Abstract
Representing musical notes as points in pitch-time space causes repeated motives and themes to appear as translationally related patterns that often correspond to maximal translatable patterns (MTPs) [1]. However, an MTP is also often the union of a salient pattern with one or two temporally isolated notes. This has been called the problem of isolated membership [2]. Examining the MTPs in musical works suggests that salient patterns may correspond more often to the intersections of MTPs than to the MTPs themselves. This paper makes a theoretical contribution, by exploring properties of patterns that are maximal with respect to their translational equivalence classes (MTEC). We prove that a pattern is MTEC if and only if it can be expressed as the intersection of MTPs. We also prove a relationship between MTECs and so-called conjugate patterns.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Meredith, D., Lemström, K., Wiggins, G.A.: Algorithms for discovering repeated patterns in multidimensional representations of polyphonic music. Journal of New Music Research 31, 321–345 (2002)
Collins, T., Thurlow, J., Laney, R., Willis, A., Garthwaite, P.H.: A comparative evaluation of algorithms for discovering translational patterns in Baroque keyboard works. In: Downie, J.S., Veltkamp, R.C. (eds.) Proceedings of the 11th International Symposium on Music Information Retrieval, Utrecht, The Netherlands, pp. 3–8 (2010)
Clifford, R., Christodoulakis, M., Crawford, T., Meredith, D., Wiggins, G.A.: A fast, randomised, maximal subset matching algorithm for document-level music retrieval. In: Lemström, K., Tindale, A., Dannenberg, R.B. (eds.) Proceedings of the 7th International Symposium on Music Information Retrieval, Victoria, Canada, pp. 150–155 (2006)
Collins, T.: Improved methods for pattern discovery in music, with applications in automated stylistic composition. PhD thesis, The Open University (2011)
Collins, T., Laney, R., Willis, A., Garthwaite, P.H.: Modeling pattern importance in Chopin’s mazurkas. Music Perception 28, 387–414 (2011)
Lemström, K.: Towards more robust geometric content-based music retrieval. In: Downie, J.S., Veltkamp, R.C. (eds.) Proceedings of the 11th International Symposium on Music Information Retrieval, Utrecht, The Netherlands, pp. 577–582 (2010)
Lubiw, A., Tanur, L.: Pattern matching in polyphonic music as a weighted geometric translation problem. In: Lomelà Buyoli, C., Loureiro, R. (eds.) Proceedings of the 5th International Symposium on Music Information Retrieval, Barcelona, pp. 154–161 (2004)
Meredith, D.: The ps13 pitch spelling algorithm. Journal of New Music Research 35, 121–159 (2006)
Meredith, D.: Point-set algorithms for pattern discovery and pattern matching in music. In: Proceedings of the Dagstuhl Seminar on Content-Based Retrieval (No. 06171). Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2006)
Meredith, D.: Computing pitch names in tonal music: a comparative analysis of pitch spelling algorithms. D.Phil. thesis, University of Oxford (2007)
Romming, C.A., Selfridge-Field, E.: Algorithms for polyphonic music retrieval: The Hausdorff metric and geometric hashing. In: Proceedings of the 8th International Symposium on Music Information Retrieval, Vienna, Austria, pp. 457–462 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Collins, T., Meredith, D. (2013). Maximal Translational Equivalence Classes of Musical Patterns in Point-Set Representations. In: Yust, J., Wild, J., Burgoyne, J.A. (eds) Mathematics and Computation in Music. MCM 2013. Lecture Notes in Computer Science(), vol 7937. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39357-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-39357-0_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39356-3
Online ISBN: 978-3-642-39357-0
eBook Packages: Computer ScienceComputer Science (R0)