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Published online by Cambridge University Press: 05 February 2021
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.
I am grateful to Mat Baskin, Stephen A. Crist, Ellen Exner, Julian Hook, Michael Marissen and Bettina Varwig for their advice.
1 For a history of speculative Bach interpretation and a consideration of its significance see Daniel R. Melamed, ‘Rethinking Bach Codes’, in Rethinking Bach, ed. Bettina Varwig (New York: Oxford University Press, in press). In that essay I acknowledge ways in which numbers apparently were significant to early eighteenth-century musicians (the personal number 14 to Bach, for example), but to a degree much less than has been asserted in our time and in ways very different from the numerical codes that have been claimed for Bach. On this subject see also Tatlow, Ruth, Bach and the Riddle of the Number Alphabet (Cambridge: Cambridge University Press, 1991)Google Scholar.
2 Tatlow, Ruth, Bach's Numbers: Compositional Proportion and Significance (Cambridge: Cambridge University Press, 2015)CrossRefGoogle Scholar; Tatlow, , ‘Parallel Proportions, Numerical Structures and Harmonie in Bach's Autograph Score’, in Exploring Bach's B-Minor Mass, edited by Tomita, Yo, Leaver, Robin A. and Smaczny, Jan (Cambridge: Cambridge University Press, 2013), 142–162CrossRefGoogle Scholar; Tatlow, , ‘Bach's Parallel Proportions and the Qualities of the Authentic Bachian Collection’, in Bach oder nicht Bach? Bericht über das 5. Dortmunder Bach-Symposion 2004 (Dortmund: Klangfarben-Musikverlag, 2009), 135–155Google Scholar; Tatlow, , ‘Collections, Bars and Numbers: Analytical Coincidence or Bach's Design?’, Understanding Bach 2 (2007), 37–58Google Scholar.
3 Tatlow, Bach's Numbers, 6–8.
4 This is the place to acknowledge that I consider Dr Tatlow a friend and that she has always been a most generous colleague.
5 This is the strongest contribution of Pieter Bakker, ‘Postmodern Numbers: Ruth Tatlow on Proportions in the Written Music of Johann Sebastian Bach’ http://www.kunstenwetenschap.nl/postmd-e.pdf (6 December 2019).
6 Tatlow, Bach's Numbers, 134.
7 Tatlow, Bach's Numbers, 134–136.
8 Bach, Johann Sebastian, Missa, Symbolum Nicenum, Sanctus, Osanna, Benedictus, Agnus Dei et Dona nobis pacem, später gennant: Messe in h-Moll BWV 232, ed. Smend, Friedrich (Kassel: Bärenreiter, 1954)Google Scholar; Bach, Johann Sebastian, Frühfassungen zur h-Moll-Messe, ed. Wolf, Uwe (Kassel: Bärenreiter, 2005)Google Scholar.
9 A relatively accessible treatment of the partition problem is Hayes, Brian, ‘Computing Science: The Easiest Hard Problem’, American Scientist 90/2 (2002), 113–117CrossRefGoogle Scholar.
10 It turns out that this sort of result could have been predicted. With twelve numbers and a limited range like this, the likelihood of a so-called perfect partition has been shown to be very high and to approach one hundred per cent under some circumstances. See Hayes, ‘Computing Science’.
11 The Dresden Missa has movements with a relatively narrow range of bar lengths compared, say, to Mass settings by Zelenka, whose movements contain – depending on how you count – much more widely varying numbers of bars that yield the sort of lopsided set we saw earlier.
12 The theory suggests that ‘Bach's notation of stile antico movements and the TS feature create a useful ambiguity to the bar count’. Tatlow, Bach's Numbers, 333.
13 From the large literature on this issue I recommend mathematician Brendan McKay's website at http://users.cecs.anu.edu.au/~bdm/dilugim/torah.html. Summarizing his tongue-in-cheek analysis of Moby Dick that supposedly predicts historical assassinations (an answer to a bible coder's challenge), he writes that the reason a result ‘looks amazing is that the number of possible things to look for, and the number of places to look, is much greater than you imagine’.
14 John Z. McKay, ‘The Problem of Improbability in Musical Analysis’, in L'analyse musicale aujourd'hui (Musical Analysis Today), ed. Xavier Hascher, Monher Ayari and Jean-Michel Bardez (Sampzon: Delatour, 2015), 77–90. He shows that an analytical ‘“extraordinary circumstance” appears to be nearly a 1 in a million occurrence, but . . . it is much more likely than not that [the analyst] would find something to satisfy’ the stated conditions (10).
15 We can note that if a strict half-and-half division of movements is not required, as it often is not in illustrations of the theory, there is no need for an even total number of bars.
16 On the surface it appears that there might be a parallel to what Emily Zazulia has called ‘false exceptionalism’, the error of ‘making interpretive claims based on the distinctiveness of features that are not unique or even unusual’. It is true that parallel proportions can be derived from almost any piece and are thus not special to the Dresden Missa or the solo violin works or indeed to compositions by Bach. But these relationships are not essentially features of Bach's music – they are products of the numbers, independent of the compositions with which they are associated. There is plenty of false exceptionalism in Bach studies, as any Telemann or Graupner scholar will tell you, but parallel proportions are not an example. Zazulia's point about the role of the Strong Law of Small Numbers, relevant to her examination of supposed proportions in DuFay's motet, does resonate with problems considered here. Zazulia, Emily, ‘Out of Proportion: Nuper rosarum flores and the Danger of False Exceptionalism’, Journal of Musicology 36/2 (2019), 131–166CrossRefGoogle Scholar.
17 In fact the theory's author has pointed to this feature in personal communication.
18 For example, if we flip two honest coins and want to know the probability that they will both come up heads, we can count the number of possible outcomes (heads/heads, heads/tails, tails/heads, tails/tails, for a total of four) and see right away that in one of four cases (0.25 probability) both coins will show heads. We can also get this by multiplying the probability of one coin coming up heads (0.5, or one out of two) by the probability for the other (also 0.5) for a probability of 0.25 of simultaneous heads. The likelihood that one or the other will show heads is 0.5 (two cases out of four), and the probability that one or both will is 0.75 (three out of four). And of course the coins do not influence each other.
19 Tatlow, Bach's Numbers, 333.
20 Tatlow, Bach's Numbers, 138–139.
21 Tatlow, Bach's Numbers, 149.
22 Tatlow, Bach's Numbers, 138–139.
23 One more aspect of the theory that is difficult to test logically, musically or mathematically is the claim that revised pieces and collections, in particular, contain round numbers of bars (800, 1,200, 1,600). Results like these depend on methods of counting, just like proportions, and these need to be examined in detail. But it is difficult to see how probability could play a role in evaluating this sort of claim. Oddball numbers (753) are just as likely to occur randomly as ones that look round to human observers (1,200). The only approach I can see to this element of the theory would be a close look at the methods of counting bars; at the least, this sort of claim needs to be investigated separately from proportions. Just because they both involve numbers does not mean they are connected.
24 For a concise summary of the method see http://mathworld.wolfram.com/HypothesisTesting.html.
25 Tatlow, Bach's Numbers, 138–139.
26 After this essay was completed I had the opportunity of seeing Alan Shepherd's work in progress on various mathematical analyses of Bach's music, including parallel proportions. We have used similar methods, but his treatment does not extend to a judgment on whether proportions are musically or historically significant. I am grateful to him for sharing this material.
27 A working copy of the spreadsheet is available at http://www.melamed.org/Calculator.xlsm.