Melodic Cluster ing: Motivic Analysis of Schumann’s Träumerei

Emilios Cambouropoulos and Gerhard Widmer

Austrian Research Institute for Artificial IntelligenceSchottengasse 3, A-1010 Vienna, Austria

{emilios, gerhard}@ai.univie.ac.at

Abstract

In this paper a formal model will be presented that attempts to organise melodic segments into ‘significant’musical categories (e.g. motives). Given a segmentation of a melodic surface, the proposed model constructs anappropriate representation for each segment in terms of a number of attributes (these reflect melodic andrhythmic aspects of the segment at the surface and at various abstract levels) and then a clustering algorithm (theUnscramble algorithm) is applied for the organisation of these segments into ‘meaningful’ categories. Theproposed clustering algorithm automatically determines an appropriate number of clusters and also thecharacteristic (or defining) attributes of each category. As a test case this computational model has been used forobtaining a motivic analysis of Schumann’s Träumerei.

Keywords: motivic analysis, musical categories, similarity, clustering

1. Introduction

Making sense of a musical work means beingable to break it down into simpler componentsand to make associations between them.Musical analysis is geared towards providingthe ‘ resolution of a musical structure intorelatively simpler constituent elements, andthe investigation of those elements within thatstructure’ (Bent, 1980: 340).

Paradigmatic analysis (Nattiez 1975, 1990) isan analytic methodology that aims atproviding a segmentation of a musical surfaceand an organisation of the derived musicalsegments into ‘significant’ musical categories(or paradigms). This methodology reliesheavily on a notion of musical similarity andaims at assisting a human analyst to explicatehis/her own similarity criteria for obtaining acertain analysis.

In relation to the above, the current paperaddresses the following problem: given asegmentation of a melodic surface, how can acomputational system be developed that canarrive at a ‘plausible’ clustering of the givensegments and at the same time provide explicitdescriptions of each category/cluster?

In the problem description stated in theprevious paragraph, melodic segmentation istaken to be a pre-requisite. This is asimplification so that we can focus on thecategorisation problem. As has been shownelsewhere (Cambouropoulos 1998)segmentation is strongly associated with

similarity and categorisation, e.g. a ‘good’categorisation of musical segments may suggest a‘preferred’ segmentation that may actuallyoverride local perceptual grouping indications.The relation between segmentation andcategorisation is a very complex topic; a study ofthis relation is currently in progress and someresults will appear in a forthcoming paper.

There has been a number of attempts to useclustering techniques for organising melodicsegments into paradigms. A brief survey andcomparison of some existing formal models ispresented in (Anagnostopoulou et al., 1999).Some difficulties of existing clusteringapproaches for musical purposes will beaddressed later on in this text.

The main topics that will be discussed in thispaper are: representation of melodic segments,pattern-processing techniques, clusteringtechniques and intensional description ofemerging clusters. As a test case the proposedcomputational model will used for obtaining amotivic analysis of the melody of R. Schumann’sTräumerei (this is the 7th piece fromKinderszenen, op. 15).

2. Motivation

The construction of a sophisticated computationalsystem for organising melodic segments into‘meaningful’ categories, apart from theoreticalinterest for the domains of musical analysis andmusical cognition, provides a potentially veryuseful tool for a number of applications such as

interactive composition and performance,musical data indexing and retrieval,expressive machine performance, and so on.The main argument is that, the moresophisticated musical computer applicationsone envisages to develop, the more‘understanding’ of musical structure acomputational system should have.

As the current study is undertaken within theframework of a project titled ‘ArtificialIntelligence models of musical expression’ wewill elaborate on the usefulness of structuralanalysis for the study of expressive features ofperformance.

The aim of the project on musical expressionis to study expressive features of musicalperformance (at this stage tempo and loudnessmicro-variations) and to develop learningalgorithms that can induce general rules ofexpressive music performance from examplesof real performances by human musicians.

