ABSTRACT

In this study, three measures of temporal pattern com-plexity were compared with regard to their perceptualvalidity. The first measure, based on the work of Tan-guiane (1993), uses the idea that a temporal pattern canbe described in terms of (elaborations of) more simplepatterns, which occur simultaneously at different levels.The second measure is based on the complexity meas-

ure for finite sequences proposed by Lempel and Ziv(1976), which is related to the number of steps in a self-delimiting production process by which such a sequenceis presumed to be generated.The third measure, newly developed here, is rooted in

the theoretical framework of rhythm perception of Poveland Essens (1985). It takes into account the ease of cod-ing a temporal pattern and the complexity of the seg-ments resulting from this coding. The perceptual validityof the three measures was evaluated in an experiment inwhich subjects judged the complexity of 35 temporal pat-terns.Correlations between the three measures and the col-

lected complexity judgments indicated that the thirdmeasure is a much better predictor of temporal patterncomplexity than the other two measures. This is probablydue to the fact that this measure, unlike the other two, isbased on an empirically tested model of rhythm percep-tion that takes into account the isochronous frameagainst which the rhythm is perceived. Reasons for thedifferences in performance between the three measuresare discussed.

INTRODUCTION

The notion of complexity has generally been stud-ied in the context of information theory and isclosely connected with concepts such as random-ness, information, regularity, and coding (Calude,1994). Classical information theory, as well asnotions of randomness, based on Shannon's con-cept of entropy (Shannon, 1948), relies on a prioriknowledge of a probability distribution. In thatrespect, it does not allow one to speak of a particu-lar object or outcome as being random or complex.In general, an object's complexity reflects the

amount of information embedded in it. The repre-sentation of the object's information is achieved viacoding. When a human being enters the equation,however, care must be taken in interpreting thenotion of complexity, which necessarily becomessubjective. Moreover, depending on the context,only certain types of codes may be perceptually sig-nificant and hence coding efficiency or complexitymust be considered within such constraints (Chater,1996). This is well known, for example, in the fieldof visual perception (Leeuwenberg, 1971). In orderto obtain complexity measurements from subjects,Pressing (n.d.) suggests equating complexity with

0929-8215/00/2901-061$15.00#Swets & Zeitlinger

Journal of New Music Research, 29 (2000), No. 1, pp. 61^69

Measures of Temporal Pattern Complexity

Ilya Shmulevich1 and Dirk-Jan Povel2

1Tampere International Center for Signal Processing,Tampere University of Technology,Tampere, Finland

2Nijmegen Institute for Cognition and Information, University of Nijmegen,Nijmegen,The Netherlands

Correspondence: Ilya Shmulevich, Signal Processing Laboratory, Tampere University of Technology, P.O. Box 553,33101 Tampere, Finland. Tel.: +358-3-365-3869. Fax: +358-3-365-3817. E-mail: ilya@cs.tut.fi

difficulty of learning, which in turn could beexpressed by recognition or production.In this work, we consider the complexity of tem-

poral patterns. Our aim is to construct a measureof complexity that corresponds to a high degreewith a human's subjective notion of complexity.Pressing (n.d.) discusses three notions of complex-ity. The first is termed hierarchical complexity,which refers to structure on several levels simulta-neously. The receiver is then able to perceive struc-ture on one or more levels, inducing an appropriatecomplexity judgement. A general approach to hier-archical structure in perception has been proposedby Leyton (1986). The divisible nature of Westernrhythms lends itself to hierarchical subdivision andreveals regularity of time organization on severallevels simultaneously (Lerdahl & Jackendoff,1983). The second type of complexity is referred toas dynamic complexity. This notion refers to thedegree of stationarity or change with respect totime, in the sensory input. A highly nonstationarystimulus would tend to be perceived as being com-plex. In music, rhythms tend to be stationary orperiodic in that events or groups of events arerepeated in time. Finally, the third type of complex-ity, called generative complexity, refers to a tendencytowards the most economical description (Hoch-berg & McAlister, 1953; Chater, 1996).In this paper, we examine three new measures of

