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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 2      March, 1995      ISSN:  1067-3040   |
  All queries to: mto-editor@boethius.music.ucsb.edu or to
AUTHOR: Demske, Thomas R.
TITLE: Relating Sets: On Considering a Computational Model of 
Similarity Analysis
KEYWORDS: similarity, REL, clustering, decision, post-tonal analysis
Thomas R. Demske
Connecticut College
Department of Music
270 Mohegan Avenue
New London, CT  06320
ACCOMPANYING FILE:  mto.95.1.2.demske.gif
ABSTRACT: A clustering program is proposed for identifying salient
similarity relationship trends within populations of pitch-class sets
abstracted in post-tonal analysis.  That program requires an
evaluation component, but attempts to define one are frustrated.  The
similarity relation itself is too abstract to imply guidelines for its
own application, and other potential criteria resist formal
implementation.  Even less rigorous applications of the more commonly
used tools in post-tonal analysis are susceptible to the concerns
raised here.
[1] A few months ago, I considered applying conceptual clustering
techniques to identify context-specific similarity relations among
pitch-class sets.  That project quickly stalled for lack of suitable
evaluation criteria, a problem which I take as the subject of this
essay.  The issues here are not necessarily new, but they are
persistent.  Recent SMT e-list interest in formal analysis suggests
the timeliness of reviewing these issues now from an applicational
[2] Many fields use clustering to establish meaningful distinctions
within arbitrary data sets.  The data are events described as n-tuples
and interpreted as points in n-dimensional space.  Points lying closer
together according to some distance specification "cluster" into
groups distinguished by boundaries.  Clustering comes in many flavors.
The conceptual clustering model in Michalski and Stepp (1983) is
interesting because it determines cluster membership, not only through
predefined metrics, but also by the resultant shape of the cluster --
its "conceptual cohesiveness."  Conceptual clusters are
context-specific, in that events cohere on the basis of their
relationships to other events in the data set.
[3] I was first attracted to conceptual clustering through ongoing
research into melodic grouping in piano scores, an early stage of
which is documented in Demske (1993).  Clustering is fruitfully
applied there to isolate and address causes of rule base misfire.
Might the technique also prove useful in post-tonal analysis, despite
the far more diffuse nature of the problem domain?  Given a collection
of pitch-class set segmentations abstracted from a composition, the
clustering implementation would "learn" the most salient, structural
trends particular to that population.
[4] Cluster membership indicates relatedness, an idea which is
fundamental in pitch-class set analysis.  Most analytical assertions
eventually require boundaries in a conceptual space to distinguish one
or more sets (or sets of sets, etc.) from others.  Mainstream
analytical set theory tends to recognize only a few types of musically
motivated boundaries.  These account for the familiar TnI set classes,
transformational partitionings, "referential" collections and their
subsets, and so forth.  In all cases, kinship and contrast between
objects disposed respectively on one or both sides of a boundary
provide the potential for a syntax.  This simple duality can spawn
sophisticated arguments about form, association, and process.
[5] "Similarity" in pitch-class set theory refers narrowly to special
inclusion-based relationships.  These are commonly understood as
reflecting degrees of context-free, "aural similitude" (Morris, 1980),
although some writers avoid any perceptual interpretation (Forte
1973).  Isaacson (1990) surveys several similarity functions proposed
over the years, while offering his own "IcVSIM" as an alternative.(1)
Most similarity functions take two sets as arguments and return a
single, usually scaled value determined through subset counts;
Morris's ASIM, Rahn's ATMEMB, and Lewin's REL are perhaps the best
known examples (Morris 1980; Rahn 1980; Lewin 1980).
1.  Isaacson has also written a handy, inexpensive, and
well-documented program which calculates the values of
various similarity functions for arbitrary sets.  DOS
and Windows versions are available.
[6] Similarity functions seem an ideal test bed for clustering because
of the apparent simplicity with which they model distance in set-class
space.  However, conceptual clustering is a learning algorithm.
Without a means for evaluating prospective clusters, the algorithm's
output will likely make little sense.  How might similarity-based
clusters be evaluated, assuming the identification of "salient trends"
as a loosely- defined goal?
