Some Group Properties of Triple Counterpoint and Their Influence on Compositions by J. S. Bach

Abstract

Invertible counterpoints in tonal music are a special and unique class of thematic repetition. Like unvaried reprises, recapitulations, and transpositional sequences, invertible counterpoints reuse thematic material in different parts of a musical form. Unlike these other procedures, which preserve interrelationships among the several melodic lines, invertible counterpoint alters these interrelationships so that the melodic components are rearranged with respect to each other upon repetition. For this reason invertible counterpoint can be a more subtle and covert means of thematic repetition than the other procedures mentioned above, and therefore can become a powerful way to unify musical structure. In the hands of J. S. Bach, who recognized new and unexplored possibilities for invertible counterpoints, it is used so often and with such imagination that the formal economies of his compositions are often extraordinarily simple in relation to the beautifully complex sound of the music. In the pages following, I will examine both the general properties of triply invertible counterpoints (triple counterpoint) and the use of this compositional technique in some of Bach's music. The reader might well ask why a study of Bach's use of double counterpoint is not prerequisite. The answer lies in the different abstract structure of these two types of invertible counterpoint and the effect these differences have on compositional procedure.