complexity of temporal patterns. The first measureis based on the work of Tanguiane (1993), and usesthe idea that a rhythmic pattern can be describedin terms of (elaborations of) more simple patterns,which occur simultaneously at different levels. Thesecond measure is based on the complexity measurefor finite sequences proposed by Lempel and Ziv(1976), which is related to the number of steps in aself-delimiting production process by which such asequence is presumed to be generated. Finally, thethird measure proposed is rooted in the theoreticalframework of rhythm perception discussed in Poveland Essens (1985). This measure takes into accountthe ease of coding a temporal pattern and the (com-bined) complexity of the segments resulting fromthis coding. The measure presupposes the existenceof a `̀ temporal grid'' or time scale consisting of iso-chronic intervals, which is selected among a set ofpossible grids according to the `̀ economy principle''

(Povel, 1984). All three measures, which will be dis-cussed in detail below, capture one or more of thetypes of complexity discussed above.

SELECTION ANDNOTATION OF RHYTHMS

Before we proceed to explain the three measures, wemust define the domain of rhythms studied. First,we restrict ourselves to quantized rhythms, i.e.,rhythms as notated in a score, without timing devia-tions due to performance. Therefore, these rhythmscan be described fully in musical notation.Withoutloss of generality and for the ensuing discussion, weuse the sixteenth note as the smallest note duration(unit of length). Furthermore, the rhythms studiedare supposed to repeat or loop infinitely and thusform an infinite sequence of events. Finally, wenotate a rhythmic pattern as a string of ones andzeros, in which the symbol `1' represents a noteonset and `0' represents no note onset. Of course,the smallest quantization level must be used for theencoding. For example, the pattern

would be represented by 1011100010011000. Weshould emphasize that in this notation, only inter-note intervals are relevant and so the pattern

is represented by exactly the same string as above.Now, we are ready to give the definitions of thecomplexity measures.

T-Measure (Tanguiane measure)The first measure we consider is based on the workof Tanguiane (1993). A basic notion in the theoryof Tanguiane is that of elaboration (Mont-Reynaud& Goldstein, 1985). Figure 1 gives an example ofthe elaboration of a quarter note. As can be seen,the quarter note is elaborated into patterns consist-ing of two notes on the second row, which in turnare further elaborated into patterns of three noteson the third row, which are all finally elaboratedinto a pattern of four notes on the last row. Thus,all patterns that are linked by a line, either directly

62 I. SHMULEVICH. ANDD.-J. POVEL.

or indirectly, to a higher pattern form elaborationsof that higher pattern.Tanguiane shows how a rhythmic pattern can be

described by rhythmic configurations at severallevels simultaneously, e.g., at the eighth note level,at the quarter note level, and so on. Of course, foreach such level, we use an appropriate partiallyordered set similar to the one shown in Figure 1.This representation on several levels accounts forthe structural redundancy inherent in divisiblerhythms, such as those found in Western music.TheT-measure intends to capture hierarchical com-plexity. As shown for the quarter note exampleabove, not all patterns are elaborations of otherpatterns.Within any given set of patterns, those pat-terns that are not elaborations of any other patternare called root patterns.These root patterns form the core of Tanguiane's

notion of complexity, where he states that the rootpatterns are `̀... irreducible to each other [and] canbe used for estimating the complexity of rhythm''(Tanguiane, 1993). It may be noted that the set ofroot patterns defines a monotone Boolean function(Yablonsky, 1989, p. 37).Tanguiane defines complex-ity as the number of root patterns required to gener-ate the given rhythmic pattern on a given structurallevel. This corresponds to the number of minimaltrue vectors of the corresponding monotone Boo-lean function. Finally, the overall complexity of therhythmic pattern is defined by taking the maximum

of all complexities over all the structural levels. It iswell known (Gilbert, 1954) that the maximum num-ber of minimal true vectors, of a monotone Booleanfunction of n variables is equal to

