"The
Reduction Graph as Analytical Tool"
The work
of John Clough is marked by an enduring interest in the hierarchical relationships
between diatonic intervals. Employing ordered pitch class intervals mod 7, he
thoroughly describes the procedures of "extrapolation" and "interpolation" and
demonstrates how they elucidate melodic sequences in the music of Beethoven, Brahms
and Mozart. In a paper with Cuciurean and Douthett, he renames "extrapolation"
"reduction," applying it to multiplicity sequences. Though Clough demonstrates
that ordered pc intervals mod 7 can be used to label the root motions in harmonic
sequences, the technique of reduction has not yet been systematically applied
to them. In this paper, I extend the procedure of reduction to harmonic sequences
whose patterns contain more than two chords. Employing a classification scheme
for sequences derived from Clough's work, I propose an analytical tool called
the reduction graph that displays all sequences that are embedded within a particular
larger sequence. The reduction graph proves to be an invaluable tool for relating
"many-chord" sequence patterns to more familiar ones, and-more significantly-for
uncovering hidden motivic parallelisms. Sequences in the music of Bach, Chopin,
Schumann, and Wagner are examined.
"Enharmonic
Systems: A Theory of Key Signatures, Enharmonic Equivalence, and Diatonicism"
Key
signatures and enharmonic equivalence are taken as points of departure for a study
of the diatonic-chromatic relationship. Key signatures are modeled as signature
vectors, seven-dimensional vectors with integer coordinates, each coordinate indicating
the number of sharps or flats assigned to one of the seven letter classes. Several
musically meaningful operations on signature vectors are studied, and a definition
of standard signature vectors (corresponding to the key signature of some major
or minor key) is formulated. These definitions do not depend on any convention
for enharmonic equivalence of pitch classes. Enharmonic equivalence (EE) conditions
may, however, also be formalized in terms of signature vectors, called in this
context EE vectors; the canonical EE vector gives rise to a familiar twelve-pc
enharmonic system, but other systems are possible. Under certain conditions, these
systems share many of the standard properties arising in diatonic set theory (maximal
evenness, cardinality equals variety, well-formedness). The usual staff notation,
including key signatures, may be realized within any enharmonic system, and various
transformations (diatonic and chromatic transposition, and signature transformations
that alter the key signature) may be applied to music thus notated. The interaction
between the EE vector defining the system and the signature vector defining a
seven-note subset thereof is subtle and sometimes unexpected. Some applications
of nonstandard signature vectors and noncanonical EE vectors are shown, and the
concepts described are illustrated with musical examples from the Twelve Microtonal
Etudes by Easley Blackwood.