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Generating Milton Babbitt's all-partition arrays

JOURNAL OF NEW MUSIC RESEARCH 2016年第2期45卷 184-204页

作者： Bemman, Brian Meredith, David Aalborg Univ Rendsburggade 14 DK-9000 Aalborg Denmark

In most of Milton Babbitt's (1916-2011) works written since the early 1960s, both the pitch and rhythmic content is organized according to a highly constrained structure known as the all-partition array. The all-partition array provides a framework that ensures that as many different forms of a tone row as possible (generated by any combination of transposition, inversion or reversal) are expressed horizontally' and that each integer partition of 12 whose cardinality is no greater than the number of lynes in a piece is expressed by exactly one vertical' aggregate. We present a greedy backtracking algorithm for generating a particular type of all-partition array found in Babbitt's works, known as a Smalley array. Constructing such an array is a difficult task, and we present two heuristics for helping to generate this type of structure. We provide the parameter values required by this algorithm to generate the specific all-partition arrays used in three of Babbitt's works. Finally, we evaluate the algorithm and the heuristics in terms of how well they predict the sequences of integer partitions used in two of Babbitt's works. We also explore the effect of the heuristics on the performance of the algorithm when it is used in an attempt to generate a novel array.

关键词： music analysis Milton Babbitt all-partition array pitch class set theory algorithms music theory

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Constraint Programming Approach to the Problem of Generating Milton Babbitt's all-partition arrays 22nd

Constraint Programming Approach to the Problem of Generating...

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22nd International Conference on the Principles and Practice of Constraint Programming (CP)

作者： Tanaka, Tsubasa Bemman, Brian Meredith, David UPMC CNRS STMS Lab IRCAM Paris France Aalborg Univ Aalborg Denmark

ISBN: (纸本)9783319449531;9783319449524

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.

关键词： Babbitt all-partition array Computational musicology Constraint programming

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