'PARALLEL PROPORTIONS' IN J. S. BACH'S MUSIC

Recent studies propose that J. S. Bach established 'parallel proportions' in his music - ratios of the lengths of movements or of pieces in a collection intended to reflect the perfection of divine creation. Before we assign meaning to the number of bars in a work, we need to understand the mathematical and musical basis of the claim. First we need to decide what a 'bar' is and what constitutes a 'movement'. We have explicit evidence from Bach on these points for Bach's 1733 Dresden Missa, and his own tallies do not agree with those in the theory. There are many ways to count, and the numbers of movements or bars are analytical results dependent on choices by the analyst, not objective data. Next, chance turns out to play an enormous role in 'parallel proportions'. Under certain constraints almost any set of random numbers that adds up to an even total can be partitioned to show a proportion, with likelihoods better than ninety-five per cent in sets that resemble the Missa. These relationships are properties of numbers, not musical works. We thus need to ask whether any apparent proportion is the result of Bach's design or is simply a statistically inevitable result, and the answer is clearly the latter. For pieces or sets with fewer movements the odds are less overwhelming, but the subjective nature of counting and the possibility of silently choosing from among many possibilities make even these results questionable. Theories about the number of bars in Bach's music and possible meanings are interpretative, not factual, and thus resistant to absolute disproof. But a mathematical result of the kind claimed for 'parallel proportions' is essentially assured even for random sets of numbers, and that makes it all but impossible to label such relationships as intentional and meaningful.