[7] I use Lewin's REL here for illustration, although other similarity
index functions would work as well.  Consider a pivot, P = {0147}, and
two target sets, T1 = {01369} and T2 = {047a}.  REL(P,T1) = 0.783 and
REL(P,T2) = 0.636.  Drawing a boundary around all sets Y such that
0.783 >= REL(P,Y) > 0.636, P and T1 cluster together on one side,
while T2 falls on the other.  This boundary might be treated like any
other in analysis. However, nothing in the interpretation of REL
compels us to draw any particular boundary.  Assume another set T3 =
{0134}, where REL(P,T3) = 0.529.  We could keep our original boundary,
or now draw a new one separating P, T1, and T2 from T3.  In fact, we
could draw a boundary separating P from all of the target sets, or
another which includes P and the targets all on one side.  Fig. 1
shows the relationships just described, with vertical bars indicating
potential boundary sites (the vee is explained later):
Figure 1.
pc set:     |  0147  |  01369  |  047a  |  0134  |
               P        T1        T2       T3
REL(P,Tn):              0.783     0.636    0.529
set class:     4-18     5-31      4-27     4-3
[8] Comparing REL values tells us only where boundaries may be drawn,
not where they should be drawn.  Barring *ad hoc* provisions in the
interpretation, all sets are similar with respect to each other.  Only
the degree of similarity varies.  This makes it difficult to evaluate
competing clusters, such as those distinguished by alternately drawing
boundaries through the third and fourth vertical bars in Fig. 1 above.
An *a priori* cutoff point X would help, whereby only targets T with
REL(P,T) >= X are considered "similar" in an absolute sense.  (TnI
"equivalence" supports analogous appeals to the absolute.)  But could
such a point be meaningfully determined here?  Statistical analysis of
how the REL values at issue are disposed around an average or mean
begs another question: are all of REL's returns with respect to a
given pivot commensurable in a meaningful way (I am not sure that they
are), or are they merely indications of a partial ordering?
[9] Other boundary ambiguities must be considered.  We can compare two
returns of REL only when an instance of the same set class appears in
both calls.  Rahn (1989) demonstrates this tangentially with his DATM
function, derived from ATMEMB, but the same applies to REL and
comparable similarity index functions.  That argument common to both
calls is the pivot, P.  Figs. 2 - 4 below show how the potential
boundaries change when our old target sets are cycled through P.
{01369} is the pivot at Fig. 2:
Figure 2.
                      V                 V
pc set:     |  01369  |  047a     0147  |  0134 |
               P         T1       T2       T3
REL(P,Tn):               0.783    0.783    0.536
set class:     5-31      4-27     4-18     4-3
Unlike the situation earlier at Fig. 1, {047a} and {0147} at Fig. 2
will always appear on the same side of a boundary.  Compared to
Fig. 1, taking {047a} as the pivot at Fig. 3 means that {047a} and
{0134} can no longer be grouped as a pair apart from {01369} and
Figure 3.
pc set:     |  047a  |  01369  |  0147  |  0134  |
               P        T1        T2       T3
REL(P,Tn):              0.783     0.636    0.363
set class:     4-27     5-31      4-18     4-3
In Fig. 4, it suddenly becomes possible to place the pairs, {0134} and
{01369}, and {0147} and {047a}, on opposite sides of a boundary:
Figure 4.
pc set:     |  0134  |  01369  |  0147  |  047a  |
               P        T1        T2       T3
REL(P,Tn):              0.536     0.529    0.363
set class:     4-3      5-31      4-18     4-27
[10] Figs. 1 - 4 together represent a tiny bubble in that
"(staggeringly complex) network of relations" revealed by functions
like REL (Rahn 1980, 494).  The downside to this attractive notion is
that it means more potential conflicts among REL-derived clusters.  Is
{0147} more related to {01369} than it is to {047a}?  Expressed
through the totality of REL returns in Figs.  1 - 4, the answer is yes
and no.  For example, {0147} and {01369} cluster together without
{047a} under the boundary drawn through the vee at Fig. 1, while
{0147} and {047a} cluster together without {01369} under the two
boundaries drawn through the vees at Fig. 2.