In a double experiment reported by G.Widmer(1996) expression rules were induced fromreal melodic performances a) at the note-leveland b) at the structural level (motivic andphrase level). The results obtained indicatedthat learning was significantly improved whenstructural aspects of the melody were takeninto account. It was therefore stronglysuggested that musical structural informationis indispensable for the development of amachine model of musical expression.

In a detailed study of expressive features inSchumann’s Träumerei Bruno Repp (1992)presents at the outset a primarily motivicanalysis of the melodic gestures of this piece(see Figure 4). This analysis is an intuitiveanalysis performed by a human and is usedthroughout the main body of the paper forstudying the micro-timing variations of 28different performances of Träumerei. Again,metric and rhythmic structural information isthought to be paramount for the study ofmusical expression.

In this paper, we have used the melody ofTräumerei as a test case to study the proposedclustering model. The segmentation providedin Repp’s paper is taken as an input in thecurrent study. The machine-generated resultsobtained herein will be compared to and tested

against the human analysis provided in Repp’sstudy.

We believe that research on musical expressioncan benefit from the development of moresophisticated machine models of musicalstructure but we hypothesise that the reverse mayalso be true, namely that the study of musicalexpression may provide cues to structuralanalytic models in order to disambiguate complexmulti-faceted analytic results and to fine-tune allsorts of parameters and preference rules in suchmodels. This will be a topic of futureinvestigations.

3. Representation of melodic segments

Let us suppose that at the lowest level ofrepresentation each melodic segment isrepresented as a string of notes. The question thatarises is whether this representation is sufficientor whether further processing of the melodicsegments is necessary before presenting them asinput to a clustering algorithm.

The reply to the above question depends on whatnotion of musical similarity one intends toemploy as a basis for pattern processing andclustering. A more detailed discussion on thistopic can be found in (Cambouropoulos et al,1999). Here, we will only present two mainstrategies:

a) Two musical segments - represented asvectors of notes or intervals - are construed asbeing similar if they share at least a certainnumber of their component elements (notesor intervals); approximate pattern-matchingalgorithms and the edit distance arecommonly used in this approach (e.g.Rolland et al, 2000).

b) Two musical segments - represented asvectors of a number of patterns for variousparameters at many levels of abstraction - areconstrued as being similar if they share atleast a certain number of patterns at thesurface level or reductions of it; this strategyrequires more sophisticated representation ofmusical segments; exact pattern-matchingtechniques and the hamming distance can beused in this approach.

Figure 1 illustrates these possibilities with asimple melodic example. Of course, there can beall sorts of variations or even combinations of theabove two main strategies.

Figure 1. These two melodic segments may beconstrued as similar either because five of their

pitches and onsets match at the surface level(approximate matching) or because they matchexactly at the reduced eighth-note metric level.

In this study a strategy very close to thesecond approach given above has beenemployed. For each melodic segment ofTräumerei the following pattern attributes arecomputed (see example in Figure 2):

a. Surface level• Pitch-intervals: exact (p_ex), scale-steps (p_ss), step-leap (p_sl), step-doublestep-leap (p_sdsl), doublestep-leap(p_dsl), and contour (p_contour).• Rhythm: exact durations (r_dur), inter-onset interval (r_ioi), short-long (r_sl),inter-onset interval ratios (r_ratio),shorter-longer-equal (r_sel)

b. Quarter-note level plus notes on boundariesof segments.• Pitch-intervals: exact (p_ex), scale-steps(p_ss), step-leap (p_sl), step-doublestep-leap (p_sdsl), doublestep-leap (p_dsl).• Rhythm: inter-onset interval (r_ioi),short-long (r_sl)

c. Quarter-note level.• Pitch-intervals: exact (p_ex), scale-steps (p_ss), step-leap (p_sl), step-doublestep-leap (p_sdsl)

Note: in r_sl, durations longer than a certainvalue (in this instance a quarter) arerepresented merely by ‘ long’ .

