With the advent of atonal music at the beginning of the 20th 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), 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 pitch-class 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. 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.
