Haskins Laboratories Status Report on Speech Research1992, SR-1111112, 227-260

Diversity and Commonality in Music Performance: AnAnalysis of Timing Microstructure in Schumann's

uTraumerei"*

Bruno H. Repp

This study attempts to characterize the temporal commonalities and differences amongdistinguished pianists' interpretations of a well-known piece, Robert Schumann's"Triiumerei." Intertone onset intervals (lOIs) were measured in 28 recorded performances.These data were subjected to a variety of statistical analyses, including principalcomponents analysis of longer stretches of music and curve fitting to series of lOIs withinbrief melodic gestures. Global timing patterns reflected the hierarchical groupingstructure of the composition, with pronounced ritardandi at the ends of major sections andfrequent expressive lengthening of accented tones within melodic gestures. Analysis oflocal timing patterns, particularly of within-gesture ritardandi, revealed that they oftenfollowed a parabolic timing function. The major variation in these patterns can bemodelled by families of parabolas with a single degree of freedom. The grouping structure,which prescribes the location of major tempo changes, and the parabolic timing function,which represents a natural manner of executing such changes, seem to be the two majorconstraints under which pianists are operating. Within these constraints, there is room formuch individual variation, and there are always exceptions to the rules. The strikingindividuality of two legendary pianists, Alfred Cortot and Vladimir Horowitz, is objectivelydemonstrated here, as is the relative eccentricity of several other artists.

INTRODUCTIONA. Diversity and commonality in music

performance

More than at any earlier period in musicalhistory, the contemporary scene in serious musicis dominated by the performer (see Lipman, 1990).Music consumers thrive on a limited repertoire ofstandard masterworks, primarily from the 19thcentury, that are offered again and again indifferent performances, both in live concerts andon recordings. The great musical events of ourtime are not the premieres of new compositions,but the appearances and reappearances ofsuperstar conductors and instrumentalists.

This research was made possible by the generosity ofHaskins Laboratories (Carol A. Fowler, President). Additionalsupport came from NIH (BRSG) grant RR05596 to theLaboratories. I am grateful to the Record Library, School ofMusic, Yale University, for providing most of the recordingsanalyzed here, and to its manager, Karl Schrom, for his help.Thanks also to Pat Shove for many stimulating discussions.

227

Young musicians compete for career opportunitiesby entering competitions in which their perfor­mances are compared and evaluated by juries (seeCline, 1985; Horowitz, 1990). While ever new ren­ditions of the standard repertoire vie for the at­tention of record buyers and concert audiences, aspate of reissues of historical recordings on CDs isoffering stiff competition. For some of the morepopular works, the Schwann catalogue listsdozens of performances; if deleted records, avail­able in libraries and private collections, arecounted, they may run into the hundreds. Therehave never been such ample opportunities to com­pare different performances of the same music.

Obviously, this remarkable diversity reflects notonly clever marketing and the promotion of su­perstars, but also the ability of music lovers to dis­tinguish and appreciate different performances.The more sophisticated among these listeners de­tect qualities in the performances of individualartists that lead them to search out these per­formers' concerts and recordings. They may voice

228 Repp

OpInIOnS about the quality of particular perfor­mances by these and other artists, describingthem as "brilliant" or "noble" or "thoughtful."Some of the more gifted professional critics excelin characterizing different performances in termsalternatingly scholarly and poetic.

While the attention of listeners and critics thusis mainly drawn to the differences among perfor­mances, there are also strong commonalities, usu­ally taken for granted and hence unnoticed.Though there is a large variety of acceptable per­formances of a given piece of music, there is aneven larger variety of unacceptable performances,which rarely make their way into the concert hallsor onto records. Unless they have the mark of in­spired iconoclasm (as do some of the performancesby the late Glenn Gould; see, e.g., Lipman, 1984),they quickly succumb to the fierce competition ofthe musical marketplace. Music teachers, how­ever, have to deal with them every day and trytheir best to mold immature and wayward stu­dents into performers that can be listened to withpleasure. Though teachers may differ considerablyin their methods and goals, and are rarely veryexplicit about what these are, they are transmit­ting the unwritten rules (though see Lussy, 1882)of a performance tradition that goes back to 19thcentury central Europe, where most of thestandard repertoire originated. Despite variouschanges in performance practices during the last200 years, most of them of a narrowly technicalnature, there are generally accepted norms of mu­sical performance, according to which the artist'sactions are largely subordinated to the musicalstructure. The artist's primary task is the expres­sion of the musical structure, so it can be graspedand appreciated by the listener, and make an im­pression on him or her. (See Lussy, 1882;Riemann, 1884; Stein, 1962.) This is presumablydone by conventional means that are adapted tothe hearers' perceptual and cognitive abilities.However, the particular doses in which thesetechniques are applied (unconsciously, for themost part) vary from artist to artist and accountfor individual differences in interpretation.

Thus there are two basic aspects of music per­formance: a normative aspect (i.e., commonality)that represents what is expected from a competentperformer and is largely shared by differentartists, and an individual aspect (i.e., diversity)that differentiates performers. The individual as­pect may be conceived of as deviations from a sin­gle ideal norm; more profitably, however, it maybe thought of as individual settings of free param­eters in the definition of the normative behavior.

That way, it is possible for different artists tomeet the norm equally and yet be discriminablydifferent. For example, two pianists may beequally adept in expressing a particular musicalstructure in their performances, but one maychoose a slow tempo and large tempo changes,whereas the other may prefer a faster tempo andsmaller deviations from the rhythmic beat. Theformer artist may then be characterized as"romantic" or "exuberant," while the latter willevoke epithets such as "restrained" and "noble,"though both may receive equal acclaim from asensitive audience.

The commonality-diversity distinction seemsobvious enough, but there is little tangible evi~

dence to substantiate it. Although volumes havebeen written about different performers and theircharacteristics, these discussions rarely go beyondgeneralities, and the vocabulary used (such as theadjectives quoted above) are not specifically linkedto particular performance properties; they alsomay be used differently by different writers. Thereis no consistent terminology, nor a well-developedand generally accepted theory of performance de­scription and evaluation, that would make possi­ble an objective characterization of performancecommonalities and differences. Music criticism isan art rather than a science, and the critic's im­pressions, accurate as they may be, are filteredthrough an idiosyncratic web of personal experi­ences, expectations, preferences, and semantic as­sociations. In fact, the belief is widespread thatperformance differences cannot be characterizedobjectively.

Such a negative conclusion, however, can onlybe justified if objective performance analysis hasbeen attempted and has failed. A total failure ishighly unlikely, however, for there are manyphysical properties of a musical performance that,without any question, can be measured objec­tively. Whether objective analysis can capture aperformance exhaustively may remain in doubt;the proper question is how much can be learnedfrom it. Surely, even a partial characterization interms of verifiable and replicable observations canmake a valuable contribution to our understand­ing of performance commonalities and differences, .which remain mostly a mystery to this day. Forexample, most music lovers would agree thatArtur Rubinstein and Vladimir Horowitz were twovery different pianists, whose performances ofsimilar repertoire (e.g., Chopin) are instantly dis­tinguishable. But what is it, really, that makesthem so different and individual? And do theyhave anything in common at all? It is all too easy

Diversitv and Commonalitv in Music Performance: An ,4na/vsis of Timmg Microstructure In Schumann's "Triiumcrc;P 229

to couch the answers to such questions in terms of"artistic personality," which do not enlighten us atall about the nature of the differences and com­monalities. Objective performance analysis has acontribution to make here.

B. Objective performance analysisThe concept of objective performance analysis

goes back to Carl Seashore (1936, 1938, 1947) andhis collaborators, who pioneered the use ofacoustic analysis techniques to derive"performance scores" that show the exactvariations of pitch, timing, and intensity producedby an artist on some instrument. A considerableamount of data was collected in Seashore'slaboratory, but their analyses remainedrudimentary and focused primarily on technicalaspects such as pitch accuracy and vibrato insinging and string playing, and chord synchronyon the piano. Although some studies compareddifferent performances of the same music, nostatistical characterization of commonalities anddifferences was attempted, nor were the resultsinterpreted with more than a passing reference tomusical structure. These limitations reflect thebehavioristic approach of American psychology atthe time, as well as the unavailability ofpsychological performance models and advancedstatistical methods. Nevertheless, Seashore (1947,p. 77) was able to reach conclusions that reinforcethe premises of the present study:

... there is a common stock of principles whichcompetent artists tend to observe; ... We shouldnot, of course, assume that there is only one wayof phrasing a given selection, but, even with suchfreedom, two artists will reveal many commonprinciples of artistic deviation. Furthermore,insofar as there are consistent differences in theirphrasing, these differences may reveal elementsof musical individuality.

Contemporary with Seashore's work, similarresearch was going on in Germany. Hartmann(1932) compared the timing patterns of twofamous pianists' performances of a Beethovensonata movement and provided detailed numericaldescriptions of the differences between them. Thesmall size of the sample, however, limits thegenerality of his conclusions. Although one of thetwo artists seemed quite eccentric, this impressioncannot be substantiated without more extensivecomparisons.

The three decades between roughly 1940 and1970 were barren years for the objective study ofmusic performance. In the last two decades,

however, several researchers have taken up thetopic again. Their work, with few exceptions, hastaken a case study approach; their focus was noton individual differences but on the instantiationof certain principles in (hopefully, representative)performance samples, mostly from pianists. Thus,Shaffer (1980, 1981, 1984, 1989; Shaffer, Clarke,& Todd, 1985) examined the timing of pianoperformances from the perspective of motorprogramming and control (see also Povel, 1977).His erstwhile students, Eric Clarke and NeilTodd, went on to make significant contributions oftheir own. Clarke (1982, 1985) conducted casestudies of piano performances of Satie's music andwrote at length on how musical structures areexpressed in timing variations (Clarke, 1983,1988). Todd (1985, 1989; Shaffer & Todd, 1987)developed a computational model of timing at thephrase level, which has been revised and extendedto dynamics in his most recent publications (Todd,1992a, 1992b). Bengtsson and Gabrielsson (1980;Gabrielsson, 1974; Gabrielsson, Bengtsson, &Gabrielsson, 1983) conducted extensive studies ofthe performance of different rhythm patterns inrelatively simple musical contexts, but withattention to individual differences. Gabrielsson(1985, 1988) has written more generally abouttiming in music performance. A provocative theoryof "composer's pulse" in performance timing hasbeen developed by Clynes (1983, 1987).

There are few studies in the literature that em­ployed what one might consider a representativesample of different performances. Gabrielsson(1987) conducted a detailed comparison of five pi­anists' performances of the first eight measures ofMozart's Piano Sonata in A major, K. 331; thestudy included measurements of timing, intensi­ties, and articulation (i.e., tone durations). Palmer(1989) studied the same music as performed by sixpianists in "musical" and "unmusical" styles, andshe analyzed the timing patterns, note asyn­chronies, and articulation. Palmer also recordedeight pianists playing the first 16 measures of aBrahms Intermezzo and studied timing patternsin relation to intended phrasing. She concludedthat "pianists share a common set of expressivetiming methods for translating musical intentionsinto sounded performance" (p. 345). These studiesstill used relatively small samples and obtainedonly limited amounts of data from each performer.

The largest amount of performance data wasanalyzed in Repp's (1990) study, which included19 complete performances by famous pianists of aBeethoven sonata movement. The analysis waslimited, however, in that it concerned primarily

230 Repp

timing patterns at the level of quarter-note beats.Also, the focus of the study was a search forClynes' (1983) elusive "Beethoven pulse" in thetiming patterns. Following the example ofBengtsson and Gabrielsson (1980), Repp appliedprincipal components (factor) analysis to thesetiming data, to determine how many independenttiming patterns were instantiated in this sizeablesample of expert performances. Two factorsemerged, the first representing primarily phrase­final lengthening, while the second factor capturedother types of expressive timing variation. Musicallisteners' evaluations of the performances werealso obtained, and some relationships between themeasured timing patterns and listeners' judg­ments were found.

The present study continued the general ap­proach taken by Repp (1990), but without the aimof testing Clynes's theory of composer's pulse. Itsmain purpose was to assemble a large sample ofperformances of a particular composition by out­standing artists, and to analyze the timing pat­terns in detail, using various statistical methods.These methods, it was hoped, would make it pos­sible to separate commonalities from differences.The common patterns would reveal how most pi­anists transmit musical structure and expressionthrough timing variations, and it was of interestto determine whether there would be a singlecommon factor or several. The characterization ofindividual differences was also of interest, espe­cially with respect to some of the legendary pi­anists in the sample, who are known for theirindividuality.

The music chosen for this investigation was awell-known piano piece from the Romantic period,"Traumerei" by Robert Schumann. It was selectedbecause it is a higWy expressive piece that permitsmuch freedom in performance parameters, and .hence much room for individual. differences ininterpretation. Also, there are numerous record­ings available.

I. THE MUSIC"Traumerei" ("Reverie," "Dreaming") is the

seventh of the 13 short pieces that constituteRobert Schumann's (1810-1856) "Kinderszenen"("Scenes from Childhood"), op. 15. This little suite,universally considered one of the masterpieces inits genre, was composed by Schumann in 1838when he was secretly engaged to Clara Wieck. Thepieces were selected from some 30 piecescomposed for Clara around that time, and theirtitles may have been added as an afterthought.They are not intended for children but rather

reflect an adult's recollection of childhood(Brendel, 1981; Chissell, 1987).