Further research is necessary to establish aplausible and quite general set of levels ofabstraction and reduction. The above set givesgradually less prominence to the more abstractreduction levels and gives a slight preferenceto parameters related to pitch as it takes intoaccount fewer attributes for gradually moreabstract reduction levels and for rhythmicaspects of the melody. The initial set ofattributes can be modified by the clusteringalgorithm - in terms of altering attributeweights - so that more important attribute-values for the specific context of the givenpiece may be highlighted (see next section).

Near-exact matching is an additionaltechnique incorporated in the currentproposal. Limited approximate matching isemployed in the following manner: twopattern attributes match if they are identical or

if they contain a common ‘significant’ sub-pattern (this of course includes the case whereone of the two is a sub-pattern of the other). Thenotion of a ‘significant’ sub-pattern is controlledby a parameter that defines the relation of the sizeof the sub-pattern to the two patterns (e.g. sub-pattern should be at least 70% of the size of eachpattern). A sub-pattern may consist only ofcontiguous elements from the given super-pattern. Exact matching techniques can be usedfor determining near-exact matches.

Melodic surface

p_ex: 1 - 4 - 3 - 5 - 0p_ss: 1 - 2 - 2 - 3 - 0p_sl: step - leap - leap - leap- 0p_sdsl: step - dstep- dstep- leap- 0p_dsl: dstep- dstep- dstep- leap- 0p_contour: U - U - U - U - 0

r_dur: 1/8- 1/8 - 1/8 - 1/8 - 1/8 - 1/2r_ioi: 1/8- 1/8 - 1/8 - 1/8 - 1/8 - 1/2r_sl: 1/8- 1/8 - 1/8 - 1/8 - 1/8 - longr_ratio: 1 - 1 - 1 - 1 - 4r_sel: equal- equal- equal- equal- longer

Notes on quarter beats and on boundaries (reduction)

p_ex: 1 - 7 - 5p_ss: 1 - 4 - 3p_sl: step - leap - leapp_sdsl step leap leapp_dsl: dstep - leap - leap

r_ioi: 1/8 - 1/4 - 1/4 - 1/2r_sl: 1/8 - 1/4 - 1/4 - long

Notes on quarter beats (reduction)

p_ex: 7 - 5p_ss: 4 - 3p_sl: leap - leapp_sdsl leap leap

Figure 2 Attribute-values for one melodic segment(surface and reductions); each row of alphanumericsymbols constitutes a single attribute-value, i.e. this

melodic pattern is represented by a vector of 22attribute-values – see text for the description of

abbreviations.

A very simple example of near-exact matching ispresented in Figure 3. This kind of partial

matching is valuable in that it allows the useof a smaller set of pattern attributes (e.g. thetwo patterns in Figure 3 could actually bematched exactly at a high enough reductionlevel such as the 2/4 metric level). It may be,however, problematic as it incorporates in therepresentation itself a degree of similaritybefore any distance metrics are applied by theclustering algorithm.

Figure 3 These two melodic segments (fromTräumerei) share common sub-patterns for both thepitch and rhythm profiles (actually the first is a sub-pattern of the second) - that is, their attribute-values

for both pitch and rhythm are near-exact.

At the current stage, the computational systemrequires as input the melodic surface and aselected segmentation, and then itautomatically generates the segment attribute-value vectors to be used by the clusteringtechnique.

4. The Unscramble cluster ing algor ithm

In this section a brief description will be givenof the Unscramble clustering algorithm thathas been developed primarily for dealing withclustering problems in the musical domain(Cambouropoulos and Smaill, 1997).

The Unscramble algorithm is a clusteringalgorithm which, given a set of objects and aninitial set of properties, generates a range ofplausible clusterings for a given context.During this dynamically evolving process theinitial set of properties is adjusted so that asatisfactory description is generated. There isno need to determine in advance an initialnumber of clusters nor is there a need to reacha strictly well-formed (e.g. non-overlapping)categorisation. At each cycle of the processweights are calculated for each propertyaccording to how characteristic each propertyis for the emergent clusters. This algorithm isbased on a working definition of similarityand category that inextricably binds the twotogether.

