Autocorrelation in meter induction: The role of accent structure a)

Autocorrelation in meter induction: The role of accent structure a) Petri Toiviainen and Tuomas Eerola Department of Music, P.O. Box 35(M), 40014 University of Jyväskylä, Jyväskylä, Finland Received 16 April 2005; revised 14 October 2005; accepted 7 November 2005 The performance of autocorrelation-based meter induction was tested with two large collections of folk melodies, consisting of approximately 13 000 melodies for which the correct meters were available. The performance was measured by the proportion of melodies whose meter was correctly classified by a discriminant function. Furthermore, it was examined whether including different melodic accent types would improve the classification performance. By determining the components of the autocorrelation functions that were significant in the classification it was found that periodicity in note onset locations was the most important cue for the determination of meter. Of the melodic accents included, Thomassen s melodic accent was found to provide the most reliable cues for the determination of meter. The discriminant function analyses suggested that periodicities longer than one measure may provide cues for meter determination that are more reliable than shorter periodicities. Overall, the method predicted notated meter with an accuracy reaching 96% for binary classification and 75% for classification into nine categories of meter. 2006 Acoustical Society of America. DOI: 10.1121/1.2146084 PACS number s : 43.75.Cd DD Pages: 1164 1170 I. INTRODUCTION Most music is organized to contain temporal periodicities that evoke a percept of regularly occurring pulses or beats. The period of the most salient pulse is typically within the range of 400 to 900 ms Fraisse, 1982; Parncutt, 1994; van Noorden and Moelants, 1999. The perceived pulses are often hierarchically organized and consist of at least two simultaneous levels whose periods have an integer ratio. This gives rise to a percept of regularly alternating strong and weak beats, a phenomenon referred to as meter Cooper and Meyer, 1960; Lerdahl and Jackendoff, 1983. In Western music, the ratio of the pulse lengths is usually limited to 1:2 duple meter and 1:3 triple meter. Meter in which each beat has three subdivisions, such as 6/8 or 9/8, is referred to as compound meter. A number of computational models have been developed for the extraction of the basic pulse from music. Modeling of meter perception has, however, received less attention. Large and Kolen 1994 presented a model of meter perception based on resonating oscillators. Toiviainen 1997 presented a model of competing subharmonic oscillators for determining the meter duple versus triple from an acoustical representation of music. Brown 1993 proposed a method for determining the meter of musical scores by applying autocorrelation to a temporal function consisting of impulses at each tone onset whose heights are weighted by the respective tone durations. A shortcoming of Brown s 1993 study is a Portions of this work were presented in The role of accent periodicities in meter induction: a classification study, Proceedings of the 8th International Conference on Music Perception and Cognition, Evanston, IL, August 2004; and Classification of musical metre with autocorrelation and discriminant functions, Proceedings of the 6th International Conference on Music Information Retrieval, London, 2005. that she does not provide any explicit criteria for the determination of meter from the autocorrelation functions. Although there is evidence that the pitch information present in music may affect the perception of pulse and meter Dawe et al., 1993; Thomassen, 1982; Hannon et al., 2004, most models of pulse finding developed to date rely only on note onset times and durations. Dixon and Cambouropoulos 2000, however, proposed a multi-agent model for beat tracking that makes use of pitch and amplitude information. They found that including this information when determining the salience of notes significantly improved the performance of their model. Vos et al., 1994 applied autocorrelation to the determination of meter in isochronous or almost isochronous music. They utilized a method similar to that proposed by Brown 1993, except for using the melodic intervals between subsequent tones instead of tone durations as the weighting factor in the autocorrelation analysis. Using a corpus of 30 compositions by J. S. Bach, they found that the maxima of the obtained autocorrelation functions matched the