Mn � nbn=2c

� �

and hence determines the maximum possible num-ber of root patterns and consequently the maxi-mum complexity under the T-measure. For theelaborations of the quarter note, as shown in Figure1, the number of root patterns can be no largerthan 3. Therefore, for this example, the T-Measureof complexity is an integer between 1 and 3.Still, such a definition of complexity is problem-

atic since some structural levels may not lendthemselves to subdivision. The reason is that theoccurrence of certain durations in the rhythmicpattern may prevent subdivision into some otherpatterns. For example, the presence of a dottedeighth note, which is essentially comprised of 3 six-teenth notes, prohibits the subdivision of the entirepattern into eighth notes. However, the definitionof complexity can be slightly modified by simplytaking the maximum over all allowable structurallevels, that is, over those levels which can be sub-divided into the duration of interest. Furthermore,it can be decided a priori, if desired, to restrictoneself to only certain structural levels. Forinstance, we may wish to ignore subdivisions intopatterns equal to the duration of 5 sixteenth notes,even though such subdivisions may be possiblesince the rhythm is assumed to be infinite. Let usnow look at a specific example illustrating theapplication of the T-Measure.Example 1: Consider the pattern 111011111001

which can be written as . In this exam-ple, we only take into account structural levels gen-erated by dotted eighth notes and quarter notes.Remember that the temporal sequence is assumedto be infinite. On the dotted eighth level, we findthe following sub-patterns: 100, 111, and 101.Note that in this example, 110 cannot be a pattern(even though it is found in the sequence) becausethe immediately following pattern always needs tostart with a 1, representing a tone onset. Clearly,111 and 101 are both elaborations of 100, which is

63RHYTHMCOMPLEXITY

Fig. 1. Elaboration of a quarter note.

the only root pattern on that level, implying a com-plexity of 1. On the quarter note level, we find:1110, 1111, and 1001. In this case, we see that1110 and 1110 are both root patterns, 1111 beingthe elaboration of both of them and hence thecomplexity on that level is equal to 2. Taking themaximum over all considered levels, we find theoverall complexity to be equal to 2.

LZ-Measure (Lempel-Ziv measure)Another approach for quantifying complexity ofrhythms is to use the popular measure proposedby Lempel & Ziv (1976). This complexity measurecaptures the number of `̀ new'' substrings discov-ered as the sequence evolves from left to right (asis the case in music). As soon as a new substring isfound, the complexity increases by 1. The measureessentially takes into account repetitions of pat-terns on all structural levels, thus capturing bothdynamic as well as hierarchical complexities. Itcan easily be shown that the Lempel-Ziv (LZ) com-plexity of a periodic binary sequence is finite. Thisis, of course, desirable for quantifying complexityof temporal patterns. It should also be pointed outthat the LZ complexity in general is not well suitedfor very short sequences and thus the assumptionof cyclical rhythms is very useful. The measure isintended to capture the multi-level redundancyembedded in the rhythmic pattern without regardto any perceptual mechanisms involved in codingit. Thus, the measure does not take into accountthe possibility that some of the informationembedded in the sequence may not be perceptuallyrelevant to a human listener. Therefore, it can beused as a reference point for other measures thatdo incorporate perceptual constraints as theyshould exhibit greater correspondence to subjectivejudgements of complexity than the LZ-Measure.Let us now look at the same example as above andcompute the LZ complexity for that rhythm.Example 2: We shall write the sequence repre-

senting the rhythm two times, since it will turnout that the LZ complexity needs the secondinstance of the sequence to reach its final value.So, the sequence is:

111011111001 111011111001

In the sequence below, new substrings are delimitedby dots.

1.110.1111.100.1 1110.11111001...

Note that after the last dot, no new substrings exist.The LZ complexity for this sequence is equal to 5,which is equal to the number of dots.