[11] Critical REL returns are interpreted directly at Fig. 1 and
indirectly at Fig. 2, suggesting that one criterion for evaluating
conflicting clusters might be to prefer the more musically intuitive
one.  "Intuition" is difficult to pin down, however.  For example,
Rahn (1980) claims special intuitive status for ATMEMB, but others may
find the notion of "mutual embedding" less compelling.  All else being
equal, ATMEMB(A,B) remains constant so long as the sum of EMB(/X/,A)
and EMB(/X/,B) remains constant, and so long as neither term equals
zero.(2) This means A and B can differ considerably in the number of X
types they separately include, and yet still return high ATMEMB
values.  REL seems more "intuitive" in this respect because it takes
the product of EMB(/X/,A) and EMB(/X/,B).  While REL(A,B) still
increases along with either one of EMB(/X/,A) and EMB(/X/,B) (assuming
nonzero values), it does so more quickly when *both* subterms
increase.  This makes sense, since we would expect -- in the abstract
-- that two sets with convergent inclusion capabilities will have more
"aural similitude" than two sets which diverge in their capabilities.
2.  The embedding function EMB(/q/,r) counts
the multiplicity of q-type sets included in set r; see Lewin 1977.
[12] The mechanics of ATMEMB, REL, and comparable functions present
other potential barriers to intuition.(3) But those pale in comparison
to the problems which arise when similarity function returns are
viewed in their totality.  Assume two sets WT1 = {0246} and WT2 =
{0248}.  Considered in isolation, REL(WT1,WT2) = 0.846 bespeaks
urgency, emphasizing the difference in "aural similitude."  That
changes by adding CHR = {0123} to the mix.  WT1 and WT2 now seem quite
similar, by virtue of their relationship to CHR: an absence of odd
interval classes binds the wholetone sets together.  Relationships are
thus colored by other relationships.  In Rahn's vast network, where
everything is connected, the result is a flat, homogeneous black (or
white, depending on perspective).  But when only a portion of the
network is considered, glimmers of color surface, forming
kaleidoscopic patterns of arbitrary complexity.  Tracking those
patterns can strain intuition mightily.
3.  Blind subset polling is a basic source of such
barriers.  Two REL calls with the same pivot may yield
identical results, and yet differ with respect to the
types of subsets counted.  Ignoring the degree of this
difference when comparing REL value spreads strikes me
as questionable.
[13] So, who is to say whether the direct relation described in
paragraph 10 above is more intuitive than, and thus necessarily takes
precedence over, the indirect one?  Rahn (1989) briefly explores the
idea of following only "paths 'of least resistance'" through a
network, while Block and Douthett (1994) offer a new way to focus on
specific paths.(4) But neither indicates whether certain paths should
generally be preferred over others, given an arbitrary collection of
sets.  Both Hindemith (1942) and Friedmann (1990) informally consider
interval class 1 content as being especially determinative of chord quality.  
Perhaps those paths reflecting aspiration toward high semitone counts -- 
REL hierarchies based on chromatic pivots -- are therefore to be preferred
*a priori*.
4.  Block and Douthett's (1994) "vector product" parts
with other similarity measures in that the pivot
argument is not a set, but rather an arbitrarily
specified standard of subset content.  What was once
merely a "staggeringly complex" network of
relationships is now potentially infinite.  Otherwise,
vector product similarities continue to admit the same
critique voiced here with respect to REL.
[14] Such concerns raise the inevitable issue of perception.
Evaluating similarity clusters with perceptual criteria is complicated
because we can attend to the same music in many different ways.
Still, there are limits.  Fig. 5 graphically displays similarity
relations between successive piano chords in the first movement of
Messiaen's *Quatour pour le fin du temps*.  The piano part is a
recurring sequence of 29 chords set to a rhythmic ostinato of 17
durations; chord 30 is thus the same as chord 1.  Higher ASIM, ATMEMB,
and REL values plotted on the graph indicate greater similarity.
(ASIM is adjusted to facilitate comparison with the other functions.)