"Traumerei" occupies a central position in the"Kinderszenen" suite, not only by its location butby its duration and structural role. It serves as aresting and turning point in the cycle, whichshows so many intricate thematic connections thatit may be considered a set of free variations (Reti,1951; Traub, 1981). Its key signature, meter, andmotivic content also single it out as the hub of thesuite (Brendel, 1981; Traub, 1981). However, it isalso often performed by itself, both by classicalartists (e.g., it was one of Horowitz's favoriteencores) and in numerous popular versions andarrangements. Indeed, "Traumerei" is perhaps themost popular Romantic piano piece. Its score isshown in Figure 1.

The melodic/rhythmic structure (or groupingstructure; see Lerdahl & Jackendoff, 1983) of"Traumerei" is depicted schematically in Figure 2,which also introduces terminology to be usedthroughout this paper. The layout of the figurecorresponds to that of the score in Figure 1, andbars (measures) are numbered. The piece iscomposed of three 8-bar periods (A, B, A'), the firstof which is obligatorily repeated. Each period issubdivided into two 4-bar phrases, which arerepresented by staff systems in Figure 1 and bylarge rectangular boxes in Figure 2. (Actually, thebeginning and end of each phrase extend slightlybeyond the four bars, overlapping with thepreceding and following phrases, respectively.)There are two phrase types, a and b, each of whichrecurs three times with slight variations(indicated by subscripts in Figure 2). Phrase al(period A) is repeated literally in period A'.Phrases b2 and b3, which constitute period B, arestructurally identical but differ in key, harmony,and some other details.

The melodic events are divided horizontally(roughly, along the dimension of relative pitch)among four registers or voices (S = soprano, A =alto, T = tenor, B = bass). Vertically (along thedimension of metrical distance), the events withinphrases are grouped into melodic gestures, whichare represented by filled boxes in Figure 2.Characteristically, they extend across bar lines(vertical solid and dashed lines in Figure 2). Amelodic gesture (MG) is an expressive unit com­posed of at least two and rarely more than sevensuccessive tones. It is defined here to begin withthe onset of its first tone and to end with the onsetof its last tone. This seems reasonable, for a pi­anist cannot influence the time course or dynam­ics of the last tone once the key has been struck.

Oiversitll and Commonalitll in lvluS1C Performance: An Anaillsis of Timing Microstructure in ScJlllmann '5 "Tniumerel" 231

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Oiversit1/ and Commonalit1/ in Music Performance: An Analllsis of Timmg Microstructure in Schumann's "Triiumerei" 233

(In Figure 2, each MG box extends one eighth-notespace beyond the metrical onset of the last tone.)Blank spaces represent time spans devoid of MGs;they may contain single tones, sustained tones, orrests. Multivoiced chords having some gesturalquality are represented by vertical ellipses inFigure 2. Arrows indicate continuity of a MGacross a line break or with a subsequent melodicevent. Thus, MG1 and MG6 continue from the endof one line to the beginning of the next, and MG3band MG5b in the soprano voice are closely linked.When an arrow points into blank space, it pointsto the onset of a single-tone event that cohereswith the MG.

MGs are divided into primary (p) and secondary(s) ones. The latter are usually shorter andaccompany primary MGs; they are represented byboxes filled in lighter shades. An exception to thisclassification is MG2i, a delayed imitation of MG2

Table 1. The artists and their recordings.

divided between the tenor and alto voices (bars 10and 14). In the following timing analyses, we willessentially be concerned only with the primaryMGs, which represent the leading voice(s) in thepolyphonic quartet. As can be seen in Figure2, in phrases of Type a the primary MGs(including the final MG6f in bar 24) are all in thesoprano, except for MG6a which is in the bass andoverlaps both MG5a and MG1. In Type b phrases,during the second half, the primary MGs cascadedown through the four voices, overlapping eachother.

II. THE PERFORMANCESThe sample of performances analyzed here

includes 28 performances by 24 outstandingpianists, selected according to ready availability(see Table 1). Two artists, Cortot and Horowitz,are represented by three different recordings each.

Code

ARGARRASHBREBUNCAPCalC02C03CURDAVOEMESCGIAHalH02H03KATKLIKRUKUBMalNEYNOVaRTSCHSHEZAK

Artist

Martha Argerich (1941-)Claudio Arrau (1903-1991)Vladimir Ashkenazy (1937-)Alfred Brendel (1931-)Stanislav Bunina

Sylvia Capovaa

Alfred Cortot (1877-1962)Alfred CortotAlfred CortotClifford Curzon (1907-1982)Fanny Davies (1861-1934)Jorg Demus (1928-)Christoph Eschenbach (1940-)Reine Gianoli (1915-1979)Vladimir Horowitz (1904-1989)Vladimir HorowitzVladimir HorowitzCyprien Katsaris (1951-)Walter Klien (1928-1991)Andre Krusta

Antonin Kubalek (1935-)Benno Moiseiwitsch (1890-1963)Elly Ney (1882-1968)Guiomar Novaes (1895-1979)Cristina Ortiz (1950-)Artur Schnabel (1882-1951)Howard ShelleyaYakov Zak (1913-1976)

Recording

DG 410 653-2 (CD) [1983]Philips 420-871-2 (CD) [1974]London 421 290-2 (CD) [1987]Philips 9500 964 [1980]DG 427315-2 (CD) [1988]Stradivari SMC-6020 (C) [T] [1987]EMI3C 153-53793M [1935]EMI3C 153-53794M [1947]EMI3C 153-53795M [1953]London LL-I009 [-1955]Pearl GEMM CD 9291 (CD) [1929]bMHS OR 400 [-1960s]DG 2535 224 [1966]'Ades 13.243-2 (CD) [1974]RCA LD 7021 [T] [1947]Columbia MS 6411 [1963]Columbia MS 6765 [1965] (live) [T]Telefunken 6.42479 AP [1980](live)[T]Allegretto ACS 8023 (C) [T]CMHS 1009 (orig. Erato) [-1960s]Dorian DOR-90116 (CD) [1988]Decca DL 710,048 [-1950s]Electrola WDLP 561 [-1935]Vox PL 11.l60 (orig. PL8540) [1954]MCA Classics MCAC-25234 (C) [1988]PatM COLH 85 [1947]CHAN 8814 (CD) [1990]Monitor MC 2039 [-1960]

abirthdate not knownboriginally Pearl CLA 1000Cdate unknown

Abbreviations: CD =compact disc; C =cassette; < =date of liner notes (recording date probably the same or preceding year); - =estimated date; T = "Traumerei" only

234 Repp

Most of the recordings (17) are on long-playingrecords, 3 are on cassettes, and 8 are on CDs(including a transfer of a 1929 recording by FannyDavies, a one-time student of Clara Schumann,from a very scratchy original). Most of theperformances are of the complete "Kinderszenen,"but five are of "Traumerei" only. Two of the latterare from live concerts; all others are studiorecordings. Table 1 includes actual or estimatedrecording dates.

The 24 artists include some of the mostrenowned pianists of this century as well as someless well-known artists. They can be groupedaccording to gender (6 female, 18 male), country oforigin (5 from Russia; 4 each from Austria andFrance; 3 from England; 2 each from Germanyand Brazil; one each from Czechoslovakia,Argentina, and Chile; one unknown), andapproximate age at the time of recording (aboutequal numbers of young, middle-aged, and old). Atleast 12 pianists are no longer alive.

III. ANALYSIS METHODSA. Measurement procedure

All tone interonset intervals (lOIs) were hand­measured by the author using a waveform editingprogram. Each performance was low-pass filteredat 4.9 kHz and digitized at a 10 kHz samplingrate. The digitized waveform was displayed on thescreen of a computer terminal, 2 s at a time. Thescreen resolution for that display was about 2 ms.A cursor was placed at each tone onset, and apermanent "label" was attached to that point inthe waveform file. The differences between thetime points of successive labels yielded the lOIs,which were noted down to the nearest millisecond.After some practice, it took about 2 hours tomeasure one complete performance.

Two problems had to be coped with. One wasthat the onset of a soft tone was often difficult todetect by eye, particularly in some of the olderrecordings, which had much surface noise. (DAVwas the worst by far.) An auditory method wasused in that case: The cursor was moved back insmall steps, and the waveform segment up to thecursor was played back until the onset of the tonein question could no longer be heard. This proce­dure was time-consuming but usually resulted inrather accurate location of tone onsets (cf.Gabrielsson, 1987). The second problem concernedonset asynchronies among simultaneous tones indifferent voices. Unintended asynchronies, whichusually were too small to be detected by eye in thewaveform, had to be ignored. Larger asynchronies,which in most cases must have been intended by

the artists, were noted (especially in performancesby COl-3, DAV, MOl, and SHE) and measured,but the label from which the 101 was computedwas placed at the onset of the major melody tone(usually the one with the highest pitch and thelatest onset). The same convention was followed inthe case of the written-out arpeggi in bars 2, 6,and 18.

B. Data representation

The data were submitted to further analysis inthe form of eighth-note lOIs. That is, intervalsinvolving grace note onsets (in bars 2, 6, 8, 16, and18) were omitted from the data and wereconsidered separately.l lOIs longer than anominal eighth-note were divided into eighth-noteintervals of equal duration. Thus, for example, thehalf-note 101 in bar 2 was represented as fourequal eighth-note lOIs. A complete performancethus yielded 254 eighth-note lOIs.

e. Measurement error

The measurement procedure would have beenprohibitively time-consuming if maximum accu­racy had been aimed for (e.g., by displayingshorter waveform segments on the computer ter­minal screen). A certain amount of speed-accuracytradeoff had to be taken into account. The magni­tude of the resulting measurement error can beestimated from three performances (BRE, C02,and the first half of CUR) that, for various rea­sons, had to be remeasured. Two of them (BRE,CUR) derived from good LPs with a moderateamount of surface noise, whereas C02 was ofolder vintage and had special problems related tothe pianist's tendency to play the left- and right­hand parts asynchronously. There were severalvery large discrepancies (in excess of 80 ms)between the two measurements of the C02performance, due to a conscious change in theauthor's criteria for treating asynchronies; thesediscrepancies were not considered truemeasurement errors and were omitted from thedata presented below, though some smallerinconsistencies of the same kind may still beincluded.

The measurement error distributions are shownin Table 2. The combined BRE+CUR datarepresent average accuracy, whereas the C02data constitute a worst-case scenario; only DAVwas even more difficult to measure. It can be seenthat over 90% of the measurement errors in theBRE+CUR data did not exceed 10 ms, which isless than 2% of the average 101. The largest errorwas under 35 ms, or about 6% ofthe.average 101,and the average measurement ,error was 4.3 mS,

DIversItv and Commonalltv In MusIc Performance: An Anail/SIS ofTImll1R MIcrostructure l1J Schumann's "TriiumereI" 235

less than 1%. In the C02 data set, about 90% ofthe errors ranged from 0 to 20 ms, and the largesterror was under 50 ms. The average error was10.4 ms, or about 2%. Measurements of thevirtually noiseless CD recordings presumablywere even more accurate than suggested by theBRE+CUR data.

Table 2. Distributions of absolute measurement errorin two sets of remeasured data.

BRE+CUR (N=412) C02 (N=245)

Range (ms) Percent Cum. Percent Cum.

0-5 77.4 77.4 24.5 24.56 - 10 13.8 91.3 40.0 64.5

11 - 15 2.7 93.9 18.8 83.316 - 20 1.7 95.6 5.7 89.021 - 25 2.2 97.8 3.7 92.726 - 30 1.7 99.5 3.3 95.931 - 35 0.5 100.0 2.0 98.036 -40 1.2 99.241 - 45 0.4 99.646 - 50 0.4 100.0

IV. RESULTS AND DISCUSSIONThe strategy in presenting the results will be to

proceed from more global properties to moredetailed aspects.

A. Overall tempoPerhaps the most obvious dimension along

which different performances vary is that ofoverall tempo. The present set of performances isno exception. Tentative estimates of global tempowere obtained by computing the first quartiles(the 25% point) of the individual eighth-note 101distributions, multiplying these millisecond valuesby 2, and dividing them into 60,000. The choice ofthe first quartile was motivated by theconsideration that expressive lengthening of lOIsis both more frequent and more pronounced thanshortening, and also by the fact that it gave tempifor two pianists, BRE and DAV, that agreedclosely, respectively, with Brendel's (1981)statement of his preferred tempo and with ClaraSchumann's (DAV's teacher's) recommendedtempo. (Repp, 1993b, will provide a more extendeddiscussion.) Table 3 shows that the tempo rangeextended from 48 to 79 quarter-notes per minute.Apart from the fact that the three fastestperformances are all old recordings, there does notseem to be any systematic relationship betweentempo and the time at which the recording was

made, nor with pianists' gender, age at the time ofrecording, or country of origin.

Table 3. Overall tempi (qpm) estimated from the firstquartile of the fOf distribution.

