81k:05023a 05B20 15A36 20K01
Bazelow, Alexander R.; Brickle, Frank
A combinatorial problem in music theory---Babbitt's partition problem.
I. (English)
With a cassette containing an original composition by Daniel Starr of
Yale University. Second International Conference on Combinatorial
Mathematics (New York, 1978), pp. 47--63,
Ann. New York Acad. Sci., 319,
New York Acad. Sci., New York, 1979.
81k:05023b 05B20 20K01
Bazelow, Alexander R.; Brickle, Frank
A combinatorial problem in music theory---Babbitt's partition problem.
II. (English)
Ars Combin. 9 (1980), 289--306.
Mathematical features in 12-tone musical composition have long been
recognized and exploited, in particular by M. Babbitt. For a survey
including references to Babbitt's writings, see, for example, Chapter
I of an article by G. D. Halsey and the reviewer (Jahresber. Deutsch
Math.-Verein. 80 (1978), no. 4, 151 - 207; MR 80g:20063). In the
papers under review, the authors take up principally three problems
posed by Babbitt that are of interest for musical composition. One of
these is the problem of finding an algorithm for determining all
m{times}m matrices with entries drawn from the set (1, 2, ... ,n) for
which all rows and columns have the sum n. Here m and n are arbitrary
positive integers. The authors point out in part II that this problem
has been solved (albeit by a not strictly constructive argument) by R.
P. Stanley (Duke Math. J. 43 (1976), no. 3, 511 - 531; MR 56 #2865).
The second and third problems of Babbitt have to do with certain
s{times}n matrices, the rows of which are the set of numbers (0, 1,
... , n - 1) in various specified orders, and how these matrices can
be decomposed into blocks having certain specified properties. The
results obtained are, by the nature of the problem, incomplete. Part I
comes with an accoutrement unique in the reviewer's experience: a
cassette on which a chromatic 12-tone scale, six interesting
hexachords, and a composition by Daniel Starr (which the reviewer
found very engaging) are recorded. (For the entire collection
containing part I see MR 80j:05003.)
Reviewed by Edwin Hewitt
© Copyright American Mathematical Society 1981, 1998