PS-Measure (Povel-Shmulevich measure)The PS-Measure is rooted in the theoretical frame-work of perception of temporal patterns discussedin Povel and Essens (1985). A basic notion of thatmodel is that a listener attempts to establish aninternal clock (beat) that segments the rhythm intoequal intervals. While there is no physiologicalexplanation of the internal clock nor reason for itsexistence, structural regularity of auditory stimuliinduce in the listener a certain response, be it tap-ping of a foot or merely recognition of the rhythmicorganization. Presumably, this temporal segmenta-tion serves to reduce the coding complexity of thestimulus, which would be consistent with theGestalt simplicity principle, implying that sensoryinput is coded in the simplest possible way (Chater,1996). This aspect of the model is concerned withgenerative complexity.The induction of the clock is determined by the

distribution of accents in the sequence (see alsoParncutt, 1994; Jones & Pfordresher, 1997). As sev-eral possible clocks will fit with any given rhythm,it is assumed that the clock which best fits the dis-tribution of accents in the rhythm is the one actu-ally induced. This clock is referred to as the bestclock. Furthermore, the ease with which the bestclock is induced depends on how well it fits the dis-tribution of accents. After the selection of the bestclock, the rhythm is represented by coding the seg-ments produced by this clock.Discussing the complexity of rhythms, the authors

state that a `̀... given temporal pattern will be ...judged complex when either no internal clock isinduced or, where it is induced, when the coding ofthe pattern is relatively complex'' (Povel & Essens,1985). In light of that, the proposed measure ofcomplexity should be a combination of the induc-tion strength of the best clock on the one hand andthe efficiency of coding the rhythm on the other.

64 I. SHMULEVICH. ANDD.-J. POVEL.

The first part of the PS-Measure thus pertains tothe induction strength of the best clock, which iscaptured by the C-score (Povel and Essens, 1985).The C-score is computed by taking into account aweighted combination of the number of clock ticksthat coincide with unaccented events and withsilence:

C �W � s� u

in which s stands for the number of clock tickscoinciding with silence and u for the number ofunaccented events. The lower the score, the higherthe induction strength of the clock; hence higherscores correspond to higher complexity.The second part of the PS-measure pertains to

the efficiency of the code. In determining codingcomplexity, we distinguish between four types ofpossible segments as shown in Figure 2: an emptysegment (E), an equally subdivided segment (Sk,where k indicates the number of equal subdivi-sions), an unequally subdivided segment (U), andfinally a segment which begins with silence (N).To compute the coding complexity, a different

weight is associated with each type of segment.Weights d1; ::::; d4 correspond respectively to thefour types of segments distinguished above. Finally,a weight d5 is used in order to account for repeti-tions of segments. Specifically, if a segment is dif-ferent from the segment following it, d5 is added tothe sum of all weights accumulated so far. Therationale behind this is that two different consecu-tive segments are likely to increase complexity.Now, the formula for the total coding complexity

is:

D �Xni�1

ci �m � d5

where ci 2 fd1; :::; d4g is the weight of the ith seg-ment, n is the number of segments, and m is the

number of consecutive segment pairs containingdifferent segments.Finally, the PS-Measure is defined as the

weighted combination of the induction strength ofthe clock and the total coding complexity:

P � � � C � �1ÿ �� �D

where C is the induction strength of the best clockand D is the total coding complexity obtained bysegmenting the rhythm with that clock.Two parameters which must be determined areW

and �.W is the weight used in the formula to com-pute C (see above). The parameter � represents therelative importance of clock induction strengthand coding efficiency. Before discussing the pro-cedure for determining all the necessary para-meters, let us first look at an example.Example 3: Consider the rhythm pattern repre-

sented by

1010 1010 1110 1100

which can be written as .Suppose that the best clock for this rhythm has

unit 4 and location 1 and a value of C = 1. Supposethat � � 0:5. The codes of the four segments arerespectively: S2, S2, 1-1-2, and 1-3. Suppose furtherthat d2 = 1, d3 = 3, and d5 = 1. Since the secondsegment is different from the third which is in turndifferent from the fourth, m = 2, and thus, the over-all coding complexity is equal to D � 1� 1�3� 3� 2 � �1� � 10. Finally, P � 0:5 � 1� 0:5 � 10� 5:5, which is the overall complexity of therhythm pattern.

