Hermann Hesse wrote:
“It was the achievement of one individual which brought the Glass Bead Game almost in one leap to an awareness of its potentialities, and thus to the verge of its capacity for universal elaboration. And once again this advance was connected with music. A Swiss musicologist with a passion for mathematics gave a new twist to the Game, and thereby opened the way for its supreme development. [...] There was a passionate craving among all the intellectuals of his age for a means to express their new concepts. They longed for philosophy, for synthesis. The erstwhile happiness of pure withdrawal each into his own discipline was now felt to be inadequate. Here and there a scholar broke through the barriers of his specialty and tried to advance into the terrain of universality. Some dreamed of a new alphabet, a new language of symbols through which they could formulate and exchange their new intellectual experiences.Testimony to the strength of this impulse may be found in the essay "Chinese Warning Cry," by a Parisian scholar of those years. The author, mocked by many in his day as a sort of Don Quixote (incidentally he was a distinguished scholar in the field of Chinese philology), pointed out the dangers facing culture, in spite of its present honorable condition, if it neglected to develop an international language of symbols. Such a language, like the ancient Chinese script, should be able to express the most complex ideas graphically, without excluding individual imagination and inventiveness, in such a way as to be understandable to all the scholars of the world. It was at this point that Joculator Basiliensis applied himself to the problem. He invented for the Glass Bead Game the principles of a new language, a language of symbols and formulas, in which mathematics and music played an equal part, so that it became possible to combine astronomical and musical formulas, to reduce mathematics and music to a common denominator, as it were. Although what he did was by no means conclusive, this unknown man from Basel certainly laid the foundations for all that came later in the history of our beloved Game.”
Throughout the whole of the novel, especially in these first few chapters, we are given glimpses of the Glass Bead Game itself. We learn from Ziolkowski's foreword that at least a prototype of the game is real enough for Hermann Hesse, as he has played it himself.
“In the idyllic poem "Hours in the Garden" (1936), which he wrote during the composition of his novel, Hesse speaks of "a game of thoughts called the Glass Bead Game" that he practiced while burning leaves in his garden. As the ashes filter down through the grate, he says, "I hear music and see men of the past and future. I see wise men and poets and scholars and artists harmoniously building the hundred-gated cathedral of Mind." These lines depict as personal experience that intellectual pastime that Hesse, in his novel, was to define as "the unio mystica of all separate members of the Universitas Litterarum" and that he bodied out symbolically in the form of an elaborate Game performed according to the strictest rules and with supreme virtuosity by the mandarins of his spiritual province.”
This being the case, Ziolkowski contradicts himself in then concluding that "the Game is of course purely a symbol of the human imagination." While I agree that it is not "a patentable Monopoly of the mind" I think the Glass Bead Game is eminently realisable in reality, and I certainly don't think it was described "to defy any specific imitation in reality" as a number of people have demonstrated since by designing playable variants of the Game. On the contrary, the connections between mathematics and music that Hermann Hesse describes as being at the origin of the game are real, and they are amenable to being applied in other areas of thought, as evidenced by philosophers and scientists in most civilisations through the ages.
And, to come back to the achievements of Joculator Basiliensis (which translates from Latin simply as "the player from Basel"), the quest to find a language which can represent ideas in abstract ways is also real, and not symbolic. Joculator Basiliensis likely represents a composite of Johann Bernoulli and Gottfried Liebniz. Bernoulli, from Basel, was associated with the early calculus of Gottfried Leibniz, which is one of the most powerful tools of symbolic representation in mathematics. (Johann Bernoulli's great great grand daughter Maria was Hermann Hesse's first wife.) As well as being a mathematician, Johann Bernoulli studied and practiced music, and the connections between the two fields (e.g. his work on "harmonic series"). He was in correspondence with Leibniz, who himself had a passion for symbols and representation, including Egyptian hieroglyphics and Chinese characters. I would suggest that the earlier "Parisian scholar" was Rene Descartes, another mathematician deeply interested in music (his first published work was a treatise on music theory and aesthetics), who provided the theoretical foundation for Leibniz's work on calculus. (Descartes is prominently cited in Joseph Knecht's letter to the Board of Educators in Chapter 11 of the novel.).
Far from the Game being "purely a symbol", this section of the history of the development of the Game demonstrates a direct analogy with the history of mathematics in the seventeenth century, as shaped by mathematicians with a deep interest in music theory, and Joseph Knecht as Magister Ludi later seeks to rebalance the Castalian focus on the seventeenth century of "Descartes, Pascal, and Froberger" with a deeper concern for "Cromwell or Louis XIV" along the lines advocated by Jacob Burkhardt (Father Jacobus in the novel).