jon b. prince & mark a. schmuckler - murdoch...

THE TO NA L-METR IC HIERAR CHY: A CORPUS ANALYSIS

JO N B. PRINCEMurdoch University, Perth, Australia

MARK A. SCHM UCKL ERUniversity of Toronto Scarborough, Toronto, Canada

DESPITE THE PLETHORA OF RESEARCH ON THE ROLEof tonality and meter in music perception, there is littlework on how these fundamental properties functiontogether. The most basic question is whether the twohierarchical structures are correlated – that is, do met-rically stable positions in the measure preferentially fea-ture tonally stable pitches, and do tonally stable pitchesoccur more often than not at metrically stable locations?To answer this question, we analyzed a corpus of com-positions by Bach, Mozart, Beethoven, and Chopin, tab-ulating the frequency of occurrence of each of the 12pitch classes at all possible temporal positions in the bar.There was a reliable relation between the tonal andmetric hierarchies, such that tonally stable pitch classesand metrically stable temporal positions co-occurredbeyond their simple joint probability. Further, the pitchclass distribution at stable metric temporal positionsagreed more with the tonal hierarchy than at less met-rically stable locations. This tonal-metric hierarchy waslargely consistent across composers, time signatures,and modes. The existence, profile, and constancy of thetonal-metric hierarchy is relevant to several areas ofmusic cognition research, including pitch-time integra-tion, statistical learning, and global effects of tonality.

Received: October 18, 2012, accepted March 17, 2013.

Key words: pitch, time, tonality, meter, hierarchy

A NEAR-UNIVERSAL CHARACTERISTIC OFWestern music is the use of tonality (Krum-hansl, 1990) and meter (Lerdahl & Jackendoff,1983). Accordingly, there is abundant research in musiccognition on these topics (cf. Krumhansl & Cuddy, 2010;London, 2004). For instance, tonality influences memoryfor musical materials (Dowling, 1978, 1991; Krumhansl,1991), discrimination of musical chords (Beal, 1985),change detection (Cohen, Trehub, & Thorpe, 1989;Lebrun-Guillaud & Tillmann, 2007; Trainor & Trehub,

1994; Trehub, Cohen, Thorpe, & Morrongiello, 1986),ratings of completion and goodness (Boltz, 1989a,1989b; Cuddy, 1991; Prince, 2011), melodic recognition(Bigand & Pineau, 1996; Schmuckler, 1997), recall (Boltz,1991), and expectation (Schmuckler, 1989, 1990;Tillmann, Janata, Birk, & Bharucha, 2008). Similarly,meter plays a pivotal role in the perception of musicalsequences, guiding attention and providing a frameworkfor organizing the events in a melody (Jones & Boltz,1989; Large & Jones, 1999; Palmer & Krumhansl,1990). Its importance as a central organizational factoris evident in its influence on recognition memory (Jones& Ralston, 1991; Keller & Burnham, 2005), complexity(Povel & Essens, 1985), categorization (Bigand, 1990;Desain & Honing, 2003; Hannon & Johnson, 2005), sim-ilarity (Creel, 2012; Monahan & Carterette, 1985),change detection (Bergeson & Trehub, 2006; Hannon& Trehub, 2005; Jones, Johnston, & Puente, 2006; Repp,2010; Smith & Cuddy, 1989), and expectancy (Huron,2006; Large & Palmer, 2002; Repp, 1992). The role ofmeter in performance deserves its own volume (includ-ing Drake & Botte, 1993; Essens & Povel, 1985; Palmer,1997; Povel & Essens, 1985; Repp, Iversen, & Patel, 2008;Shaffer, Clarke, & Todd, 1985; Sloboda, 1983), but isbeyond the purview of the current study.

Literature on how tonality and meter work together,however, remains surprisingly small. Experimentalwork manipulating the alignment of the tonal and met-ric hierarchies supports the notion that listeners aresensitive to both hierarchies when listening to melodies(Bigand, 1997; Palmer & Krumhansl, 1987a, 1987b; Ser-afine, Glassman, & Overbeeke, 1989). Interestingly,these authors reported that the tonal and metric hierar-chies functioned independently, implying that thesehierarchies need not necessarily co-occur. Otherresearchers report interactive relations between tonalityand meter (Boltz, 1989b, 1991; Dawe, Platt, & Racine,1993, 1994, 1995; Schmuckler, 1990; Schmuckler &Boltz, 1994). A substantial related literature has exam-ined the independence versus interaction of pitch andtime (see for reviews, Ellis & Jones, 2009; Krumhansl,2000; Prince, Thompson, & Schmuckler, 2009),although these investigations have been confined toaspects of pitch and temporal information other thantonality and meter per se (but see Prince, Thompson,et al., 2009, for an exception). Deutsch (1980) found

Music Perception, VOLUME 31, ISSUE 3, PP. 254–270, ISSN 0730-7829, ELEC TRONIC ISSN 1533-8312. © 2014 BY THE REGENTS OF THE UNIVERSIT Y OF CALIFORNIA ALLRIGHTS RESERVED. PLEASE DIRECT ALL REQUESTS FOR PERMISSION TO PHOTOCOPY OR REPRODUC E ARTICLE CONTENT THROUGH THE UNIVERSIT Y OF CALIFORNIA PRESS’S

RIGHTS AND PERMISSIONS WEBSITE, HT TP://WWW.UCPRESSJOURNALS.COM/REPRINTINFO.ASP. DOI: 10.1525/MP.2014.31.3.254

254 Jon B. Prince & Mark A. Schmuckler

that using temporal grouping in musical sequencesinfluences the encoding of tonal information (see alsoLaden, 1994). She found that recall accuracy (melodicdictation) declined when temporal segmentation cuesconflicted with pitch sequencing cues, which decreasedfurther when sequences did not conform to a simplehierarchical structure. However, more recent work inthe perception of probe events (Prince, Thompson,et al., 2009) found mixed evidence, specifically an asym-metric interaction between pitch and time. Whenreporting if probe events occurred on or off the beat,tonally stable pitches were more likely to be reported asbeing on the beat, showing an interference of pitchstructure (tonality) on temporal perception. Howeverthere was no complementary effect of temporal positionwhen reporting if the probe was in or out of the key.

The music theory literature also explores the relationbetween tonality and meter. Combining concepts fromcognitive psychology and Schenkerian analysis, Lerdahland Jackendoff (1983) created a model that uses well-formedness and preference rules to describe how listen-ers apprehend the structure in a musical sequence. Well-formedness rules provide a number of structuraldescriptions of the musical piece in question, whereasthe preference rules function to determine whichdescription most accurately corresponds to the listener’sexperience. Their model includes rules for deriving met-rical structure from the musical surface; this informa-tion then feeds into a time-span reduction that focuseson the pitch structure. Interestingly, in this model pitchand temporal structures are not treated entirely inde-pendently – for example, metric preference rule 7 spe-cifically incorporates tonal function into deriving themetrical structure (‘‘Strongly prefer a metrical structurein which cadences are metrically stable,’’ p. 88). Simi-larly, time-span reduction preference rules incorporaterhythmic structure (rule 1: ‘‘prefer a choice that is ina relatively strong metrical position,’’ p. 160) and meter(rule 5: ‘‘prefer a choice that results in more stablechoice of metrical structure,’’ p. 165). Benjamin (1984)also acknowledges the importance of tonal structure inhis theory of musical meter, arguing that tonal structureinfluences the derivation of the meter at several levels ofmetrical analysis. Temperley (2001) discusses a numberof possible factors in key identification, but does notconsider metrical stability; the success of his model not-withstanding casts doubt on the idea that metricalstrength and tonal stability are aligned in any consistentway.

