Evening in the Palace of Reason. James Gaines

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 and no other voice must be added …

      Another of Bach’s contemporaries imagined the moment when the first contrapuntist, stumbling on a perpetual canon, found “the beginning and end bound together” and discovered “the eternal unending origins as well as the harmony of all eternity.”

      From such a celestial height, perhaps it is possible to look again at a young boy copying music from his brother’s notebook on a moonlit night and see what he is doing a bit more clearly. The composers in the “moonlight manuscript”—Kerll, Froberger, Pachelbel—were the reigning masters of counterpoint, men who knew about the great design, who plied its strings and levers. To a boy so recently an orphan, simply the belief that there was such a design—that God was present in an orderly universe—must have been as comforting as it was elusive. His brother’s notebook was the closest Sebastian had ever come to such an idea of life and music, and the gesture of putting his hand through the grille of that cupboard was about more than the desire for a musical education: He was reaching for answers. Christoph had been Pachelbel’s (the sorcerer’s) apprentice, so the secrets in his notebook were worth any amount of lost sleep to Sebastian. But in this light, Christoph’s attitude is no less understandable: What gave his little brother, a schoolboy, the right to such precious and hard-won knowledge? Of course his brother took away Sebastian’s copy of the notebook, and of course he would have forbidden anyone to copy it in the first place.

      Such a reading of this anecdote requires no great psychoanalytic reach. Sebastian’s worldview was profoundly allegorical, like that of his time and culture. The favored allegories at the time were Lutheran, of course, but not exclusively so. After all, Kepler had read horoscopes, and both Newton and Leibniz still had hopes for alchemy. Efforts to advance the education of Sebastian’s day could hardly be called enlightened, confounded as they still were by ignorance and superstition. The seventeenth-century educational reformer Jan Amos Comenius, for example, introduced physics into the curriculum, but it was a physics in which the world’s qualities were exactly three: “consistence, oleosity and aquasity,” the attributes of salt, sulfur, and mercury, a decidedly medieval stew. So Sebastian’s education was as thoroughly theological, or, more broadly, mythological, as his father’s and grandfather’s had been.

      In tertia, where Sebastian began his Lyceum studies in Ohrdruf, there was some reading of classics and history, but it was carefully edited for religious content; most of his class time was spent on Scripture and the catechism. In secunda and the first part of prima, he got lots of Latin—especially rhetoric and oratory—as well as some Greek, math, history and science, but his hardest work was on Leonhard Hutter’s exhaustive and exhausting exegesis on Lutheran doctrine, the Compendium locorum theologicorum, hundreds and hundreds of pages in Latin, great chunks of which he was expected to memorize.

      Given his class standing, he obviously mastered it, but it is difficult to see how, even for someone trying to throw himself into work. Outside of class, the lovely singing voice that got him his scholarship made him an anchor not only of the chorus musicus but also of a smaller group that did advanced works for the church as well as weddings and funerals. At the same time he must have been practicing the organ and harpsichord for long hours every day. He credited his brother Christoph with giving him his first keyboard lessons, but by the time he left Ohrdruf, after only five years, his technical mastery was already prodigious. The transformation of a novice into a budding virtuoso in five years would have been a remarkable feat even without all his other work, an accomplishment for which even the best teacher could not take credit.

      TEN DAYS BEFORE his fifteenth birthday, Sebastian put his clothes in a bag, strung a violin over his shoulder, and set out, on foot, for a new school more than two hundred miles away. Nobody in his family had ventured so far from their Thuringian heartland, but of course Sebastian was not anybody else. The move was forced on him. Christoph’s home was becoming overcrowded as his family grew, and Sebastian could no longer pay for his keep because for unknown reasons he had lost his job as a tutor to the children of wealthy citizens. He may have felt he was being orphaned yet again, but in fact the offer of a choral scholarship to St. Michael’s Lyceum in Lüneburg, a town almost four times the size of Ohrdruf, was providential. His brother would probably have told him not to go there but to apprentice himself to a master as Christoph had done at this age. None of Sebastian’s siblings or ancestors had gone as far in school as he had gone already. But there was a wonderful library at St. Michael’s, with a famous collection of all the contrapuntal art of Europe; and Lüneburg was not far from Hamburg, the largest and most musical city in Germany.

      For a boy who had known only Eisenach and Ohrdruf, Lüneburg was another country, with large open public squares, Renaissance architecture, and a distractingly robust musical life. We know that Sebastian graduated St. Michael’s, but there are no claims for his academic excellence there. When his voice changed soon after he arrived, he kept his music scholarship by singing bass and playing keyboards and violin, but apart from choral service the plainest trail he left in Lüneburg was extravagantly extracurricular.

      While St. Michael’s tried to teach him yet more Latin, Sebastian taught himself French and Italian, which he needed to make his way through the music library. As Leo Schrade noted in his deceptively small book, Bach: The Conflict Between the Sacred and the Secular, the richness of the library at St. Michael’s must have been dazzling to Sebastian but also somewhat disorienting: There was a great deal of German church music in the collec-tion, but none of it was as German as it was the work of German composers writing in French or English or Italian or Dutch, the problem that sent Handel off to Italy and Telemann into the opera. After serving as a battleground for the great powers, Germany had not developed its own musical (or any other) traditions since the Thirty Years War so much as it absorbed them, a fact which Bach’s foremost predecessor, the great composer Heinrich Schütz, bemoaned in his late life even as he continued