respective bar lengths as indicated in the musical score. As the majority of music is nonisochronous, reliance on mere melodic interval structure in meter induction is probably a special case. In a general case, it can be expected that meter determination be based on both temporal and pitch structure. To address the question of the relative importance of temporal and pitch structure in meter determination, it would be necessary to use a rhythmically more variable set of stimuli. According to a commonly adopted notion, meter can be inferred from phenomenal, structural, and metrical accents Lerdahl and Jackendoff, 1983. Phenomenal accents are the primary source of meter and are related to the surface structure of music and arise from changes in duration, pitch, timbre, and dynamics. Pitch-related phenomenal accents, more 1164 J. Acoust. Soc. Am. 119 2, February 2006 0001-4966/2006/119 2 /1164/7/$22.50 2006 Acoustical Society of America

commonly referred to as melodic accents, arise from changes in pitch height, pitch interval or pitch contour Huron and Royal, 1996. Using a correlational study, Huron and Royal 1996 investigated the extent to which different types of pitch-related accent correspond to respective metrical positions as indicated by musical notation. Of eight different types of melodic accent, they found that the empirically derived accent by Thomassen 1982 was the only one to correlate significantly with metric position. The lack of correlation of the other accent types however, does, not exclude the possibility that various metric accents exhibit periodic structure that may serve as an additional cue for meter induction. For instance, different types of accents may contain periodicities of equal length, while being phase shifted with respect to each other. This is the case, for instance, with the accents 1, 4, and 5 in Fig. 3. It must be noted that temporal structure and pitch information are not the sole determinants of meter, as it is also influenced by other features such as phrasing and lyrics as well as accents introduced by the performer. These aspects are, nonetheless, beyond the scope of the present study. A shortcoming of meter induction models presented to date is that they have not been evaluated with large sets of musical material. While Vos et al. 1994 utilized a corpus of 30 compositions, Brown 1993 presented only a handful of short musical excerpts to visualize the performance of her model. The present study investigated the performance of autocorrelation-based meter induction with large collections of folk melodies consisting of thousands of items, for which the notated meters are available. The performance was assessed by the proportion of melodies that were correctly classified in terms of their meter. The components of the autocorrelation function that are significant in the classification were determined. Moreover, a number of different types of melodic accent and combinations thereof were applied to the classification to assess the significance of each of them in the induction of meter. Finally, confusions made by the algorithm between different types of meter were investigated in detail. II. AUTOCORRELATION AND METER Below, the method for constructing the autocorrelation function for meter induction is described. For the original description, see Brown 1993. Let the melody consist of N notes with onset times t i,i=1,2,...,n. Each note is associated with an accent value a i,i=1,2,...,n; in Brown 1993, a i equals the duration of the respective note. The onset impulse function f is a time series consisting of impulses of height a i located at each note onset position: N f n = a i i n, n = 0,1,2,..., 1 i=1 where = i n 1, n = t i/dt, 2 0, otherwise, where dt denotes the sampling interval and denotes rounding to the nearest integer. FIG. 1. Excerpt from a melody, its onset impulse function weighted by durational accents, f, and the corresponding autocorrelation function, F. The maximum of the autocorrelation function at the lag of 4/ 8 indicates duple meter. Autocorrelation refers to the correlation of two copies of a time series that are temporally shifted with respect to each other. For a given amount of shift or lag, a high value of autocorrelation suggests that the series contains a periodicity with length equaling the lag. In the present study, the autocorrelation function F was defined as F m = n f n f n m n f n 2, where m denotes the lag in units of sampling interval; the denominator normalizes the function to F 0 =1 irrespective of the length of the sequence. Often, the