There are several algorithms that derive musicalmeter based on coding melodic (pitch) information,although these models rely on pitch repetition rather

than tonal stability. For instance, Collard, Vos, andLeeuwenberg (1981) determine the meter by compilingdistance between pitch pattern repetitions. Theapproach of Steedman (1977) is similar, in that he takesrepetitions in melodies to indicate points of metricaccent (expanding on Longuet-Higgins, 1976). Theperiod of these accents then forms the metric groupingand thus indicates the metrical framework. It is note-worthy that Steedman asserts that key identification isindependent from knowledge of meter, and that scalehas no influence on metric structure. The success ofthese techniques suggests that composers (at least Bach)use melodic repetition – but not necessarily tonality –and metric hierarchies in a congruent manner. Otheralgorithms extract metrical structure with purely tem-poral factors (particularly onsets and grouping), with-out including any parameters related to pitch (Desain,1992; Desain & Honing, 2003; Povel & Essens, 1985).Similarly, the most common keyfinding algorithm doesremarkably well despite ignoring all temporal informa-tion (Krumhansl, 1990; Krumhansl & Schmuckler,1986; Schmuckler & Tomovski, 2005; but see Schmuck-ler, 2009); however, others contend that note order,recency, and possibly other temporal factors are criti-cally important (Brown, 1988; Brown, Butler, & Jones,1994; Huron & Parncutt, 1993; Temperley & Marvin,2008). For example, Aarden (2003), using reaction timesto the seven diatonic tones of the major scale, found thatstability of these tones varied as a function of placementin a musical passage. Specifically, he observed that thediatonic major tonal hierarchy values as defined byKrumhansl and Kessler (1982) best characterized listen-ers’ responses at points of closure (i.e., cadences), butwere less descriptive of tonal percepts at other points ina passage.

Overall, the body of theoretical literature that specif-ically examines the relation between tonality and meteris miniscule. Similarly, the extant empirical literature onhow tonality and meter combine focuses principally onperceptual processes, and not on determining whethertonal and metric structures actually co-occur in music.Thus, there appears to be virtually no information as towhether or not tonality and meter are systematicallyaligned. The only related research on this topic comesfrom Järvinen’s (1995) work examining 18 examples ofbebop style jazz improvisation. He calculated a weightedaverage of the number of times each pitch class occurredon a given temporal position (down to the eighth notelevel). Järvinen found that tonally stable pitchesoccurred more often at metrically stable temporal posi-tions, and tonally unstable pitches (i.e., nondiatonic)occurred more often at less stable temporal positions.

Tonal-Metric Hierarchy 255

In this study we examine this issue with regard tothe common practice period of Western music byasking the following questions. First and foremost,is there an alignment of tonal and metric informationin typical tonal music, such that tonally stable notesoccur on metrically stable temporal positions, andvice versa? Alternatively, tonally important pitchesmight be equally prevalent across temporal position(i.e., not selective for metric stability), or similarly,metrically stable positions may not favor any partic-ular pitch. To investigate this question, the most obvi-ous starting point is to examine the tonal and metrichierarchies in corpus data. With occasional devia-tions, perceptual studies of both hierarchies concurwith frequency of usage and theoretical predictions(Krumhansl, 1990; Palmer & Krumhansl, 1990, butsee Aarden, 2003); similar hierarchies also arise inmusic production studies (Palmer & Pfordresher,2003; Schmuckler, 1989, 1990). Accordingly, it is ofinterest to determine whether the frequency of co-occurrence of each pitch class and temporal position(within a measure) conforms to perceptual studies ofthe tonal-metric hierarchy (e.g., Prince, Thompson,et al., 2009).

The next most fundamental question is whether therelationship between tonality and meter varies acrossmusical genre, musical mode, and/or time signature.For instance, this hypothetical relationship may varyacross musical genre, particularly between styles thatare largely diatonic (e.g., Baroque music) relative tostyles containing increasing levels of chromaticism(e.g., Classical to Romantic and beyond). Similarly,musical mode is of interest, given the evidence thatminor keys are somewhat more perceptually ambiguousthan major keys (Delzell, Rohwer, & Ballard, 1999; Har-ris, 1985; Vuvan, Prince, & Schmuckler, 2011; Vuvan &Schmuckler, 2011). Lastly, research examining in detailthe frequency of occurrence of tones across temporalpositions within different time signatures is virtuallynonexistent; therefore, it is unknown whether any exist-ing alignment between tonal and metric informationvaries for different metric structures.

Answering these questions is the main goal of thepresent research. Towards this end, we quantified thefrequency of occurrence of each scale degree at eachtemporal position in 365 works from representativecomposers of tonal Western music (Bach, Mozart,Beethoven, and Chopin), totaling 721,293 notes. Withina given piece, we assumed one metric hierarchy and onetonal hierarchy (i.e., ignoring modulation) throughout;we return to the implications of this latter assumption inthe General Discussion.

Method

MATERIALS

Sequenced (i.e., not performed) MIDI files of a selectionof works by Bach, Mozart, Beethoven, and Chopin weredownloaded from http://kern.ccarh.org/ (CCARH,2001) and http://www.piano-midi.de/ (maintained byBernd Krueger; see Appendix A for full list of pieces).The corpus consists of works for piano, with the excep-tion of Beethoven’s Symphony 5, 1st movement.

All pieces were transposed to have a tonic of C majoror A minor (as appropriate), and sorted into time sig-nature categories of 2/4, 3/4, 4/4, 6/8, 9/8, and 12/8. Thekey indicated in the title of each piece was used toclassify the mode and key signature. For pieces whosetitle provided no key information, the key signature ofthe score, in addition to a visual inspection of the firstand last eight measures of the piece, were used to deter-mine the key (tonic and mode). Again, one key wasassigned for the entire piece, ignoring modulations.Note onset times were used to determine the positionof each note within the metric hierarchy of the appro-priate time signature, correcting for any pickup beats.Custom scripts written in MATLAB 7.0 were used toanalyze the corpus.

PROCEDURE

The first step in analyzing the tonal-metric hierarchywas to examine the independent tonal and metric hier-archies of these pieces. Accordingly, we tabulated howoften each scale degree occurred, separately for themajor and minor mode pieces.1 Quantification of themetric hierarchy was accomplished by counting howoften a note was sounded at each temporal position;this count was done separately for different time signa-tures. The tonal and metric counts were then comparedto perceptual measures of tonal and metric stability,taken from Krumhansl and Kessler (1982) and Palmerand Krumhansl (1990), respectively. The purpose ofthese comparisons was to determine if the tonal andmetric information of these pieces conformed to theperceptually derived profiles for these two dimensions.

The second step of the analysis was to determine thefrequency of occurrence of each pitch class (after trans-position) at each temporal position, separately for allpieces in the corpus. These tonal-metric distributionswere grouped within composer, time signature, and

1 In addition to measuring frequency of occurrence, the cumulativeduration of each pitch class was also calculated. These two calculationsgave identical results (r > .99). For continuity of calculation techniquesacross dimensions, frequency of occurrence was used.


mode, then compared with the perceptual profiles andacross composers.

