“Scope,
Method, and Goal of Scale Theory, and Notes on 'Cardinality Equals Variety for
Chords'” “Diatonic
Transformations in the Music of John Adams”
In their 1985 Journal of Music
Theory article, “Variety and Multiplicity in Diatonic Systems,” John
Clough and Gerald Myerson explored the following property of the usual diatonic
system: if melodic lines are sorted into categories according to the number of
diatonic steps spanned between adjacent notes in the melody, then the number of
varieties of such melodies is equal to the number of distinct diatonic pitch classes
in the line. For instance, arpeggiated triads come in three varieties: major,
minor, and diminished, while arpeggiated seventh chords and four-note scale segments
each are found in four varieties: major, minor, dominant, and half-diminished;
tone-tone-semitone, tone-semitone-tone, semitone-tone-tone, and tone-tone-tone,
respectively. Clough and Myerson proved that this property, “cardinality
equals variety for lines,” holds for a class of scales that they called “diatonic
systems.” The property “cardinality equals variety for chords,”
however, holds in some diatonic systems (including the usual diatonic), but not
in others. Clough and Myerson made a conjecture about which chords (unordered
subsets) would fail to exhibit cardinality equals variety. The principal concern
of this paper is to determine precisely under what conditions cardinality equals
variety for chords holds or fails.
The purposes
of diatonic theory or scale theory have sometimes been misunderstood, especially
for mathematical results such as those set forth in this paper. Accordingly, the
paper is framed by a discussion of the “scope, method, and goal” of
scale theory that provides a context for this work.
In
a significant number of John Adams's pieces, the number of tones held in common
between adjacent diatonic areas, and also between the musical events occurring
within these areas, indicates an important facet of the relationship between the
corresponding musical passages. By varying or alternatively by preserving the
number of common tones held between the sound events and their implicit or explicit
diatonic contexts in adjacent passages, Adams transforms strongly established
diatonic areas, along with their constituent musical elements, into new diatonic
areas, creating an ebb and flow within his music based on the relative smoothness
of the relationships. The presentation develops a formal model for describing
common-tone relationships between chords (triads or seventh chords), sonorities
(all sounding pitch classes), and fields (diatonic collections inferred by the
musical context).
The diatonic transformations
discussed in this paper will be shown to be related to similar transformations
familiar from tonal music of the eighteenth and nineteenth centuries. However,
the new formal approach taken to this topic, the repertoire used to illustrate
these transformations, the characteristic shimmer of both Adams’s orchestration
and his sonority construction, and especially the distinctive harmonic and diatonic
relationships that ensue from his style, all suggest that this paper will provide
a fresh perspective on this topic. The presentation will be illustrated by examples
drawn from four important pieces in Adams’s early development and maturation
as a composer, presented both in written form and aurally.