In speculating about my aim in her letter of response, Ruth Tatlow wonders whether my article is ‘designed to take the discussion forward, or to discredit the theory of proportional parallelism’. I do not think that these are the only two choices or that they are mutually exclusive.
The author's book on parallel proportions does devote space to eighteenth-century understandings of bars and other elements, but this discussion contributes little when it comes time to assign numbers of bars and to add them up. There are multiple ways to count, sometimes invoked in the same analysis, and the matter is complicated by the composer's own ambiguous counting. The study of eighteenth-century writings does not fix these problems, and I suggested not that the author was unaware of them, but rather that she sidesteps them in the theory's application.
The response maintains that the theory of proportional parallelism is supported by the recent ‘discovery’ that Chopin used Bach's proportional ordering in his own preludes. But if this sort of relationship is mathematically inevitable in Bach, it is equally inevitable in Chopin. The law of large numbers applied in the nineteenth century as well as in the eighteenth, and points to the near certainty of a particular result in both. There is no evidence that Bach intentionally established proportions, none that Chopin found them in Bach's music, none that he purposely created them himself, and none that the practice was ‘handed down verbally and in writing from teacher to pupil’, as is claimed.
I was indeed fortunate to see Alan Shepherd's work after my article was completed, but it did not change my view. Shepherd ran randomized tests similar to the ones I performed on the Dresden Missa but using the Well-Tempered Clavier Book 1. In reporting the results, he mentions in passing that of 100,000 tests, every one had a solution – a 100 per cent probability of there being a proportion. But he then goes on to calculate that the ‘probability of finding a 1:1 proportion by chance’ is, on average, 0.146 per cent (page 122 of pre-publication version). I am not exactly sure what he means by the probability of ‘finding a proportion’, but the letter echoes this language in speaking of the improbability of Bach's ‘finding a proportion among all the possible combinations’. Perhaps this means that it would have been difficult for Bach to spot the proportions, but there is no evidence that he did, or even knew they existed. The modern analyst has found them, not Bach, and assigned significance to them.
Or maybe it relates to the likelihood of hitting on a particular proportional combination, but it is difficult to see why we might care about the odds of finding a specific proportion in any event. If I drop a hook and worm into water teeming with hungry fish, chances are really good that I will catch one; that's what it means to say a spot is a good place to fish, not that I have a certain (tiny) probability of landing a particular fish from among the many filling the waters of the seas.
Overall, the response reiterates the claim that proportions exist in this music, but it is trivial that they do, given their mathematical inevitability. The matter is non-trivial only if they can be shown to mean something, and I still do not see any evidence that Bach himself thought so. (Shepherd explicitly does not offer an opinion on Bach's ‘intentionality’.)
Finally, it is difficult to miss the curious tone of rebuke in the response. Suggesting that in my article I not only misunderstand but also mislead and misrepresent, the letter clearly implies an ethical lapse in my failure to withdraw the work. But most puzzling is the letter's fundamental concern, expressed in distinctly religious terms, that what I wrote could ‘sow doubt for those who have found [the theory] inspiring’. I have no interest in interfering with anyone's inspiration, just in asking questions about what lies behind any interpretative claim, using well-established tools and the evidence of the musical sources.
