J. S. Bach, Multiple “Parallel Proportions,” and US Tax Enforcement

Abstract: “Parallel proportions” in J. S. Bach’s music—the division of movements or pieces into groups with the same total number of measures, in 1:1 and other simple proportions—are mathematically inevitable; they are not the meaningful compositional acts that recent claims have suggested. But adherents to this theory have further argued that the existence of multiple proportions—not just in a whole work but among subsets of pieces or collections as well—is so unlikely that it proves that they were Bach’s intention. This claim can also be shown to be incorrect. It is notoriously difficult to estimate the probability of something after the fact, largely because it is so hard to define the entire range of things that might have occurred. Such problems, like estimating the probability of a proportion already identified, are challenging to define precisely enough to give a meaningful result. “Commonsense” estimates of very small probabilities are misleading at best. When compounded by the overwhelming likelihood that a satisfying result will arise randomly, claims of unlikelihood (and thus of compositional intent) do not hold up. Examples from the news, including the enforcement of US tax laws and the results of a medical school examination, illustrate the problem in ways that closely match the questionable analytical claims. Arguments about multiple proportions are not conclusive and do not demonstrate successfully that Bach composed proportional relationships into his works. There are many reasons to admire Bach’s construction of his music, but “parallel proportions” are not among them.