Fig. 5 shows the three functions diverging in their values, but mostly
agreeing about peaks, valleys, and plateaus.  The relation expressed
in moving from chord 1 to 2 appears above point 2 on the x axis; that
from chord 2 to 3 appears above point 3; etc.  The middle chord in
each three-chord succession is the pivot, while the two flanking
chords are the targets.
[15] The graph resembles a "harmonic fluctuation" analysis (Hindemith
1942), and should capture a comparable aspect under the most obvious
interpretation of the similarity relationship.  However, and despite
great effort in attending, I can discern no correspondence at all
between musical transitions in the *Quatour* and those represented
symbolically on the graph.  For example, the strong contrast that
Fig. 5 reports between chords 15 and 16 simply does not conform to
Messiaen's smooth realization.  Taking this perception into account
suggests preferring a boundary through the vee at Fig. 6:
Figure 6.
chord:       |    16    |    17    |    15    |
                  P          T1         T2
REL(P,Tn):                   0.710      0.356
set class:        5-21       6-15       5-35
This is not entirely satisfactory, however, even allowing for the wide
disparity in REL values.  To my ear, the move between Messiaen's
chords 16 and 17 is in fact more striking than that between his chords
15 and 16.  Perhaps a boundary should therefore separate chords 15 and
16 from chord 17.  The REL scheme of Fig.  6 cannot accomodate that
boundary, so another pivot is required:
Figure 7.
chord:       |  18   |   17    |   16    |   15    |
                P        T1        T2        T3
REL(P,Tn):               0.784     0.460     0.379
set class:      6-21     6-15      5-21      5-35
[16] The pivot in Fig. 7 no longer coincides with the original point
of transition, a fact which obscures the harmonic fluctuation
interpretation of similarities.(5) Perhaps this shows harmonic
fluctuation to be an insufficient standard for perceptual cluster
evaluation.  That in turn suggests another complication when appealing
to perceptual criteria.  Exactly how similarity relationships might
inform perception is speculative.  Gradations of "smooth" chord
succession, strength of motivic association, and degree of harmonic
contrast in form delineation are only the first potential
manifestations which spring to mind.  The idea that any formal
evaluation procedure could embrace all of the possibilities seems
untenable.  On what bases would a partial set of possibilities be
selected for implementation?  Since the different criteria may address
different -- and possibly conflicting -- aspects of perception, how
would the application of one criterion be coordinated with that of
5.  Hindemith (1942) types chords according to
intrinsic intervallic properties, which is very roughly
analogous to holding one particular set of ideals as a
constant pivot argument to REL.  Consequently, variance
between pivot and point of transition would not
conflict with Hindemith's absolutist conception of
harmonic fluctuation.  The relativistic conception
pursued here in the interpretation of Fig. 5 conforms
more closely to that implied in Rahn's (1989) "theory
of chord progression."
[17] Apart from the manner in which similarities shape perception,
there is the further question of the extent to which they do so.  For
example, strong contrast is perceived in moving between Messiaen's
chords 6 and 7, despite the initial plateau of Fig. 5.  This implies
drawing a boundary through the vee at Fig. 8, even though that is not
an allowed boundary site:
Figure 8.
chord:       |     7    |     8          6     |
                   P          T1         T2
REL(P,Tn):                    0.618      0.618
set class:         7-20       7-35       7-35
Nothing in the micro-network of REL relationships among Messiaen's 29
chords will support this boundary, since chords 6 and 8 are members of
the same set class.  Clearly, competing factors inaccessible to
similarity functions and the pitch-class set model in general
influence perception enormously.(6) In order to realistically evaluate
similarity clusters according to perception, we must first have a
means of isolating and gauging the influence of these factors.
6.  Rahn (1989) presupposes a "theory of instances" to
account for discrepancies between the heard surface and
DATM's predictions.  That theory remains incomplete.
[18] Such a means does not appear to be forthcoming at present, given
current knowledge about music perception.  Implementing practical
event structures and distance functions whose resultant clusterings
satisfy general perceptual criteria is thus but a dim hope.  We cannot
expect a simple automaton processing high-level abstractions to have
much to do with perception.  For now at least, the place for
perceptual criteria remains in cautiously selecting input and very
carefully weighing output.
