Collections, bars and numbers : Analytical coincidence or Bach's design?

This paper describes the results of a long-term search for a historically-consistent theory of mathematical procedures in Johann Sebastian Bach’s compositions. The mathematical nature of Bach’s works has entered popular myth. Comments about the supposed numerical bases of his compositions date back to his lifetime, and were perpetuated in the obituary and later biographies. Theorists contemporary with Bach strongly imply that numerical ordering and the use of proportions were important to the composer, but tantalisingly they do not say how, nor with what a composer should order a composition or how he should create proportions. Was there a well-tried and tested numerical method underlying the general guidelines described by early eighteenth-century theorists? Did Bach use proportions to organise his works? And if so would it be possible to find them in his scores? Working from historical sources and the autograph scores the initial negative results of my research were overturned by an unexpected breakthrough which led to the formulation of the theory of proportional parallelism. The technical nature of the new theory is demonstrated in four parallel levels of numerical structure found in Sei Solo a Violino senza Basso accompagnato, the six violin sonatas and partitas, BWV 1001-1006. That these parallel proportions are found in all compositions Bach published or transcribed in Fair Copy raises important issues for the Bach scholar and editor, and demonstrates the far-reaching scope of the new theory