lag corresponding to the maximum of the autocorrelation function provides an estimate of the meter. This is the case for the melody depicted in Fig. 1. Sometimes the temporal structure alone is not sufficient for deducing the meter. This holds, for example, for isochronous and temporally highly aperiodic melodies. In such cases, melodic structure may provide cues for the determination of meter. This is the case, for instance, with the melody depicted in Fig. 2. With this isochronous melody, the autocorrelation function obtained from the duration-weighted onset impulse function fails to exhibit any peaks, thus making it impossible to determine the meter. Including information about pitch content in the onset impulse function leads, however, to an autocorrelation function with clearly discernible peaks. III. STUDY 1 A. Material The material used in the first study consisted of folk melodies in MIDI file format taken from two collections: the Essen collection Schaffrath, 1995, consisting of mainly European folk melodies, and the Digital Archive of Finnish Folk Tunes Eerola and Toiviainen, 2004a, subsequently referred to as the Finnish collection. For the present study, all melodies in either duple 2/4, 4/4, 4/8, etc.; 2 n eighth notes per measure or triple/compound 3/8, 3/4, 6/8, 9/8, 12/8, etc.; 3n eighth notes per measure meter were chosen. Consequently, a total of 5507 melodies in the Essen collection were used in the study, of which 3121 56.7% were in duple and 2386 43.3% were in triple/compound meter. From the 3 J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006 P. Toiviainen and T. Eerola: Autocorrelation in meter induction 1165

FIG. 3. a Onset impulse functions constructed from a melodic excerpt using the six accent types described in the text; b the respective autocorrelation functions. As can be seen, the melodic accents frequently fail to co-occur either with each other or with the durational accents. All the autocorrelation functions, however, have maxima at lags of either 6/8 or 12/8, indicating triple or compound meter. FIG. 2. Excerpt from an isochronous melody. a Onset impulse function weighted by durational accents, f, and the corresponding autocorrelation function, F, showing no discernible peaks. b Onset impulse function weighted by interval size, f, and the corresponding autocorrelation function, F. The maximum of the autocorrelation function at the lag of 12/ 8 indicates triple or compound meter. Finnish collection, 6861 melodies were used, of which 5518 80.4% were in duple and 1343 19.6% were in triple/ compound meter. B. Method For each of the melodies in the two collections, we constructed a set of onset impulse functions weighted by various accent types Eqs. 1 and 2. In each case the sampling interval was set to 1/16 note. The accents consisted of 1 durational accent a i equals tone duration, 2 Thomassen s 1982 melodic accent, 3 interval size in semitones between previous and current tone see, e.g., Vos et al., 1994, 4 pivotal accent a i =1 if melody changes direction, a i =0 otherwise, and 5 gross contour accent a i =1 for ascending interval, a i = 1 for descending interval, a i =0 otherwise. Since the note onset times alone, without regard to any accent structure, provide information about metrical structure, we further included 6 constant accent a i =1. The analysis was carried out using the MIDI Toolbox for Matlab Eerola and Toiviainen, 2004b. For each melody, each of the onset impulse functions was subjected to autocorrelation. The components of the obtained autocorrelation functions corresponding to lags of 1, 2,, 16 eighth notes were included in the subsequent analyses. Figure 3 depicts the onset impulse functions and the respective autocorrelation functions constructed from a melodic excerpt using each of the accent types described above. The extent to which these autocorrelation functions could predict the meter of each melody was assessed by means of stepwise discriminant function analyses, in which various subsets of autocorrelation functions were used as independent variables and the meter duple versus triple/ compound as the dependent variable. The leave-one-out cross-validation scheme Lachenbruch and Mickey, 1968 was utilized. The measures observed in the analyses were the percentage of correctly classified cases, the order in which variables entered into the discriminant function, and the discriminant function coefficients. C. Results The analysis was started by considering the autocorrelation function based on durational accents. First, only the components