48 ESC4950 KAT, NEY5152 ZAK53 CAP54 KLI5556 CUR57 BUN, NOY58 DEM, KUB59 ARR, KRU, MOl, SCH6061 H0262 ARG63 ASH64 H0365 HOI, SHE66 C0367 BRE,ORT68 GIA69707172 COl737475 C0276777879 DAY

B. Repeats

The first step in the data analysis was anattempt to eliminate redundancy and reducerandom measurement error by averaging acrossrepetitions of the same (or highly similar) musicalmaterial. Of course, such averaging is meaningfulonly if there are no systematic differences intiming microstructure across repeats. The primetarget for averaging was the first period, bars 1-8,which was repeated literally, according toSchumann's instructions in the score. All but twopianists (DAV, KRU) observed this repeat. 2

To compare the first and second repeats for allpianists, a grand average timing pattern was firstobtained by computing the geometric mean ofcorresponding lOIs across all 28 performances.(For DAV and KRU, the data of bars 1-8 were

236

simply duplicated for the missing second repeat.)The geometric mean was preferred over thearithmetic mean because it compensated for anytendency of slower performances to show moreexpressive variability, which would havedominated in an arithmetic average. In this grandaverage, the two repeats of bars 1-8 were found tohave extremely similar timing profiles, with only aslight tendency for the second repeat to be playedslower. The correlation between these twoaverages was 0.987, which indicates that anyvariations across repeats for individual pianistswere either random or idiosyncratic. Thecorrelations between the two repeats in individualperformances are shown in Table 4 (column 2);most of them were quite high.

The close similarity of timing patterns across re­peats has been noted in virtually every study in

Repp

which such a comparison was made (see, e.g.,Palmer, 1989; Repp, 1990; Seashore, 1938). Thus,it seemed justified to average across the two re­peats of bars 1-8 in all further analyses, except asnoted.

There were other instances of identical or highlysimilar musical material in the piece. Their timingpatterns may be compared in Figure 3, whichplots the grand average eighth-note lOIs. Alogarithmic ordinate scale is used here toaccommodate the longest values; it also makes thescale comparable to the percentage scale used byauthors such as Gabrielsson (1987). The abscissarepresents metrical distance (or "score time") ineighth-note steps.3 Longer notes appear as"plateaus" of multiple eighth-notes; that way, alllOIs are represented on the same proportionalscale.

Table 4. Correlations of the timing profiles for repeated or similar sections (R = repeat; GM = geometric mean).

Bars

Artist 1-8/1R-8R 1-4117-20 9-12/13-16

ARG 0.781 0.861 0.630ARR 0.933 0.938 0.850ASH 0.922 0.951 0.927BRE 0.953 0.888 0.880BUN 0.510 0.633 0.632CAP 0.935 0.922 0.846COl 0.859 0.717 0.837CO2 0.865 0.736 0.851cm 0.750 0.741 0.571CUR 0.937 0.911 0.921DAY * 0.863 0.863DEM 0.938 0.924 0.828ESC 0.890 0.734 0.802GIA 0.777 0.809 0.914HOI 0.918 0.958 0.697H02 0.811 0.861 0.822H03 0.826 0.738 0.770KAT 0.825 0.901 0.867KLI 0.804 0.893 0.861KRU * 0.875 0.940KUB 0.951 0.913 0.888MOl 0.805 0.855 0.493NEY 0.920** 0.904 0.872NOY 0.931 0.906 0.908ORT 0.677 0.609 0.707SCH 0.676 0.574 0.748SHE 0.868 0.831 0.859ZAK 0.839 0.875 0.909GM 0.987 0.986 0.950

*no repeat**bars 1-4 only (see Footnote 2)

DlVersitu and Commonalitu In Music Performance: An Analysis of Timing Microstructure in Schumann 5 "Triiumerei" 23;""

2500

-- a, (bars 0·4) -- b1 (bars 5·8)

2000......... -e- a, (bars 17-20) -e- bz (bars 9-12)Ul

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BAR/EIGHTH-NOTE

MG5bI I

MG4bITl

MG3b

BAR/EIGHTH-NOTE

Figure 3. Grand average lOIs (geometric means across all 28 performances). The two panels show the timing profilesfor phrases of Type a and b, respectively. Primary melodic gestures (d. Figure 2) are indicated by brackets. The barnumbers on the abscissa refer to bars 0-8 only; for the later phrases, an appropriate constant must be added.

The left-hand panel of Figure 3 compares thetiming profiles for the three Type a phrases. Bars17-20 (phrase al) are musically identicalwith bars1-4 (see Figure 1), and it can be seen that theirtiming patterns are highly similar (r = 0.986),though bars 17-20 tend to be played at asomewhat slower tempo. The individualcorrelations for these four bars (Table 4, column 3)are comparable to the correlations between thetwo repeats of bars 1-8. The timing pattern forbars 21-24 (phrase a2) is initially similar butdiverges soon, due to the fermata (long hold) inbar 22 and the progressive ritardando (slowing oftempo) towards the end of the piece.

The right-hand panel of Figure 3 compares thetiming patterns of the three Type b phrases. Onceagain a striking similarity can be seen, due to thesimilarity or identity of the melodic structure (see

Figure 2). The timing profiles of phrases b2 (bars9-12) and b3 (bars'13-16) are especially close. Onlythe (prescribed) ritardando in bar 16 is muchmore pronounced than that in bar 12 (which is notnotated). If the final halves of bars 12 and 16 (theritardandi) are excluded, correlations betweenthese two timing patterns are again extremelyhigh (see Table 4, last column). Phrase bl (bars 5­8) also shows a rather similar profile; differencesoccur precisely where its musical content deviatesfrom phrases b2 and b3 (especially in bar 6).

The correlations in Table 4 indicate whichartists were highly consistent in their timingpatterns, and which of them were less consistent.The consistent group is led by ASH, CUR, andNOV and includes many others; the lessconsistent group includes ARG, BUN, C03, MOl,ORT, and SCH.4

238

C. Melodidrhythmic structure and theaverage timing pattern

Let us examine now the average timing patternand its relation to the musical structure in greaterdetail. Although the grand average timing profileis not necessarily an optimal performance timingpattern, it captures. features that many individualperformances have in common, whereas it sup­presses random and idiosyncratic timing devia­tions that differ from performance to performance.

In Figure 3 it can be seen that there are globaltemporal trends within phrases, particularly thoseof Type b. Their timing profiles follow a concavefunction on which various local peaks aresuperimposed. Bars 21-24 (phrase a2), too, seemto follow such a curve, though it is grosslydistorted by the fermata and the final grandritardando. Bars 1-4 and 17-20 (phrase al) have aflatter global timing shape. The curvilinear trendsin Type b phrases are reminiscent of the paraboliccurves hypothesized by Todd (1985) as the basicphrasal timing pattern. Todd's model5, however,operates on the durations of time units equivalentto whole bars, whereas the present timing profilesare across eighth-note units and thus providemuch finer resolution. The curves that "cradle" thepresent timing patterns (i.e., the "bottom lines" ofthe Type b timing profiles) are, in fact, poorly fitby quadratic functions. However, they dorepresent the general principle formalized byTodd and observed by many others, that there is aslowing of the tempo at major structuralboundaries, in proportion to the importance of theboundaries. Thus the most extreme ritardando(prescribed in the score) occurs at the end of thepiece; a pronounced slowing down (alsoprescribed) occurs at the end of the second period(bar 16); next come the end of the first period (bar8) and the end of phrase bl (bar 12), which showabout the same amount of ritardando (not explicitin the score); the end of phrase al (bars 4 and 20)shows the smallest, but still quite noticeableslowing down.

The piece begins with a brief melodic gesture(MG1) that ascends from the dominant to thetonic, with accent on the latter (hence theintervening bar line, which indicates accent on thefollowing beat). MGl recurs in bars 4-5, 8-9,12-13,16-17, and 20-21, though the upbeat is shortenedin bars 8, 12, and 16 (cf. Figure 1). It thuscontains a single 101 of variable nominal length.Its realization was strongly context-dependent:The quarter-note upbeat at the beginning of thepiece (bar 0) was short relative to the following

Repp

101, which seemed fairly stable across all phrases.The quarter-note upbeats at the ends of bars 4and 20, which coincided with MG6a, wererelatively longer. The eight-note upbeats in bars 8and 12, which coincided with MG6b, were evenlonger. Finally, although the grace-note upbeat ofbar 16 is not shown explicitly in Figure 3, the 101accommodating it was lengthened enormously,due to the major ritardando at the end of periodB. (More information about the grace-note upbeatis provided later on.)

The chord following MG1 enriches the harmonicand rhythmic texture but is not really part of themelody; however, it may be thought of as a kind ofecho of the tonic and thus could be linked withMG1. The chord bisects the inter-gesture intervalbetween MGl and MG2. Of the resulting two lOIs,the first (nominally shorter) one was consistentlylonger relative to the second (nominally longer)one, as if pianists tried to equalize the two. Thismay reflect final (and/or accent-related)lengthening of the tonic in MG1, which isabsorbed by the 101 preceding the chord.

MG2 is undoubtedly the most salient melodicgesture of the piece. It comprises five eighth-notesthat ascend in increasingly larger pitch steps to afinal note which constitutes the apex of the four­bar melodic arch that constitutes each phrase.Variants ofthe gesture occur in bars 1-2, 5-6, 9-10,13-14, 17-18, and 21-22. MG2 is unusual becauseit culminates on the second beat of a bar, negatingentirely the customary accent on the first beat. Ascan be seen in Figure 3, the average temporalpattern of the five lOIs is quite similar in alloccurrences of the gesture: The first 101 is close tothe proportional duration of the preceding 101, butthe next two lOIs are much shorter. The fourth101 is longer again, similar to the first, and thefifth 101 is the longest; it is clearly visible as a

. sharp peak in the timing profile. The duration ofthat fifth 101 varies with position in the piece: It isleast extended in bars 10 and 14, longer in bars 2,6, and 18, where it accommodates the two gracenotes of the left-hand chord (cf. Figure 1), andlongest in bar 22 where it leads to the climacticfermata. In fact, this eighth-note 101 in bar 22tended to be lengthened relative to the fermatachord itself (the plateau following the peak). Thespecific accelerando-ritardando shape of MG2 willbe examined more closely later on.

The 101 following MG2 in bars 2, 6, 18, and 22(plateau in Figure 3) represents another inter-ges­ture interval, nominally three or four eighth-noteslong. Relative to the' last 101 of MG2, this intervalwas much shorter proprortionally, even in bar 22,

Diversitll and Commonalitll in Music Performance: An Ana/lIsls of Timln\; MIcrostructure In Schumann's "Triiumerei" 239

where it received the fermata. The fermata had itsmaximal effect on the inter-gesture interval, dou­bling its duration relative to bars 2 and 18, but italso affected the whole preceding MG2. In bars 10and 14, the inter-gesture IOI is bridged by the in­ner-voice imitation gesture MG2i, which has atiming pattern that is almost the mirror image ofMG2 in that it accelerates during the first threeIOIs and slows down only at the end. The first IOIis long because it also represents the last 101 ofMG2; the relative length of the second IOI may betransitional, or it may reflect final lengthening ofMG2 which was absorbed by MG2i.6

The remaining MGs, which descend in a chainfrom the melodic apex reached by MG2, arepatterned differently in phrase Types a and b. Thegrouping structure of the Type a chain is indicatedwith slurs in the score (Figure 1). MG3a consistsof four (a1) or five (a2) eighth-notes in the sopranovoice, with accent on the penultimate; a shorteraccompanying gesture occurs in the tenor voice(al only). It can be seen in Figure 3 thatlengthening on the accented note occurred inperformance. MG4a is similar in rhythmicstructure to MG3a and exhibits a similar timingpattern, except that the final unstressed eighth­note 101 was lengthened as well, especially inphrase a2, where the final ritardando tilted thetiming profile. MG5a, reinforced by shortersecondary MGs in the bass voice, is similar toMG4a, except that in phrase al it leads into afinal half-note that coincides with the onset ofMG6a. Its timing pattern is a progressivelengthening, similar to that observed in MG4a. Inphrase a2, the final ritardando is in full control.The final gesture in phrase aI, MG6a, is in thebass voice and leads back to the low-pitched tonicwhich doubles the tonic of the next MG1. Its firstIOI coincides with the final lengthened IOI ofMG5a, and the last two lOIs coincide with thequarter-note upbeat of MG1 and are bothlengthened as well. The lOIs in between areconsiderably shorter. The total timing profile ofMG6a is not unlike that of MG2, which itresembles rhythmically. The final gesture inphrase a2, MG6f, is a truncated version of thepreceding MGs and shows an enormous slowdownin tempo; note, however, the discontinuity in theritardando between MG5a and MG6f, which is amanifestation of final lengthening in MG5a.

The melodic chain of type b is also divided intofour MGs, but they are organized differently. Theslurs in the score are not entirely consistent here;what distinguishes the gestures is the fact thatthey occur in different voices, imitating and

overlapping each other. Simultaneously, othervoices articulate shorter two-note cadentialgestures which, as can be seen in Figure 3,essentially govern the timing pattern. In eachMG, both lOIs constituting these accompanyingmotifs were lengthened, the first, unaccented oneusually more than the second, accented one. Thisis explained by the harmonic function of the tonecluster initiating the unaccented 101, which isthat of suspense followed by resolution. In part,however, it may also be due to final lengthening ofthe preceding, overlapping MG.

