Bach and compositional numbers
Written by: Ruth Tatlow
It was during my undergraduate years in the 1980s that I stumbled across numbers in music. It was fashionable at the time to ridicule anything that smacked of number symbolism, and I joined the fun. However, the analyst in me decided that a rational destruction of the evidence could have a more lasting effect than derision. It all boiled down to whether or not Bach knew the number alphabet, and this is the question I decided to address at doctoral level.
Beginning with the writings and private letters of German theologian and musicologist Friedrich Smend, who kick-started the number alphabet craze, I saw very quickly that his idea had almost no historical basis and even less internal logic. Momentarily it looked as though the demolition would be swift and decisive. But it was not. In what I expected to be a vain search, I discovered over thirty-seven different number alphabets and an entire tradition of their use in paragrams. These were unknown to Smend and to modern scholars of baroque literature, but known to Bach and his contemporaries. Furthermore, I discovered that poets in Bach’s circle wrote about how to use these poetical paragrams to stimulate creative ideas.
Riederer’s Paragramma Cabbalisticum Trigonale, Nuremberg, 1718
Illustration 9, Bach and the Riddle, p. 97
All this new information went into my doctoral thesis, which was rewritten to become Bach and the Riddle of the Number Alphabet.
Rather than the wished-for closure to number symbolism, my research had, very unsatisfactorily, opened up many new possibilities. The next logical step was to examine the foundations of Bach’s use of a number alphabet, or of any kind of compositional numbers. More easily formulated than executed, this task involved solving a complex set of overlapping problems, including developing a methodology that would reflect Bach’s thinking and not my own; exploring which number systems Bach knew, and whether or not the Golden Section and Fibonacci numbers would feature; using evidence from Bach’s autograph scores and examining printed and manuscript sources of the time to reconstruct what Bach would have counted, or thought of counting, in order to introduce numerical patterns. That is IF Bach had introduced numerical patterns. It must be remembered that I was still trying to prove a negative.
After several years of working with Bach’s vocal compositions it was plain that these questions were unlikely to be resolved. There were too many variables in the data, the theoretical framework was not as solid as I had hoped, and looking for what I did not think existed was beginning to take an intellectual toll. Out of the blue I received an invitation to speak on the structure of Bach’s Solos for violin (BWV 1001–1006). I decided to accept the invitation, while making a personal pact that this would be my swan song presentation on Bach and numbers.
The Fates had other ideas! For the first time in years the numerical results were overwhelmingly perfect, and supported with a devastating clarity by the theoretical framework that had been emerging from surveys of theory sources. Bach’s collection of six violin solos had exactly 2400 bars, in which four of the solos had a total of precisely 1600 bars, and two exactly 800 bars. Not a bar more, not a bar less. Equally persuasive was evidence showing that Bach had revised and expanded this work from an imperfect numerical state, to make the perfected structure. He then copied the corrected collection onto beautifully lined manuscript paper as a calligraphic autograph score, the equivalent of a publication.
The next stage was to examine the structures of his other calligraphic scores and published collections. To my consternation the proportional pattern was repeated: the majority of publications had a bar total that was a whole round multiple of 100 or 1000, within which were rational units forming perfect 1:1 or 1:2 proportions. It was time to formulate a testable theory, and to come to terms with having changed sides. This was not the destruction of numbers in Bach’s music I had been working for.
I named the theory ‘proportional parallelism’. In Bach’s Numbers: Compositional Proportion and Significance the reader will find numerous new sources and tables to help weigh up the numerical evidence and test the validity of the theory.
The repeated use of large-scale 1:1 and 1:2 proportions in Bach’s revised works raised the all-important question of what motivated him to use such a time-consuming method. One answer lies in his Lutheran beliefs in the theory and practice of music that still leant heavily on universal harmony.
Puttenham, The Arte of Englishe Poesie (London, 1589),
Figure 3.1 Bach’s Numbers, page 76.
The appendix of Bach’s Numbers gives a selection of sources in parallel German-English translation as a taster of the theology of musical proportions in Lutheran Germany in Bach’s time.
The evidence and conclusion in Bach’s Numbers generated almost more questions than answers. Was Bach the only composer to count bars and proportion them in this way? Did the practice die out with him? Did he teach it to his sons and students? And if so, did they continue the practice? If so, what would it have meant to them in the changing philosophical climate of the second half of the eighteenth century? I am currently working on many of these questions, and plan to publish answers, and no doubt raise further questions, in a third monograph on Bach and numbers.