTS 3.4,6,8

V*-.-.

SUBJECTS 2,5,7

Figure 7. Mean rating profiles for three groups of listeners in Experiment 2, which used puresinusoidal tones and included additional test tones interpolated halfway between the previous 13chromatic tones.


Subjects 2, 5, and 7. The ratings bythese subjects showed a still stronger dis-tance effect, which held throughout the oc-tave range of test tones, with higher ratingsfor low tones in ascending contexts and forhigh tones in descending contexts. Althoughthere was no discernible difference betweenscale and nonscale tones, the upturn in thegraphs at the end of the range indicated thepresence of some degree of octave equiva-lence. Again, no difference was found be-tween the chromatic tones and the interpo-lated quarter tones.

Discussion

Listeners' judgments of scale completionin Experiment 2 provide no evidence of apreference for the standard musical tonesover the quarter tones falling halfway be-tween the musical tones. The ratings for theinterpolated quarter tones generally wereclose to the average of the ratings for thetwo neighboring musical tones. This resultsuggests that the listeners were unable todistinguish clearly between tones differing byno more than a 24th part of an octave—atleast under the conditions of this experi-ment, in which the tones themselves werefree of harmonics and in which the test tonewas presented after the context sequenceand within a different octave. As Dowling(1978) suggests, there may be cognitive-psychological reasons why melodies in mosthuman cultures are constructed using a re-stricted set of discrete pitches. For, as Dowl-ing notes, this set typically contains fromfive to seven scale-step categories within eachoctave, as might be expected on the basis ofMiller's (1956) well-known generalizationconcerning the limitation of the human ca-pacity for categorical judgment to 7±2 cate-gories. Moreover, a number of investigatorshave found evidence for categorical percep-tion of musical pitches that is similar to thatfound for speech sounds (Burns & Ward,1978; Locke & Kellar, 1973; Siegel & Siegel,1977a, 1977b; Blechner, Note 7; Halpern &Zatorre, Note 8).

The rating profiles showed many of thesame characteristics as those of Experiment

1, with ratings of scale completion indicatingagain that pitch height, octave equivalence,and the tonal functions of the pitches areimportant factors underlying the listeners'judgments. Despite the use in Experiment 2of the relatively less musical tones containingno overtone structure, the results of mostlisteners showed some degree of octave equi-valence and the results of at least some of thelisteners provided evidence for the hier-archy of tonal functions.

Although the subjects in Experiment 2had on the average a moderate level of musi-cal sophistication, the distance effect wasgenerally stronger whereas the more struc-tural factors of octave equivalence and tonalhierarchy were weaker than in Experiment1. (This distance effect was particularly no-ticeable for the lower tones in the test range,where the absence of overtone structure maybe more salient.) This finding is consistentwith our general supposition that these struc-tural factors tend to emerge only to the ex-tent that the tones are perceived as musical.It is also possible that the inclusion of theunconventional quarter tones tended to con-fuse the listeners and hence weaken theirmaintenance of an orderly correspondencebetween the test tones in Experiment 2 andthe listeners' discrete internal tonal system,which presumably makes no provision forsuch interpolated tones. We tentatively con-clude that the harmonics usually contained inmusical pitches may contribute to, althoughare probably not necessary for, the extrac-tion by the listener of the musically impor-tant relations between tones.

Again, individual differences were foundin the profiles, although these differenceswere not consistently related to the reportedmusical training of the subjects. The rela-tively small number of listeners who partici-pated in Experiment 2 and the relativelygross measure of musical sophisticationused, as well as the ambiguities introducedby the elimination of any overtone structureand the inclusion of quarter tones, may haveobscured any relationship between the musi-cal background of the subjects and the pat-tern of scale completion judgments, such asthat found in Experiment 1.


General Discussion

The results of Experiments 1 and 2, takentogether with previously reported work, ap-pear to us to be consistent with the follow-ing generalizations: To the extent that tonesdiffering in frequency are not interpreted asmusical stimuli—because they are presentedin isolation from a musical context, becausethe tones themselves are stripped of har-monic content, or because they are playedto musically unsophisticated listeners—themost potent factor governing their perceivedrelations is simply their separation alonga one-dimensional continuum of pitch height.To the extent that tones are interpretedmusically—because they are embedded in amusical context, because they are rich inovertones, and because they are playedto musically sophisticated listeners—simplephysical separation in log frequency givesway to structurally more complex and cog-nitive factors, including octave equivalenceor its psychological counterpart, tone chroma,and a hierarchy of tonal functions specificto the tonality induced by the context.

