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A Structural Theory of Rhythm Notation Based on Tree Representations and Term Rewriting Read chapter Read first chapter

2015 | OriginalPaper | Chapter

Describing Global Musical Structures by Integer Programming on Musical Patterns

Authors : Tsubasa Tanaka, Koichi Fujii

Published in: Mathematics and Computation in Music

Publisher: Springer International Publishing

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Abstract

Music can be regarded as sequences of localized patterns, such as chords, rhythmic patterns, and melodic patterns. In the study of music generation, how to generate sequences that are musically adequate is an important issue. In particular, generating sequences by controlling the relationships between local patterns and global structures is a difficult and open problem. Whereas grammatical approaches, which examine global structures, can be used to analyze how a piece is constructed, they are not necessarily designed to generate new pieces by controlling the characteristics of global structures, such as the redundancy of a sequence or the statistical distribution of specific patterns. To achieve this, we must overcome the difficulty of solving computationally complex problems. To deal with this problem, we take an integer-programming-based approach and show that some important characteristics of global structures can be described only by linear equalities and inequalities, which are suitable for integer programming.

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previous chapter Genetic Algorithms Based on the Principles of Grundgestalt and Developing Variation
next chapter Improved Iterative Random Walk for Four-Part Harmonization
Footnotes
1
The study [8] focuses on the redundancy of musical sequence, and models the musical style by referring to the Lempel-Ziv compression algorithm. We extend this perspective and pay attention to the redundancies of multiple levels simultaneously.
 
2
In this paper, in order to avoid the explanation from being complicated, we only treat the stereotype examples of musical pieces whose groupings are always combinations of two consecutive elements in every level. However, in practice, such structures should vary depending on the specifications of the pieces that the user wants to create. For example, we can think of the case where the number of combination in the groupings are different between the levels. We can also think of the case where the consecutive patterns can be overlapped as is mentioned in [9]. How to formulate such cases is an important future issue.
 
3
In practice, it is not necessary to stick to existing pieces.
 
4
For example, they can be set depending on user’s preference or statistics of the original piece. If \(L_{j,i} \le 1\) and \(H_{j,i}\) is larger than or equal to the length of the sequence, these equations give no limitation to the number of each pattern that appears in the sequence.
 
5
However, at the current moment, we do not know an alternative way to implement the constraints for state transitions and frequencies of each pattern in the high levels.
 
6
Here, we can also represent the pitches and the intervals based on the chromatic scale. However, we use the scale degrees and the interval numbers based on the diatonic scale because that is more efficient.
 
Literature
1.
Pachet, F.: The continuator: musical interaction with style. J. New Music Res. 32(3), 333–341 (2003)CrossRef
2.
Lerdahl, F., Jackendoff, R.: A Generative Theory of Tonal Music. MIT Press, Cambridge (1983)
3.
Hamanaka, M., et al.: Implementing “A generative theory of tonal music”. J. New Music Res. 35(4), 249–277 (2006)CrossRef
4.
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)CrossRefMATH
5.
Tanaka, T., Fujii, K.: Melodic pattern segmentation of polyphonic music as a set partitioning problem. In: Proceedings of International Congress on Music and Mathematics (to be published)
6.
7.
Herremans, D., Srensen, K.: A variable neighbourhood search algorithm to generate first species counterpoint musical scores. Working Paper, University of Antwerp Faculty of Applied Economics Operations Research Group ANT/OR (2011)
8.
Lartillot, O., et al.: Automatic Modeling of Musical Style. 8èmes Journées d’Informatique Musicale 2001, 113–119 (2001)
9.
Mazzola, G., et al.: The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Birkhäuser, Basel (2002)CrossRefMATH
Metadata
Title
Describing Global Musical Structures by Integer Programming on Musical Patterns
Authors
Tsubasa Tanaka
Koichi Fujii
Copyright Year
2015
Book
Mathematics and Computation in Music

Print ISBN: 978-3-319-20602-8

Electronic ISBN: 978-3-319-20603-5

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-20603-5_5

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