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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
                  mto-manager@boethius.music.ucsb.edu
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AUTHOR: Robison, Brian
TITLE: Category structures and fuzzy sets
KEYWORDS: Bruckner, Zbikowski, fuzzy sets, fuzzy algorithms
REFERENCE: mto.95.1.4.zbikowski.art
Brian Robison
Department of Music
Lincoln Hall
Cornell University
Ithaca, NY  14850
bcr2@cornell.edu
[1] Prior to his examination of the categorical structure of the first
main theme in the first movement of Bruckner's Sixth Symphony,
Lawrence Zbikowski comments that "The complexity of the phenomenon of
typicality may be one reason Zadeh's theory of fuzzy sets(1) has met
with only limited success in characterizing the structure of Type 1
categories."(2) The following look at Prof. Zbikowski's analysis
highlights (in a non-technical manner) those aspects of it which lend
themselves more readily to "fuzzy" rather than "crisp" models, the
complexity of their interaction notwithstanding.  In particular, I
focus on the problems of describing formally those intuitions which
enable us to associate different musical phrases in order to group
them as members of the same category.  
========== 
1. Lotfi A. Zadeh,
"Fuzzy sets," Information and Control 8 (1965): 338-353.
2. Lawrence Zbikowski, "Theories of categorization and
theories of music," Music Theory Online 1.4 (1995): 11.
==========
[2] For the benefit of those persons unacquainted with fuzzy sets,
algorithms, arithmetic, measures, etc., I'd like to emphasize that I
use the word "fuzzy" as a connotationally neutral term which refers
simply to matters of "to what degree?" (graded membership) rather than
"yes, or no?"  (exclusive membership).(3) Whenever I refer below to
aspects of "fuzziness" in Zbikowski's analysis, I'm *not* using the
word in its colloquial, pejorative sense, to criticize the vague
handling of ideas which one ought to render precisely.  Rather, I
elaborate on various points where, implicitly or explicitly, he
touches on the intrinsically imprecise character of certain musical
ideas.  Thus, I'm not taking issue with his analysis; on the contrary,
I merely offer observations geared toward an appropriate formalization
of some of the comparatively informal insights which he offers (all of
which I believe to be valid).
==========
3. Zadeh, "Fuzzy sets," 339.
==========
[3] At first glance, Zbikowski seems to be dealing with a classic
fuzzy set: for elements in a universe of "crisply" defined segments of
the Bruckner, he tests for membership in the set of *typical*
instances of the first theme.  The set is fuzzy because it admits
degrees of membership: we don't ask simply
  "*Is* this a typical instance of the first theme
  (or not)?",
but rather,
  "*To what degree* is this instance of the first
  theme typical?"
Accordingly, Zbikowski ranks these instances: taking mm. 2-6 as
typical, mm. 8-12 are "somewhat less typical"(4), mm. 26-28 and 32-34
are "even less typical,"(5) and so on.  It's a simple step to
formalize these rankings by translating them into membership values
between 0 and 1, which reflect not only their relative order but the
subjective distances among them as well.(6)
==========
4. Zbikowski, "Theories," 17.
5. Ibid., 18.
6. Zadeh, "Fuzzy sets," 339.
==========
[4] However, as Zbikowski points out, the process by which we assign
these degrees of membership is anything but linear.  Discussing the
fragmentary gestures in mm. 35-36, he notes that our judgments of
typicality are not simple tallies of the numbers of structural
propositions met (and the degree to which a musical segment fulfills
them).(7) Instead, some propositions may carry more weight than
others, such as his P4, by which texture can signal thematic
potential, and these relative weights may well vary according to
context.(8) Such factors suggest fuzzy algorithms(9) to model our
intuitive calculations of typicality, for example:
  "If the texture is *highly* complex, then
  assign *much less* weight to correspondences
  of rhythm and contour."(10)
==========
7. Zbikowski, "Theories," 18.
8. Ibid., 25.
9. Lotfi A. Zadeh, "Fuzzy algorithms," Information and
Control 12 (1968): 94-102.