PARAMETER ESTIMATION

The parameters were determined by utilizing theresults of an experiment reported in Essens (1995).In Experiment 3 of that work, twenty subjectswere asked to make complexity judgements on 24rhythmic patterns, on a scale of 1 to 5. All para-meters were optimized so as to increase the corre-lation between the average judged complexity col-lected in that experiment and the PS-Measure.More formally, let ~� � ��;W ; d1; :::; d5� be the vec-

65RHYTHMCOMPLEXITY

j... j Empty (E)

j. j. j Equally subdivided (S2)

j.. j j Unequally subdivided (U)

. j.. j Starting with silence (N)

Fig. 2. Four types of segments.

tor of parameters used in the PS-Measure and let~CJ and ~CPS be 24-length vectors containing com-plexities from human listeners (averaged) and thePS-Measure respectively. Then, the goal is to findthe parameter �̂ so that

�̂ �~�

arg max ���~CJ ; ~CPS��

where ���; �� is the correlation.To achieve this, simplex search as well as quasi-

Newton search methods were used. The parametersmaximizing the correlation are shown in Table 1.The resulting correlation between the averagejudged complexities and the PS-Measure complex-ities computed with these parameters is �

�~CJ ;

~CPS

�= 0.83.

As shown by Essens (1995), the C-score had nosignificant effect in predicting complexity. This issupported by the fact that �, the weight given to C,is low, implying that the coding complexity plays amuch more important role1. As expected, empty (E)segments contribute less to complexity than equallyand unequally subdivided segments, as is shown bythe value of d1.What is surprising, however, is thatthe weight associated with equally subdivided seg-ments (d2) is not significantly lower than the weightassociated with unequally subdivided segments (d3).The weight associated with segments that do notbegin with a note onset is close to zero. This condi-tion occasionally occurs because the optimal valueof W is different from the one used by Essens (W =4), consequently changing the best clock. Finally,the weight d5 indicates that variations between seg-ments do play a role in determining complexity,which again supports the findings of Essens (1995).

EXPERIMENT

The purpose of this experiment was to determinehow well the PS-Measure predicts the judged com-plexity of a set of temporal patterns not used in thecomputation of the measure. Also the predictivepowers of the T-measure and the LZ-measure weredetermined. For that purpose subjects judged a setof 35 temporal patterns as described below.

MethodParticipants. Twenty-five subjects, graduate, under-graduate students, and faculty at the University ofNijmegen, participated in the experiment. Themedian age of the subjects was 20 years. All ofthem were musicians, with an average of 9.2 yearsof practical musical experience.Stimuli. The stimuli used in the experiment con-sisted of rhythmic patterns, each containing fourdifferent intervals, namely 200 ms, 400 ms, 600 ms,and 800 ms (which may respectively be notated as1, 2, 3, 4). The patterns were all permutations of thecombination 1 1 1 1 1 2 2 3 4. The 800 ms intervalwas always at the end of a pattern. The 35 patternsused in the experiment are displayed inTable 2.Stimulus presentation. The patterns were generatedon a Rhodes model 760 MIDI-synthesizer, usingthe middle C (261,6 Hz) of the Marimba sound,which was emitted via a Kawai KM-20 activespeaker. There were no differences in intensity orpitch between the tones. Stimulus presentation wascontrolled by a Power Macintosh 8200/120. Eachpattern was repeated three times, so one entirestimulus had a length of four patterns. As each pat-tern lasted for 3.2 seconds, an entire stimulus hada duration of 3.2 seconds * 4 (repeats) = 12.8 sec-onds. Patterns were presented in random order, dif-ferent for each subject.Procedure. Subjects were seated in front of the com-puter screen. After the presentation of each stimu-lus, subjects were required to judge the complexityof the stimulus on a 5-point scale (1 = simple, 5 =complex), displayed on the screen. Subjects wereasked to imagine how difficult it would be to repro-duce the rhythms. Subjects could listen to asequence more than once, and change their judg-ment before proceeding to the next stimulus. Eachsubject first trained with five practice trials to getacquainted with the procedure. The entire experi-ment lasted approximately twenty minutes.

66 I. SHMULEVICH. ANDD.-J. POVEL.

Table 1. Estimated parameter values used in the PS meas-ure.