4.1 A Working Formal Definition ofSimilar ity and Categor isation

Let T be a set of entities and P the union of allthe sets of properties that are pertinent for thedescription of each entity. If d(x,y) is the distancebetween two entities x and y, and h is a distancethreshold we define similarity sh(x,y) as follows:

1 iff d(x,y) ≤ h (similar objects)sh(x,y) {

0 iff d(x,y) > h (dissimilar objects)(I)

In other words, two entities are similar if thedistance between them is smaller than a giventhreshold and dissimilar if the distance is largerthan this threshold.

The above definition of similarity is brought intoa close relation with a notion of category. Thatis, within a given set of entities T, for a set ofproperties P and a distance threshold h, acategory Ck is a maximal set:

Ck={x1,x2,...xn/xi∈T} with the property:∀i,j∈{1,2,...n}, sh(xi,xj)=1 (II)

In other words, a category Ck consists of amaximal set of entities that are pairwise similarto each other for a given threshold h.

A category, thus, is inextricably bound to thenotion of similarity; all the members of acategory are necessarily similar and a maximalset of similar entities defines a category.

As the similarity function sh is not transitive, theresulting categories need not be disjoint (i.e.equivalence classes). In other words, overlapbetween categories is permitted.

The distance threshold may take values in therange of 0≤h≤dmax where the distance dmax isdefined as the maximum distance observedbetween all the pairs of entities in T. For h=0every object in T is a monadic category; forh=dmax all the objects in T define a singlecategory.

4.2 The Unscramble algor ithm

The above definitions of similarity and categorycan be restated in the terminology of graphtheory as follows: objects are represented byvertices in an undirected graph, similaritybetween similar objects is represented by edges,and categories are defined as maximal cliques (amaximal clique is a maximal fully connectedsub-graph). We will use this terminology below

for the description of the Unscrambleclustering algorithm.

It should also be noted that in the context ofthis paper ‘properties’ are taken to mean‘binary features’ that correspond to aparticular ‘attribute-value’ pair.

4.2.1 Algor ithm input

The input to the Unscramble algorithm is aset of N objects each described by an m-dimensional property vector (e.g. object x:[p1,p2,...pm] and object y: [q1,q2,...qm] ). Eachproperty has a corresponding initial weightwp=1. The distance between two objects isgiven by the following function (based on theHamming distance): md(x,y)= Σwpi

·wqi·δ(pi,qi) (III)

i=1

where: δ(pi,qi) = 0 if pi=qi

δ(pi,qi) = 1 if pi≠qi

4.2.2 The algor ithm

The algorithm proceeds in cycles; in eachcycle, firstly, all the possible thresholds arecalculated, then for each threshold anundirected graph is constructed (edgesconnect similar objects), then for each graphmaximal cliques are enumerated, then foreach clustering a ‘goodness’ value iscomputed and finally the clustering with thehighest ‘goodness’ value is selected and newweights for all the properties are computed. Amore detailed description is given below:

Step 1. All the possible threshold values h arecalculated. The number of thresholds l isequal to the number of possible distancesbetween the N objects of set T; lmax = N·(N-1)/2 - it often is smaller as some entities areequidistant.Step 2. For each of these thresholds, all thesimilar objects are computed according todefinition (I) and (III) and an undirectedgraph for each threshold is created whereedges connect similar objects.Step 3. All the maximal cliques (II) arecomputed for each of these graphs, resultingin l different clusterings.Step 4. For each of the l clusterings a‘goodness’ value is calculated according tofunction (IVa,b).Step 5. The clustering that rates highestaccording to the ‘goodness’ function is

selected and new weights are calculatedaccording to function (V).Step 6. The algorithm is repeated from step 1 forthe new weights.Step 7. The algorithm terminates when the newlycalculated ‘goodness’ value is less or equal to thevalue that resulted during the immediatelypreceding run.

