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Pairs
of Chords as Objects Illuminating Set-Class Similarity: Some Viewpoints and a
Computer-Assisted Procedure to Create Test Materials for Listener Experiments
by
Marcus Castrén
Theorists have been
studying ways of relating pitch-class sets (pcsets) or set-classes (SCs) already
for several decades. This has resulted in a variety of approaches aiming to
reveal some sort of equivalence, family membership, resemblance, association or
closeness between pairs or within groups of these objects. In turn, the
approaches have been based on concepts such as transposition, inversion,
complementation, inclusion, exclusion, interval-class contents, interval
successions, subset-class contents, etc.
In a number
of cases the validity and the descriptive powers of a certain approach, be it a
similarity measure, a set-complex or a construct of some other type, have been
tested with the help of an analysis, or at least of an analytic fragment.
Typically these analyses present a number of segmentations, process the
identified SC materials with the help of the approach that is being tested, and
conclude with discussions about its usefulness. SCs, it would seem, are
treated as entities with which one can illuminate abstract harmonic cohesion and
concealed principles of pitch organization in some musical context.
In my
opinion, however, it is evident that these analyses also share an important,
tacit assumption about set-classes as analytical objects: abstractly identified
SC relatedness can be identified also at other, more immediate levels of
experiencing the music. Thus, for example, if two SCs have near-identical
subset-class contents and are therefore considered to be highly similar at some
abstract level, the tacit assumption would have it that the similarity can be
evident also at an aural level. Registral, textural and other aspects might
vary, but in a context like this we might say to ourselves, for example: “The
similarity measure suggests close resemblance between the set-class 6-35
{0,2,4,6,8,10}, being represented by these pitch formations, and set-classes
5-33 {0,2,4,6,8} and 4-21 {0,2,4,6}, being represented by those other pitch
formations. I indeed hear the shared whole-tone quality to be a highly
discernible and unifying element in this passage.”
In the
theoretical literature, the correlation between abstractly assessed similarity
between SCs and aurally perceived similarity between pitch formations
representing them has not received a great deal of attention. Writers, when
discussing not only similarity assessments in particular, but various possible
relations between pitch-class-based entities in general, come to different
conclusions about their perceptual validity. Deutsch, being skeptical, questions
whether the transformations of a 12-tone row, being theoretically equivalent to
one another, can result in “perceptual equivalences” as well (Deutsch
1982:283). She continues by referring to some pcset-theoretical constructs (the
successive-interval array discussed in Chrisman 1971 and the similarity
relations introduced in Forte 1973), stating that “The extent to which the
structures defined by such theories are processed by the listener remains to be
determined” (Deutsch 1982:285). Listener experiments arranged by different
researchers have produced differing results, from suggesting reserve, or at
least caution, to showing positive correlation between experimental data and
some similarity measures.
Some commentators see the validity of pcset/SC relations to be tied first and
foremost to contextual considerations (Beach 1979:13), while others are
discussing the possibility of acquiring the ability to hear the relations or,
generally, to train one’s aural sensibility to pitch-class sets.
At the optimistic end of the spectrum is Morris, who, when discussing the
equivalence relation between the member sets of a single Tn/I-type
set-class, states: “Presumably, sets that are so related will be aurally as
well as logically related somewhat independently of their compositional
realization” (Morris 1979-80:445). When discussing the SIM similarity measure,
he establishes his belief in the correlation between theoretical and estimated
similarity in exceptionally strong terms: “...[the measure] provides a
rationale for the selection of sets that ensure predictable degrees of aural
similitude” (Ibid., 446). He continues by saying that if two Z-related SCs are
“comparably represented in a musical setting, they will have a good deal of
sonic similarity” (Ibid., 447).
I share the
opinion that the correlation between abstractly assessed and aurally perceived
similarity can indeed exist at a meaningful level. I also think that the topic
merits systematic investigation. Out of the several possibilities that this
correlation opens for research, this study aims to concentrate on one,
deliberately limited, aspect: how the correlation might be made evident in the
simplest of circumstances, namely, between two chords that are distinct from any
musical context.
Among the
questions charting this territory, for example, are the following: If some ideal
similarity measure would suggest a close resemblance between two SCs X and Y,
would it be possible, or perhaps even certain, that we would find two chords,
one representing X and the other Y, so that the pair would produce an experience
of close aural resemblance as well? If the measure suggested a strong
dissimilarity between X and Y, would chords derived from X and Y necessarily
bring about a sensation of strong aural dissimilarity, even if we tried our best
to find registral settings that were as comparable as possible? In the latter
case, could there be a sudden dramatic exception or exceptions from the
otherwise consistent dissimilarity? Generally, is SC similarity among the
necessary preconditions without which a sensation of aural similarity cannot
exist between two chords?
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