10. Construction of an explicit fuzzy model would then
entail mapping the imprecise quantifiers "highly" and
"much less" to ranges of appropriate numerical values.
(see Zadeh, "Fuzzy algorithms," 96.)
==========
[5] Additionally, examination of the structural propositions reveals
varying degrees of specificity.  Consider Zbikowski's P3, which
describes the contour of the theme's first statement.  As Zbikowski
compares it to subsequent statements, many of the contours stated as
specific (chromatic) intervals seem better served by generic
(diatonic) intervals, as in his Figure 1:
        specific intervals:
mm. 2-6:   0 -7  -2 +2 +1 -1 -2 +2  +8 -1
mm. 8-12: +1 -7  +1 -1 -2 +2 +1 +1  +7 +1
                 --------------
        generic intervals:
mm. 2-6:   0 -4  -1 +1 +1 -1 -1 +1  +5 -1
mm. 8-12: +1 -4  +1 -1 -1 +1 +1 "0" +4 +1
                 --------------
[6] But though these generic intervals clarify the diatonically
precise inverse relation of the underlined segments, they fail to
convey other essential information: the directed specific interval of
-7 between the first two structural tones (Zbikowski's P2), the
semitone at the very end, and the presence of a semitone in the turn
figure, the absence of which, as Zbikowski mentions, diminishes the
typicality of the statements in mm. 159-66.(11) Furthermore, one is
left with the task of accounting for the discrepancies between the
first, eighth, and ninth intervals of the two series: are we to admit
any "off-by-one" surrogate as an acceptably similar variant?  If so,
are we to admit any number of such surrogates, or is there a point at
which the accumulated alterations would disrupt our sense of
parallelism?  In his pursuit of larger issues, Zbikowski declines to
treat these details, yet they are decidedly nontrivial in the creation
of an explicit formalization.
==========
11. Zbikowski, "Theories," 22.  Zbikowski doesn't comment
on whether he feels that the semitones in the statements of
mm. 167-74 somehow impart greater typicality to those
variants in relation to mm. 159-66 and 175-82.  My own
intuition is that the presence or absence of a semitone is
actually subordinate to the distinction between a strictly
stepwise turn (mm. 2-6 and mm. 8-12) and one which
embellishes by step-above and third-below (mm. 159-82).
==========
[7] So, in addition to (*not* "instead of") Zbikowski's P3, we might
propose a slightly more general (and thus more robust) version of the
theme's contour:
        P3'a) the theme contains four structural notes,
        P3'b) the first (preceded by a sixteenth-note
                pickup) descends to the second by a
                perfect fifth,
        P3'c) a stepwise diatonic turn figure
                (incorporating a semitone) embellishes
                the second structural note, proceeding
                to the third,
        P3'd) the theme then ascends by leap to the
                penultimate note, which neighbors the
                final one (i.e. the fourth structural
                note) by a semitone.
[8] Like Zbikowski's original P3, this new description (P3') of the
theme's initial contour doesn't rely on a specification of scale step.
However, P3' incorporates both more and less detail: more, in that it
invokes a local diatonic frame of reference and (implicitly)
considerations of harmony (structural and embellishing tones); less,
in that it doesn't specify the interval of the pickup in P3'b, the
direction of the turn in P3'c, nor the size of the leap in P3'd.  I
offer P3' as a version of how one might generalize from Zbikowski's P3
after hearing mm. 8-12: mm. 2-6 are still *more* typical, but P3'
subsumes the points by which the second statement deviates from the
first.  The modified description thus captures somewhat more
explicitly the ways in which the statement of mm. 8-12 is more typical
than some of the subsequent variants, but without taking on the full
complications of a "binary basis for typicality."(12) Furthermore, it
does so in a way that distinguishes among the various stepwise
motions: some are intrinsic to the theme's schematic contour, while
others appear as the results of local harmonic changes.
