Abstract
Milton Babbitt (1916–2011) was a composer of twelve-tone serial music noted for creating the all-partition array. One part of the problem in generating an all-partition array requires finding a covering of a pitch-class matrix by a collection of sets, each forming a region containing 12 distinct elements and corresponding to a distinct integer partition of 12. Constraint programming (CP) is a tool for solving such combinatorial and constraint satisfaction problems. In this paper, we use CP for the first time to formalize this problem in generating an all-partition array. Solving the whole of this problem is difficult and few known solutions exist. Therefore, we propose solving two sub-problems and joining these to form a complete solution. We conclude by presenting a solution found using this method. Our solution is the first we are aware of to be discovered automatically using a computer and differs from those found by composers.
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Notes
- 1.
Maximal diversity is the presentation of as many musical parameters in as many different ways as possible [2].
- 2.
- 3.
We denote an integer partition of an integer, L, by \(\mathrm {IntPart}_L(s_1,s_2,\ldots ,s_I)\) and define it to be an ordered set of non-negative integers, \(\langle s_1,s_2,\ldots ,s_I\rangle \), where \(L = \sum _{i=1}^{I} s_i\) and \(s_{1} \ge s_{2} \ge \cdots \ge s_{I}\).
- 4.
We define an integer composition of a positive integer, L, denoted by \(\mathrm {IntComp}_L(s_1,s_2,\ldots ,s_I)\), to also be an ordered set of I non-negative integers, \(\langle s_1,s_2,\ldots ,s_I\rangle \), where \(L = \sum _{i=1}^{I} s_i\).
- 5.
Examples of this matrix can be found in Babbitt’s My Ends are My Beginnings (1978) and Beaten Paths (1988), among others.
References
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Acknowledgments
The work of Tsubasa Tanaka reported in this paper was supported by JSPS Postdoctoral Fellowships for Research Abroad. The work of Brian Bemman and David Meredith was carried out as part of the project Lrn2Cre8, which acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET grant number 610859.
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Tanaka, T., Bemman, B., Meredith, D. (2016). Constraint Programming Approach to the Problem of Generating Milton Babbitt’s All-Partition Arrays. In: Rueher, M. (eds) Principles and Practice of Constraint Programming. CP 2016. Lecture Notes in Computer Science(), vol 9892. Springer, Cham. https://doi.org/10.1007/978-3-319-44953-1_50
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