Exact Cover Problem in Milton Babbitt’s All-Partition Array

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9110)


One aspect of analyzing Milton Babbitt’s (1916–2011) all-partition arrays requires finding a sequence of distinct, non-overlapping aggregate regions that completely and exactly covers an irregular matrix of pitch class integers. This is an example of the so-called exact cover problem. Given a set, A, and a collection of distinct subsets of this set, S, then a subset of S is an exact cover of A if it exhaustively and exclusively partitions A. We provide a backtracking algorithm for solving this problem in an all-partition array and compare the output of this algorithm with an analysis produced manually.


Babbitt Knuth All-partition array Exact cover Computational music analysis 



The work reported in this paper was carried out as part of the EC-funded collaborative project, “Learning to Create” (Lrn2Cre8). The Lrn2Cre8 project 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.


  1. 1.
    Babbitt, M.: Set structure as a compositional determinant. J. Music Theor. 5, 72–94 (1987)CrossRefGoogle Scholar
  2. 2.
    Bazelow, A.R., Brickle, F.: A partition problem posed by Milton Babbitt. Perspect. New Music 14(2), 280–293 (1976)CrossRefGoogle Scholar
  3. 3.
    Bemman, B., Meredith, D.: From analysis to surface: generating the surface of Milton Babbitt’s Sheer Pluck from a parsimonious encoding of an analysis of its pitch-class structure. In: The Music Encoding Conference, Charlottesville, VA, 20–23 May 2014Google Scholar
  4. 4.
    Eger, S.: Restricted weighted integer compositions and extended binomial coefficients. J. Integer Seq. 16(13.1.3), 1–25 (1997)MathSciNetGoogle Scholar
  5. 5.
    Donald, K.: Dancing links. 22 February 2000. uno/musings.html
  6. 6.
    Mead, A.: An Introduction to the Music of Milton Babbitt. Princeton University Press, Princeton (1994)CrossRefGoogle Scholar
  7. 7.
    Scott, D.S.: Programming a combinatorial puzzle. Technical report No. 1, Princeton University Department of Electrical Engineering, Princeton, NJ, 10 June 1958Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Architecture, Design and Media TechnologyAalborg UniversityAalborgDenmark

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