Results

DISTRIBUTION OF PITCH CLASSES AND TEMPORAL POSITIONS

Composers’ use of the 12 pitch classes in the corpuscorrelated highly with the perceptually derived majorand minor tonal hierarchies (see Table 1). However,correlations between composers were higher (meanr ¼ .94) than those with the tonal hierarchies (meanr ¼ .87). Similarly, the observed metric distributionswithin the measure correlated well with the Palmer-Krumhansl metric hierarchies (mean r ¼ .87), but againwere lower than the inter-composer correlations (meanr ¼ .92, see Table 2). Overall, these tables reveal thatthere was a remarkable agreement across composers intheir use of the tonal and metric hierarchies, despitedifferences in mode, musical genre, and time signature.

Figure 1 shows three different versions of both themajor and minor tonal hierarchies – the frequency ofoccurrence observed in the corpus (limited to the down-beat metric position), the overall total frequency ofoccurrence (summing across all metric positions), andthe Krumhansl-Kessler (1982) goodness of fit ratings.Figure 2 compares the metric hierarchies in the corpusto the Palmer and Krumhansl (1990) goodness of fitratings (for time signatures 2/4, 3/4, 4/4, and 6/8).

TONAL-METRIC DISTRIBUTION ANALYSIS

For ease of depiction, the tonal-metric hierarchy countspresent pitch classes and temporal positions by decreas-ing levels of frequency of occurrence instead of the con-ventional method of sorting these dimensionschromatically and chronologically, respectively (as in Fig-ures 1 and 2). For example, Table 3 depicts the tonal-metric frequency of occurrence values for the corpuspieces in a major tonality (all transposed to C major) inthe most common time signature of 4/4. The pitch classes(rows) are sorted as C, G, E, D, F, A, B, G#, D#, F#, A#, andC#, an ordering based on the observed pitch class distri-bution occurring on the downbeat (not across all metricpositions) for all major-mode pieces in the corpus.2

TABLE 1. Correlations Between the Pitch Class Distributions of the Current Corpus and the Tonal Hierarchy Values of Krumhansl and Kessler(1982).

Major modality Minor modality

Bach Mozart Beethoven Chopin Bach Mozart Beethoven Chopin

K&K 1982 .83 .91 .91 .95 .82 .87 .86 .84Bach .96 .92 .90 .95 .94 .96Mozart .99 .98 .99 .99Beethoven .99 .99MEAN .90 .96 .95 .95 .92 .95 .95 .95

TABLE 2. Mean Intercorrelation of the Metric Distributions of the Current Corpus (for Each Composer) and the Matching Metric HierarchyValues of Palmer and Krumhansl (1990).

P&K 1990 Bach Mozart Beethoven Chopin

2/4 .83 .86 .89 .88 .793/4 .95 .94 .94 .94 .904/4 .88 .95 .97 .97 .966/8 .82 .92 .93 .93 .909/8 .96 .9612/8 .88 .88MEAN .87 .92 .93 .93 .89

Note: All hierarchies are restricted to the sixteenth note level.

2 It is important to note that this ordering of pitch classes does notcorrespond exactly with their perceived psychological stability asindicated by the classic tonal hierarchy values of Krumhansl andKessler (1982). Probably the most significant point of divergence in thisregard arises in the ordering of the diatonic scale tones, which ranked D,F, A, and B in frequency of occurrence in our corpus, as opposed to F, A,D, and B in the Krumhansl and Kessler tonal hierarchy values. Of lessernote, the non-diatonic tones in our corpus appear in the order G#, D#, F#,A#, and C#, as opposed to F#, G#, D#, A#, and C#, in Krumhansl andKessler. Although the convergence with the Krumhansl and Kessler majorprofile is strong (r ¼ .96), our data also mirror Aarden’s (2003) rankordering of the seven diatonic major scale tones (based on reaction times)more strongly at points of closure (cf. p. 52; r ¼ .71) than for continua-tions (cf. p. 75; r ¼ .29). Separately, for the minor pieces the pitch classdistribution on the downbeat correlates with the Krumhansl and Kesslerminor tonal hierarchy at r ¼ .91.


The columns of Table 3 represent temporal positionswithin the measure, sorted as beats 1 (downbeat), 3,2þ4, eighth notes, sixteenth notes, and Other, groupedin accordance with theoretical predictions of beatstrength (e.g., Lerdahl & Jackendoff, 1983). That is, thefrequency of occurrence of temporal positions is aver-aged (not summed) within a category of equal metricstrength, such that beats 2 and 4 are averaged, becoming2þ4, then the remaining 4 eighth note (off-beat) posi-tions, followed by the next lower metric hierarchy level(off-beat sixteenth notes), then ‘‘Other’’ (designatingthirty-second notes, sixteenth note triplets, and anynon-quantized metric positions).

Finally, Table 4 provides the same data for pieces inthe minor mode (4/4 time signature, all composers),also with the rows sorted according to the pitch classdistribution of the minor mode on the downbeat. Thefull data set of all time signatures, modalities, and com-posers is available upon request from the first author.

Figures 3-8 depict the tonal-metric hierarchies of eachtime signature as 3-dimensional histograms, separatelyfor major and minor (collapsing across composer). They axis shows the frequency of occurrence of each com-bination of pitch class (x axis) and temporal position(z axis). These graphs use the same arrangement asTables 3 and 4; namely, they sort the axes by frequencyof occurrence on the downbeat, as well as averagingacross temporal positions with equal metric stability.These graphs demonstrate that the most commonoccurrence is a tonally stable pitch (e.g., the tonic) at

a metrically stable temporal position (e.g., the down-beat), and that the frequency of occurrence decreasesat lower levels of tonal and metric stability. At firstglance, the overall tonal-metric hierarchy looks likea replication of the tonal hierarchy at different temporalpositions. Closer inspection, however, reveals that thefavoring of tonally stable pitch classes is more discern-ible at the downbeat than at lower levels of metricstability. Indeed, the correlation of the Krumhansl-Kessler major tonal hierarchy with the pitch class dis-tribution decreases across metric category stability:r ¼ .96 for the downbeat, then r ¼ .92, .90, .88, and.87 down to the ‘‘Other’’ level. For the same metriccategories in the minor mode, the pattern is r ¼ .91 forthe downbeat, then r ¼ .83, .84, .85, and .79. Thischange is also evident in Figure 1, as it compares thetonal hierarchy for only downbeat occurrences to theoverall total (summing all metric positions).

These figures also demonstrate the remarkable con-sistency of the tonal-metric hierarchy across time sig-nature and mode. Despite these changes in the metricalframework and major/minor modality, the profile of thetonal-metric hierarchy remains largely the same. Thisshape is slightly less reliable for Figures 7 and 8, likelydue to the small sample size (note the markedly lowerscale of the y axis compared to other figures).

Because Figures 3-8 show that notes rarely occurredat fine subdivisions, and in the interest of simplicity,further analyses of the tonal-metric hierarchy used the16 most common temporal positions (e.g., sixteenth note

FIGURE 1. Tonal hierarchy values: frequency of occurrence (on downbeat, and all temporal positions) and Krumhansl-Kessler (1982) goodness of fitratings, all standardized to maximum of 1. (a) Major mode hierarchy, (b) Minor mode.


level for 4/4).3 Note that the exclusion of the least com-mon temporal positions represents a more conservativeanalysis – the uniformly rare use of the positions of lowermetric stability might otherwise artificially inflate corre-lations between the tonal and metric hierarchies.

Table 5 shows the results of correlating the resulting12 (pitch class) by 16 (metric stability) tonal-metricmatrices across composer, separately for each time

signature and modality. Values are the average corre-lation coefficients of each composer with all othercomposers. All Table 5 values below .70 are the resultof cells with the minimum possible small sample size(N pieces ¼ 2).

