Computer Algebra Assisted Analysis of Atonal Music
 
Paul Lombardi and Michael Wester
University of New Mexico, USA
 
With the advent of atonal music at the beginning of the twentieth century,
composers drastically changed the way they thought about pitches.  Octave
equivalence became the prominent idea, in which pitches with the same relative
placement in different octaves were considered to be functionally equivalent
(leading to the definition of _pitch classes_ [PCs]), as well as sets of pitch
classes arranged in any order in the same or different octaves (_set classes_).
Common pitch classes between pitch-class sets are considered to be
_invariances_ of the music.  In some types of atonal music, the order of the
pitch classes is of little consequence, while in others, the music is based on
the pitch-class order.  In this latter kind, the music is said to be serial,
and a pitch-class series can be represented by its _prime_ form P, its
_retrograde_ R (P written in reverse order), its _inversion_ I (the negative of
P mod 12 and then transposed to start at the same PC as P), and its _retrograde
inversion_ RI.  These forms can be combined into matrix representations such as
twelve-tone matrices and rotational arrays.  We have developed a Maple program
that, given the generating series, finds all the invariances of any size within
a single matrix or set of matrices, as well as in slices of higher dimensional
objects such as cubes and four-dimensional hypercubes.  The list of invariances
within a twelve-tone matrix is useful because one can quickly search the list
for significant musical qualities within a twelve-tone composition.  We use
statistics from the complete list of invariances within the 16 rotational
arrays from Igor Stravinsky's _Requiem Canticles_ to consider Stravinsky's
serial mistakes.  Lastly, we search for specific invariances within the
twelve-tone hypercube from Pierre Boulez's _Structures 1a_ to compare the
seemingly arbitrary assignment of the various serialized musical elements.