corresponding to lags of 4 and 6 eighth notes were used as independent variables. This yielded a correct classification rate of 80.9% for the Essen collection and 84.6% for the Finnish collection. Thus, a significant proportion of the melodies was misclassified, suggesting that periodicities of 4/8 and 6/8 in durational accent are not sufficient for reliable classification. This can be clearly seen in Fig. 4, where the values of these components are displayed as scatter plots, showing a significant overlap between melodies representing the two types of meter. Next, all the components of the durational-accent-based autocorrelation function were entered into the analysis. This yielded a correct classification rate of 90.5% for the Essen collection and 93.1% for the Finnish collection. Inclusion of all 16 components as independent variables thus considerably improved classification performance. The first components to enter into the discriminant function were lags of 8/8, 12/8, and 16/8 in this order for the Essen collection, and 4/8, 12/8, and 16/8 for the Finnish collection. This suggests that periodicities longer than one bar may provide cues for meter determination that are more reliable than shorter periodicities. Subsequently, discriminant function analyses were carried out with the autocorrelation functions obtained using each of the remaining accents, one at a time. The results are summarized in Table I. As can be seen, a significant proportion of the components of the autocorrelation functions that entered first in the stepwise analysis correspond to relatively long time lags. In particular, for all accent types lag 12/8 is among the three most important components. For lag 16/ 8 the same holds true for all but one accent type. This again suggests that periodicities exceeding the span of one bar 1166 J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006 P. Toiviainen and T. Eerola: Autocorrelation in meter induction

FIG. 4. Scatter plot of the values for lags 4/8 and 6/8 of the durationalaccent-based autocorrelation function for the melodies in the Essen collection and the Finnish collection that were used in the present study. Circles and crosses represent melodies in duple and triple meter, respectively. seem to offer highly important cues for meter induction. Somewhat unexpectedly, the autocorrelation function obtained by ignoring any accent structure constant accent yielded for both collections the highest classification rate. In light of this result, the onset function constructed using a constant accent may be slightly more efficient in meter induction than the function originally introduced by Brown 1993. In terms of correct classification rate, the constant accent was followed by durational accent and Thomassen s melodic accent, in this order. Finally, all autocorrelation functions were used together as independent variables. This yielded a correct classification rate of 95.3% and 96.4% for the Essen collection and the Finnish collection, respectively. The first variable to enter into the discriminant function was lag 8/8 with durational accent for the Essen collection and lag 4/8 with constant accent for the Finnish collection. The next three variables to enter were the same for both collections: lag 12/8 with constant accent, lag 16/8 with constant accent, and lag 6 with melodic accent, in this order. This analysis contained a total of 96 independent variables 16 for each of the 6 accent types ; due to the large number of cases, a large number of variables were entered into the stepwise analyses. To obtain a simpler model for meter classification, we performed a further discriminant function analysis in which the six most prominent components for two most prominent accent types from the previous analysis were used. These corresponded to the values of the autocorrelation functions at lags of 3/8, 4/8, 6/8, 8/8, 12/8, and 16/8 using constant accent and Thomassen s melodic accent. Using both collections together N=12 368, we obtained the following discriminant function, = 1.042 + 0.318F const 3/8 + 5.240F const 4/8 0.630F const 6/8 + 0.745F const 8/8 8.122F const 12/8 + 4.160F const 16/8 0.978F mel 3/8 + 1.018F mel 4/8 1.657F mel 6/8 + 1.419F mel 8/8 2.205F mel 12/8 + 1.568F mel 16/8, 4 where F const and F mel denote autocorrelation functions ob- TABLE I. Proportions of correctly classified melodies classification rate and components of the autocorrelation function that entered first in each analysis primary components; numbers refer to lags in units of one eighth note for both collections and each accent type. Accent type 1 Duration 2 Thomassen 3 Interval 4 Pivot 5 Contour 6 Constant Classification rate % Primary Components Classification rate % Primary Components Essen Collection N=5507 90.5 86.8 85.9 78.5 73.1 91.5 8, 12, 16 16, 12, 8 16, 12, 4 16, 12, 6 16, 12, 8 8, 12, 16 Finnish collection N=6861 93.1 90.1 87.6 79.4 77.9 94.7 4, 12, 16 8, 12, 4 8, 12, 16 6, 16, 12 12, 6, 16 4, 12, 16 J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006 P. Toiviainen and T. Eerola: Autocorrelation in meter induction 1167