4.2.3 Additional fundamentals

The following definitions are also necessary forthe algorithm:

4.2.3.1 ‘Goodness’ of cluster ing

As the Unscramble algorithm (section 4.2.1)generates a large number of clusterings (one foreach possible similarity threshold) it is necessaryto define some measure of ‘goodness’ for eachclustering so as to select the best. Two suchmeasures have been considered:

a. Overlap FunctionOne simple criterion for selecting preferredclusterings is a measure for the degree by whichclusters overlap. The less overlapping betweenclusters the better. An overlap function OL couldbe defined as:

kOL = 1 – N / Σni where: (IVa)

i=1k = number of clustersN = number of objects in Tni = number of objects in cluster Ci

The problem with such a measure is that in theextreme cases where each object is a cluster ofitself and where all the objects form a singlecategory overlapping is necessarily zero. It isthus necessary either to set ad hoc limits forminimum and maximum allowed number ofclusters or to multiply the overlapping value by afunction that has values close to 1 for a preferredrange of number of clusters and values close tozero for the extremes of either too many or onlyone cluster. This ad hoc parametric bias isavoided in the next measure.

b. Category UtilityCategory Utility (Gluck & Corter, 1985; Fisher,1986) is a measure that rates the homogeneity ofa clustering. Given a universe of entities T, a setof (binary) properties P={p1,...,pm} describing theentities, and a grouping of entities in T into kclusters C={c1,...,ck}, category utility is definedas:

k m m

ΣP(ci) ·[ΣP(pj/ci)2 - ΣP(pj)2] i=1 j=1 j=1CU({c1,...,ck}) = k

where:(IVb)

P(ci) is the probability of an entity belongingto cluster ci,

P(pj) is the probability that an entity hasproperty pj, and

P(pj/ci) is the conditional probability of pj,given cluster ci.

Probabilities are estimated by relativefrequencies.

CU favours categorisations with highuniformity (in terms of properties) withinindividual clusters (‘ intra-class similarity’ )and strong differences between clusters(‘ inter-class dissimilarity’ ).

Another way of interpreting this is thatcategory utility measures the predictionpotential of a categorisation: it favoursclusterings where it is easy to predict theproperties of an entity, given that one knowswhich cluster it belongs to, and vice versa.The main advantages of this measure are itsfirm grounding in statistics, its intuitivesemantics, and the fact that it does not dependon any parameters.

Both of these ‘goodness’ functions have beenapplied and compared in the example insection 5 (see also Appendices I and II).

4.2.3.2 Weighting function.

When a clustering is selected, then the initialweights of properties can be altered inrelation to their ‘diagnosticity’ , i.e. propertiesthat are unique to members of one categoryare given higher weights whereas propertiesthat are shared by members of one categoryand its complement are attenuated. Afunction that calculates the weight of a singleproperty p could be:

w = m/n-m’/(N-n) where: (V)

m = number of objects in category Ck thatpossess property p

m’ = number of objects not in category Ck thatpossess property p (i.e. objects in T-Ck)

n = number of objects in CkN = number of objects in T

The weights of each property calculated for eachcategory can then be averaged and normalisedfor a given clustering. The whole process may berepeated for the new set of weighted propertiesuntil the terminating conditions of Unscrambleare met.

4.2.4. Complexity issues and mer its ofUnscramble

The enumeration of maximal cliques in anundirected graph is known to be an NP-completeproblem; an extended overview of algorithms formaximum and maximal clique finding algorithmsis presented in (Bomze et al, 1999). However, forsmall graphs this need not be a serious problem.Most musical categorisation tasks would involvetens or maybe a few hundreds of musicalsegments rather than thousands. It is alsopossible to consider using a semi-incrementalversion of the algorithm whereby objects areclustered in small chunks rather than all at once(this option is considered for further research).

An additional problem is that in each cycle ofUnscramble all maximal cliques have to becomputed for all the graphs that correspond toeach threshold (maximum number of thresholds:lmax = N·(N-1)/2 where N = number of objects) -i.e. the maximal clique enumeration algorithmhas to be applied ~N2 times in the worst case! Asolution to this problem has been given byE.Bomze and his colleagues (at the departmentof Statistics, University of Vienna) that is basedon the observation that this problem is equivalentto finding all the maximal cliques in a singlegradually evolving graph. According to thisdescription of the problem edges are added one-by-one in an empty graph of N vertices until afully connected graph is reached (or the reverse);each newly added edge is examined as to how itmodifies the previously determined cliques. Thisevolving clique finding algorithm provides anefficient solution to the aforementioned problem.

