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       M U S I C          T H E O R Y         O N L I N E
                     A Publication of the
                   Society for Music Theory
          Copyright (c) 1995 Society for Music Theory
| Volume 1, Number 5    September, 1995    ISSN:  1067-3040   |
  All queries to: mto-editor@boethius.music.ucsb.edu or to
AUTHOR: Zbikowski, Lawrence M.
TITLE: Response to Robison, "Category structures and fuzzy sets"
KEYWORDS: Robison, categorization, musical categories, empirical
REFERENCE: mto.95.1.5.robison.tlk
Lawrence M. Zbikowski
University of Chicago
Department of Music
Goodspeed Hall
1010 East 59th Street
Chicago, IL  60615
[1] I am gratified by Brian Robison's response to my essay, not
the least because his references are drawn exclusively from the
end of the alphabet, a place dear to my heart.  I also value
Brian's close and sensitive reading of the essay, and his
extension of some of its informal formalizations through further
formalization and fuzzification.  Readers familiar with the SMT
list will remember the considerable amount of interest in fuzzy
set theory expressed on the list somewhat over a year ago, and
may even have at hand the fine bibliography Betsy Marvin
assembled and distributed through the list.  Because I think of
the theoretical methodology I proposed in the essay as a flexible
one, I have no argument with Brian's extension.  My response is
consequently geared toward a brief exploration of some of the
implications of Brian's response, as well as a few points of
[2] In my original propositions I offered a fairly loose
characterization of categorical structure, intended to model the
understanding of a listener moderately familiar with the
repertoire but not necessarily well-acquainted with Bruckner's
Sixth.  As I mentioned in the essay, my assumption is that a
listener would arrive at these characterizations without recourse
to even the informal formalizations I offered (Zbikowski, 1995:
[25])--the propositions instead reflect listeners' "intuitions"
about musical organization.
[3] Of course, some people's intuitions are considerably more
refined than others, and so one would expect the structure of
their categorizations of musical events to be rather more
detailed, and perhaps embody the specificity of Brian's P3'c ([7]
above).  Figuring out which is the more accurate characterization
would seem to be a matter for empirical verification:
recompositions of the relevant passages might be played for
listeners, each passage emphasizing different aspects of the
hypothesized categorical structure, with the intent of revealing
just what musical attributes are most relevant for determinations
of similarity.  A model for this sort of investigation is Lucy
Pollard-Gott's 1983 study.  Interestingly enough, initial
determinations of thematic similarity by most of her listeners
had little to with pitch structure, but were instead based on
what are oftentimes thought of as "secondary" parameters:
register, dynamics, and the like (Pollard-Gott, 1983: 92-93).
[4] The aspect of contour is another area where Brian and I had
slightly different approaches.  Here I am guilty of a little bit
of sloppiness in my prose (what Kerry Snyder, within the context
of looking for typos, referred to as a "thinko").  When I claimed
there was an "exact mirror" of contour pattern, my thoughts were
firmly in Robert Morris's c-space (Morris, 1987).  Brian's
accommodation of my observation is entirely appropriate, and
clarifies what I had expressed unclearly.  His discrimination
between large and small intervals (and his comment about the
problem of large steps and small leaps) fits well with Eugene
Narmour's recent work on melodic intervals (see Narmour, 1990;
Zbikowski, 1993; and Krumhansl, 1995), which offers a more-or-
less explicit formalization of this discrimination.  And Brian's
fuzzy corollary relating to the importance of the beginnings of
thematic statements [10] is borne out by any number of
psychological studies of in-time processing.
[5] What remains, of course, is a specific implementation of the
fuzzy logic approach sketched in Brian's response.  The advantage
of this approach is that it will introduce a type of
formalization about which there is a good deal of interest into
the theoretical methodology I proposed, and use this
formalization to create somewhat more precise characterizations
of categorical structure.  I look forward to seeing further work
in this area.
Krumhansl, Carol L. 1995. "Music psychology and music theory:
     Problems and prospects." *Music Theory Spectrum* 17
Morris, Robert D. 1987. *Composition with pitch classes: A theory
     of compositional design*. New Haven: Yale University Press.
Narmour, Eugene. 1990. *The analysis and cognition of basic
     melodic structures: The implication-realization model.*
     Chicago: University of Chicago Press.
Pollard-Gott, Lucy. 1983. "Emergence of thematic concepts in
     repeated listening to music." *Cognitive Psychology* 15
Zbikowski, Lawrence. 1993. "Review of Eugene Narmour *The
     Analysis and Cognition of Basic Melodic Structures*."
     *Journal of Music Theory* 37 (Spring):177-206.
Zbikowski, Lawrence. 1995. "Theories of categorization and
     theories of music. *Music Theory Online* 1.4.

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