Frederick the Great had been on his throne for seven years when he summoned Johann Sebastian Bach to Potsdam in 1747. Music had been a passion of Frederick’s since his youth, when he secretly played flute and lute duets with his sister – and hid this from his father. According to witnesses, there was agitation in his voice when he heard of the 62-year-old master’s arrival at the gates one Sunday evening.

Once Bach was installed at the keyboard, he was given a challenge: a short theme, 21 notes in all, on which Bach was to improvise a three-part fugue. Bach did so to the “astonishment” of all present and was given another challenge. Could he now play a six-part fugue on the subject? To improvise a fugue of such complexity was beyond even Bach, but within a month he had published his response. The “Musical Offering,” a series of movements – including a six-voice fugue – on the royal theme is still considered one of his great works.

Traditionally, the encounter between Bach and Frederick has been painted as the felicitous meeting of two great minds who reflected and enhanced each other’s glory. A new book by James R. Gaines, “Evening in the Palace of Reason” (Fourth Estate, 336 pages, $23.95) sees it as a clash between opposing world views and presents the “Musical Offering” as nothing less than a musical gauntlet.

Mr. Gaines elegantly sketches parallel biographies of the two protagonists. Bach is the devout follower of Luther in both faith and music, the pater familias whose intricately contrapuntal compositions are rooted in the belief in, and intended to mirror, a divinely ordained, harmonious universe. Frederick, a bisexual atheist, suffered an abusive childhood at the hands of a father who was in equal measure autocratic and insane. As crown prince, Frederick’s love for all things French and for his flute, which he called “Principessa,” earned him public beatings from his father. By the time he ascended the throne, at the age of 28, his enthusiasm for the art and philosophy of the Enlightenment was only matched by that for military expansion and duplicitous diplomacy.

If their love for music united Bach and Frederick, their view of its mission in the world could not have been more different. Bach, who spent most of his career in the service of the church, believed that music “can be for nothing else but the glory of God and the restoration of the heart. Where this is not the case, there is no real music but only a demonic noise.” For the young Frederick, music had been an escape, a balm to soften the countless acts of paternal humiliation. As king, he had assembled around him composers who rejected counterpoint for the new galant style, which favored a single accompanied melodic line. Its only aim: the enjoyment of the listener.

The meeting in Potsdam, then, was charged with tension. The theme Frederick proposed to Bach, with its jarring downward jump followed by a chromatic, snaky descent, was an open taunt. As Mr. Gaines writes, “These were twenty-one notes that, if they were not calculated to make the task as difficult as possible, had been thrown together in an accident of anti-contrapuntal genius.” Bach was being set up for failure.

Bach resoundingly rebuked the king’s worldview in his “Musical Offering.” Frederick hated canons in particular – counterpoint created by setting a theme against itself – yet Bach presents him with no fewer than 10 of them. One canon, prefaced by the words, “As the keys ascend so may the glory of the king also ascend,” has the melody rising in whole tone steps, until it arrives back where it started – in other words, it doesn’t appear to ascend at all. In Bach’s oeuvre, the figure 10 often stands for the Ten Commandments, and there can be no accident to the number of canons he presents to the king: Beware of worldly glory, he seems to be saying, and heed only the law of God.

“Evening in the Palace of Reason” leaves many questions about the “Musical Offering” and its genesis unanswered, as any conscientious author is bound to do. Thankfully, Mr. Gaines resists the temptation to turn this story into another “Da Vinci Code,” although there is no shortage of esoteric riddles in Bach’s music. There could be more on the actual piece and a little less on the admittedly entertaining antics of the Hohenzollern dynasty, but this would have made the book more technical and less generally appealing. Mr. Gaines overindulges in his conversational style, but his enthusiasm for his subject is infectious.

Whether Bach and Frederick were really children of such different ages should remain a matter of debate. Certainly Bach, for all his conservative faith, was not a backward-looking composer. Mr. Gaines spends a good deal of the earlier parts of his book exploring the connection between Baroque composition and alchemy, but Bach also showed a very modern, scientific delight in exploring the limits of counterpoint into the darkest harmonic recesses of the scale. His “Well-Tempered Clavier,” in this sense, is closer in spirit to Diderot’s Encyclopedia than to a medieval alchemist’s cookbook.

As a new generation of composers today turns away from the rigors of serial music, which, paradoxically, sounds most random when it is at its most structured, the words of the galant composer Johann Mattheson sound oddly prophetic. “Do not expect after all that quill-chewing and toiling to be rewarded for your pains,” Bach’s contemporary wrote in an attack on canons and learned counterpoint. “There will probably be not a single one among 2,000 listeners who will notice your finesse, unless he be alerted to it beforehand.”

The struggle between music that is pleasing to the ear and that which is the pure expression of an inner architecture remains a dilemma in contemporary music. In the music of Bach, they were united.

Miss da Fonseca-Wollheim is a writer living in New York.