The most useful characteristics of theUnscramble algorithm - depending on the task athand - are the following:• there is no need to define in advance a number

of categories• the prominence of properties is discovered by

the algorithm• categories may overlap.

Some examples of the usefulness of suchclustering characteristics will be presented in themusical test case in the next section.

5. Motivic analysis of Träumerei

A detailed melodic and rhythmic analysis ofR.Schumann’s Träumerei is presented in(Repp, 1992). In this study we limit ourselvesstrictly to the soprano voice ignoring melodicgestures that appear in other voices. Thesoprano part of Träumerei, along with thesegmentation provided by Repp, is depicted inFigure 4.

Table 1 presents the clustering of melodicsegments that is used in Repp’s study andTables 2 and 3 present the clustering producedby the proposed computational system. Thetables in this section are taken to be schematicrepresentations of the score in Figure 4. In thebold outlined tables, rows correspondgraphically to staffs in Figure 4; vertical linescorrespond to segmentation points in staffs;table entries correspond to melodic segments;alphabet symbols correspond to names ofmelodic clusters. The first column, outside thebold part of the tables, indicates the structureat the level of melodic periods, i.e. 8-barsections, as proposed by Repp; the secondcolumn indicates the phrase structure, i.e. 4-bar sections.

In this experiment, the Unscramble algorithmhas been applied on the melodic segments inFigure 4; each segment is represented byattributes of equal initial weight at the surfacelevel and reductions of it for various pitch andrhythmic parameters as described in section 3.The number of relevant clusters is computedby the algorithm – there’s no need topredefine the number of clusters. Both of the‘goodness’ functions described in section4.2.3.1 have been used yielding in this casethe same results (see appendices I and II).

When the attributes for each segment arecalculated employing only exact matchingthen the results given in Table 2 are obtained.The main two clusters (a and b) emergesuccessfully along with clusters e and f;cluster c is selected because its members sharean identical rhythmic pattern of 4 eighth-noteswhich is characteristic of the category. Thisclustering captures some of the main strongmelodic categories that are given in Repp’sanalysis but over-emphasises the rhythmicaspects of cluster c and also totally missessome other important categories (see emptytable entries).

A1 a b c d eA B1 a b f g

B2 a b f gB B3 a b f g

A1 a b c d eA’ A2 a b c d e h

Table 1 Organisation of Träumerei’s melodicsegments according to Repp (1992).

A1 a b c c eA B1 a b

B2 a b fB B3 a b f

A1 a b c c eA’ A2 a b c c

Table 2 Machine organisation of Träumerei’smelodic segments when exact matching is employed atthe surface level and at reduced levels (empty entries

signify monadic categories).

A1 a b c d eA B1 a b

B2 a b f gB B3 a b f g

A1 a b c d eA’ A2 a b c d d d,e

Table 3 Machine organisation of Träumerei’smelodic segments when near-exact matching is

employed at the surface level and at reduced levels.

When near-exact matching is introduced then theresults given in Table 3 are obtained. In this case,there are some additional clusters: cluster c isnow the same as the one in Table 1; cluster dincludes the last two segments of the melody andalso overlaps on the last segment with cluster e;cluster g is also discovered; there are only twosegments which remain monadic.

