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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- 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.
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Acknowledgments
This work was supported by JSPS Postdoctoral Fellowships for Research Abroad.
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© 2015 Springer International Publishing Switzerland
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Tanaka, T., Fujii, K. (2015). Describing Global Musical Structures by Integer Programming on Musical Patterns. In: Collins, T., Meredith, D., Volk, A. (eds) Mathematics and Computation in Music. MCM 2015. Lecture Notes in Computer Science(), vol 9110. Springer, Cham. https://doi.org/10.1007/978-3-319-20603-5_5
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DOI: https://doi.org/10.1007/978-3-319-20603-5_5
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