TABLE II. Confusion matrices between notated and predicted meters for both collections. Each element shows the number of melodies with the respective notated and predicted meter. For each meter, the recall, precision, and F score values are given. Essen Collection N=5592 Predicted meter 2/4 3/2 3/4 3/8 4/1 4/2 4/4 6/4 6/8 Total Recall Precision F Notated meter 2/ 4 1130 0 21 9 0 0 124 0 1 1285 0.88 0.86 0.87 3/ 2 0 65 11 0 0 8 16 0 0 100 0.65 0.92 0.76 3/ 4 52 0 930 30 0 0 91 106 6 1215 0.77 0.90 0.83 3/ 8 23 0 9 168 0 0 0 0 91 291 0.58 0.53 0.55 4/ 1 0 1 0 0 36 2 0 0 0 39 0.92 0.75 0.83 4/ 2 0 4 0 0 11 148 10 0 0 173 0.86 0.87 0.86 4/ 4 98 1 32 0 1 12 1452 1 1 1598 0.91 0.85 0.88 6/ 4 0 0 21 0 0 0 9 80 0 110 0.73 0.43 0.54 6/ 8 15 0 12 109 0 0 1 1 643 781 0.82 0.87 0.84 Total 1318 71 1036 316 48 170 1703 188 742 Finnish Collection N=7351 Predicted meter 2/ 4 3/ 2 3/ 4 3/ 8 4/ 4 5/ 2 5/ 4 6/ 4 6/ 8 Total Recall Precision F Notated meter 2/ 4 2439 17 62 0 740 12 14 5 4 3293 0.74 0.69 0.71 3/ 2 5 45 4 0 15 3 0 2 0 74 0.61 0.44 0.51 3/ 4 98 23 693 2 16 2 2 65 1 902 0.77 0.86 0.81 3/ 8 9 0 17 47 0 0 0 1 55 129 0.36 0.48 0.42 4/ 4 958 7 5 0 1203 25 0 5 2 2205 0.55 0.60 0.57 5/ 2 0 0 0 0 11 26 1 1 0 39 0.67 0.38 0.49 5/ 4 32 6 1 0 0 0 374 0 0 413 0.91 0.95 0.93 6/ 4 4 1 9 0 26 0 0 38 0 78 0.49 0.32 0.39 6/ 8 14 3 16 48 0 0 1 2 134 218 0.61 0.68 0.65 Total 3559 102 807 97 2011 68 392 119 196 tained with constant and Thomassen s melodic accent, respectively. With this discriminant function, the correct classification rates were 92.9% and 94.8% for the Essen and the Finnish collections, respectively. This simpler discriminant function thus yielded correct classification rates that were merely 2.4% and 1.6% lower than those obtained with 96 predictive variables. Variables that received the largest coefficients in the discriminant function were F const 12/8, F const 4/8, F const 16/8, F mel 12/8, and F mel 6/8, in this order. In concordance with the aforementioned results, this suggests that temporal structure above the bar level produces important cues for meter determination. Further, the fact that the most significant components of the melodic accent autocorrelation function correspond to multiples of 3/8 lags suggests that, especially for triple and compound meters, melodic accent structure provides additional cues. IV. STUDY 2 The aim of study 2 was to assess the capability of the autocorrelation-based meter induction method to carry out a more detailed classification. More specifically, instead of mere classification as duple versus triple, the dependent variable used in this study was the actual notated meter. In the analysis, special attention was paid to the pattern of confusion between meters. A. Material As in study 1, the material was taken from the Essen collection and the Digital Archive of Finnish Folk Tunes. From each collection, melodies that consisted of a single notated meter were included. Moreover, for each collection only meters that contained more than 30 exemplars were included. Consequently, a total of 5592 melodies in the Essen collection were used, representing nine different notated meters 2/4, 3/2, 3/4, 3/8, 4/1, 4/2, 4/4, 6/4, 6/8. From the Finnish collection, 7351 melodies were used, representing nine different notated meters 2/4, 3/2, 3/4, 3/8, 4/4, 5/2, 5/4, 6/4, 6/8. For each collection, the number of melodies representing each notated meter is shown in Table II. B. Methods The classification of meters was based on the discriminant function obtained using the autocorrelation functions obtained using all the accent types. The performance was assessed by means of a confusion matrix. Furthermore, for both collections the precision and recall values as well as the F score were calculated for each meter Salton and McGill, 1983. For a given meter, precision is defined as the number of melodies having the meter and being correctly classified divided by the total number of melodies being classified as representing the meter. A high value of precision thus indi- 1168 J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006 P. Toiviainen and T. Eerola: Autocorrelation in meter induction

