In Harmony

Emblems of Mind: The Inner Life of Music and Mathematics.
by Edward Roth-Stein.
Times Books. 263 pp. $25.00.

Music is the most radically elusive of the arts. It is conceived in the mind, “made” in the physical world, and experienced through the senses, but its essence is not to be found in any of these domains. Those who view music as part of the world of ideas—a notion intrinsic to all music criticism worthy of the name—inevitably fail to do justice to its fundamentally nonintellectual nature. Yet discussions of music couched solely in terms of formal language have a circular quality that fails no less inevitably to speak to our instinctive belief that music has meaning.

Thus, although we may be intrigued when Igor Stravinsky assures us that “music is, by its very nature, essentially powerless to express anything at all,” we should be more inclined to believe Schopenhauer when he tells us that music

does not express a particular and definite joy, sorrow, anguish, horror, delight, or mood of peace, but joy, sorrow, anguish, horror, delight, peace of mind themselves, in the abstract, in their essential nature, without accessories. . . .

These issues are the subject matter of what is sometimes called the philosophy of music. The term is not in general use (most English-speaking musicologists prefer “musical aesthetics”), but it is both logical and to the point. Philosophy, according to the Concise Oxford English Dictionary, is “the use of reason and argument in seeking truth and knowledge of reality,” and that is what philosophers of music do: they seek the truth about music. Such quests are by definition unending, and so there is no shortage of important books dealing with the meaning of music, some by scholars (Victor Zuckerkandl’s Sound and Symbol, Leonard B. Meyer’s Emotion and Meaning in Music) and some by major composers (Stravinsky’s Poetics of Music, Paul Hindemith’s A Composer’s World).

Now Edward Rothstein, chief music critic of the New York Times, has written a book of comparable interest that seeks to illuminate music by examining its relationship to mathematics. This approach is, of course, as old as Pythagoras, but Rothstein comes at it from a distinctive angle: Emblems of Mind, as its subtitle indicates, is not about the specifically mathematical aspects of music but about the “inner lives” of the two disciplines, and in particular about the way in which they both rely on “metaphor, abstraction, and comparison.” Rothstein, who studied pure mathematics before going into music, is unusually qualified for this task, and Emblems of Mind is full of the kinds of fresh, unforced comparisons that spring naturally to the double consciousness of a man conversant with two fields.

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Rothstein begins by exploring the styles of thought that are characteristic and common to both mathematics and music, showing how each discipline “keeps stepping back from the arguments it has made, discovering ever more elaborate abstractions that reveal ever deeper similarities.” From there he goes on to explore the more complex questions of what music does, and why it matters. In examining these questions, Rothstein appears at first glance to side more with Stravinsky than Schopenhauer:

There are no discoveries in music that illuminate our physical universe. Instead, through music, we learn more about musical objects—phrases, chords, gestures. . . . Once music is detached from function, once it becomes a repertory art, it explicitly strives to define itself, out of itself, to become “mathematical”—that is to say, to begin from premises and proceed to conclusions by interpreting its own universe, finding its own laws. . . .

But the seeming chilliness of this vision is warmed by a constant awareness of the fact that the goal of music—as well as of mathematics—is beauty. Innumerates are typically nonplussed when they hear mathematicians speak of the “beauty” or “elegance” of a particular argument (though not Edna St. Vincent Millay, who assured her readers that “Euclid alone has looked on beauty bare”). In fact, the concept is easy to grasp. “The beauty” of mathematics, Rothstein explains,

is in the simplicity of the reasoning when faced with the complexity of the expected calculations. . . . Beauty is experienced as a form of knowledge because it is through the archetypical rational act—that of analogy and metaphor—that we come to know the beautiful.

As for beauty in music, though more familiar to the average reader/listener, it is no less subtle to capture. What do we mean when we say the Mozart G-Minor Symphony is “beautiful”? Do we have in mind a property that the music of Mozart has in common with Rigoletto? Is that same property also to be found in Stravinsky’s The Rite of Spring—or Arnold Schoenberg’s Moses and Aron, a work whose melodic and harmonic materials are not organized according to the traditional “laws” of tonality (but whose rhythmic materials, by contrast, are utterly conventional)?

According to Rothstein, beauty in music is directly analogous to beauty in mathematics:

We can have such an immediate reaction to the aesthetic creation lying before us, such awe at its internal workings and the powers they exercise upon us, that the work seems something not made but simply presented; it requires no analysis of purpose nor any reflection about meaning; it appears to our senses as something completely natural.

Furthermore, music can suggest a “truth” comparable to the truth of a mathematical equation: “Musical truth involves something about the ways a composition is consistent, complete, and open to mappings in our various worlds of sense and thought.” By contrast, “we think of music as false when it violates its own premises.”

Rothstein does not shy away from the final implication of this insight. For him, music and mathematics “remain mysteries, seeming too close to Truth to be merely human, too close to invention to be divine.” Having invoked the ultimate source of truth and beauty, he properly concludes Emblems of Mind with twin inscriptions: Q.E.D., or Quod erat demonstrandum (“That which was to be proved”), the traditional conclusion of a mathematical proof; and S.D.G., or Soli Deo gloria (“To God alone the glory”), with which Bach frequently inscribed his compositions.

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Regular readers of Edward Roth-stein’s criticism will not be surprised by the thoughtfulness of his approach to the problem of beauty in music, or by the bracing austerity of his taste. Bach, Beethoven, and Mozart are the composers most often cited in Emblems of Mind (though there is also an interesting discussion of that most enigmatic of Romantic piano pieces, Chopin’s A-Minor Prelude). Yet Rothstein assures us that the power of music to “find its own laws”

allows us to comprehend many compositions written during the last 30 years by composers like Pierre Boulez, Milton Babbitt, and Elliott Carter, whose relationships and variations are built not upon tonal laws and relations but upon laws that may have been invented for a particular composition.

One may doubt that Rothstein’s explanation of how music works can, in fact, be convincingly extended to the experience of listening to a work like the Elliott Carter Double Concerto: after all, an internally consistent, seemingly elegant mathematical proof can easily be erected on the foundation of a false premise. But even a reader who differs with Rothstein on matters of taste will find Emblems of Mind a highly persuasive essay in the philosophy of music. Though he may not succeed in explaining why most listeners prefer Poulenc and Prokofiev (neither of whom appears in the index) to Carter and Schoenberg, Rothstein has nonetheless succeeded in framing a “general theory” of musical meaning that is not only stimulating but also quite strikingly poetic. Few books have taught me more about the essential nature of music, and fewer still have done so with comparable grace and clarity.

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Emblems of Mind is in no way a chatty or anecdotal book. “My intention,” Rothstein says in his introduction, “has been to write for the literate layman, the curious listener and student who might never have studied math beyond high school or music beyond elementary school.” Laymen will certainly be able to follow Rothstein’s line of argument—the mathematics looks more frightening than it actually is—but this book demands, and rewards, the kind of sustained concentration that our bite-sized musical culture has sought to render superfluous. Anyone for whom classical music remains a central, indispensable part of high culture will find Emblems of Mind a powerful and provocative work.