That the data from some of the musicallysophisticated listeners in Experiment 2 pro-vided evidence for such an underlying hier-archy even when the tones were reduced tosimple sinusoids argues against theories,widely held since Helmholtz (1863/1954),according to which the perceived relation-ships between tones is to be explained prin-cipally in terms of coincidences between fre-quencies of the harmonics (overtones or par-tials). Further evidence that cognitive fac-tors must be invoked in addition to suchphysical factors is Krumhansl's (1979;Krumhansl, Note 5) demonstration that theperceived similarities between tones formingan interval of a fixed size depend on the re-lation of those tones to the contextually es-tablished tonal center.

In retrospect, we are inclined to use tworelated propositions to interpret the indica-tion from Experiment 2 that the hierarchyof tonal function does not extend below thestandard division of octaves into the semi-tone intervals of the chromatic scale. Thefirst proposition is that there is a definite cog-

nitive-psychological constraint on the num-ber of categories that can be successfully dis-tinguished by absolute judgments with re-spect to a single dimension (Dowling, 1978;Miller 1956)—here, presumably, the circu-lar dimension of tone chroma (Balzano,1977; Shepard, 1964). The second is thatjust as the perceptual interpretation ofsounds as speech depends on the categoricalassimilation of those continuously variablesounds to an underlying discrete set ofphonemes (Liberman, Cooper, Shankweiler,& Studdert-Kennedy, 1967), the perceptualinterpretation of sounds as music dependsupon the categorical assimilation of thosecontinuously variable sounds to an under-lying discrete set of tones (Burns & Ward,1978; Siegel & Siegel, 1977a, 1977b;Blechner, Note 7; Halpern & Zatorre, Note8) arranged in what we have been callingthe tonal hierarchy.

Finally, our results reinforce previous in-dications that individual listeners differwidely in the extent to which they interpretmusical tones in terms of such an under-lying tonal system. From this we concludethat the indiscriminate pooling of data fromsuch different listeners prior to analysis canobscure important patterns in those data.We suggest instead that the systematic pur-suit of these individual differences by suchdata analytic techniques as hierarchical clus-tering (which we used here) or Carroll andChang's (1970) individual differences scal-ing (which we have applied to this problemelsewhere; see Shepard, Note 9, Note 10)can help to separate out such underlyingperceptual components as pitch height, tonechroma, and tonal function with respect to acontextually established tonality.

Reference Notes1. Krumhansl, C. L. Component factors in judg-

ments of scale completion. In R. N. Shepard(Chair), Cognitive structure of musical pitch.Symposium presented at the meeting of theWestern Psychological Association, San Fran-cisco, April 1978.

2. Dowling, W. J. Musical scales as cognitivestructures. In R. N. Shepard (Chair), Cogni-tive structure of musical pitch. Symposium pre-sented at the meeting of the Western Psycho-logical Association, San Francisco, April 1978.


3. Balzano, G. J. The structural uniqueness ofthe diatonic order. In R. N. Shepard (Chair),Cognitive structure of musical pitch. Sym-posium presented at the meeting of the WesternPsychological Association, San Francisco,April 1978.

4. Cohen, A. J. Inferred sets of pitches in melodicperception. In R. N. Shepard (Chair), Cogni-tive structure of musical pitch. Symposium pre-sented at the meeting of the Western Psycho-logical Association, San Francisco, April 1978.

5. Krumhansl, C. L. The psychological represen-tation of musical pitch in a tonal context. Paperpresented at the meeting of the PsychonomicSociety, San Antonio, Texas, November 1978.

6. Balzano, G. Personal communication, 1978.7. Blechner, M. J. Musical skill and the categori-

cal perception of harmonic mode. (Status Rep.on Speech Perception SR-S1/S2). New Haven,Conn.: Haskins Laboratories, 1977. Pp. 139-174.

8. Halpern, A. R., & Zatorre, R. J. Categoricalperception and selective adaptation of simul-taneous musical intervals. Manuscript in prepa-ration, 1978.

9. Shepard, R. N. The double helix of musicalpitch. In R. N. Shepard (Chair), Cognitivestructure of musical pitch. Symposium pre-sented at the meeting of the Western Psycho-logical Association, San Francisco, April 1978.

10. Shepard, R. N. Geometrical approximations tothe structure of musical pitch. Manuscript inpreparation, 1979.

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Received August 28, 1978 •

Correction to Kirsner and Sang

In the article "Visual Persistence and Code Selection in Short-TermMemory" by Kim Kirsner and D. L. Sang (Journal of Experimental Psy-chology: Human Perception and Performance, 1979, Vol. 5, No. 2, pp. 260-276), Figures 2 and 3 were mislabeled. The figure on page 264 should belabeled Figure 3, Experiment 2, and the figure on page 265 should belabeled Figure 2, Experiment 1.

quantification of the hierarchy of tonal functions within a diatonic context€¦ · ·...

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