==========
12. Zbikowski, "Theories," 23.
==========
[9] The utility of such vague terms as "leap" becomes apparent when we
consider the thematic variants beginning in mm. 159-62.  Again, a
comparison based on directed specific intervals (Zbikowski's Fig. 2)
seems to indicate as many differences as similarities:
        specific intervals:
   mm. 2-6: 0  -7  -2  +2  +1  -1  -2  +2   +8 -1
mm. 159-62:   +12  +2  -2  -3  +3  +2  -2   -9 +1
Although Zbikowski describes the relation between the contours of
mm. 2-6 and mm. 159-62 as an "exact mirror"(13), we see from the
specific intervals that the only "exact" aspect of mirroring is with
regard to interval *direction*, not size.  Again, conversion to
diatonic intervals doesn't resolve all discrepancies:
        generic intervals:
mm. 2-6:    0  -4  -1  +1  +1  -1  -1  +1   +5 -1
mm. 159-62:    +7  +1  -1  -2  +2  +1  -1   -5 "0"
How, then, can we model the intuition of an "exact mirror"?  One
answer lies on a still greater level of generalization, in a
distinction between large (L) and small (s) intervals--or,
colloquially, between (L)eap and (s)tep-or-(s)kip:(14)
        fuzzy-generic intervals:
mm. 2-6:     0 -L  -s1 +s1 +s2 -s2 -s1 +s1  +L -s
mm. 159-62:    +L  +s1 -s1 -s2 +s2 +s1 -s1  -L +s
The designations "s1" and "s2" (ideally with subscripted numerals)
provide a simple but flexible means to preserve the turn's diatonic
character: that is, although the specific manifestations of s1 and s2
may vary among theme statements, they must adhere to fixed intervals
within a single statement.(15)
==========
13. Zbikowski, "Theories," 20.
14. Although not needed within the limits of the present
narrow example, a full-fledged fuzzy model would treat
explicitly the vague transition between "large steps" and
"small leaps."
15. These restrictions thus prevent inappropriately
chromatic realizations of < +s -s -s +s +s -s >
                  such as < +3 -2 -2 +1 +2 -1 >
--a perfectly good turn for Ligeti or Birtwistle, but
obviously not characteristic of Bruckner's material at hand.
==========
[10] Not surprisingly, the variants in mm. 163-82 require further
adjustment to our description:
        specific intervals:
mm. 159-62:   +12   +2  -2  -3  +3  +2  -2    -9 +1
mm. 163-66: 0 +12   +2  -2  -3  +3  +3  -1   -11 -1
mm. 167-70: 0 +12   +1  -1  -4  +4  +1  -1    -7 -1
mm. 171-74: 0 +12   +1  -1  -4  +4  +1  -1    -7 +5
mm. 175-78: 0 +12   +2  -2  -3  +3  +2  -2    -9 +1
mm. 179-82: 0 +12   +2  -2  -3  +3  +2  -1    -9 +1
        generic intervals:
mm. 159-62:    +7   +1  -1  -2  +2  +1  -1    -5 "0"
mm. 163-66: 0  +7   +1  -1  -2  +2  +2  -1    -6 -1
mm. 167-70: 0  +7   +1  -1  -2  +2  +1  -1    -4 -1
mm. 171-74: 0  +7   +1  -1  -2  +2  +1  -1    -4 +3
mm. 175-78: 0  +7   +1  -1  -2  +2  +1  -1    -5 "0"
mm. 179-82: 0  +7   +1  -1  -2  +2  +1  "0"   -6 +1
        fuzzy-generic intervals:
mm. 159-62:    +L   +s1 -s1 -s2 +s2 +s1 -s1  -L +s
mm. 163-66: 0  +L   +s1 -s1 -s2 +s2 +s2 -s   -L -s
mm. 167-70: 0  +L   +s1 -s1 -s2 +s2 +s1 -s1  -L -s
mm. 171-74: 0  +L   +s1 -s1 -s2 +s2 +s1 -s1  -L +L
mm. 175-78: 0  +L   +s1 -s1 -s2 +s2 +s1 -s1  -L +s
mm. 179-82: 0  +L   +s1 -s1 -s2 +s2 +s1 -s   -L +s
Although relaxing the "s1" restriction on the antepenultimate interval
resolves some discrepancies, the anomalous endings of mm. 163-66 and
mm. 171-74 suggest the need for a fuzzy corollary: "Correspondences
toward the beginnings of statements are more important than those
toward their ends."(16) Indeed, applying this fuzzy corollary to the
generic-interval descriptions enables us to recover these as
sufficiently general descriptions of the statements in mm. 159-82, and
thus to invoke the fuzzy-generic level of description only as the
means by which we relate these to the theme's initial (and subsequent)
manifestations.