In summary, these observations illustrateseveral general principles of performance timing:(1) Whole phrases tend to show global tempocurves characterized by an initial accelerando anda more pronounced final ritardando whose degreereflects the degree of finality of the phrase in thehierarchical grouping structure. (2) "Riding" onthe global pattern, individual MGs often show asimilarly curved local pattern, though accentplacement, harmonic factors, and the influence ofoverlapping MGs in other voices may modulatethat pattern in various ways. Gesture-finallengthening is commonly observed.

D. Principal components analysisAn alternative way of capturing commonalities

among different performances is offered by thestatistical technique of principal components(factor) analysis. This method decomposes thedata matrix (N performances by M lOIs) into anumber of independent components or factors,each of which resembles a timing profile (i.e., Mstandardized "factor scores"). The original dataare approximated by a weighted sum of these fac­tors; the weights, which differ for each perfor­mance, are called "factor loadings" and representthe correlations between the performance timingprofiles and the factor score profiles. The degree ofapproximation (the percentage of the "variance ac­counted for" or VAF) depends on the number offactors that are considered significant; a commoncriterion employed here is that they should have"eigenvalues" greater than 1. (Eigenvalue = Ntimes an individual factor's proportion of VAF.)The first factor always accounts for the largestamount of variance and represents a kind of cen­tral tendency or most common pattern. Additionalfactors account for increasingly less variance andthus represent patterns shared by fewer perfor­mances. A standard technique for simplifying thefactor loadings and thereby increasing the inter­pretability of the factors is called "Varimax rota­tion." It increases large factor loadings and re-

240 Repp

duces small ones without changing the number offactors or the total VAF; however, the VAF is re­distributed among the rotated factors.

Principal components analysis thus can revealwhether there is more than one shared timingpattern represented in the sample of 28performances. If there is essentially only one wayof performing the piece, then a single factorshould explain most of the variance in the data. Ifthere are several radically different timingpatterns, then several orthogonal factors shouldemerge. Idiosyncratic timing patterns will notlead to a significant factor and will constitute partof the variance not accounted for. Since all timingprofiles are standardized and intercorrelated inthe course of the analysis, differences in overalltempo are automatically disregarded.

Although it would seem desirable to conduct theanalysis on the complete performances, the largeritardandi observed universally at the ends ofphrases a2, bI, b2, and b3, as well as the largeslowdown at the fermata in phrase a2, dominate

the overall timing pattern and cause highcorrelations among the performances, with theresult that the interesting variation among theshorter lOIs is lost. In fact, a principalcomponents analysis conducted on the completedata yielded only a single factor, indicating thatall pianists observed the major ritardandi, whichis not very surprising. Therefore, it was decided toanalyze only the data for bars 0-8 (phrases al andbI), which did not include any extremelengthenings (cf. Figure 3). Because of theparallelism of the timing profiles of these phrasesto those of similar type (see Figure 3), such arestricted analysis essentially captures the wholeperformances minus the major ritardandi.

This analysis yielded four significant factors.Together they accounted for 76% of the variance.For individual performances, the VAF rangedfrom 60% to 90%. The timing profiles representingthe first three factors are shown in Figure 4, andthe factor loadings of the 28 performances arelisted in Table 5.

OIJ

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~ 400-hji-,TTM-rrrrrn-TlTl+n-rrrr-.+nlTTTrmTTrrrr+TTrITTT+-rnT1'Trh-rTTTn-I..1000rr--,---,---,----,----,---,r---,...-----,

400, , .... .~-M~~-M~~_M~~_M~~-M~~_M~~_M~~_M~_

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Figure 4. Timing patterns of the first three principal components for bars 0-8. The standardized factor scores wereconverted into milliseconds by multiplying them with the average within-performance 101 standard deviation andadding this product to the grand mean 101. Bar lines and MG brackets are added for orientation.

Diversitu and Commonality in Music Performance: An Analuszs of Timzng Mzcrostnlcture In Schumann', "Triiumerez" 241

Table 5. Sorted rotated factor loadings of the 28performances, bars 0-8. (Loadings smaller than 0.4 areomitted.)

II III IV

SCH 0.844ASH 0.760 0.423BRE 0.738DAV 0.695CAP 0.695 0.438 0.41 ISHE 0.692KAT 0.685KUB 0.672ARR 0.664 0.463ZAK 0.644 0.460aRT 0.637CUR 0.615 0.429KRU 0.593 0.495 0.445DEM 0.551 0.495HOI 0.888H03 0.886H02 0.837ARG 0.770NEY 0.665 0.524ESC 0.482 0.626GIA 0.595 0.491BUN 0.588KLI 0.523 0.570NOV 0.509 0.549 0.464C03 0.876Cal 0.875CO2 0.850Mal 0.501 0.582

Factor I (VAF = 30%) represents a timingpattern shared (partially) by a large number ofperformances; half the pianists had their largestloading on this factor, with SCH, ASH, and BRE

. leading the group. This pattern has the followingfeatures: a relatively long initial upbeat (MGl)and pronounced lengthening of the last 101 ofMG6, which accompanies the upbeat to the nextMGl (positions 4-8 and 8-8); general lengtheningof long lOIs (plateaus in the profile); a relativelylong accelerando phase in MG2 followed by apronounced lengthening of its last 101; and smallbut regular gestural accent peaks for MG3-MG6,except for special emphasis at the end of MG5b(positions 8-3 to 8-5). This pattern reflects themelodic/rhythmic structure even more clearlythan the average timing profile displayed inFigure 3.

Factor II (VAF = 25%) may be called the"Horowitz factor," since the three performances bythat artist showed the highest loadings, though

ARG, NEY, ESC and a number of otherperformances shared features of this pattern. Dueto the statistical orthogonality of the factors, thistiming pattern is radically different from that ofFactor I. It is characterized by a very short initialupbeat (essentially reduced to an eighth-note,though it was rarely that extreme in the actualperformances; see below); relatively short longlOIs (i.e., low plateaus); no accelerando but apronounced ritardando during MG2; pronouncedlengthening in MG3a, MG3b, and MG5b, withextreme prolongation of the IOI in position 8-4,which accommodates a grace note.

Factor III (VAF = 15%) is the "Cortot factor"; thethree performances by this artist were the onlyones that showed substantial loadings. Its timingpattern is most unusual: a relatively short upbeatin MGl; a short final lOI in MG2 (positions 2-2and 6-2); marked speeding up during MG3 with afollowing ritardando extending through MG4 tothe endofMG5; and rushing during the end ofMG5b, followed by a pronounced ritardandoduring MG6b.

Factor N (VAF =6%) seems less important andwill not be discussed in detail, as no artist showeda high loading on it (Table 5).

The three factor timing patterns shown inFigure 4 do not correspond exactly to any real per­formance, though they are similar to certain per­formances, as indicated by the factor loadings(Table 5). Most artists' performances are best de­scribed by a weighted combination of several fac­tors; thus, for example, the CAP timing profile is acombination of Factors I, II, and III. Such a com­bination will of course result in an attenuation ofextreme features. About one fourth of the variancewas not explained by the four factors extracted,and most of that variance was probahly nonran­dom, representing the artists' individuality.

It had to be asked at this point whether theHorowitz and Cortot factors emerged only becauseeach of these artists was represented by threedifferent performances. 7 The principal compo­nents analysis was repeated with only a singleperformance of each of these pianists included(COl and H02). The three factors that emerged(VAF = 71%) were highly similar to the first threefactors of the earlier analysis. The second andthird factors still had their highest loadings inH02 and COl, respectively. The patterns offactorloadings for the three factors in the two analysescorrelated 0.96, 0.98, and 0.95, respectively. Thus,the Horowitz and Cortot factors are not artifactsdue to the over-representation of these artists inthe sample. They represent two true alternatives

242 Repp

to the "standard" pattern of performance timinginstantiated by Factor I-alternatives that onlyHorowitz and Cortot dared to choose in nearlypure form, but that several other pianists incorpo­rated partially into their timing strategies.

E. Detailed analyses

In this section, we examine how melodicgestures and other details of the score wereexecuted by using, as it were, a magnifying glasson the data. While the emphasis has so far beenprimarily on commonalities, the analyses are nowdirected increasingly towards uncovering artisticdiversity. They reveal the different ways in whicha gesture may be shaped by pianists of greatauthority. They also reveal some unusual, andoccasionally questionable, interpretations ofnotational details in the score. In addition todemonstrating the range of individual differences,these analyses illustrate what temporal patternsare preferred by a majority of the artists, andwhat patterns are never used, and hencepresumably unacceptable.

1. MG1. This gesture consists of merely twotones forming the pitch interval of an ascendingfourth. It is followed, however, by a chord whosetiming, though not strictly part of the melody, isalso of interest. MG1 appears six times: Threetimes with a nominal quarter-note upbeat (bars 0,4, and 20) and three times with a nominal eighth­note upbeat (bars 8, 12, 16), one of which (bar 16)is written as a grace note and thus is open to theassignment of an even shorter value. Two of theinstances of the gesture are relatively uncon­strained because the upbeat is unaccompanied(bars 0 and 16); in the other cases the upbeat coin­cides with eighth-notes in the bass voice (MG6).

Figure 5 presents the individual data in terms oftwo 101 ratios, N(BC) and B/C, where A is theduration of the intragesture 101 (i.e., the upbeat),B is the intergesture 101 preceding the chord(nominally two eighth-notes) and C is theintergesture interval following the chord(nominally three eighth-notes). If the score wereplayed literally, N(BC) should be either 0.2 or 0.4,depending on whether the upbeat is notated as aneighth-note or a quarter-note, and B/C should be0.67. In Figure 5, however, the ratios have beennormalized with respect to the underlying eighth­note pulse (assumed to be isochronous for thispurpose), so that the expected "literal" ratios(effectively, tempo ratios) are equal to 1 in allcases.

Let us consider first the ratio plotted on theabscissa. The top row of panels in Figure 5 shows

the three instances in which the MG1 upbeat isnominally a quarter-note. The leftmost panelshows the unconstrained situation at thebeginning of the piece (bar 0). It can be seen thatvery few pianists played this initial upbeat-orthe following intergesture interval, as the casemay be-"as written" (ratio of 1). The largemajority played the upbeat quarter-note shortrelative to the intergesture interval. Severalpianists (NEY, COl, BUN) come very close toplaying it as if it were an eighth-note (ratio of 0.5),and one (ARG) does so without question. This mayhave been a deliberate anticipation of the(notational) shortening of the upbeat in laterincarnations of MGl, but it is a true deviationfrom the written score and is perceived as such.

Later, however, when the quarter-note upbeatoccurs together with two eighth-notes in the bassvoice (top center and right-hand panels in Figure5), pianists are more evenly divided between thosewho shorten the upbeat and those who lengthen itrelative to the intergesture interval. While theaverage ratio is close to 1, there is enormousvariation among individual artists, and some whoseverely shortened the initial, unconstrainedupbeat (e.g., ARG, BUN) do not repeat thistendency later on.

The lower row of panels in Figure 5 showsinstances where the upbeat is nominally aneighth-note. In two cases (left-hand and centerpanels) the upbeat is accompanied by a bass-voiceeighth-note that marks the approaching end ofMG6 and of a whole phrase, so here theoverwhelming tendency is to lengthen the upbeat,sometimes to an extent corresponding to anominal quarter-note (ratio of 2; BUN and ARR inbar 12). In bar 16, where the upbeat is notated asa grace eighth-note, a significant minority ofpianists plays the upbeat shorter than an eighth­note, in one case (KAT) shorter than a sixteenth­note (ratio of 0.5). The majority, however, stilllengthen it, and at least one (ASH) plays the gracenote as if it were a quarter-note (ratio of 2).

There is little indication of clustering orbimodality in these data. Although there are some"outliers," the values of the ratio seem to be ratherevenly distributed over a wide range. Correlationsof individual timing patterns across the differentinstances of MG1 are not high, though someconsistencies can be seen. Thus ARR and BUNalways tend to lengthen the upbeat, whereasARG, DEM, and COI-3 tend to shorten it. Somepianists, perhaps to be regarded as "literalists,"consistently avoid extremes (e.g., ESC, KRD,MOI,ZAK).

Figure 5. Timing patterns of the six instances of MG1 and the following chord. (See text for explanation of ratios.) Ratios of 1 represent equal underlyingbeats (Le., equal tempo). The upper panels plot the three instances where the upbeat.is nominally a quarter-note, while the lower panels plot the threeinstances where the upbeat is nominally an eighth-note (a grace note in bar 16). The dashed lines indicate a doubling or halving of the nominal duration inexecution.

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244 Repp

With regard to the BIC ratio, plotted on theordinate, there is more consistency. In all sixinstances, which are notationally equivalent,there is a strong tendency to lengthen the restpreceding the chord, sometimes to the extent thatthe onset of the chord occurs in the middle of theintergesture interval (ratio of 1.5). Among thosewho tend to play the chord very late are ARG,ARR, BUN, and NOV. Nevertheless, there aresome pianists who do not show that pattern, mostnotably Cortot.