cates that, of the melodies classified as being notated in a given meter, a large proportion is correctly classified. Similarly, for each meter, recall is defined as the number of melodies being notated in the meter and being correctly classified divided by the total number of melodies being notated in the meter. A high value of recall thus indicates that of the melodies being notated in a given meter, a large proportion is correctly classified. The F score is defined as the harmonic mean of precision and recall and is regarded as an overall measure of classification performance see, e.g., Salton and McGill, 1983. C. Results Overall, 83.2% of the melodies from the Essen collection and 68.0% of those from the Finnish collection were correctly classified. These proportions being lower than the corresponding figures in the first study is due to the larger number of classes used in this study nine than in the first two. The notably low correct classification rate for the Finnish collection can be mainly attributed to the fact that a large proportion 43.4% of the melodies representing 4/4 meter were classified as being 2/4 see below. To obtain a more detailed view of the classification performance, we calculated the confusion matrices for both collections. They are displayed in Table II. The table also shows the precision and recall values as well as the F scores for each meter. In terms of the F score, the most accurately classified meters were 4/4 and 2/4 for the Essen collection and 5/4 and 3/4 for the Finnish collection. Similarly, the least accurately classified meters were 6/4 and 3/8 for both collections. Table II reveals that the most frequent confusions were made within the groups of duple and triple/compound meters, as defined in study 1, whereas confusions across these groups where significantly less frequent. For both collections, meters 2/4 and 4/4 displayed the highest mutual confusion rate, followed by meters 3/4 and 6/4. A large proportion of these misclassifications can probably be attributed to the effect of tempo on the choice of notated meter cf. London, 2002. Certain confusions imply more severe misattributions by the algorithm. For instance, 11.7% of the melodies in the Essen collection notated in 3/4 meter were misclassified as representing binary meter 4/4 or 2/4, the corresponding figure for the Finnish collection being 12.6%. In general, duple meters were less frequently misclassified as representing triple/compound meter as vice versa. This asymmetry is in line with the results obtained in study 1. Further research would be needed to account for this phenomenon. As the confusion matrices contain an abundance of numbers, the relationship between meters may be difficult to see. Therefore we visualized the relations between meters by performing separate hierarchical cluster analyses for both collections. To this end, we calculated the distance between each meter from the confusion matrix according to the formula FIG. 5. Dendrograms obtained from the confusion matrix using the similarity measure of Eq. 5. The leftmost column displays the average note durations in quarter notes for the melodies representing each meter. d ij =1 c ij + c ji c ii + c jj, where d ij denotes the distance between meters i and j, and c ij is the number of cases where a melody in meter i has been classified as being in meter j. By definition, the larger the proportion of melodies confused between meters, c ij +c ji,to the number of melodies correctly classified for both meters, c ii +c jj, the smaller the distance d ij between the meters. Figure 5 displays the dendrograms obtained from the clustering algorithms. In the dendrograms of Fig. 5, the stage at which given meters cluster together reflects the algorithm s rate of confusion between the meters. For both collections, the meters to first cluster together are 3/8 and 6/8. For the Essen collection, this is followed by the clustering of the meters 3/4 and 6/4 as well as 2/4 and 4/4, in this order. Also for the Finnish collection these pairs of meters cluster next, albeit in reverse order, that is, the clustering of 2/4 and 4/4 precedes that of 3/4 and 6/4. A further similar feature between the two dendrograms is that the last clustering occurs between the cluster formed by the meters 3/8 and 6/8 and the cluster formed by all the other meters. This suggests that, in terms of the autocorrelation functions, meters 3/8 and 6/8 are most distinct from the other meters. One peculiar feature of the dendrogram for the Essen collection is the relatively late clustering of meters 4/1 and 4/2 with meters 2/4 and 4/4. In particular, the former two meters cluster with meter 3/2 before clustering with the latter two meters. A potential explanation for this is the difference in the average note durations between the meters, shown in the leftmost column of Fig. 5. More specifically, the average note durations for meters 4/1, 4/2, and 3/2 exceed those of meters 2/4 and 4/4 by a factor of 2. This anomaly, however, is not significant, as meters 4/1, 4/2, and 3/2 constitute merely a minor proportion of the whole collection. 