The strikingly high inter-composer tonal-metric cor-relations of Table 5 motivated including additionalcomposers, including some from more modern compo-sitional periods; we chose Schubert, Brahms, Liszt, andScriabin (see Appendix B for the list of included piecesof these composers). These data are not included in theearlier analyses because our corpus had too few pieces ofthese composers for valid inter-composer comparisons.

FIGURE 2. Metric hierarchy values: frequency of occurrence and Palmer-Krumhansl (1990) goodness of fit ratings, all standardized to maximum of 1.(a) 2/4 time signature, (b) 3/4 time signature, (c) 4/4 time signature, (d) 6/8 time signature.

3 We used the 16 most common temporal positions from all timesignatures even if it did not correspond to the sixteenth note level (e.g.,3/4) because we were sorting by stability as indexed by frequency ofoccurrence rather than beat strength.


Smaller sample size can lead to unstable relationships inthe data (Knopoff & Hutchinson, 1983), as observedfor isolated cases in Table 5. However, collapsing acrosstime signature before testing inter-composer correla-tions allows a more reliable comparison, and is justi-fied given the similarity of the tonal-metric hierarchyacross time signature (cf. Figures 3-8). Table 6 showsthe average inter-composer correlations of these eightcomposers, after collapsing across time signaturesbefore correlating. Major and minor modes remainseparate.

Although intriguing, none of the aforementionedanalyses establish definitively whether the tonal andmetric hierarchies are truly correlated. Combining anypitch class distribution with a set of temporal positions

would result in the commonest pitch class occurringmost frequently at the commonest temporal location(i.e., approximately the shape of Figures 3-8), regardlessof if there is any true connection between the two. Oneway to establish quantitative proof of a link between thetwo hierarchies is to compare the average metric stabil-ity of each pitch class with the tonal hierarchy, andcompare the average tonal stability of each temporalposition with the metric hierarchy. The first proceduretests if tonally stable pitches are more likely to have highmetric stability (and tonally unstable pitches have lowermetric stability). The second approach tests if metricallystable positions have high tonal stability (and metricallyunstable positions are more likely to have lower tonalstability). Thus, although these two procedures are

TABLE 3. Tonal-metric Hierarchy of Major Tonality 4/4 Pieces, Summed Across Composer.

Major mode, Time signature 4/4, all composers

Pitch Downbeat Beat 3 Beats 2þ4 8th level 16th level Other

C 3356 2562 2154 1322 530 36G 3365 2681 2161 1528 612 41E 2149 1877 1650 1140 423 30D 1859 1978 1723 1123 481 31F 1588 1475 1358 813 308 22A 1586 1474 1248 895 344 26B 1598 1468 1302 819 356 26G# 517 500 451 254 77 13D# 401 359 334 176 75 12F# 531 601 614 384 149 13A# 307 376 319 215 80 11C# 320 374 339 214 70 10

Note: The four rightmost columns represent the average frequency of occurrence at the relevant temporal positions. Pitches are sorted by frequency of occurrence at thedownbeat across all major mode pieces in the corpus, transposed to C major.

TABLE 4. Tonal-metric Hierarchy of Minor Tonality 4/4 Pieces, Summed Across Composer.

Minor mode, Time signature 4/4, all composers

Pitch Downbeat Beat 3 Beats 2þ4 8th level 16th level Other

A 2134 1591 1489 1020 333 22E 2280 1804 1721 1160 422 29C 1462 1239 1280 738 257 20D 1079 1106 1084 716 279 17B 1108 1169 1097 721 291 19G 1000 775 870 567 164 14F 942 922 922 529 194 15G# 521 633 524 313 105 11C# 255 261 259 184 60 5F# 290 314 325 220 84 8D# 367 354 340 172 63 7A# 230 256 269 150 66 4

Note: The four rightmost columns represent the average frequency of occurrence at the relevant temporal positions. Pitches are sorted by frequency of occurrence at thedownbeat across all minor mode pieces in the corpus, transposed to A minor.


conceptually related, they test separate hypotheses.More importantly, by using the average tonal or metricstability, these calculations are not biased by the simplejoint probability of the two independent distributions.

The average metric stability of each pitch class hasa strong positive correlation with the tonal hierarchy(r ¼ .77 when collapsing across mode; r ¼ .68 for majorpieces, and .58 for minor). There is a similar relation-ship between the average tonal stability of each tempo-ral position and the metric hierarchy (r ¼ .68; .70 for

major pieces, and .55 for minor). Thus, the tonal andmetric hierarchies are indeed connected beyond theextent predicted by their joint probability, yet slightlyless so for minor than major.

Additional inter-composer analyses repeated thesecalculations (comparing the metric stability of eachpitch class and the tonal stability of each metric posi-tion) for each composer separately, collapsing furtheracross mode (see Table 7). Even though the correlationswere always positive, they varied across composer. In

FIGURE 3. Tonal-metric hierarchy for 2/4 time signature, for major (a) and minor (b) mode.



Bach’s compositions, all pitch classes had similar levelsof metric stability, but metrically stable temporal posi-tions had higher tonal stability. One way to conceptu-alize this result is to say that in these pieces, tonallystable pitch classes could occur at any time in the mea-sure, but on the downbeat, only pitches with high tonalstability occurred. The Mozart pieces showed the oppo-site pattern – metrically stable locations could havepitches of any tonal stability, while tonally stable pitcheswere more limited to stable metric locations. Chopin

moved further in this direction, whereas Beethovenrespected both relationships more equally.

Discussion

The current study analyzed how central composersfrom a variety of Western tonal music periods usedpitch classes and temporal positions, and in particular,if this information was used congruently (i.e., were thetonal and metric hierarchies positively correlated). In




general, both distributions were strongly – but not per-fectly – related to the perceptually derived tonal andmetric hierarchies of Krumhansl and Kessler (1982) andPalmer and Krumhansl (1990). Although this finding ishardly surprising on one level, it is notable in that itemerged consistently in a corpus of music that coversa wide range of musical styles, including styles that con-tain high levels of chromaticism and rhythmic deviation.As such, the demonstration that the distribution of tonaland metric information in such works nevertheless

conforms to a prototypical hierarchy of stability is a strik-ing finding.

The primary goal of this study was to test if tonal andmetric information aligned in Western tonal music.Indeed, they correlated strongly throughout this corpus,such that tonally stable pitch classes were more likely tooccur on metrically stable temporal positions. More-over, and mirroring the findings with the independenttonal and metric distributions, this relation was rela-tively stable across time signature and modality.




The reality of a correlated tonal-metric hierarchy maynot be much of a revelation to some, yet it is far froma trivial finding. By themselves, tonally strong pitchesand metrically stable temporal positions function asimportant cognitive reference points for the listenerwhen constructing a mental representation of the musicthey are hearing (Krumhansl, 1990; Palmer & Krum-hansl, 1990). But regardless of its intuitive nature, dem-onstrating this alignment between hierarchies issignificant, particularly given the frequent assumptionsof its existence. For example, Serafine et al. (1989) explic-itly state that ‘‘all theorists would agree’’ (p. 404) with thenotion that tonally stable pitches occur on metricallystrong positions. Similarly, in the related (yet arguablyseparate) context of harmony in melodic anchoring,Bharucha (1984) simply asserts that ‘‘chord tones tendto occur in metrically stressed positions more often thannonchord tones’’ (p. 492). Both of these authors makethese assumptions explicitly, yet based on no actual evi-dence for this relation. More cautious approaches cite analignment between tonal and metric hierarchies asa ‘‘generally held belief ’’ (Palmer & Krumhansl, 1987a).