Finally, it should be noted that the two ratiosare uncorrelated across different artists.Evidently, the timing ofMG1 is quite independentof the timing of the chord in the intergestureinterval.s

2. MG2. This is the signature melodic gesture ofthe piece, and its manner of execution is crucial tothe impression of a performance of "Traumerei." Itoffers a unique opportunity to investigate thetemporal shaping of a gesture, for several reasons:(1) It comprises six notes (i.e., five IOIs)-asufficient number of degrees of freedom for anyconstraints on temporal shape to emerge veryclearly; (2) it is unidirectional in pitch anduninterrupted by metric accents-the accent thatwould normally occur on the note following thebar line is clearly suspended in this case; (3) itrecurs six times, with slight variations; and (4)pianists are likely to give special attention to itsexecution.

MG2 occurs in bars 1-2, 5-6, 9-10, 13-14, 17-18,and 21-22. (We will not consider the imitationgesture MG2i in detail.) The versions in bars 1-2and 17-18 are identical; during the last 101, twograce notes (a written-out arpeggio) occur in theleft-hand accompaniment of the melody. In bars 5­6 and 21-22, the penultimate pitch interval isextended from a fourth to a major sixth; inaddition, the melody in bars 21-22 leads to afermata and also lacks the grace notes during thelast 101. In bars 9-10 and 13-14, there are nograce notes, the penultimate melodic interval isreduced to a minor third in the first instance,there is a change of key (to B-flat major) in thesecond instance, and the occurrence of MG2i inthe middle voices creates a forward movementthat is absent in the other variants. Therefore,timing differences are to be expected reflectingthese factors.

The average temporal shapes of the six versionsof MG2 are shown in Figure 6. The five lOIs foreach version, representing the geometric meansacross all 28 performances, are plotted here on alinear scale. The five data points for each version

have been fitted with a quadratic function (i.e., aparabola), which seems to describe their temporalshape rather wel1. 9 Each gesture acceleratesinitially and then slows down at the end. Thisfinal ritardando is least pronounced in bars 9-10and 13-14; it is more evident in bars 1-2, 5-6, and17-18; and in bars 21-22, preceding the fermata, itis dramatic. Except in this last instance, there is atendency for the penultimate IOI to rise slightlyabove the parabolic trajectory. This was evidentlydue to including in the average performances suchas Cortot's which, as we have seen in Figure 4(Factor III), showed a pronounced tendency toshorten the last 101.

1400

0 Bars 1/2

• Bars 5/61200

A Bars 9/10

'" Bars 13/14---.(J) 1000 0 Bars 17/18

E Bars 21/22---- •0 800

600

4001-6 1-7 1-8 2·1 2-2

BAR/EIGHTH-NOTE

Figure 6. Geometric mean lOIs for the six versions ofMG2, with best-fitting quadratic functions. The abscissalabels refer to bars 1 and 2.

The temporal shapes of the six versions of MG2were examined and fitted with quadratic functionsin each of the 28 individual performances (a totalof 168 instances). It emerged that most individualartists' timing patterns (87% of all instances)could be described well by parabolas, with fitsranging from good to excellent. What variedbetween artists and between different instances ofMG2 were the curvature and elevation of theparabolic functions, but not so much theirgoodness of fit (however, see Footnote 9). The dataof six individual artists are shown in Figure 7.

Diversit1/ alld Commonahtliln l'vll1SIC Performance: An Anal1/sls of Timin" Microstmctllre 111 5chumaml's "Triillmere:" 245

1200 -,----,-----,--.,.----,---.m

1000

800 -4\-\--t----t--.;---It--rl'-i

1-6 1-7 1-8 2-1 2-2

1000--.(f)

.s 800 +---+---+---+-:>.<.+-,I4-;----i

o

400-!--~~f=:::::=-+--+-_I1-6 1-7 1-8 2-1 2-2

1400-.----,--,---,-----r----,

1200

1000-+---+----;---++-~____1

800-+--'rl-----;---f-+h~_,f_-__1

1-6 1-7 1-8 2-1 2-2

1200..,......--r---,--,---r-.,-,

1000

800 -t---4---1---+-_++--"'''''''''

1-6 1-7 1-8 2-1 2-21600....---,---,-----,--.,-----,-.....,

1400

1200-t---+---f---+---ff---;

1000-j-l,---+---f---+--r-t---;

800-t-~-+---+----l-,4~+---i

400-!---+-~-T~-+--f----i

1-6 1-7 1-8 2-1 2-21600..,......--,----r----,,.---.,---.....,

1400

1200-i---+---+---if---+-,f---i

1000 -t---t---+-+-I-f--,.<p>o"6-"71

400 +--+--=+=:::::l!!::::::PL-~-l1-6 1-7 1-8 2-1 2-2

BAR/EIGHTH-NOTE

Figure 7. MG2 timing data for six individual artists, with best-fitting quadratic functions. Where the pattern deviatedmarkedly from a parabolic shape, the data points are connected with straight lines. See Figure 6 for the legend ofsymbols.

The two panels on top illustrate two individualcases that were fit well by parabolas but differedin curvature: The highly modulated timing curvesof ARR contrast with the flatter ones of SCH. Thecenter panels show one representative perfor­mance of each of the two great individualists,

Horowitz and Cortot. Horowitz's curves are fitfairly well by parabolas but generally lack theinitial acceleration shown by most other pianists.Cortot shows a truly deviant pattern: In all threeperformances (which spanned 18 years the last101. This seems to indicate that he grouped the

246 Repp

penultimate tone with the following long tone,treating it like an upbeat; in fact, the first fourIOIs are fit well by a parabola. This idiosyncratictiming shape held only for the first five instancesof MG2, however; in bars 21-22, preceding thefermata, Cortot produced a beautiful parabolictiming shape in all three performances. Finally, ascan be seen in the bottom panels of Figure 7, ARGand BUN, two highly variable pianists, intermit­tently adopted the alternative timing pattern in­stantiated by Cortot; one other pianist who did sowas CUR. The only other type of deviant timingpattern was shown by ORT who, in bars 9-10 and13-14 only, lengthened the third IOI above theparabolic trajectory, which led to a W-like shape.

The quadratic functions (y = C + Lx + Qx2) thatfit the large majority of individual phrasal shapesare characterized by three parameters: a positiveconstant C that reflects vertical displacement,related to overall tempo; a generally negativecoefficient L that reflects horizontal-verticaldisplacement of the curve in x-y coordinatespace;10 and a positive coefficient Q that reflectsthe degree of curvature of the concave (U-shaped)parabola. A question of great theoretical interestwas whether there are any constraints among thethree parameters; in principle, of course, they arequite independent of each other. The coefficients Land Q were plotted against each other for all 146individual gestures that followed a parabolicshape. There was a remarkably tight linearrelationship between these two parameters, withcorrelations ranging from 0.93 to 0.98 across thesix versions of MG2. The reason for this result isthat the location of the minimum of the function isrelatively fixed. Therefore, as the curvature of theparabola increases, the negative slope of thetangent at xO also increases.

If the x coordinate of the minimum is relativelyfixed, then the constant C must be correlated withthe y intercept of the function and should increaseas a function of curvature Q. There were indeedpositive linear correlations between C and Q,ranging from 0.79 to 0.93. 11 Thus, both Land Care fairly predictable from Q, which leads to theconclusion that the expressive timing patterns ofthe large majority of the performances of MG2 canbe characterized by a single family of parabolas,varying in Q only. This family of functions isshown in Figure 8; it was generated by increasingQ from 20 to 140 in steps of 20, and by setting Land C according to their average linearregressions on Q. Figure 8 embodies a constrainton the expressive timing shape of MG2 that wasobeyed by the majority of pianists. With few

exceptions, any individual execution of this phrasecan be characterized by a single curvatureparameter, with some additional freedom in globaltempo (vertical displacement), which is notrepresented in the figure.

1400

0 Q = 20

• Q= 401200

A Q = 60

A. Q= 80..--...(j) 1000 0 Q = 100E--- • Q = 120

0 800 \I

600

4001-6 1-7 1-8 2-1 2-2

BAR/EIGHTH-NOTE

Figure 8. Family of quadratic functions that describes thetiming constraint on MG2 observed by the large majorityof pianists. The functions vary in curvature (Q), roughlyover the observed range. The other coefficients of thequadratic functions were obtained by the averageregression equations. Vertical displacement should beconsidered an additional free parameter.

To give some impression of individualdifferences in tempo modulation during MG2,Figure 9 plots the average Q coefficient for bars 1­2,5-6, and 17-18 (which were generally executedsimilarly; cf. Figure 6) against that for bars 21-22,with individual artists identified. (Cortot isexcluded from this plot.) Most pianists employedmoderate modulation in the earlier instances ofMG2, but gave a very expressive shape to the lastinstance in bars 21-22, which led to the fermata.Some, most notably DEM, used strong modulationin all instances. Horowitz's three performancesare at the other extreme. They appear relativelyunmodulated despite a substantial ritardandobecause he did not show any initial accelerando;therefore, his timing patterns were fit byparabolas with low curvature.

DiversitVi1nd Commonalitv in Music Performance: An Analusis of Timll1\i Microstructure in Schumann's °Triilin/erei" 24;"

tempo, the slower and the more inflected was MG2(see also Footnote 8).

3. Grace notes accompanying MG2. Only three ofthe six instances of MG2 have grace notes (reallya written-out arpeggio) in the left hand during thelast 101; they occur in bars 2, 6, and 18. Since allsix instances of MG2 followed a parabolictrajectory, on the average (cf. Figure 6), the gracenotes as such were not responsible for thelengthening of the final 101 and the systematicconstraints that it followed. Rather, they seemedto be fitted into whatever temporal shape thepianists chose for the primary melodic gesture. Wenow turn our attention to the execution of thesegrace notes, which exhibited surprisingvariability.

In the score, the grace notes are represented asnominal sixteenth-notes. However, since thepreceding notes in the left hand are nominally(tied-over) quarter-notes, there is literally no timefor the grace notes in the rhythmic scheme; thusthey must be "taken away" from the preceding orfollowing note values. The standard way ofexecution suggested by the spatial placement ofthe grace notes in the score is to play them withinthe last 101 of MG2-that is, after the onset of thepenultimate melody tone in the soprano voice, butbefore the onset of the final long tone and theaccompanying two-tone chord. However, only 12 ofthe 24 pianists consistently played the passageaccording to this conventional interpretation ofthe notation. (The two repeats of bars 1-8 wereexamined separately in this analysis.) It seemsthat this brief passage offered pianists ampleopportunity to indulge in "deviations from thescore."

Most of the variants are illustratedschematically in Figure 10, which uses horizontaldisplacement of the critical notes in bar 2 toindicate the actual succession of tones. Thestandard version is shown in (a). As shown in (b),some pianists delayed the last note in the lefthand until after the octave chord in the righthand: ARR (slightly, bar 6 only), DAV (bar 2 only),MOl (bar 6, once in bar 2), NOV (only the firsttime in bar 2, but very dramatically), and ZAK(consistently). Others (c) played all notes of theleft hand before those of the right hand: ARG(except the second time in bar 6), HOI and H02(bar 2). In a variant of that pattern, the right­hand chord was arpeggiated (DEM, consistently)or mysteriously interleaved with the broken chordin the left hand (Horowitz, in bar 6 only, but in allthree performances).

I I• DavNa~

De_

·Ash •ru Ki~ Zak Cap

Esc •• ·She ArgKa~An

Cur 0Neye

Sre

SCh

Gia Ie Kli·

Un!.

•HaZ.Ha3

Hal II,

20

40

(\J

~ 100,,-(\J(f) 80

0:« 600)

120

140

Figure 9, Individual variations in tempo modulationduring MG2. The average curvature (Q) for threeinstances of the gesture is plotted against the curvaturefor the last instance (bars 21-22). Only cases followingthe parabolic timing shape are included.

oo 20 40 60 80 100 120 140 160

BARS 1/2, 5/6, 17/18

160

This "parabolic constraint" on the timing shapeof an expressive gesture is of great theoreticalinterest, as it supports Todd's (1985) suggestion ofthe parabola as a basic timing function. It alsoappears to be consistent with his more recentmodelling (Todd, 1992b), even though he proposedthat a linear function can account for most of thetiming variation; clearly, a quadratic function dideven better. The parabolic constraint may thus beunderstood as an allusion to physical motion (cf.Todd, 1992b), which most performers aim at, andwhich listeners with some musical education findaesthetically pleasing (Repp, 1992b).