� W d1 d2 d3 d4 d5

0.2223 1.1695 0.0235 1.2722 1.2955 0.0736 0.7931

1It should be noted that the C-score did play a role in a rhythm reproduction task reported by Essens (1995).

RESULTS

The averages of the scores for each item were com-puted over all subjects and are displayed next toeach stimulus in Table 2. To assess the reliability ofthe average of the responses, we computed theCronbach alpha measure. This measure providesus with a lower bound for the correlation betweenthe average scores of the performed experimentwith another identical experiment containing thesame number of subjects. For our experiment, a =0.88, which indicates a high degree of internal con-sistency.The correlations between the average judgements

and the values as predicted by the complexitymeasures discussed herein were computed. Thecomplexities given by the PS-Measure were com-puted by using the estimated parameters given inTable 1. The correlations between the judged com-plexities and the complexities predicted by the T-Measure, LZ-Measure, and the PS-Measure are: r= 0.02, r = 0.15, and r = 0.75 respectively.

DISCUSSION

Essens (1995) showed that the C-score had no sig-nificant effect in predicting complexity, although itdid play a role in a rhythm reproduction task.Because of the nature of the instructions given tosubjects, linking complexity with ease of reproduc-tion, we expected the C-score to contribute at leastpartially to complexity judgements. This indeedturned out to be the case insofar as the mean com-plexity judgements follow a general trend upwardswith respect to the ordering of the stimuli, whichare arranged by increasing values of their C-scores.This is illustrated in Figure 3. However, we hypoth-esized that the coding complexity of temporal pat-terns played a more significant role in determiningthe overall complexity judgements. This was con-firmed, in part, during the so-called `̀ trainingphase'' of the PS-Measure, which assigned a weightof approximately 80% to the coding complexity.The correlation between the average responses

for each item in the above experiment and thevalues as predicted by the PS-Measure with theparameters given in Table 1, was computed andfound to be r = 0.75. The value obtained indicatesthat the PS-Measure is a robust measure of com-plexity of temporal patterns in the sense that theparameters estimated from one set of patterns yieldhigh predictive power for another set. Furthermore,it could be inferred that this measure captures theessential components in rhythm that directly con-

67RHYTHMCOMPLEXITY

Table 2. The 35 patterns used in the experiment, togetherwith their mean judged complexity. Each vertical line re-presents a tone. The smallest interval between two toneonsets is 200 ms.The dots have no physical meaning what-soever; their function is to represent the relative durationof the intervals.

No. Pattern Comp. No. Pattern Comp.

123456789101112131415161718

jjjjj..jj.j.j...jjj.j.jjj..jj...j.jjj.jjj..jj...j.j.jjjjj..jj...j..jj.j.jjjjj...jjj.jjj.jj..j...j.jjjj.jj..jj...jj..jjjjj.j.j...jj..j.jjj.jjj...j.jjj.jjjj..j...jjj.jj..jj.jj...jj.jjjj.j..jj...jj.jj.jjjj..j...jj..jj.jj.jjj...j..jjj.jjj.jj...jj.jjjj.jj..j...jj.jjj.jjj..j...jj.jjj..jj.jj...

1.562.122.081.881.802.442.202.563.002.042.762.723.003.162.042.882.602.60

1920212223242526272829303132333435

jj..jj.jjjj.j...jj..jj.jjj.jj...jjjjj.jj.j..j...jjjj.j..jjj.j...jjj..jj.jjj.j...j.jjj..j.jjjj...j.j..jjjj.jjj...jjjj.j.j..jjj...jj.jjj.j..jjj...jj.j..jjj.jjj...j.jjjj.j..jjj...j..jjjjj.jj.j...jjjj.jjj..j.j...jjjj..jj.jj.j...jj.jjjj..jj.j...jj.j..jjjjj.j...j.j..jjj.jjjj...

