Automated extraction of motivic patterns and application to the analysis of Debussy s Syrinx

Transcription

1 Automated extraction of motivic patterns and application to the analysis of Debussy s Syrinx Olivier Lartillot University of Jyväskylä, Finland lartillo@campus.jyu.fi 1. General Framework 1.1. Motivic pattern extraction The basic principle of motivic pattern extraction consists of detecting repeated motives, i.e. identifying several short extracts or subsequences as instances, or occurrences, of a same series of description called pattern. The approach is focused here on monodic sequences: music is considered as a series of notes without superpositions. Patterns are formalised as chains of states called pattern chains. As patterns can accept multiple alternative continuations, patterns chains can be extended by multiple branches. Hence, patterns are aggregated into one single tree, called pattern tree, and each pattern chain is as a branch of the tree. Similarly, each pattern occurrence can accept multiple alternative continuations. Hence, the set of all pattern occurrences that are initiated by one note forms a tree, called pattern occurrence tree Musical dimensions Patterns are detected along multiple musical dimensions, are indicated below. See (Lartillot and Toiviainen, in press), for a detailed explanation of the dimensions.

2 Figure 1. Descriptions of a monody. Repeated sequences of values, forming patterns, are enclosed in boxes. The musical dimensions are grouped into two distinct categories: - note-based descriptions: (absolute chromatic and diatonic pitches, pulsation and duration) are associated to each note individually. - interval-based descriptions: (pitch interval, gross contour and inter-onset interval) are related to intervals between successive notes. Different interval-based descriptions may be associated with one same note, depending on the choice of the second note defining the interval Matching strategy In a fuzzy definition of pattern matching, a numerical distance is defined, and a matching is made when the similarity distance is lower than a pre-specified threshold. Yet no heuristic for precisely fixing this value has been proposed. Hence, the determination of the threshold value relies entirely on the user s intuitive choices. Another solution consists of restricting more simply to exact matching along multiple musical dimensions (Conklin and Anagnostopoulou, 2001). We propose a generalisation of this multiple viewpoint approach that allows some variability in the set of musical dimensions used during the construction of each musical pattern. This enables us to take into consideration a more general type of pattern, called heterogeneous pattern, which despite its structural complexity seems to catch an important aspect of musical structure.

3 Interesting examples of heterogeneous patterns can be found in Debussy s Syrinx (cf. Fig 3): For instance, the first pattern, indicated by a dotted rectangle covering the whole first line, and the beginning of the second line, features a complete melodico-rhythmic description of its first bar (exactly repeated in the two occurrences), and a solely rhythmic description of the reminder of the pattern (one dotted eighth-note and two 32th-notes, associated to different melodic lines in each occurrence). 2. Controlling the Combinatorial redundancy The core pattern extraction process results in a large set of candidates, which, due to its large size and its poor quality, does not offer direct interest for musicology or music information retrieval. In order to control the combinatory explosion, filtering heuristics are generally added that select a sub-class of the result based on global criteria such as pattern length, pattern frequency (within a piece or among different pieces), etc. 1 The main limitation of this method comes from the lack of selectivity of these global criteria. Another approach is based on the search for maximal patterns, i.e. patterns that are not included in any other pattern (Zaki, 2005; Agraway et al., 1995). This approach still leads to an excessive filtering of important structures. Closed patterns, finally, are patterns whose support is higher than the support of the pattern in which they are included (Zaki, 2005). A filtering of non-closed patterns ensures a compact representation of the pattern configuration without any loss of information Formal Concept Representation of Patterns We propose an application of the closed pattern paradigm to the multidimensionality of music based on formal concept analysis (Ganter and Wille, 1999). The pattern description of a sequence of notes S is expressed as a formal context (N(S),, I) (Ganter and Wille, 1999) where : - the set of objects is N(S): the set of notes in S, - the set of attributes is : the set of elementary musical descriptions of all the intervals preceding each note, 1 See (Lartillot and Toiviainen, in press) for a complete review.