Since MG1, the following inter-gesture interval,and MG2 may form a coherent half-phrase that isplanned and executed as a single expressive unit,their timing relationships acrosS the 28performances were examined (in bars 0-2 only).None of the three coefficients characterizing theparabolic shape of MG2 correlated significantlywith either of the two 101 ratios considered inconnection with MG1 (Figure 5). However, thetotal interval preceding the onset of MG2correlated moderately (0.59-0.72) with all threeMG2 coefficients. Thus, the slower the initial

rn all instances mentioned so far, the onset ofthe last melody tone of MG2 did follow the twograce notes, so the grace notes were within; thelast r01 of the gesture. This made it possible toask about the relative timing of the grace noteswithin the ror (see below). rn some further cases,however, one or both of the grace notes coincidedwith the final melody tone. Thus, SCH consis­tently played the right-hand octave tones simul­taneously with the first grace note (d). BUN (bars2 and 18) and KUB (bars 2 and 6, second repeatonly) played the right-hand tones simultaneouslywith the second grace note (e). Finally, Cortot (bar6, in all three performances; also bar 18 in COl)played all tones simultaneously as a single chordCD, as prescribed only in bar 22 of the score. (BUNalso did this once in bar 6.)12

To investigate the relative timing of the gracenotes within the last ror of MG2, provided theydid fall within that lOr (cases a-c in Figure 10),the r01 was divided into three parts (A, B, and C),defined by the respective tone onsets. Two intervalratios were calculated: A/(BC), which indicates therelative delay of the onset of the first grace note,and B/C, which represents the duration of the firstgrace-note ror relative to the second. If the firstmelody tone and the two grace notes were playedas a triplet, for example, the two ratios would be0.5 and 1, respectively; if the grace notes wereplayed as thirty-second notes following asixteenth-note rest, the ratios would both be 1.

Figure 11 plots A/(B+C) against B/C for all in­stances where these ratios could be determined.(Bars 2 and 6 are each represented by a single ra­tio, averaged across the two repeats.) A widerange of ratios was found, but most data pointsform a cluster in the lower left quadrant, suggest­ing that the majority of pianists agreed on a par­ticular timing pattern. The central tendency ofthat cluster is around A/(B+C) = 0.4 and B/C = 0.5.This means that the grace notes were not playedevenly: The first grace note started about 40% intothe ror, and the second grace note was abouttwice as long as the first, which means it startedabout 60% into the ror, on the average.Considering the compressed scale of the ordinatein the figure, however, it is clear that there waswide variation in the relative durations of the twograce notes, even within the cluster of relativeconformity. (Some of this variation undoubtedlyreflects measurement error magnified by ratio cal­culation.) What was almost always true, however,is that the first grace note started during the firsthalf of the lOr (A/(B+ C) <1), and the second gracenote ror was longer than the first (H/C < 1).

*

*

*

*

*

Jl

Jl

Jl

Jl

Jl

I

~

I.J

~

r1J

'3M.r1J

~

r1J,

•

•

(a)

(c)

(d)

(e)

(f)

Figure 10. Schematic illustration of six observed mannersof execution of the grace notes accompanying MG2. Thenotation for bars 2 and 18 is shown, but it stands for bar6 as well. Spatial displacements of the notes symbolizetemporal displacements.

Diversitv and Commonalihl In Music Performance: An Anal\fsls of Timing Microstructure 11l Schumann's "Triiumerei" 249

I0 Bar 2

-Bar 6 V\ Zak-f---

0t::. Bar 18 Zak

She.6

Co2L •Bun

~o let::. 0 She Co10t::. •oJ - 0 t::..~_O ~ t::. o Kli

~I a-o_t ~ CaoL...J

~u tR~ -t::.0 :e~

2.5

2

1.5

o--CO1

0.5

aa 0.2 0.4 0.6 0.8 1

A/(B+C)1.2 1.4 1.6

Figure 11. Relative timing of the grace notes within the last 101 of MG2, for cases (aHc) in Figure 10. The abscissashows the ratio of the time before the onset of the first grace note (A) and the remainder of the 101 (Be); the ordinateshows the ratio of the time between the two grace note onsets (B) and the time from the onset of the second grace noteto the end of the 101 (C). Ratios for bars 2 and 6 are averaged over the two repeats.

There were some notable exceptions to thispattern, though no individual pianist wasconsistently deviant. Assuming that theseinstances are not simply slips of motor control, itwould be interesting to know what led individualartists to vary their timing patterns in these ways.

4. MG3a-MG6a. We turn now to a closerexamination of the MG chains making up thesecond half of each phrase. The Type a chainoccurs in identical form in bars 2-4 and in bars 18­20 (see Figure 2). It also occurs in varied andabbreviated form in bars 22-24. We will not dealspecifically with MG3a and MG3b in bars 22-23,

which on the whole were played as in bars 2-3 and18-19. MG5a and MG6f in bars 23-24, whichcarried the final large ritardando, will beconsidered separately later on.

As noted earlier, the average timing profiles forbars 2-4 and 18-20 were extremely similar, thoughbars 18-20 were played at a somewhat slowertempo overall (see Figure 3). In the principalcomponents analysis conducted on bars 0-8, thefirst factor (see Figure 4, top panel) exhibited aregular sequence of peaks for MG3a-MG6a, whichreflect lengthening of accented tones and MG-finallOIs. Detailed examination of the individual

250 Repp

timing patterns, however, revealed enormousvariety, though not without constraints. No twopianists' interpretations of this MG chain werequite the same, and some artists also changedtheir patterns significantly between bars 2-4 andbars 18-20. (The two repeats of bars 2-4 will not beconsidered separately here, though some pianistsplayed even these differently.)

To explore this variability in a reasonablyeconomical way, principal components analysiswas used once again, but this time only on the 2 x18 36 IOIs comprising MG3a-MG6a in bars 2-4and 18-20, concatenated. From this analysis, six

significant factors emerged-more than from theearlier analysis of bars 0-8, where the largertiming deviations in MG1 and MG2 dominated thecorrelation structure. Thus, there were at least sixdistinct (i.e., uncorrelated) timing patternsunderlying the variation in the data; thesepatterns are shown in Figure 12. None of them isfully representative of any individualperformance, however, and together they accountfor only 81% of the variance. The factor scores forbars 2-4 and 18-20 were highly similar and aresuperimposed in the figure. The factor loadings ofthe individual performances are shown in Table 6.

1000 1000 -.- Bars 2-4

Q]MG6a

900 900 -e- Bars 18-20

800 800

700 700

600 600

500 500

400 400.... (") If) .... (") If) .... .... (") If) .... ~ (") If) ....

....--.. c\J M M M M ~ ~ ~ ~ c\J M M M M ~ ~ ~ ~E1000 1000

.........

OJD ~(f) 900 900Wa: 800 8000() 700 700(f)

a: 600 600

0 500 500I-()« 400 400u.. .... ~ (") If) .... ~ (") If) .... .... (") If) .... ~ (") If) ....

c\J M M M M ~ ~ ~ ~. c\J M M M M ~ ~ ~ ~

1000 1000

900 ~ 900 @J800 800

700 700

600 600

500 500

400 400.... ~ (") If) .... <'? If) .... .... C') If) .... <'? "( r-;-c\J M M M M ~ ~ ~ ~ c\J M M M M ~ ~ ~ ~

BAR/EIGHTH-NOTE

Figure 12. Rescaled factor scores (underlying timing patterns) for the MG chain of Type a in bars 2-4 and 18-20. Theabscissa labels refer to bars 2-4.

Diversitll and Commonalitll m ivfusic Performance: An Anaillsis o(Timing Microstructure m Schumann's "Triiumerei" 251

Table 6. Sorted rotated factor loadings from the principal components analysis ofMG3a-MG6a. (Loadings below 0.4are omitted.)

II III IV V VI

ARR 0.773NEY 0.758BRE 0.744ASH 0.708KAT 0.671 0.437ESC 0.606 0.416H03 0.908H02 0.893HOI 0.830ARG 0.629 0.402COl 0.873CO2 0.832cm 0.784ORT 0.446 -0.593KUB 0.823NOV 0.808SCH 0.786CUR 0.676 0.541MOl 0.864SHE 0.564 0.693DAV 0.456 0.574GIA 0.410 0.535 0.564KLI 0.576 0.602DEM 0.457 0.436 0.565ZAK 0.427 0.515 0.540BUN 0.431 0.410 0.491KRU 0.414 0.500 0.436 -0.435CAP 0.455

Factor I (VAF = 18%) is characterized by accel­eration during MG3a; a fast traversal of MG4a; adramatic ritardando within MG5a with maximallengthening of the final, unaccented note (position4-2), further augmented in bars 18-20; and apronounced lengthening of the last 101 in MG6a(position 4-8). The pianists whose interpretationscome closest to this pattern are ARR, NEY, BRE,and ASH.

Factor II (VAF = 17%) is the "Horowitz factor."It is characterized by significant lengthening ofthe accented tone in MG3a as well as of the pre­ceding unaccented tone; slight deceleration duringMG4a; final lengthening in MG5a; and a ritar­dando during MG6a, followed by a sudden short­ening of the last 101. Some of these tendencieswere exaggerated in bars 18-20 relative to bars 2­4. Horowitz's actual performances (especially H02and H03) resemble this pattern, except that theexaggeration in bars 18-20 is much more dra­matic. H03 differs in that bars 2-4 and 18-20 areexecuted very similarly, both showing dramaticlengthening in positions 4-2 and 4-3.

Factor III (VAF = 14%) is the "Cortot factor." Itis characterized by a fast start; progressive slow­ing down during MG4a and MG5a, with final

lengthening in MG5a; and some shortening of thefinal 101 in MG6a. Some additional peaks appearin bars 18-20. Cortot's real performances are simi­lar, especially COl and C02, except that the ten­dencies in bars 18-20 are much more exaggerated.In C03, there is much lengthening in position 20­2 instead of 20-3 (read 4-2 and 4-3 on the abscissain Figure 12). Note the sizeable negative loading ofORT.

Factor N (VAF = 14%) shows pronounced finallengthening in both MG3a and MG4a; a "flat"MG5a with slight final lengthening; and a smoothritardando through MG6a, with a very long last101. Bars 2-4 and 18-20 are executed very simi­larly. This beautifully regular pattern comes clos­est to the actual interpretations of KUB, NOV,and SCH.

Factor V (VAF = 11%) shows a rather flatpattern, with initial acceleration and pronouncedfinal deceleration, but with the final 101 shorterthat the penultimate one. This pattern is the leastdifferentiated, as if the whole MG chain werethought of as a single gesture. This pattern comesclosest to MOl's performance, though he doesshow some small local peaks in positions 3-5 and4-1.

Repp

Factor VI (VAF = 7%), finally, begins with apattern for MG3a and MG4a that is very nearlythe opposite of Factor II; MG5a shows strikingfinal lengthening; and MG6a shows only a smalllengthening of the final 101. There is an overalltrend to accelerate during the MG chain. Thispattern is not representative of any individualperformance and occurs only in mixed patterns,such as those ofKLI, DEM, and ZAK, all of whichexhibit pronounced lengthening in position 4-3.

In summary, the information contained inFigure 12 and Table 6 offers only a moderateamount of data reduction. The timing patterns ofindividual artists were remarkably diverse.Nevertheless, there are many possible patternsthat never occurred, such as lengthening inposition 3-4 (the second 101 of MG4a), whichclearly was treated as transitional by all pianists.Nor were the patterns random in any way; themajority of the artists produced very similartiming patterns in bars 2-4 and 18-20, and also inthe two repeats of bars 2-4 (which were averagedhere). The performances with the highestconsistency between bars 2-4 and 18-20 wereARR, CAP, HOI, and KUB, whereas the mostvariable ones were BUN, Cal, C02, H02, H03,and aRT. That Cortot's and Horowitz's within­performance variations were carefully planned issuggested by the fact that they were replicated indifferent performances (only HOI retained thepattern of bars 2-4 for bars 18-20). WhetherBUN's and aRT's inconsistencies were similarlyplanned is not known, since only a singleperformance by each pianist was available forexamination.

5. MG3b-MG6b. The Type b version of the MGchain occurs three times in the piece: first in bars6-8, and then in bars 10-12 and 14-16. Because ofthe substantial ritardandi that occur duringMG6b, it was decided to treat the last six lOIs ofeach chain separately. The remaining 3 x 12 = 36lOIs were subjected to principal components anal­ysis, which yielded six significant factors, just asfor the Type a passage. They accounted for 80% ofthe variance. Their timing profiles are shown inFigure 13, and the matrix of factor loadings isshown in Table 7. As for the Type a chain, thefactor patterns are only rarely representative ofindividual artists' patterns, which more often area combination of several factor patterns. It may berecalled that the grand average timing profileshowed a fairly regular lengthening of pre-ac­cented and accented lOIs in each MG, whichreally represents final lengthening of theaccompanying secondary MG, which ends with an

accented tone. The factor profiles essentiallyrepresent a varying focus on individual MGs inthe chain.

Factor I (VAF = 22%) shows substantiallengthening in MG3b, much less in MG4b, andsomewhat more in MG5b. The timing profiles forthe three instances of the MG chain are verysimilar. The pianist most closely associated withthis pattern is ARR, and a number of other artists,including Horowitz, show substantial loadings inthis factor. (In contrast to the Type a chain, therewas no "Horowitz factor" here.)

Factor II (VAF = 21%) is characterized by totalabsence of lengthening in MG3b and variabilityduring MG4b and MG5b, usually with more pre­accentuation lengthening than accentuationlengthening. Two performances showing this pat­tern in relatively pure form are Cal and KRU,and several other performances (including C02)show substantial loadings. Thus this was again akind of "Cortot factor," though it was not unique toCortot. However, C03 deviates from this patternand is responsible for a unique factor, after all(Factor VI).

Factor III (VAF = 14%) is characterized by a flatpattern (bars 6-7) or pronounced accelerando (bars10-11 and 14-15) during MG3b, variable lengthen­ing in MG4b, and pronounced lengthening inMG5b. NEY, KAT, and DAV are the best repre­sentatives.