The unique profile of the tonal-metric hierarchyderives in part from changes in the use of pitch classes

across temporal position. Indeed, pitch class distribu-tions correlated more highly across composers thanthey did with the perceptually derived tonal hierarchy(particularly for the minor mode, likely because of ourtreating all three minor modes as one). The fact that thetonal hierarchy did best at metrically stable locationsconcurs with Aarden’s (2003) findings – that the dia-tonic members of the original major tonal hierarchy aremost representative of pitch expectations at points ofclosure. Because the most expected closure is a cadenceon a metrically stable location, it follows that the shapeof the tonal hierarchy would change slightly based onmetric position. This interpretation accords with Hur-on’s (2006) proposal of four pitch schemas in Westernmusic (major/minor modes both for cadences as well ascontinuations). Additionally, the fact that composersdeliberately violate schematic expectations (Meyer,1957) means that their use of pitch classes and temporalpositions should not be a perfect match to the schematichierarchies.

The current findings also converge with the only (toour knowledge) other existing corpus analysis of therelation between the tonal and metric information: Jär-vinen (1995). The extension and generalization provided

TABLE 6. Inter-composer Correlations Including More Modern Composers, Collapsed Across Time Signature and Mode.

Bach Mozart Beethoven Schubert Chopin Brahms Liszt Scriabin

Major .84 .92 .93 .92 .88 .91 .80 .86Minor .86 .93 .93 .89 .87 .90 .86 .85

TABLE 5. Average Inter-composer Correlation.

Major modality Minor modality

Bach Mozart Beethoven Chopin Bach Mozart Beethoven Chopin

2/4 .86 .91 .92 .72 .80 .79 .633/4 .82 .91 .91 .85 .79 .87 .88 .854/4 .86 .92 .90 .83 .88 .90 .89 .896/8 .69 .87 .84 .83 .74 .65 .75 .749/8 .82 .82 .78 .7812/8 .72 .72MEAN .81 .90 .88 .84 .77 .80 .80 .78

Note: Each value represents the average correlation of that composer with all other composers for that time signature and modality. Missing values mean no data were available.

TABLE 7. Variations Across Composer in the Correlations of Pitch Class Average Metric Stability with the Tonal Hierarchy, and TemporalPosition Average Tonal Stability with the Metric Hierarchy.

Calculation Bach Mozart Beethoven Chopin

r(pitch class metric stability, tonal hierarchy) .35 .66 .71 .78r(temporal position tonal stability, metric hierarchy) .67 .37 .67 .44


by the current findings is critical given that the metrichierarchies in jazz may function differently than in othermusical genres. Indeed, Palmer and Pfordresher’s (2003)distribution analysis of 1930’s swing jazz (well before thebebop period analyzed by Järvinen) showed that inbinary meter, events occurring on beat 4 were most com-mon, followed by beats 2 and 1. One of the more intrigu-ing aspects of Järvinen’s study is that his weightingtechnique found that non-diatonic (i.e., tonally unstable)pitches were actually more common off the beat than onthe beat, thus observing a crossover effect. His weightingtechnique involved multiplying the frequency of occur-rence of notes at the quarter-note level by 2, those at thehalf-note level by 4, and the whole-note level by 8. Thisprocedure corrected the imbalance in how often a givenmetric position could occur within one bar (i.e., eighth-notes could occur at 8 different points, whole notes onlyat one). Using Järvinen’s weighting technique with thecurrent data did not yield a similar crossover, possiblyindicating how a major stylistic difference (classicalmusic to bebop jazz) may actually change the tonal-metric hierarchy. Nevertheless, the advantage of tonallystable pitches decreased at lower levels of metric stability,thus the pattern in the current findings is qualitativelysimilar to Järvinen’s, only not to the point of observinga full crossover.

Comparisons of the tonal-metric hierarchy acrosscomposer yielded particularly intriguing results. Theoverall shape of the tonal-metric hierarchy is surpris-ingly consistent across composer, even when includingmore modern composers still within the tonal tradition(Brahms, Liszt, Schubert, Scriabin). Despite all inter-composer correlations remaining above .8, the moremodern composers tended to show relatively lowercoefficients, but the fact that Bach is comparable toScriabin in this analysis undermines this interpretation.Despite this commonality, there were variations in howthe composers used the tonal-metric hierarchy. Specif-ically, correlations of the average metric stability of pitchclasses with the tonal hierarchy, as well as the averagetonal stability of temporal positions with the metrichierarchy, did not remain entirely constant across com-poser. Table 7 suggests that Bach placed tonally stablepitches in multiple places throughout the measure,while reserving metrically stable positions for tonallystable pitches. Conversely, Chopin was more likely tolimit tonally stable pitches to metrically stable points,but was less restrictive on what pitch classes occurred atthese times (i.e., tonally unstable pitches could also occurat strong metric locations). One possible mechanism forthis shift in the use of the tonal-metric hierarchy couldinvolve changes in composers’ use of appoggiatura and

grace notes, but such interpretations are speculative atthis point.

The fact that composers correlate the tonal and metrichierarchies in their compositions raises the question ofwhy they would do so in the first place. One possibilityis that a correlated tonal-metric hierarchy offers a wayto maximize the strength of musical expectancies, whichare of central importance to communicating musicalemotion (Huron, 2006; Meyer, 1957). By themselves,the tonal hierarchy influences expectations of what(pitch) event might occur, and the metric hierarchy con-cerns when. Used independently (uncorrelated), theyboth offer a means of creating and fulfilling/denyingexpectancies. But when used in a correlated manner, thestrength of the resulting expectancies go beyond the sim-ple joint probability of what and when towards exactlywhat happens at exactly when. In other words, the tonal-metric hierarchy may provide a mechanism of strength-ening the communicated musical emotion, analogous tomaking the whole (expectancy) greater than the sum ofthe parts (tonal and metric hierarchies).

Demonstrating the existence of a tonal-metric hierar-chy is relevant to research on how pitch and time com-bine in music perception. This question has receivedconsiderable treatment over the years, yet there has beenlittle consensus (for reviews, see Ellis & Jones, 2009; Jones& Boltz, 1989; Krumhansl, 2000; Prince, 2011). Percep-tual work examining the relation between tonality andmeter (as opposed to pitch and time) found an asym-metric relationship: tonality affected metric judgments,but meter had no influence on key membership judg-ments (Prince, Thompson, et al., 2009). In fact, an effectof rhythmic regularity on pitch height comparisons onlyoccurred in the context of atonal sequences (Prince,Schmuckler, & Thompson, 2009). Moreover, Prince,Thompson, et al. (2009) found that goodness of fit rat-ings for probe events consisting of any one of 12 pitchclasses at any one of eight temporal positions (eighthnote level) were predictable based on a linear combina-tion of both the tonal and metric hierarchies. These hier-archies explained 89% of the variance in one experiment,and 86% in another, but a multiplicative interaction termnever accounted for any additional variance.