[19] Analytical music theory usually holds a less naive view of
context than that described in the chord- by-chord scenario above.
Relationships not immediately perceptible can nevertheless inform
musical structure.  This suggests another possible evaluation
criterion: goodness of fit between a boundary and an analytical goal.
The clustering program could take as input segmentations together with
an analytical paradigm.  For example, a short movement strongly
projects the large-scale succession of pitch-class sets shown at
Fig. 9:
Figure 9.
form:              A          B           C
pc sets:           469b       01256       048a
The analyst wants to emphasize that the movement is through-composed.
A REL clustering solution appears at Fig. 10, with two boundaries
indicating progressive motion away from the opening set in "similarity
Figure 10.
                          V           V
pc set:        |   469b   |   01256   |   048a   |
                   P          T1          T2
REL(P,Tn):                    0.371       0.181
set class:       4-23         5-6         4-24
[20] The boundaries of Fig. 10 are a good fit to the analytical
description, but they are not a completely honest fit.  Fig. 11 shows
another REL clustering of the same sets:
Figure 11.
pc set:        |   01256   |   469b     048a   |
                   P           T1       T2
REL(P,Tn):                     0.371    0.371
set class:                     4-23     4-24
The single boundary here at least suggests a different form, the
symmetrical, ternary structure distinguished by outer sections equally
distant from B in similarity space (Fig. 12):
Figure 12.
form:              A          B           A'
                -------->             ---------->
pc set:            469b       01256       048a
The goodness of fit criterion now appears suspicious, in that the
analyst receives only supportive solutions, and is shielded from
possibly damaging ones.
[21] REL and comparable similarity functions are not used much in
analysis today.  Still, my basic concerns extend beyond a particular
function and its potential for rigorous application in a computer
model.  Post- tonal analyses typically involve countless decisions
about boundaries, ranging from relatively low-level segmentation
selections to high-level identifications of referential collections in
refractory textures.  The decisions interact and motivate one another
freely across all levels of description, and often depend on abstract
criteria whose perceptual relevance and provability is speculative.
We all slip, sometimes conspicuously: dubious examples of "composed
out" sets in undergraduate textbooks; untempered enthusiasm for a
single octad class underlying a composer's entire output.  Exuberant
flexibility is essential to the post-tonal analytical dialectic.  But
I suggest checking it occasionally by asking what constraints we would
impose on a machine approaching the same decisions.
Block, Steven and Jack Douthett. 1994. "Vector Products and
Intervallic Weighting." Journal of Music Theory 38/1:21-41.
Demske, Thomas R. 1993. "Recognizing Melodic Motion in Piano Scores:
Rules and Contexts." Ph.D. diss., Yale University.
Forte, Allen. 1973. *The Structure of Atonal Music*.  New Haven: Yale
University Press.
Friedmann, Michael L. 1990. *Ear Training for Twentieth-Century
Music*. New Haven: Yale University Press.
Hindemith, Paul. 1942. *The Craft of Musical Composition*,
v. 1. Trans. Arthur Mendel. New York: Associated Music.
Isaacson, Eric J. 1990. "Similarity of Interval-Class Content Between
Pitch-Class Sets: The ICVSIM Relation."  Journal of Music Theory
Lewin, David. 1980. "A Response to a Response: On Pcset Relatedness."
Perspectives of New Music 18/2:498-502.
Lewin, David. 1977.  "Forte's Interval Vector, My Interval
Function, and Regener's Common-Note Function."  Journal of Music
Theory 21/2:194-237.
Michalski, Ryszard S. and Robert. E. Stepp. 1983.  "Learning from
Observation: Conceptual Clustering." In *Machine Learning: An
Artificial Intelligence Approach*, R. Michalski, J. Carbonell, and
T. Mitchell, eds. Palo Alto, CA: Tioga Publishing Company.
Morris, Robert. 1980. "A Similarity Index for Pitch- Class Sets."
Perspectives of New Music 18/2:445-460.
Rahn, John. 1980. "Relating Sets." Perspectives of New Music
Rahn, John. 1989. "Toward a Theory for Chord Progression." In Theory
Only 11/1,2:1-10.
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