Mathematics and Computation in Music

MCM 2015: Mathematics and Computation in Music pp 237-242 | Cite as

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)

Abstract

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.

Keywords

Babbitt Knuth All-partition array Exact cover Computational music analysis 

References

  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. http://www-cs-faculty.stanford.edu/ 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

Personalised recommendations