In contrast, the shape of the tonal hierarchy variedacross levels of metric stability in the current study. Themost obvious explanation of the divergence betweenthese findings and earlier research is that it involves com-paring a corpus analyses to perceptually derived results(goodness of fit ratings to probe events) from perceptualexperiments. Accordingly, this result might representa divergence between perceptual and theoretical levelsof analysis. However, humans demonstrate exquisite


sensitivity to the statistical regularities of the environ-ment across an array of domains and modalities (Gold-stone, 1998), with music being no exception (forexamples specifically in pitch and time, see Loui, Wessel,& Kam, 2010; Tillmann, Stevens, & Keller, 2011). Con-sequently, this discrepancy between theoretical and per-ceptual findings on the tonal-metric hierarchy isunexpected. Future research should explore this issue,as it provides a possible exception to this principle ofperceptual learning of statistical information.

Another interpretation of this discrepancy betweenperceptual results and our corpus analysis stems fromthe largest limitation of this study; namely, assigningone tonal centre to an entire piece ignores the fact thatpieces often modulate to new keys. Previous corpusanalyses take the same approach (Knopoff & Hutchin-son, 1983; Krumhansl, 1990; Youngblood, 1958), withthe notable exception of Temperley (2007). Despite thislimitation, it is worth noting that composers typicallymodulate to tonally proximate keys, and then return tothe original key. Given the similarity in the use of pitchclasses between related keys, short-term modulationsshould not influence the overall shape of the tonal hier-archy when compiled across the entire piece. In fact,effects of modulation would primarily serve to flattenthe distribution of the tonal hierarchy; if this flatteninghas occurred in our data it would only reduce thechanges of the tonal hierarchy across metric position.Overall, the consistent shape of the tonal-metric hierar-chy demonstrates that on the aggregate, modulation isa local effect, and does not radically influence the dis-tribution of frequency of occurrences.

Nevertheless, determining how modulation mayaffect this corpus is the most urgent next step in thisresearch. Methodological approaches could be to exam-ine only pieces that do not modulate, or extractexcerpts that stay in the original key, or interpret mod-ulated sections in terms of their new key (such thatduring a modulation from C to G, all G pitches wouldcount as instances of the tonic). We are currentlyexploring these approaches; in tandem we are planningbehavioral work to test if see if listeners’ perceptions ofmodulation are similarly local. For shorter choraleexcerpts, at least, listeners are indeed sensitive to the

extent of modulation (Thompson & Cuddy, 1989,1992); thus, once established, the tonal hierarchy mayfunction as a robust schema that persists through evenlonger time spans.

Overall, the analyses reported in this paper substan-tiate the correlation of tonal and metric information intypical tonal music, thus providing evidence for theexistence of a tonal-metric hierarchy, similar to the pre-viously identified and well-studied tonal and metrichierarchies. The observed tonal-metric hierarchyremained remarkably consistent across compositionalperiod, time signature, and mode. Thus it is indeed truethat these composers used tonally stable pitches atpoints of high metric stability. Importantly, the tonal-metric hierarchy is not a simple linear combination ofthe separate hierarchies of tonality and meter. Instead, itrepresents an interactive relationship between the twohierarchies: the tonal hierarchy is most pronounced onmetrically stable temporal positions, and less distinct atpositions of lower metric stability. The current findingstherefore offer several useful contributions to the musiccognition literature. We have validated and elaboratedon a critical yet tacit assumption regarding the use ofthe tonal and metric hierarchies, extended theoreticalresearch on hierarchical organization in musical pitchand time, and revealed an interesting discrepancybetween perceptual and theoretical research on tonalityand meter. Hopefully these findings will stimulate futureresearch on the complex issue of pitch, time, and struc-tural hierarchies in music.

Author Notes

Preparation of this manuscript was supported by theWalter Murdoch Distinguished Collaborator Schemeand a grant from the Natural Sciences and EngineeringResearch Council of Canada to the second author. Theauthors wish to thank Carol Krumhansl for her encour-agement of this line of inquiry, and Davy Temperley forhis insightful comments.

Correspondence concerning this article should beaddressed to Jon B. Prince, School of Psychology, Mur-doch University, 90 South Street, Murdoch, WA 6150.E-mail: [email protected]

References

AARDEN, B. J. (2003). Dynamic melodic expectancy.(Unpublished doctoral dissertation). Ohio State University.Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num¼osu1060969388

BEAL, A. L. (1985). The skill of recognizing musical structures.Memory and Cognition, 13, 405-412.

BENJAMIN, W. E. (1984). A theory of musical meter. MusicPerception, 1, 355-413.


http://rave.ohiolink.edu/etdc/view?acc_num=osu1060969388http://rave.ohiolink.edu/etdc/view?acc_num=osu1060969388http://rave.ohiolink.edu/etdc/view?acc_num=osu1060969388

BERGESON, T. R., & TREHUB, S. E. (2006). Infants’ perception ofrhythmic patterns. Music Perception, 23, 345-360.

BHARUCHA, J. J. (1984). Anchoring effects in music - The reso-lution of dissonance. Cognitive Psychology, 16, 485-518.

BIGAND, E. (1990). Abstraction of two forms of underlyingstructure in a tonal melody. Psychology of Music, 18, 45-59.

BIGAND, E. (1997). Perceiving musical stability: The effect oftonal structure, rhythm, and musical expertise. Journal ofExperimental Psychology: Human Perception and Performance,23, 808-822.

BIGAND, E., & PINEAU, M. (1996). Context effects on melodyrecognition: A dynamic interpretation. Cahiers de PsychologieCognitive/Current Psychology of Cognition, 15, 121-134.

BOLTZ, M. G. (1989a). Perceiving the end: Effects of tonalrelationships on melodic completion. Journal ofExperimental Psychology: Human Perception andPerformance, 15, 749-761.

BOLTZ, M. G. (1989b). Rhythm and good endings - Effects oftemporal structure on tonality judgments. Perception andPsychophysics, 46, 9-17.

BOLTZ, M. G. (1991). Some structural determinants of melodyrecall. Memory and Cognition, 19, 239-251.

BROWN, H. (1988). The interplay of set content and temporalcontext in a functional theory of tonality perception. MusicPerception, 5, 219-249.

BROWN, H., BUTLER, D., & JONES, M. R. (1994). Musical andtemporal influences on key discovery. Music Perception, 11,371-407.

CCARH. (2001). THE MUSEDATA DATABASE (http://www.Musedata.Org). Stanford, CA: Center for Computer AssistedResearch in the Humanities.

COHEN, A. J., TREHUB, S. E., & THORPE, L. A. (1989). Effects ofuncertainty on melodic information-processing. Perceptionand Psychophysics, 46, 18-28. doi: 10.3758/bf03208070

COLLARD, R., VOS, P., & LEEUWENBERG, E. (1981). What melodytells about meter in music. Zeitschrift Fur Psychologie, 189,25-33.

CREEL, S. C. (2012). Similarity-based restoration of metricalinformation: Different listening experiences result in differentperceptual inferences. Cognitive Psychology, 65, 321-351.

CUDDY, L. L. (1991). Melodic patterns and tonal structure:Converging evidence. Psychomusicology, 10, 107-126.

DAWE, L. A., PLATT, J. R., & RACINE, R. J. (1993). Harmonicaccents in inference of metrical structure and perception ofrhythm patterns. Perception and Psychophysics, 54, 794-807.

DAWE, L. A., PLATT, J. R., & RACINE, R. J. (1994). Inference ofmetrical structure from perception of iterative pulses withintime spans defined by chord changes. Music Perception, 12,57-76.

DAWE, L. A., PLATT, J. R., & RACINE, R. J. (1995). Rhythmperception and differences in accent weights for musicians andnonmusicians. Perception and Psychophysics, 57, 905-914.

