SECTION
IV: CHOOSING A SELECTION FROM A LARGE GROUP OF CHORD PAIRS
In
the case described in examples 1-4, the referential chord was picked at random.
As the number of 5-21B chords fulfilling the criteria was 5, it is obvious that
if each of the 120 5-21A chords was to serve in its turn as the referential
chord, the number of all 5-21A/5-21B chord pairs fulfilling the criteria would
be quite large (the exact number, in fact, would be 268 pairs, meaning that the
referential chord above participated in more chord pairs than an average chord
would).
In my experience, creating hundreds or even thousands of pairs is not an
end in itself. Given the task of assessing all of them in such detail, it
seems likely that a smaller sample could produce a sufficient enough overall
impression. When two SCs
of small cardinalities are examined, the situation is naturally different. The
amount of pairs fulfilling the criteria is typically such that all pairs can be
meaningfully examined.
Example
5 shows pairs derived from the 4-pc SC pair 4-Z29B / 4-Z15A. The RECREL value
between the two, 8, is the lowest value belonging to any 4-pc pair where the SCs
are not inversions of each other. The number of pairs fulfilling the
comparability criteria is 16, meaning that the nine pairs shown in the example
already represent more than half of the total.
EXAMPLE
5: Nine chord pairs fulfilling the comparability criteria. The chords are
derived from SCs 4-Z29B and 4-Z15A.
At
this point it is not my intention to adopt additional criteria with which to
limit the outcome in cases where the number of chord pairs fulfilling the
comparability criteria is very large (typically concerning SC pairs of
cardinality 5 or more). With such additional criteria one could ether
select a limited amount of referential chords, or produce all pairs and then
filter out some or most of them. The reason not to define more criteria is
obvious. The approach with which the limited materials would be created could be
seen as one reflecting some aspect of chordal similarity itself. In other words,
the material could be deemed “pre-filtered” before the observer had even
started to assess the chord pairs.
I will, however, discuss some possible limiting solutions below. Some of
them are also implemented in the demonstrational program. They are to be
understood as notions separate from the comparability criteria. It is left to
the observer to assess what sort of effects the sorting and/or filtering
mechanisms might have on the resulting chord pairs.
A neutral way of selecting referential chords from a chord class is to
choose them randomly, or at least so that one deliberately avoids paying
attention to their individual properties. Thus, for example, taking simply every
tenth chord from a chord class belonging to a (transpositionally non-symmetric)
6-pc SC produces a group of 72 referential chords. In a given situation these
could constitute a sufficient referential-chord sample.
Another obvious solution is to adopt width constraints. The chord class
of a 5-pc set-class, for example, contains 55 chords whose width is more than
one octave but less than two (see Castrén 1997:15). Under some circumstances
these could alone serve as the referential-chord sample.
If we decide to adopt viewpoints that observe the properties of the
chords, we might, for instance, focus on pairs whose chords share some common
intervallic feature. Pair a in Ex. 6 and pair b in Ex. 8, for example, contain
chords whose INTs are permutations of each other. In the first pair the INTs are
<4,3,4,4> and <4,4,3,4>, in the second <7,4,4,5> and
<7,4,5,4>.
Another possibility is to observe the spans of the linear intervals that
are created between non-common pitches of the two chords. When we have, for
example, chord pairs derived from two 5-pc SCs that share one or more 4-pc
subset classes, we could concentrate only on pairs where the intervals between
the two non-common pitches are as narrow as possible. Example 6 displays six
pairs of chords, derived again from 5-21A and 5-21B, with increasingly widening
intervals between the non-common pitches. In 6a the width of this interval is
one semitone; in b, three; in c, five; in d, nine; in e, 13; in f, 17.
EXAMPLE
6: Pairs of chords with increasingly widening intervals between non-common
pitches. The chords are derived from SCs 5-21A and 5-21B.
An
approach of an entirely different nature can be seen in Ex. 8, being once more
exemplified with the help of chords from chord classes 5-21A and 5-21B. The
overtone series is first modified so that each partial is rounded to the nearest
tempered pitch. Ex. 7 shows the result with 28 partials, having MIDI pitch 24,
three octaves below the middle C, as the fundamental.
EXAMPLE
7: An overtone series modified so that each partial is rounded to the nearest
tempered pitch.
Each
of the 120 5-21A chords is then compared in its turn to the modified overtone
series, the latter being at the transpositional level shown in Ex. 7. The
purpose is to find out the lowest subchord position of the 5-21A chord, that is,
what is its lowest transposition so that all of its pitches also belong to the
larger pitch formation. After this, a 5-21B chord fulfilling the comparability
criteria with the referential 5-21A chord is transposed to the same
transpositional level. It is then tested if the 5-21B chord is a subchord of the
modified overtone series as well. If it isn’t, it is transposed until its
lowest subchord position is found out. (At this stage, of course, it no longer
fulfills the criteria).
Example 8 shows six pairs where the chords both fulfill the criteria and
are subchords of the modified overtone series at its Ex. 7 transpositional
level.
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