2.643.243.083.043.042.562.562.843.602.683.283.083.523.603.042.883.08

Fig. 3.

tribute to the perception of complexity. The trainingphase of the measure, which consisted of findingthe optimal parameters corresponding to variousfeatures of the rhythm, succeeded in determiningthe relative importance of these components.It should also be mentioned at this time that the

T-Measure and the LZ-Measure correlated withcomplexity judgements from the Essens (1995)experiment with values of r = 0.13 and r = 0.12respectively. The correlations with the judgementsfrom the experiment discussed here are r = 0.02and r = 0.15 for the T-Measure and the LZ-Measure, respectively. The low correlation valuesimply that these two measures are unlikely to beperceptually reliable measures of complexity oftemporal patterns.The T-Measure's poor performance is most likely

due to the lack of perceptual validity supporting itsuse. This measure is essentially based on theassumption that root patterns are the major con-tributing factors of complexity of temporal patterns.We are not aware of any studies supporting thisassumption. Moreover, the T-Measure does notpossess enough resolution to be able to accuratelypredict complexity judgements. For example, asshown above, the elaborations of a quarter notewould only permit the complexity to be 1, 2, or 3.The LZ-Measure is likely to perform much better

if applied to longer rhythms, since the LZ complex-ity is not well suited for short sequences, evenwhen they are assumed to be cyclical. After all,there is very little information embedded in a shortsequence and the assumption of it being cyclicaldoes not add any new information. It would thusbe appropriate to consider the LZ-Measure for usewith much longer rhythms.It is not surprising that the PS-Measure outper-

formed the T-Measure as well as the LZ-Measure,because the PS-measure incorporates perceptualinformation and is based on an empirically testedmodel of rhythm perception.While the PS-Measure has not been tested on

temporal patterns with timing deviations due toperformance, such as those found in real musicand which undoubtedly add another dimension tocomplexity, it nevertheless seems to be able to cap-ture the basic structural components of musicalrhythm that contribute to the perception of com-

plexity. We would like to emphasize, however, thatthe factors that we expected to contribute to thesubjects' notions of complexity are very much afunction of their cultural background. As is impli-citly assumed in the T-Measure and the PS-Measure, the temporal patterns we consider reflectWestern divisible rhythms. It is to be expected thata listener used to additive rhythms with complexduration ratios will have significantly differentjudgements of complexity for the same set of stimu-li than aWestern listener.In Shmulevich and Povel (1998), it is argued that

a successful and perceptually salient measure ofrhythm complexity can be used in a music patternrecognition system (Coyle & Shmulevich, 1998,Shmulevich et al., 1999) by allowing it to determinerelative weights of pitch and rhythm errors. ThePS-Measure seems to be an appropriate startingpoint for further development in this direction.

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Ilya ShmulevichSignal Processing LaboratoryTampere University of TechnologyP.O. Box 55333101 TampereFinland

Ilya Shmulevich received his Ph.D. degree in Electrical andComputer Engineering from Purdue University,West Lafay-ette, IN, USA, in 1997. In 1997-1998, he was a postdoctoralresearcher at the Nijmegen Institute for Cognition and Infor-mation at the University of Nijmegen in The Netherlands,where he studied computational models of music perceptionand recognition. Presently, he is a researcher at the TampereInternational Center for Signal Processing at the Signal Pro-cessing Laboratory in Tampere University of Technology,Tampere, Finland. His research interests include MusicRecognition and Perception, Nonlinear Signal and ImageProcessing, and Computational Learning Theory.

Dirk-Jan PovelNijmegen Institute for Cognition and Information (NICI)Nijmegen UniversityP.O. Box 91046500 HE NijmegenThe Netherlands

Dr. Dirk-Jan Povel is a senior researcher at NICI. He hasdone both theoretical and applied research in speech percep-tion and speech production, the perception of temporal pat-terns including musical rhythms, and the production of seri-al motor patterns. The applied work concerned thedevelopment of the `Visual Speech Apparatus' ö a speechteaching system for hearing impaired children. His currentresearch is in the field of music cognition, studying the on-line processes of music perception He has been a researchfellow in USA at Indiana University, Bloomington, Univer-sity of California, San Diego, University of California, SantaCruz and has taught at Northwestern University in Chicago.

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