DELZELL, J. K., ROHWER, D. A., & BALLARD, D. E. (1999). Effectsof melodic pattern difficulty and performance experience onability to play by ear. Journal of Research in Music Education,47, 53-63. doi: 10.2307/3345828

DESAIN, P. (1992). A (de)composable theory of rhythmperception. Music Perception, 9, 439-454.

DESAIN, P., & HONING, H. (2003). The formation of rhythmiccategories and metric priming. Perception, 32, 341-365. doi:10.1068/p3370

DEUTSCH, D. (1980). The processing of structured and unstruc-tured tonal sequences. Perception and Psychophysics, 28,381-389.

DOWLING, W. J. (1978). Scale and contour: Two components ofa theory of memory for melodies. Psychological Review, 85,341-354.

DOWLING, W. J. (1991). Tonal strength and melody recognitionafter long and short delays. Perception and Psychophysics, 50,305-313.

DRAKE, C., & BOTTE, M.-C. (1993). Tempo sensitivity in audi-tory sequences: Evidence for a multiple-look model. Attention,Perception, and Psychophysics, 54, 277-286. doi: 10.3758/bf03205262

ELLIS, R. J., & JONES, M. R. (2009). The role of accent salienceand joint accent structure in meter perception. Journal ofExperimental Psychology: Human Perception and Performance,35, 264-280.

ESSENS, P. J., & POVEL, D. J. (1985). Metrical and nonmetricalrepresentations of temporal patterns. Perception andPsychophysics, 37, 1-7.

GOLDSTONE, R. L. (1998). Perceptual learning. Annual Review ofPsychology, 49, 585-612. doi: 10.1146/annurev.psych.49.1.585

HANNON, E. E., & JOHNSON, S. P. (2005). Infants use meter tocategorize rhythms and melodies: Implications for musicalstructure learning. Cognitive Psychology, 50, 354-377.

HANNON, E. E., & TREHUB, S. E. (2005). Metrical categories ininfancy and adulthood. Psychological Science, 16, 48-55.

HARRIS, R. W. (1985). Perceived relatedness of musical tones inmajor and minor tonal contexts. American Journal ofPsychology, 98, 605-623. doi: 10.2307/1422513

HURON, D. (2006). Sweet anticipation: Music and the psychologyof expectation. Cambridge, MA: MIT Press.

HURON, D., & PARNCUTT, R. (1993). An improved model oftonality perception incorporating pitch salience and echoicmemory. Psychomusicology: Music, Mind and Brain, 12, 154-171. doi: 10.1037/h0094110

JÄRVINEN, T. (1995). Tonal hierarchies in jazz improvisation.Music Perception, 12, 415-437.

JONES, M. R., & BOLTZ, M. G. (1989). Dynamic attending andresponses to time. Psychological Review, 96, 459-491.

JONES, M. R., JOHNSTON, H. M., & PUENTE, J. (2006). Effects ofauditory pattern structure on anticipatory and reactiveattending. Cognitive Psychology, 53, 59-96.


http://www.Musedata.Orghttp://www.Musedata.Org

JONES, M. R., & RALSTON, J. T. (1991). Some influences of accentstructure on melody recognition. Memory and Cognition, 19,8-20.

KELLER, P. E., & BURNHAM, D. K. (2005). Musical meter inattention to multipart rhythm. Music Perception, 22, 629-661.doi: 10.1525/mp.2005.22.4.629

KNOPOFF, L., & HUTCHINSON, W. (1983). Entropy as a measureof style: The influence of sample length. Journal of MusicTheory, 27, 75-97.

KRUMHANSL, C. L. (1990). Cognitive foundations of musical pitch.New York: Oxford University Press.

KRUMHANSL, C. L. (1991). Memory for musical surface. Memoryand Cognition, 19, 401-411.

KRUMHANSL, C. L. (2000). Rhythm and pitch in music cognition.Psychological Bulletin, 126, 159-179.

KRUMHANSL, C. L., & CUDDY, L. L. (2010). A theory of tonalhierarchies in music. In M. R. Jones, R. R. Fay, & A. N. Popper(Eds.), Music perception. Springer handbook of auditoryresearch. New York: Springer Science.

KRUMHANSL, C. L., & KESSLER, E. J. (1982). Tracing the dynamicchanges in perceived tonal organization in a spatial represen-tation of musical keys. Psychological Review, 89, 334-368.

KRUMHANSL, C. L., & SCHMUCKLER, M. A. (1986, July). Key-finding in music: An algorithm based on pattern matching totonal hierarchies. Paper presented at the 19th Annual Meetingof the Society of Mathematical Psychology, Cambridge, MA.

LADEN, B. (1994). Melodic anchoring and tone duration. MusicPerception, 12, 199-212.

LARGE, E. W., & JONES, M. R. (1999). The dynamics of attend-ing: How people track time-varying events. PsychologicalReview, 106, 119-159.

LARGE, E. W., & PALMER, C. (2002). Perceiving temporal regu-larity in music. Cognitive Science, 26, 1-37.

LEBRUN-GUILLAUD, G., & TILLMANN, B. (2007). Influence ofa tone’s tonal function on temporal change detection.Perception and Psychophysics, 69, 1450-1459.

LERDAHL, F., & JACKENDOFF, R. (1983). A generative theory oftonal music. Cambridge, MA: MIT Press.

LONDON, J. (2004). Hearing in time: Psychological aspects ofmusical meter. Oxford, UK: Oxford University Press.

LONGUET-HIGGINS, H. C. (1976). Perception of melodies.Nature, 263, 646-653.

LOUI, P., WESSEL, D., & KAM, C. (2010). Humans rapidly learngrammatical structure in a new musical scale. MusicPerception, 27, 377-388.

MEYER, L. B. (1957). Emotion and meaning in music. Chicago, IL:University of Chicago Press.

MONAHAN, C. B., & CARTERETTE, E. C. (1985). Pitch andduration as determinants of musical space. Music Perception, 3,1-32.

PALMER, C. (1997). Music performance. Annual Review ofPsychology, 48, 115-138.

PALMER, C., & KRUMHANSL, C. L. (1987a). Independent tem-poral and pitch structures in determination of musical phrases.Journal of Experimental Psychology: Human Perception andPerformance, 13, 116-126.

PALMER, C., & KRUMHANSL, C. L. (1987b). Pitch and temporalcontributions to musical phrase perception - Effects of har-mony, performance timing, and familiarity. Perception andPsychophysics, 41, 505-518.

PALMER, C., & KRUMHANSL, C. L. (1990). Mental representationsfor musical meter. Journal of Experimental Psychology: HumanPerception and Performance, 16, 728-741.

PALMER, C., & PFORDRESHER, P. Q. (2003). Incremental planningin sequence production. Psychological Review, 110, 683-712.doi: 10.1037/0033-295x.110.4.683

POVEL, D. J., & ESSENS, P. (1985). Perception of temporal pat-terns. Music Perception, 2, 411-440.

PRINCE, J. B. (2011). The integration of stimulus dimensions inthe perception of music. Quarterly Journal of ExperimentalPsychology, 64, 2125-2152. doi: 10.1080/17470218.2011.573080

PRINCE, J. B., SCHMUCKLER, M. A., & THOMPSON, W. F. (2009).The effect of task and pitch structure on pitch-time interac-tions in music. Memory and Cognition, 37, 368-381. doi:10.3758/MC.37.3.368

PRINCE, J. B., THOMPSON, W. F., & SCHMUCKLER, M. A. (2009).Pitch and time, tonality and meter: How do musical dimen-sions combine? Journal of Experimental Psychology: HumanPerception and Performance, 35, 1598-1617. doi: 10.1037/a0016456

REPP, B. H. (1992). Probing the cognitive representation ofmusical time - Structural constraints on the perception oftiming perturbations. Cognition, 44, 241-281.

