Evening in the Palace of Reason: Bach Meets Frederick the Great in the Age of Enlightenment. James Gaines

job was to attempt to replicate in earthly music the celestial harmony with which God had joined and imbued the universe, and so in a way to take part in the act of Creation itself. Understanding what possessed young Sebastian to spend his nights trying to steal his brother’s notebook (after very long days at school and more daily hours at his music practice) requires understanding how the practice of threading musical voices into the fabric of counterpoint could have been endowed with such metaphysical power.

      The key is music’s relation to number, a connection that was as old as Plato and as new as Newton, dating from the mythic day in the sixth century B.C. when Pythagoras heard a hammer strike an anvil. In his Textbook of Harmony of the second century A.D., Nichomachus of Gerasa recorded the moment:

      One day he was out walking, lost in his reflections [when he] happened by a providential coincidence to pass by a blacksmith’s workshop … and heard there quite clearly the iron hammers … giving forth confusedly intervals which, with the exception of one, were perfect consonances. He recognized among these sounds the consonances of the diapason (octave), diapente (fifth) and diatessaron (fourth).… Thrilled, he entered the shop as if a god were aiding his plans.…

      It is a lovely and dubious story that later gets a bit loopy, perhaps through centuries of retelling. As a historical figure, Pythagoras is irretrievably lost in myth, in part because he forbade his disciples to write down anything he said. There is little reason to believe he did not exist, but it may have been someone else, perhaps one of his followers, who figured out Euclid’s “Theorem of Pythagoras,” which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. That was one of the Pythagoreans’ more useful ideas. They also posited the existence of a “counter-earth” because they could make out only nine planetary bodies and there had to be ten because ten was the perfect number.

      For Western music, the most important discovery attributed to Pythagoras was that halving a string doubles its frequency, creating an octave with the full string in the proportion of 1:2. A little further experimentation showed that the interval of a fifth was sounded when string lengths were in the proportion of 2:3, the fourth in that of 3:4, and so on. This congruence was taken to have great cosmic significance. As elaborated over a few centuries around the time B.C. became A.D., the thinking (much oversimplified) was that such a sign of order had to be reflective of a larger, universal design—and sure enough, the same musical proportions were found in the distances between the orbits of the planets. Further, since such enormous bodies could not possibly orbit in complete silence, they must be sounding out these intervals together, playing a constant celestial harmony. Certified by Plato’s Republic and Timaeus, where the celestial music is said to be sung by sirens seated aboard their respective planets, the mathematical-cosmic nature of music was transmitted to Baroque composers and their predecessors by the Roman scholar Ancius Manlius Severinus Boethius, whose sixth-century writings constituted the most widely read treatise on music theory for the next thousand years.

      Of course such a perfectly ordered universe could only be the work of God, the all-encompassing One (represented by the unison in the proportion 1:1), and the unswerving reliability of this order was taken as proof of His continuing presence in the world. Despite that, the early church fathers continued to oppose anything but plainsong in the liturgy, hearing the work of the devil in more elaborate music. Saint Augustine finally resolved the question in favor of liberating music to glorify God, but not without torment.

      I waver between the danger that lies in gratifying the senses and the benefits which, as I know from experience, can accrue from singing. Without committing myself to an irrevocable opinion, I am inclined to approve the custom of singing in church, in order that by indulging the ears weaker spirits may be inspired with feelings of devotion. Yet when I find the singing itself more moving than the truth which it conveys, I confess that this is a grievous sin, and at those times I would prefer not to hear the singer.

      Of critical importance for Bach and his time, Martin Luther sided with the Platonic idea of music as evidence of divine order, and he set out to rehabilitate Pythagoras as a servant of God. In his commentary on Genesis he laments the fact that “we have become deaf toward what Pythagoras aptly terms this wonderful and most lovely music coming from the harmony of the spheres.” No less than the seventeenth-century astronomer Johannes Kepler gave Luther’s position the stamp of scientific certainty in his great work, Harmonices Mundi, where he correlates the orbits of the planets to the intervals of the scale and finds them to be “nothing other than a continuous, many voiced music (grasped by the understanding, not the ear).” This last point was debated: Some thought the celestial music was abstract, an ephemeral spiritual object, but others insisted it was real, inaudible to us only because it had been sounding constantly in the background from the time of our birth. In either case music was a manifestation of the cosmic order. “Now one will no longer be surprised,” Kepler wrote, “that man has formed this most excellent order of notes or steps into the musical system or scale, since one can see that in this matter he acts as nothing but the ape of God the Creator, playing, as it were, a drama about the order of celestial motions.” One of his chapters is titled “There Are Universal Harmonies of All Six Planets, Similar to Common Four-Part Counterpoint.”

      Never were allegories packed into music more enthusiastically than in Bach’s time. Andreas Werckmeister was far from alone in attaching specific integers, for example, to the Trinity: 1 stood for the Father, 2 for the Son, 3 for the Holy Spirit, the last being the sum and proportion of Father and Son (1:2, a unison at the octave). Elsewhere we find that 4 represented the four elements of matter and the four seasons, that 5 meant justice (because it stands at the center of the first ten numbers) and humanity (five senses and the five appendages of arms, legs, and head). Saint Augustine favored the number 6 (creation took six days, so God must have found 6 to be a perfect number), 7 stood for the planets, virtues, liberal arts, deadly sins, and ages of man (although sometimes it is said not to stand for anything, since on the seventh day God rested). Twelve covered apostles, months, prophets, etc. Then there were the combinations, for example: 7 being 3 (the Trinity) plus 4 (elements of matter), and 12 being 3 (ditto) times 4 (ditto), it followed (trust me) that 7 and 12 were perfect numbers. And so on. All this from Pythagoras, whose followers thought the best number was 10 because it was the sum of 1, 2, 3, and 4, elemental components of the “figured number” known as the Pythagorean Tetraktys:

      It is easy to have fun with number theory, and some of the best such fun is in the distant reaches of the Bach literature, where one can read, for example, that he left a prophecy in musical code of the date of his death on the Rosicrucian calendar. On the other hand, it is only the extent to which Bach’s music contains meanings coded in numbers that is hotly debated. The fact that it contains such coded meanings is not.

      Cosmological harmony was actually one of the few ideas on which the philosophers, scientists, and theologians of Bach’s time were agreed. Newton, for example, could not imagine that a world as orderly as this one could have occurred by “natural Cause alone.” A “powerful, ever-living Agent … governs all things,” he concluded, “not as the soul of the world, but as Lord over all.” Newton’s Agent was Luther’s Celestial Contrapuntist, whose woven voices were like so many planetary orbits.

      We marvel when we hear music in which one voice sings a simple melody, while three, four, or five other voices play and trip lustily around … reminding us of a heavenly dance.

      This heavenly dance was nowhere more sublime than in canon, the strictest form of contrapuntal writing, in which an entire piece of music is built from a single melodic phrase playing over itself at different intervals of time and key, in varying rhythms and tempi, sometimes appearing backward, inside out, or upside down, and sometimes continuing, at least theoretically, forever. Andreas Werckmeister drew the analogy of cosmos to counterpoint even more explicitly than Luther had done:

      The heavens are now revolving and circulating steadily so that one [body] now goes up but in another time it changes again and comes down.… We also have these mirrors of heaven and nature in musical harmony, because a certain voice can be the highest voice, but can become the lowest or middle voice, and the lowest and middle can again become the highest.…[In the case of canon] one voice can become all the other voices