Some comments regarding the differences withRepp’s melodic/rhythmic organisation of thismelody will now be presented. Firstly, Reppchooses to group the second-to-last melodicsegment with ‘similar’ segments in cluster ealthough this segment is identical – at least inregard to the melodic/rhythmic properties that areexamined in this study - to segments in cluster d;we conjecture that this choice is taken by Reppbecause of top-down considerations – forinstance

Figure 4 The soprano part of Schumann’s Träumerei, along with segmentation provided by B. Repp (1992)

phrases 1, 5 and 6 may be seen a being overall‘parallel’ (A1, A2, A1) so their last parts arealso considered as being similar. TheUnscramble algorithm looks only at theselected internal attributes of segments soplaces this segment in cluster d. Secondly, inRepp’s analysis the last segment is regardedas an independent monadic cluster h whereasUnscramble places it with clusters d and e.Thirdly, Unscramble fails to group the two‘ left-over’ segments (see empty entries intable 3) with clusters f and g respectively (amore sophisticated representation couldcapture the similarity for cluster f – however,it seems that for cluster g higher-levelconsiderations of similarity at the phrase-levelmay be necessary). Overall the two analysesare quite similar; one may even claim that

Unscramble yields some new ‘ intuitions’ about themelodic/rhythmic organisation of this melodic partsuch as the overlap of the two clusters on the lastsegment of the piece or the grouping of the second-to-last segment with members of cluster d.

The Unscramble algorithm not only clustersmelodic segments but also highlights theircharacteristic or defining properties. The weightingfunction described in section 4.2.3.2 reinforces‘diagnostic’ properties for discovered clusters andattenuates properties that are less characteristic. Forinstance, members of cluster b share the samedefining step-leap pitch attribute-value (‘step-leap-leap-leap-equal’ ) at the surface level (common toall members of this cluster and shared by nomembers of other clusters) and thus this attribute-value is a defining attribute (weight=1); this clusteris also characterised by the doublestep-leap

attribute-value ‘dstep-dstep-dstep-leap-equal’which is shared by all members except thethird instance (‘dstep-dstep-dstep-dstep-equal’ ) and receives a strong weight w=0.83; asimilar description for rhythm is given byUnscramble. As another example,Unscramble finds that members of cluster cshare the same pattern attribute for thedoublestep-leap representation (the first twoinstances are sub-patterns of the last); theyalso share the same rhythmic pattern but thisreceives a much lower weight as this pattern isalso shared by members of cluster d.

Such a weighting mechanism can be usedeffectively for the description of clusters interms of defining and characteristic attributes,which in turn can be used for further melodicpattern prediction tasks (not as yet fullyinvestigated).

Conclusion

In this paper a formal model that organisesmelodic segments into ‘significant’ musicalcategories was presented. Given asegmentation of a melodic surface, theproposed model constructs an appropriaterepresentation for each segment in terms of anumber of properties (these reflect melodicand rhythmic aspects of the segment at thesurface and at various abstract levels) and,then, the Unscramble algorithm organisesthese segments into ‘meaningful’ categories.The Unscramble clustering algorithmautomatically determines an appropriatenumber of clusters, allowing someoverlapping between clusters and alsohighlights the characteristic or definingattributes of each category.

As a test case, this computational model wasused for obtaining a melodic and rhythmicanalysis of the soprano-part of Schumann’sTräumerei. This machine-generated analysiswas compared with an analysis performed bya human music analyst; it was shown that theproposed model is capable of producing an‘acceptable’ analysis that has a significantamount of resemblance to the human analysis

Acknowledgements

This research is part of the project Y99-INF,sponsored by the Austrian Federal Ministry of

Education, Science, and Culture in the form of a STARTResearch Prize. The Austrian Research Institute forArtificial Intelligence is supported by the AustrianFederal Ministry of Education, Science, and Culture.

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Appendix I Graphs of the clustering ‘goodness’ values that occur during the first cycle of the Unscramblealgorithm when applied to the melodic segments of Träumerei. The x-axis indicates threshold values (that

correspond to individual clusterings); the y-axis indicates ‘overlap’ and ‘category utility’ values. During the firstcycle, the two ‘goodness’ value functions select different clusterings (different peaks of the graphs).

Appendix I I Graphs of the clustering ‘goodness’ values that occur during the last cycle of the Unscramblealgorithm when applied to the melodic segments of Träumerei. The x-axis indicates threshold values; the y-axisindicates ‘overlap’ and ‘category utility’ values. During the last cycle, the two ‘goodness’ value functions select

the same final clustering and have a very similar overall outlook (peaks at the same positions).