SECTION II: “COMPARABLE” REGISTRAL SETTINGS
The
theoretical concept upon which this study will be built is the set-class-based
chord classification examined in Castrén (1997): the permutations of a
pitch-class set belonging to some set-class X are interpreted as chords in a
certain simple manner, constituting together the chord class of X. The chord classes of all set-classes, in turn,
constitute together a universe of more than one hundred million chord
types.
In
principle, the manner in which the chord classification could be used to study
pairs of chords could be a straightforward one. Suppose, for example, that we
are testing a similarity measure. First, the measure would suggest some degree
of abstract structural resemblance between SCs X and Y. Then, the chord classes
of X and Y would be generated. After this, each element in chord class X would
be paired with each element in chord class Y. Finally, the group of individuals
testing the similarity measure would listen to the chord pairs, and it would be
analyzed if something in their listening experiences seemed to corroborate or
contradict the suggested degree of set-class resemblance between X and Y.
There are, however, obvious reasons to discourage us from adopting a test
arrangement of such simplicity. One is the sheer quantity of possibilities,
especially when two SCs of reasonably large cardinalities are examined. The
chord class of a 7-pc set-class, for example, contains 5040 chords. When two
such entities are studied, the pairing of each element in the first chord class
to each one in the second would produce more than 25 million chord pairs.
Another
reason is the great diversity among the possible registral settings that two SCs
can provide. While many observers would probably agree that it is possible to
derive two highly similar chords from two highly similar SCs, they would also
probably agree that some other registral settings might prevent the SC
similarity from being aurally evident. Comparing, for example, a narrow,
cluster-like chord to a wide one could produce a situation where the different
outer appearances dominate the observer’s perception entirely. Or, comparing a
chord in a low register to another in a high one could, in a similar fashion,
direct the observer’s attention so strongly to one aspect that the
contribution of others could be masked altogether.
A
notion that seems to suggest itself as a suitable starting point, then, is that
of “comparable” registral settings, that is, chords selected and placed so
that as many corresponding features of them as possible are in as similar an
arrangement as possible. (Informally, the pairs are to contain as few
“attention-stealing” components as possible). Before examining what sort of
criteria could be evoked when searching for such comparability, let us first
observe a few aspects pertinent to the topic.
First,
we shall never be able to reach circumstances under which we can observe and
compare chords or other pitch formations from the point of view of only their SC
identities. Other factors necessarily intervene, contributing distracting
information.
Second,
a set-class has no specific registral way of being. A given chord is not a
better or worse representative of the class than any other.
Third,
an ideal chord pair that would somehow illuminate SC similarity more truthfully
than others does not exist.
Fourth,
the concept of similarity between two chords is a complex one, as there are many
potentially relevant factors involved: sizes, widths and registers; pitch
distributions between outer pitches; SC identities; interval successions between
adjacent pitches (INTs); intervals between the lowest pitch and the rest of the
pitches (the “Above Bass” structure); the total interval and interval-class
contents; numbers of common pitches and/or pitch classes; etc. These aspects are
merely ones that might arise from typical pcset-theory-oriented discussions.
Other approaches, say, psychoacoustical ones, would naturally provide additional
viewpoints. Timbre, for instance, may play a major role in the aural comparison
of two chords.
These
observations make it evident that we cannot have perfect laboratory conditions
within which to perform absolutely controlled tests of SC similarity. To me, the
conclusions to be drawn from this are of a practical nature. I suggest an
approach that consists of two components: (a) It offers straightforward criteria
for defining comparable registral settings; and (b) It produces many chord pairs
from one SC pair comparison. This is done in order to illuminate the case from
many different sides: competent observers, I believe, can identify information
that “shines through” diverse test materials even if this information was
difficult to conceptualize or verbalize.
I
will present the comparability criteria mentioned in (a) as a simple set of
steps. I do not claim that these steps are the only possible or relevant ones;
another approach could produce different criteria. A test arrangement involving
some musical context, for example, could suggest altogether different types of
pitch formations to be selected.
As far as (b) is concerned, it is obvious that a chord-pairing procedure built
upon a computer-generated chord classification needs a computer to be feasible.
A demonstrational computer program is therefore an essential part of the study.
With it the user can generate chord pairs fulfilling the comparability criteria
and listen to them with the help of a MIDI instrument.
Within
the limits of this study, it is not my intention to conduct actual tests
concerning the correlation between SC similarity and chordal similarity. Nor
will I go into the other, closely intertwined topic mentioned above, namely,
testing the descriptive powers of similarity measures or other means of
establishing SC relatedness.
In Example 9 below I will, however, provide examples of chord pairs derived from
pairs of SCs that are deemed highly similar in their comparison groups
by the RECREL similarity measure (see Castrén 1994). On the one hand, this is
to give the reader a better general impression of what kind of results the
suggested chord pairing principles produce. On the other hand, the RECREL values
associated with the pairs give the reader an informal, small-scale opportunity
to assess whether SC similarity as identified by one approach can indeed
correlate with experiences of aural similarity.
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SECTION III
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