REPP, B. H. (2010). Do metrical accents create illusory phe-nomenal accents? Attention, Perception, and Psychophysics, 72,1390-1403. doi: 10.3758/app.72.5.1390

REPP, B. H., IVERSEN, J. R., & PATEL, A. D. (2008). Tracking animposed beat within a metrical grid. Music Perception, 26,1-18. doi: 10.1525/mp.2008.26.1.1

SCHMUCKLER, M. A. (1989). Expectation in music - Investigationof melodic and harmonic processes. Music Perception, 7,109-150.

SCHMUCKLER, M. A. (1990). The performance of global expec-tations. Psychomusicology, 9, 122.

SCHMUCKLER, M. A. (1997). Expectancy effects in memory formelodies. Canadian Journal of Experimental Psychology, 51, 292.

SCHMUCKLER, M. A. (2009). Components of melodic processing.In S. Hallam, I. Cross, & M. Thaut (Eds.), Oxford handbook ofmusic psychology (pp. 93-106). Oxford, UK: Oxford UniversityPress.

SCHMUCKLER, M. A., & BOLTZ, M. G. (1994). Harmonic andrhythmic influences on musical expectancy. Perception andPsychophysics, 56, 313-325.


SCHMUCKLER, M. A., & TOMOVSKI, R. (2005). Perceptual tests ofan algorithm for musical key-finding. Journal of ExperimentalPsychology: Human Perception and Performance, 31,1124-1149.

SERAFINE, M. L., GLASSMAN, N., & OVERBEEKE, C. (1989). Thecognitive reality of hierarchic structure in music. MusicPerception, 6, 397-430.

SHAFFER, L. H., CLARKE, E. F., & TODD, N. P. (1985). Meter andrhythm in piano playing. Cognition, 20, 61-77.

SLOBODA, J. A. (1983). The communication of musical meter inpiano performance. Quarterly Journal of ExperimentalPsychology, 35, 377-396.

SMITH, K. C., & CUDDY, L. L. (1989). Effects of metric andharmonic rhythm on the detection of pitch alterations inmelodic sequences. Journal of Experimental Psychology:Human Perception and Performance, 15, 457-471.

STEEDMAN, M. J. (1977). Perception of musical rhythm andmeter. Perception, 6, 555-569.

TEMPERLEY, D. (2001). The cognition of basic musical structures.Cambridge, MA: MIT Press.

TEMPERLEY, D. (2007). Music and probability. Cambridge, MA:MIT Press.

TEMPERLEY, D., & MARVIN, E. W. (2008). Pitch-class distributionand the identification of key. Music Perception, 25, 193-212.

THOMPSON, W. F., & CUDDY, L. L. (1989). Sensitivity to keychange in chorale sequences: A comparison of single voicesand four-voice harmony. Music Perception, 7, 151-168.

THOMPSON, W. F., & CUDDY, L. L. (1992). Perceived key move-ment in four-voice harmony and single voices. MusicPerception, 9, 427-438.

TILLMANN, B., JANATA, P., BIRK, J. L., & BHARUCHA, J. J. (2008).Tonal centers and expectancy: Facilitation or inhibition ofchords at the top of the harmonic hierarchy? Journal ofExperimental Psychology: Human Perception and Performance,34, 1031-1043. doi: 10.1037/0096-1523.34.4.1031

TILLMANN, B., STEVENS, C., & KELLER, P. E. (2011). Learning oftiming patterns and the development of temporal expectations.Psychological Research, 75, 243-258.

TRAINOR, L. J., & TREHUB, S. E. (1994). Key membership andimplied harmony in Western tonal music: Developmentalperspectives. Perception and Psychophysics, 56, 125-132.

TREHUB, S. E., COHEN, A. J., THORPE, L. A., & MORRONGIELLO,B. A. (1986). Development of the perception of musical rela-tions - Semitone and diatonic structure. Journal ofExperimental Psychology: Human Perception and Performance,12, 295-301.

VUVAN, D. T., PRINCE, J. B., & SCHMUCKLER, M. A. (2011).Probing the minor tonal hierarchy. Music Perception, 28,461-472. doi: 10.1525/mp.2011.28.5.461

VUVAN, D. T., & SCHMUCKLER, M. A. (2011). Tonal hierarchyrepresentations in auditory imagery. Memory and Cognition,39, 477-490.

YOUNGBLOOD, J. E. (1958). Style as information. Journal of MusicTheory, 2, 24-35. doi: 10.2307/842928


Appendix A

Appendix B

Schubert Brahms Liszt Scriabin

13 Variations on a Theme byAnselm Hüttenbrenne

Ballade No. 1 in D minor, Op. 10 Hungarian Rhapsodies,No. 12, 15

Etude, Op. 2, No. 1

String Quartet No. 10 in Eb Major,Op. 125, No. 1, Mvmt. 1-4

String Quartet in C minor, Op. 51,No. 1 (1873)

Grandes Etudes dePaganini, No. 3, 4

12 Etudes, Op. 8,No. 1-6

Trio for violin, viola and cello,Sept. 1816

Waltz No. 1 in B major Liebestraume Notturno,No. 3

24 Preludes, Op.11, No. 4, 15

String Quartet D 804, Mvmt. 2 Waltz No. 2 in E major Transcendental Etudes,No. 5 Feux Follets

Impromptu in Gb major Waltz No. 8 in Bb MinorWaltz No. 9 in D minorScherzo in Eb minor for piano, Op. 4Piano Sonata No. 1 in C major, Op. 1

(2.) AndanteFantasia, Op. 116, No. 2, 5, 6Intermezzi, Op. 117, No. 1, 2Rhapsodie from 4 Piano pieces,

Op. 119

Bach Mozart Beethoven ChopinWell-Tempered Clavier Piano sonatas Piano Sonatas Preludes, Waltzes, Mazurkas

24 Preludes and K.279 Op. 2, No. 1-3 24 Preludes Op. 28, No. 1-24Fugues, Book 1 K.280 Op. 7 Op. 6, No. 1-424 Preludes and K.281 Op. 10, No. 1-3 Op. 7, No. 1-5Fugues, Book 2 K.282 Op. 13 Op. 10, No. 2, 5, 9

K.283 Op. 14, No. 1-2 Op. 17, No. 1-4K.284 Op. 22 Op. 24, No. 1-4K.309 Op. 26 Op. 30, No. 1-4K.310 Op. 27, No. 1-2 Op. 33, No. 1-4K.311 Op. 28 Op. 41, No. 1-4K.330 Op. 31, No. 1-3 Op. 50, No. 1-3K.331 Op. 49, No. 1-2 Op. 56, No. 1-3K.332 Op. 53 Op. 59, No. 1-3K.333 Op. 54 Op. 63, No. 1-3K.357 Op. 57 Op. 64, No. 1-2K.545 Op. 78 Op. 67, No. 1-4K.570 Op. 79 Op. 68, No. 1-4K.576 Op. 81 Op. 69, No. 2

Op. 90 B. 82Op. 101 B. 134Op. 106 B. 140Op. 109 B. 150Op. 110Op. 111 Scherzo, Op. 81

Ballade, Op. 52, No. 4Piano Rondo Op. 129Symphony 5, Op. 67, 1st Mvmt.


jon b. prince & mark a. schmuckler - murdoch...

Documents