The remaining three factors account for muchless variance (8% in the case of Factors IV and V,6% in that of Factor VI), and each of them islargely associated with one particular artist.These three artists, which earlier analyses havealready revealed to be somewhat eccentric, areaRT, BUN, and C03. Thus, aRT put increasingemphasis on MG4b in successive renditions, BUNshowed a very atypical peak in position 7-7 andtremendous lengthening in MG4b in bar 7 only,and C03 showed no lengthening in MG3b andMG4b, and a very variable execution of MG5b.

On the whole, within-performance variabilityacross the three instances of the Type b chainseemed larger than for the Type a chain, whereasbetween-performance variability seemed some­what lower. The former observation is notsurprising, for the two MG chains of Type a areidentical, whereas the three chains of Type bdiffer harmonically and melodically. No pianistshowed nearly identical patterns for all threeinstances of Type b. Some pianists were markedlymore variable than others, however, not to sayerratic. BUN leads the list, which also includesARG, C03, DAV, and MOL The relatively most

Diversitll and Commonality in Music Performance: An Anaillsis of Timing Microstructure in Schumanrz's "Triiumerei" 253

consistent pianists were ASH, DEM, H02, KLI,NEY, and ZAK. Some pianists (BRE, KRD, SCB)were notable for their relatively flat anduninflected rendering of the whole Type b chain. 13

6. The ritardandi. As is evident from the grandaverage timing pattern plotted in Figure 3, majorritardandi occurred at the ends of periods andphrases. The largest ritardando is in bars 23-24,

at the end of the piece. A substantial slowdownoccurs also in bar 16, at the end of period B. Lesspronounced ritardandi occur in bar 8, at the endof period A, and in bar 12, at the end of phrase b2.Thus there are four locations where majorritardandi are observed. It may be asked whetherthe temporal shapes of these ritardandi followedany common pattern.

-+- Bars 6·8750

650[I]

550 550

450 450

350 3506·7 7-3 7-5 7-7 8-1 6-7 7-1 7-3 7·5 7-7 8-1-(J) 750 750

EDE]----(J)

650 650Wa:0 550<-) 550(J)

a: 450 4500f-

~ 350 350I

LL 6·7 7-1 7-3 7-5 7·7 8-1 6-7 7·1 7-3 7-5 7·7 . 8-1

750 750

650 650 C2TI

550 550

450 450

350 3506-7 7-1 7-3 7-5 7·7 8-1 6-7 7-1 7-3 7-5 7-7 8-1

BAR/EIGHTH-NOTE

Figure 13. Rescaled factor scores <underlying timing patterns) for MG chain of Type b in bars 6-8, and 10-12, and 14-16.The abscissa labels refer to bars 6-8.

254 Repp

Table 7. Sorted rotated factor loadings from the principal components analysis ofMG3b-MG5b. (Loadings below 0.4are omitted.)

II III IV V VI

ARR 0.923HOI 0.788ARG 0.745H03 0.740KLI 0.715MOl 0.701ASH 0.648 0.521DEM 0.589 0.582

ESC 0.553 0.490

GIA 0.531 0.455CAP 0.515 0.438 0.450

COl 0.860KRU 0.829BRE 0.733CO2 0.699NOV 0.418 0.691SHE 0.675 0.407CUR 0.664 0.488SCH 0.629 0.404KUB 0.530 0.416NEY 0.814KAT 0.746DAV 0.721ZAK 0.407 0.642 0.416ORT 0.893BUN 0.756C03 0.863H02 0.413 0.452

This question was addressed previously bySundberg and Verrillo (1980) and Kronman andSundberg (1987) with regard to the ritardandiobserved at the ends of performances of "motormusic," mostly by Bach. Sundberg and Verrilloplotted an average "retard curve" in terms ofinverse lOIs and observed that it could bedescribed in terms of two linear functions: agradual slowdown followed by a more rapid one.The latter phase often included only three datapoints, however, and tended to coincide with thelast melodic gesture in the music. Kronman andSundberg abandoned the bilinear model andinterpreted the average retard curve as a singlecontinuous function. That function was describedwell by expressing local tempo (lIIOl) as thesquare root of the normalized distance from a

. hypothetical zero point located one beat beyondthe onset of the final tone. Kronberg andSundberg speculated that this function may be anallusion to physical deceleration in naturalactivities such as walking.

That the average ritardando followed aquadratic function is intriguing given the present

results concerning the temporal shape of MG2.The decelerando in that ascending melodic gesturemay be considered an instance of a' localritardando, and the principles involved may bequite the same. A square-root (convex) functionfitted to reciprocal lOIs implies a quadratic(concave) function fitted to the original lOIs.However, it is difficult to reach strong conclusionsfrom Sundberg's examination of timing patternsaveraged across performances of different music.The melodic grouping structure must be takeninto account, which was done only to a verylimited extent by Sundberg and his colleagues.

This is much easier, of course, when dealingwith multiple performances of a single piece ofmusic, as in the present case. Inspection of thegrand average timing profile in Figure 3 suggeststhat ritardandi progress smoothly within melodicgestures but are interrupted between gestures.Thus, the final ritardandi in bars 12 and 16 (readbar 8 on the abscissa) can be seen to begin withthe second 101, but a "reset" occurs after thefourth 101, which corresponds to a gesturalboundary. The final lengthening of MG5b causes

Diversitll and Commonalitllll1 lvfusic Performance: All Allah/sis of Timzlll: Microstructure III Scizwnann 's "Traumae:" :!:1:'

Figure 14. Ritardando functions: Quadratic functionsfitted to the grand average (geometric mean) lOIs inMG6b (bars 12 and 16) and in MG5a (bars 23-24).

BAR/EIGHTH-NOTE

IV

/

//

~

IV

//'

/

~/-.-

12-8

16-8

24-2

12-7

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12-6

16-6

23-8

12-5

16-5

23-7

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1/~

I I

600

900

600

800

1000

1200

1400

---.(f) 800E--0 700

600

500

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the initial ritardando to "overshoot" its trajectory,which is resumed with the onset of MG6b. Theseritardandi thus are divided into two sequences of3 and 4 IOIs, respectively. Similarly, in the largefinal ritardando in bars 23-24, which begins asearly as position 23-3 or 23-4, two resets occur,corresponding to the ends of MG4a and MG5a,which in this instance are unmistakably indicatedin the score by "commas." This grand ritardandothus is divided into three groups of IOIs: 3(4), 4,and 2. Because meaningful fitting of quadraticcurves requires at least four data points, thefollowing analyses examine one sequence of fourcoherent lOIs from each of the three ritardandi,forming either part of MG6b or of MG5a. Theanalyses were analogous to those conducted onMG2.

In each case, it was found that a quadratic curvefit the average 101 pattern quite well, perfectly soin the case of MG5a. These curves are shown inFigure 14. Quadratic functions were subsequentlyfitted to all 28 individual performance timingpatterns of each ritardando section. Most fitsranged from excellent to acceptable (see Footnote9). In the case of MG6b (bar 12), there were fiveclearly deviant cases (ARG, H03, KUB, and ORT,all of whom shortened the last IOI, and KLI, whoshowed a rather flat pattern); for MG6b (bar 16)there was none; for MG6a there was one (ARG,who shortened the last 10l).

The coefficients of the quadratic equationsdescribing the acceptable (and better) fits werefound to be highly correlated in all instances, justas for MG2. Thus it was again possible toconstruct families of parabolas that capture amajor portion of the individual variation in theshapes of the ritardandi. These curves are shownin Figure 15. Taking into account the differenttime scales on the abscissa, it seems that eachritardando has its distinctive range of variation,but all share the property of being largelyparabolic in shape. These results for strictlyintragestural ritardandi support the observationson more heterogeneous materials by Sundbergand Verrillo (1980) and Kronman and Sundberg(1987).

A final analysis was conducted on the last twolOIs of the piece. While it did not make sense to fitany function to them, their correlation could beexamined, which was 0.87. Thus, the longer thepenultimate 101, the longer the final 10I.l4 Notonly was the relationship quite linear, but theregression line passed almost through the origin,suggesting that the two lOIs were in a constantproportion (1:1.8).

256 Repp

any other artist's, even if just one physicaldimension (timing) is considered. Even the sameartist's performances on different occasions, whiledemonstrably similar, are sufficiently different tobe considered distinct and individual events. Yet,there are also significant commonalities thatapparently reflect constraints on performance, atleast on those performances that have beendeemed suitable for commercial distribution. Theindividual variations among performances largelytake place within these constraints, althoughthere are always exceptions, representingconscious or unconscious transgressions of theboundaries established by musical convention.

How should those boundaries be characterized?And are they purely conventional (i.e., arbitrary),or do they represent more general laws of motorbehavior and perception that music performancemust conform to in order to be naturallyexpressive?

It is probably futile to attempt to characterizethe boundaries of acceptable performance practice.They are manifold and are likely to contract andexpand as a function of many factors.- It istheoretically more parsimonious to conceive of aperformance ideal (norm, prototype) that lies atthe center of the hypothetical space enclosed bythe boundaries. This ideal may be thought of as arelatively abstract specification that contains freeparameters, so that a multiplicity (if not aninfinity) of concrete performances can begenerated that all more or less satisfy the norm.Because of the enormous complexity of seriousmusic, any concrete performance ideal is verydifficult to attain; and, if one were attained, theartist would probably change it the next time,because art thrives on variety. However, theunderlying abstract ideal may remain constant, asit embodies generally accepted rules ofperformance practice. Concrete realizations areconceived, perceived, and judged with reference tothe underlying norm, but deviation from the norm(within certain limits of acceptability) can-tosome extent, must-be an artistic goal. That is,diversity is as necessary as is commonality: Bothuniformity and lack of an aesthetic standard aredetrimental to art.

One obvious free parameter is tempo. Eventhough there may be an "ideal tempo" for a pieceof music, this is again an abstraction. There is infact a range of acceptable tempi, and individualsmay differ considerably in what they consider"the" ideal tempo. This is illustrated by the per­formance sample examined here, which representsa very wide range of tempi, nearly all of which

Figure 15. Families of parabolic timing functions for theritardandi in MG6b (bars 12 and 16) and in MG5a (bars23-24).

v. GENERAL DISCUSSIONThe present comprehensive analysis of

expressive timing patterns in 28 performances ofSchumann's "Traumerei" provides an objectiveview of the commonalities and differences amonggreat artists' interpretations of one of themasterpieces of the piano literature. At firstglance, the differences are perhaps more strikingthan the commonalities. There is ample materialhere to support the view that every artist'sperformance is, in some sense, unique and unlike

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600 T 0=120

23·7 23-8 24-1 24-2

BAR/EIGHTH-NOTE

Diversity and Commonalitv in Music Performance: An Analysis of Timing Microstructure in Schumann's "Triiumerei" 257

seem acceptable to a musical listener (the authormust rely on subjective judgment here), eventhough some sound clearly slow, while otherssound fast, To the author, only ESC really sounds"too slow" (though perhaps even that judgmentmight change if his performance were heard in thecontext of the whole "Kinderszenen" suite), whilethat of DAV sounds "too fast," though apparently(Clara) Schumann wanted it that way (see Repp,1992d). An unusually fast "Traumerei" indeedseems to come across better than an unusuallyslow one.

The present investigation substantiates two ab­stract timing constraints embodied in the hypo­thetical performance ideal. One of them concernsthe temporal marking of the melodic/rhythmicstructure, the other one has to do with the tempo­ral shaping of individual melodic gestures, partic­ularly of the ritardandi within them. The firstconstraint is by now well known and is imple­mented in Todd's (1985) model of expressive tim­ing at the phrase leveL The principle is thatboundaries in the hierarchical grouping structureare generally marked by ritardandi whose extentis roughly proportional to the "depth" of theboundary. Thus the most extensive ritardando oc­curs at the end of the piece, where boundaries atall levels coincide; substantial ritardandi occur atthe ends of major sections, such as 8-bar periods;smaller ritardandi occur at the ends of individualphrases and gestures. A performance system suchas that devised by Todd may come close to theprinciples embodied in the performance ideal,though it is not known at present whether a per­formance strictly following such rules would infact be perceived as "ideal" by listeners. There issome perceptual evidence, however, that listen­ers-even those without much musical educa­tion-expect phrase-final and gesture-finallengthening to occur (Repp, 1992a).

What is clear from the performances examinedhere is that variability increases at lower levels ofthe structural hierarchy. Virtually every per­former observes the major ritardandi at the endsof major sections, though in different degrees.When the complete performance timing profileswere subjected to principal components analysis,only a single factor emerged, which reflected thequalitative conformity in that regard. When theanalysis was restricted to the first 8 bars only,four factors emerged. When the analysis focusedon half a phrase, omitting major ritardandi, sixfactors emerged. It seems paradoxical that thenumber of independent factors increases as thenumber of data points decreases. What this

reflects is the increasing pattern variability atlower levels of the structural hierarchy.