==========
16. The corollary is fuzzy in that it does not specify hard-
and-fast boundaries for which notes do or do not belong to
the "beginning" and "end" of a segment--instead, the
saliences of correspondences decrease gradually in sequence.
==========
[11] Returning to the verbal descriptions of the theme's contour, we
see that just as Zbikowski's P3 provides a basis for generalization to
P3' in order to encompass the theme's first two statements, P3' can in
turn serve as the basis for an even more general version which bridges
the gap between Zbikowski's first and second categories (17):
        P3"a) the theme contains four structural notes,
        P3"b) the first (normally preceded by a
                sixteenth-note pickup) moves to the
                second by leap,
        P3"c) a diatonic turn figure embellishes
                the second structural note, proceeding
                to the third,
        P3"d) the theme then moves by leap (opposite
                the direction of the initial leap) to
                the penultimate note, which normally
                neighbors the final one (i.e. the fourth
                structural note) by a semitone.
P3" thus incorporates a high level of generalization of the theme's
contour across its various incarnations, to provide a description
which one can fine-tune to produce either Zbikowski's P3n
(mm. 159-182) or my own P3' (mm. 2-6, 8-12), which one can further
refine to arrive at Zbikowski's original P3 (mm. 2-6).  The nesting of
generalities within generalities suggests that a successful numerical
formalization of Zbikowski's analysis would not only require fuzzy
algorithms, but that some of the quantities involved might themselves
be fuzzy: that is, not a unitary value between 0 and 1, but a weighted
*range* of values between 0 and 1.  And, as Zbikowski emphasizes, the
dynamic process by which we interpret Bruckner's "system of
approximate correspondences *and* exact correspondences" (18) dictates
that these values (and, indeed, their relative degrees of precision)
would vary over time in order to model appropriately our aural
cognitions.
==========
17. Zbikowski, "Theories," 21.
18. Ibid., 25.
==========
[12] Again, none of the above is to dispute Zbikowski's findings;
rather, I wish to focus attention on the many different degrees of
precision that come into play in the course of his analysis.  In
particular, we see that there is no single "proper" or "correct" level
of specificity in describing our intuitions about thematic character,
especially those by which we associate disparate material.  It seems
self-evident that, as competent listeners, we perceive music neither
as a great blur of shadowy patterns nor as an object of real-time,
exhaustively detailed note-for-note (interval-for-interval,
duration-for-duration, chord-for-chord...) analysis.  Instead, we
infer chunks of versatile structure, such that we not only register
exact matches but also distinguish among transformations which run the
gamut from the slyly (or even cryptically) subtle to the boldly (and
even shockingly) dramatic.  For these reasons, formalisms based on
fuzzy sets and fuzzy algorithms seem ideally suited to model the
respects in which, as Zbikowski notes, "categorical processes adequate
to music must deal with a large amount of auditory information
streaming by in real time...[requiring] a model of categorization that
is extremely rapid (at least in its gross aspects) and highly
flexible."(19)
==========
19. Zbikowski, "Theories," 26.
==========
References
==========
Zadeh, Lotfi.  "Fuzzy sets."  *Information and Control* 8
        (1965): 338-353.
Zadeh, Lotfi.  "Fuzzy algorithms."  *Information and Control*
        12 (1968): 94-102.
Zbikowski, Lawrence.  "Theories of categorization and
        theories of music."  *Music Theory Online* 1.4
        (1995): 1-26.
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