4 - and I is the binary relation between N(S) and, called incidence, defined by: (n i, ) belongs to I if and only if the description is correct. The derived description C' of a set of notes C from N(S) is defined as the common description of all these notes: C' = { in, such that, for all n in C, (n, ) belongs to I} The derived class D' of a complex description D of is dually defined as the set of notes complying with this description: D' = {n in N(S), such that, for all in D, (n, ) belongs to I} The pattern discovery task consists in finding exhaustive class D' sharing a same description D. The trouble is, lots of different descriptions D i may lead to same classes D i '. These derivators operations establish a Gallois connection between the power set lattices on N(S) and (Ganter and Wille, 1999). The Gallois connection leads to a dual isomorphism between two closure systems, whose elements, called formal concepts of the formal context (S(S),, I) corresponds exactly to the close patterns P=(C,D), verifying: C in N(S), D in, C'=D and D'=C. For a close pattern P=(C,D), C is called the extent of D and D the intent of C. We may simply call C and D respectively the class and the description of P. Hence, for a set of patterns P i = (D i,d i ) of same class D i = C, the close pattern P=(C,D) is described using the derived operator C' defined in equation: it contains all the elementary descriptions common to all notes of the class C. In other words, closed patterns are described as precisely as possible Specificity relations Closed patterns, or formal concepts, are naturally ordered by the subconceptsuperconcept relation (C 1,D 1 ) < (C 2,D 2 ) corresponding to an inclusion of C 1 into C 2 or, equivalently, an inclusion of D 2 into D 1 (Ganter and Wille, 1999). This subconcept-superconcept relation can also be called specificity relation: (C 1,D 1 ) is more specific than (C 2,D 2 ). These specificity relations can be drawn directly between patterns in the pattern tree, forming a directed edge from the node related to the less specific pattern to the node related to the more specific one. The set of all these edges form a specificity graph called pattern specificity graph.

5 These specificity relations can also be applied to pattern occurrences. The algorithm implemented in our model (Lartillot, 2005) is funded on a single chronological pass through the whole musical sequence. Hence, for each successive note n i, all the occurrences that it concludes are considered altogether. This set of pattern occurrences can be ordered along the specificity relation, by copying the subset of the pattern specificity graph associated with the patterns related to these occurrences. For each successive note n i, the specificity graph hence constructed is called pattern occurrence specificity graph and is denoted G i. Figure 2. The rhythmic pattern afghi is less specific than the melodico-rhythmic pattern abcde Cyclic patterns Combinatory explosions can be caused by successive repetitions of a same pattern (Cambouropoulos, 1998). The redundancy problem induced by the periodicity of patterns can be resolved in a simple and efficient manner by directly integrating a concept of cyclic pattern in the multidimensional closed pattern discovery framework. This requires a generalisation of specificity relations to cyclic patterns. This justifies also the incremental approach described in the previous paragraph. 3. Analysis of Debussy s Syrinx. The algorithm has been applied to the analysis of Debussy s Syrinx. Figure 3 shows the analysis of the beginning of the piece. Each different style of lines, rectangles or ovals indicates a particular motivic pattern. Cyclic patterns are represented by graduated lines, each graduation indicating the beginning of a new period.

6 4. From monody to polyphony Figure 3. Motivic analyis of Debussy s Syrinx. Our previous works were focused on the extraction of repeated sequence of contiguous descriptions from a single chain of multi-dimensional descriptions. In order to take into account more complex musical transformations, it is necessary to generalize the problem and tolerate in particular note insertions and deletions. For this purpose, the initial chain of descriptions commonly called syntagmatic chain in linguistics is transformed into a syntagmatic graph showing all the possible connections (or syntagmatic relations) between neighbouring notes. New patterns are formed through a progressive traversal of the syntagmatic graphs. We plan to generalize our approach to polyphony following the syntagmatic graph principle. We are developing algorithms that construct, from polyphonies, syntagmatic chains representing distinct monodic streams. These chains may be intertwined, forming complex graphs along which the pattern discovery algorithm will be applied. The additional factors of combinatorial explosion resulting from this

7 generalized framework will require further adaptive filtering mechanisms. Patterns of chords may also be considered in future works. References Agrawal, R., & Srikant, R. (1995). Mining sequential patterns, International Conference on Data Engineering (pp. 3-4). IEEE Computer Society Press. Cambouropoulos, E. (1998). Towards a general computational theory of musical structure. Unpublished doctoral dissertation, University of Edinburgh, UK. Cambouropoulos, E. (2006). Musical parallelism and melodic segmentation: A computational approach. Music Perception, 23(3), Conklin, D., & Anagnostopoulou, C. (2001). Representation and discovery of multiple viewpoint patterns. International Computer Music Conference (pp ). International Computer Music Association. Ganter, B., & Wille, R. (1999). Formal concept analysis: Mathematical foundations, Springer-Verlag. Lartillot, O. and Toiviainen, P. (in press). Motivic matching strategies for automated pattern extraction. Musicae Scientiae. Meredith, D., Lemström, K., & Wiggins, G. (2002). Algorithms for discovering repeated patterns in multidimensional representations of polyphonic music. Journal of New Music Research, 31(4), Zaki, M. (2005). Efficient algorithms for mining closed itemsets and their lattice structure, IEEE Transactions on Knowledge and Data Engineering, 17,