This variability is not likely to reflect a decreaseof control over precision in timing, though some ofit may. Pianists at the level of accomplishmentstudied here can control their timing patternsdown to a very fine grain. What the variabilitypresumably reflects is a relaxation of the perfor­mance constraints imposed by the groupingstructure at lower levels of the hierarchy. Not onlyare the boundaries between individual melodicgestures perhaps less definite than those betweenlarger units, but they are weaker determinants ofthe timing pattern and compete with other localfactors including harmonic progression, melodicpitch contour, and texture. It also seemS that themajor source of timing variation is not artists'choice of unexpected locations for expressivelengthening (though some of this occurred, too)but varying degrees of emphasis on expectedlocations. Thus, pianists often omittedlengthening where the detailed grouping structuremight have predicted it, whereas theyoveremphasized other grouping boundaries, as ifto compensate. Sometimes this resulted in thecreation of hypergestures, which combined two orthree elementary melodic motifs. In other words,the lowest level of the grouping hierarchy isstructurally flexible; it permits the marking ofgroup boundaries but does not prescribe it. Artistschoose from among the possibilities in a mannercomparable to varying focus in a spoken sentence.

The other constraint observed in the presentdata is more novel and more tentative. It is that,within melodic gestures requiring a ritardando forwhatever reason, this local tempo change is bestexecuted such that successive lOIs follow aparabolic function. This constraint was not onlyobserved in the majority of performances at fourdifferent locations in the music (nine, if thedifferent occurrences of MG2 are countedseparately), but it also is in agreement with theobservations of Kronman and Sundberg (1987) onthe shape of final ritardandi, as originallydescribed by Sundberg and Verrillo (1980).Sundberg and Verrillo also provided perceptualdata suggesting that listeners prefer ritardandicorresponding to the original (quadratic) timingcurves over other possible timing profiles, and arecent study by this author (Repp, 1992b) on theperceptual evaluation of different timing patternsfor MG2 led to similar conclusions.

It is not clear at present whether the parabolictiming constraint can also be applied to melodicgestures that do not include any pronounced

258 Repp

ritardando; certainly other factors, such as accentlocation, would have to be taken into account. Thelarger the number of successive tones in a melodicgesture, the stronger the constraint is likely to be;a minimum of five tones (4 lOIs) is required. Italso remains to be seen whether a similar timingconstraint operates at higher levels of thegrouping hierarchy. Todd (1985) postulated aparabolic timing function as the prototype for .within-group timing at the phrase level, thoughapparently more for the sake of convenience thanfor any stringent theoretical or empirical reason.Kronman and Sundberg (1987) hint at agrounding of the parabolic timing constraint inelementary principles of (loco)motion, but withoutelaborating on this hypothesis. Todd (1992b) hasclaimed that a piecewise-linear function issufficient to describe local timing changes duringa performance, but his own (rather limited) datasuggest that a quadratic function provides a betterfit. At this point, the general hypothesis may bestated that a parabolic timing profile is in somesense "natural" for both performer and listener,and probably not for purely conventional reasons.A better understanding of the origins of thisconstraint may elucidate the meaning of themotion metaphor that is often applied to musicand its performance (cf. Todd, 1992a, 1992b;Truslit, 1938).

The present performance analyses revealed noclustering of individual artists according to sex,age, or national origin. Their individualityapparently transcended these other factors, whoserelevance to music performance may bequestioned in any case. There is no doubt,however, that in this sample of highlyaccomplished artists, perhaps precisely because oftheir level of artistry, some performances seemedunusual and even deviant. Subjective impressionswill have to suffice for the time being: Long beforeobjective measurements confirmed the actualdeviations in their timing patterns, theperformances by Argerich, Bunin, and Cortot, andto a lesser degree those by Horowitz and Ortiz,struck the author as eccentric and distorted.Interestingly, some of these pianists (especiallyBunin) also turned out to be inconsistent withintheir own performance, as if they had no fixedconcept and were exploring possibilities. Otherartists, however, were remarkably consistent, forexample Arrau. The author's subjectiveimpressions were not only highly reliable onrelistening, but they also correspond to whatprofessional critics have to say about some ofthese pianists' performances. Certainly, Horowitz

and Cortot are two of the most unusual pianists ofthis century, and their greatness may lie preciselyin their individuality, which challenges thelistener. However, in the context oflistening to 28different performances in sequence, which height­ens one's sensitivity to differences and perhaps in­creases one's reliance on an internalized perfor­mance ideal, the unusual performances do not fareso well. The author's favorites are the perfor­mances by Curzon, Brendel, and Ashkenazy,which usually were in the middle field in the vari­ous analyses reported above, and hence werementioned only rarely. The author's aestheticideal seemed to correspond to the central tendencyof the sample examined here. Whether this idealrepresents a stable representation of traditionalperformance norms or whether it is a form of psy­chophysical adaptation to the range of the perfor­mance sample is an intriguing question that war­rants further investigation. Also, the absence ofthe context of the preceding and following piecesof "Kinderszenen" must be acknowledged; inter­pretations that sound unusual in isolation maysound more convincing in context. Finally, it mustbe noted that the author's impressions derived notonly from the timing patterns of these perfor­mances, but also from their intensity patterns(both "horizontal" and "vertical"), their articula­tion and pedaling, and their overall sound quality.There are many parameters that contribute to thesubjective impression of a performance, but thetiming pattern is probably the most importantone. Nevertheless, future performance analyseswill have to consider these other parameters(which are much more difficult to measure inrecorded performances) as well as the subjectiveimpressions of more than one experienced listener.Ultimately, we would like to know why individualperformers play the way they do, and what mes­sage they are sending to listeners (cf. Kendall &Carterette, 1990).

The present research illustrates one of twocomplementary approaches to the objectiveinvestigation of musical performance. The otherapproach is represented by the work of Sundbergand his colleagues on a system of performancerules (e.g., Friberg, 1991; Sundberg, 1988;Sundberg, Friberg, & Fryden, 1989) and by Todd's(1985, 1992a, 1992b) modelling of musical motion.These models are ingenious and important, butthey need to be tested on large performance databases, which do not exist at present. In this study,a very modest beginning was made towardsaccumulating and organizing data of sufficientdepth and variety (though still restricted to a

Diversity and CommonaLitv in Music Performance: An AnaLvsis of Timing Microstructure in Schumann's "Triiumerei" 259

single composition and a single parameter,timing) to present a challenge and testing groundfor emerging models of music performance. 15

Soon, of course, much more extensive data basesshould become available with the help oftechnological marvels such as the MIDI-controlledgrand piano. There are exciting times ahead forresearch on music performance, one of the mostadvanced and culturally significant human skills.

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Clarke, E. F. (1983). Structure and expression in rhythmicperformance. In P. Howell, I. Cross, & R. West (Eds.), Musicalstructure and cognition (pp. 209-236). London: Academic Press.

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Cline, E. (1985). Piano competitions: An analysis of their structure,value, and educational implications. Doctoral dissertation, IndianaUniversity.

Clynes, M. (1983). Expressive microstructure in music, linked toliVing qualities. In J. Sundberg (Ed.), Studies ofmusic performance(pp. 76-181). Stockholm, Sweden: Publication issued by theRoyal Swedish Academy of Music No. 39.

Clynes, M. (1987). What can a musician learn about musicperformance from newly discovered microstructure principles(PM and PAS)? In A. Gabrielsson (Ed.), Action and perception inrhythm and music (pp. 201-233). Stockholm, Sweden: Publicationissued by the Royal Swedish Academy of Music No. 55.

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Gabrielsson, A. (1987). Once again: the theme from Mozart'sPiano Sonata in A major (K. 331). In A. Gabrielsson (Ed.), Actionand perception in rhythm and music (pp. 81-103). Stockholm,Sweden: Publication issued by the Royal Swedish Academy ofMusic No. 55.

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Kendall, R. A, & Carterette, E. C. (1990). The communication ofmusical expression. Music Perception, 8, 129-163.

Kronrnan, U., & Sundberg, J. (1987). Is the musical ritard anallusion to physical motion? In A. Gabrielsson (Ed.), Action andPerception in Rhythm and Music (pp. 57-68). Stockholm:Publications issued by the Royal Swedish Academy of MusicNo. 55.

Lerdahl, F., & Jackendoff, R. (1983). A generative theory of tonalmusic. Cambridge, MA: MIT Press.

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FOOTNOTES'Journal of the Acoustical Society of America, 92, 2546-2568 (1993).I An effort is made in this manuscript to distinguish between notes,which are graphic symbols, and tones, which are sound events.However, when it comes to grace notes, the distinction cannoteasily be made, since "grace tone" is not an acceptable term and"grace-note tone" is awkward. It should be understood, then,that "grace note" refers either to the notated symbol or theresulting sound, depending on the context. The same ambiguityholds for "chord," although "tone cluster" is an acceptable termfor the sound event that may be used on occasion.

21n a third case (NEY), it was discovered that the two repeats werevirtually identical from bar 5 on, suggesting duplication by therecording engineers (see Repp, 1993a).3In the designation for a "position," such as 5-8, the two

numbers stand for the bar and the eighth-note 101 within it,respectively. In an expression such as "bars 5-8," however, thetwo numbers stand for the first and last bars in the rangereferred to.

4Most of the performances in the latter group strike the author asmannered. It seems that these pianists deliberately tried to playdifferently from the norm, but were not willing or able to do soconsistently. Perhaps they intended to convey animprovisatory quality.

5Todd (1992a, 1992b) has recently revised and extended hismodel of expressive timing. An application of this model to thepresent data would be most interesting but exceeds the scopeof this paper.

6That is to say, there may be an implicit, expressively modulatedeighth-note pulse going through the longer lOIs. As suggestedby the extent of the MG boxes in Figure 2, the first of thesepulses "belongs to" ·the preceding MG, but its actual durationcannot be determined unless it is marked by some tonal eventin another voice. MG2i, which breaks up the inter-gestureinterval in the soprano voice, may track the implicit pulseinduced by the primary MG2 and thus may reveal its finallengthening.

7Although these performances were by no means identical, theywere more similar than almost any pair of performances by

different artists. Cortot's three performances intercorrelatedbetween 0.80 and 0.82, and Horowitz's between 0.81 and 0.92.Only three other values in the 28 x 28 intercorrelation matrixexceeded 0.80: CAP /ZAK (0.86), KAT/SHE (0.82), andCUR/KRU (0.81). On the other hand, some of the lowestcorrelations were observed between Cortot's and Horowitz'sperformances (0.15 to 0.44), and also between Cortot andseveral other artists (ARG, BUN, ESC, KLI, NEY, NOV). Onlytwo other correlations fell below 0.30: BUN/MOl (0.26) andH01/SCH (0.26). Most correlations were between 0.4 and 0.7.Thus Cortot and Horowitz were rather extreme cases in thepresent sample, which agrees with their general reputation ashighly individual artists.

8The possible covariation of adjacent as well as distant lOIs wasinvestigated in a principal components analysis on the large(190 x 190) matrix of intercorrelations among all lOIs,computed across the 28 performances. Its purpose was to"retrieve" the melodic grouping structure through the factorsextracted, on the expectation that lOIs would covary morestrongly within MGs than between. This expectation was notfulfilled; the analysis yielded a very large number of significantfactors, none of which captured much of the variance. They didreflect correlations among some adjacent lOIs and especiallybetween lOIs in structurally analogous positions across phrasesof the same type. However, for example, there was nosignificant relationship between the timing of MG1 and MG2,nor even between the beginning and the end of MG2.

9Unfortunately, the software used for curve fitting (DeltaGraph)did not prOVide a measure of goodness of fit. Therefore,statements about this aspect of the data will remainimpressionistic in this paper.

lothe L coefficient can be understood as follows: If the constant C(without loss of generality) is assumed to be 0, the parabolamust pass through the origin, the (0,0) point. L represents theslope of a tangent through the origin. It is zero when theminimum of the parabola is located at the origin. It becomesnegative when the minimum of the parabola moves to apositive value along the abscissa, but since the left branch ofthe parabola must still pass through the origin, this implies anegative value along the ordinate for the minimum.

lIthe correlations varied systematically across the six instances ofMG2: The highest correlations (0.91-0.93) were obtained forbars 1-2 and 17-18; lower correlations (0.85-0.86) hold for bars9-10 and 13-14; and the lowest correlations (both 0.79) werefound for bars 5-6 and 21-22. The less tight relationship heremust be due to variations in global tempo among the pianists,which may be only weakly related to the degree of tempomodulation in MG2.

12For that matter, the fermata chord in bar 22, which is clearlynotated as reqUiring simultaneity of all tones, was not playedby all pianists in that way: DAV and NOV played a grandarpeggio, NEY played a partial arpeggiO (two grace notes in theleft hand, as in bars 2, 6, and 18), and ASH played only thelowest tone in advance. These variants may have beenoccasioned by small hands (note that three of the pianists arefemale), for the chord requires a large span.

l30ne detail skipped in this analysis is the timing of the melodygrace note during the last 101 of MG5b (position 8-4). Its onsetgenerally occurred near the middle (40-60%) of the 101. A fewpianists played it a little earlier (CAP, C02) or later (ARG, ESC,SCH, H02, ARR). Two artists (C03, DAV) omitted italtogether.

14In a final display of eccentricity, BUN produced a terminal 101of nearly 5 seconds duration, with ORT not far behind.

15The data matrix is available from the author.