5 J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006 P. Toiviainen and T. Eerola: Autocorrelation in meter induction 1169

V. CONCLUSIONS We studied the classification performance of the autocorrelation-based meter induction model, originally introduced by Brown 1993. Using discriminant function analysis, we provided an explicit method for the classification. Furthermore, we applied the algorithm to investigate the role of melodic accent structure in meter induction. In conformance with the general view, we found that periodicity in note onset locations was the most important cue for the determination of meter. The results also imply that periodicities longer than one measure provide additional information for meter induction. A somewhat unexpected finding was that ignoring the accent structure in the onset impulse function yielded the best classification rate. This result is difficult to explain and calls for further investigation. Furthermore, the results suggest that periodicity in melodic accent structure may serve as an additional cue in the determination of meter. In particular, including Thomassen s 1982 melodic accent was found to improve classification performance to some extent. This, however, does not necessarily imply that melodic and durational accents need cooccur but rather that they exhibit similar periodic structure with eventual mutual phase shift. An additional finding that calls for further study was the significant difference between the correct classification rates for melodies in duple and triple/compound meter. More specifically, melodies in duple meter were more often correctly classified than melodies in triple/compound meter. A detailed investigation of misclassified melodies could provide insight into this question. The material used in the present study consisted of melodies predominantly in duple, triple, and compound meters, although study 2 utilized a few hundred Finnish melodies in either 5/4 or 5/2 meter. To obtain a deeper insight into the role of accent structure in meter induction, a study with nonregular meters such as those present in the folk music of the Balkan region could be carried out. An apparent limitation of the method presented in this article is its inability to deal with melodies that contain changes of meter. For a melody that, say, starts in 2/4 meter and changes to 3/ 4 meter, the algorithm gives unpredictable results. This is due to the fact that the algorithm considers the melody as a whole. The limitation may be overcome by applying a windowed analysis analogical to algorithms used in pitch estimation from acoustical signals, in which the autocorrelation is applied to short windowed segments of the melody, with the window moving gradually throughout the melody. The present study utilized melodies that were represented in symbolic, temporally quantized form. The choice of stimuli was mainly based on the availability of correct notated meters for the melodies in the collections. In principle the method could, however, be applied to performed music in acoustical form as well, at least with a monophonic input. This would require algorithms for onset detection e.g., Klapuri, 1999, pitch estimation e.g., Brown and Puckette, 1994; Klapuri, 2003, and beat tracking e.g., Dixon, 2001; Large and Kolen, 1994; Toiviainen, 1998, 2001. ACKNOWLEDGMENTS This work was supported by the Academy of Finland Project No. 102253. Brown, J. C. 1993. Determination of meter of musical scores by autocorrelation, J. Acoust. Soc. Am. 94, 1953 1957. Brown, J. C., and Puckette, M. S. 1994. A high resolution fundamental frequency determination based on phase changes of the Fourier transform, J. Acoust. Soc. Am. 94, 662 667. Cooper, G., and Meyer, L. B. 1960. The Rhythmic Structure of Music Univ. of Chicago, Chicago. Dawe, L. A., Platt, J. R., and Racine, R. J. 1993. 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