American Scientist

MUSIC BY THE NUMBERS: From Pythagoras to Schoenberg. Eli Maor. 148 pp. Princeton University Press, 2018. $24.95.

All musicians are subconsciously mathematicians. —Thelonious Monk

Many people have said that mathematics and music are closely related. But ask why the two subjects are related, and the answers you receive will likely disappoint. You may be told that both mathematics and music rely on counting. Or both have universal appeal. Or you will hear that both subjects produce child prodigies. All these statements are true, but they serve only as evidence of a close relationship, not reasons for it.

Music by the Numbers, by Eli Maor, an emeritus professor of the history of mathematics at Loyola University, joins a steady succession of publications about the relationship between music and mathematics. Maor’s exploration seeks to “survey the musical–mathematical affinity from a historical perspective, highlighting not only the facts, but the people behind them—the scientists, inventors, composers, and occasional eccentrics.”

History’s March and Meander

The opening chapter of Music by the Numbers sets the stage. The year is 1900, a revolutionary time for science, music, and art. In that year, physicist Max Planck introduced the notion of quanta of energy, composer Gustav Mahler premiered his Symphony no. 1 in D major, and artist Pablo Picasso made his first trip to Paris and would soon begin his Blue Period. The Second International Congress of Mathematicians was held in Paris that year, and there Germany’s celebrated mathematician David Hilbert proposed 10 unsolved problems, a list he later expanded to 23—problems that continue to influence the direction of mathematics.

During the following decades, comfortable viewpoints would be abandoned. Physical frames of reference, musical keys, and artistic perspective would be set adrift. Music by the Numbers entertains the notion that Albert Einstein’s abandonment of fixed frames of reference at the turn of the 20th century influenced composers such as Igor Stravinsky and Arnold Schoenberg to abandon traditional tonality. “Was Schoenberg’s music influenced by Einstein’s theory of relativity?” Maor asks. “There is no hard evidence to suggest such a connection, and yet one wonders.”

Indeed, many people have wondered about the influence of physics on the arts during the early part of the 20th century. In her 1975 dissertation, The Artist, “the Fourth Dimension,” and Non-Euclidean Geometry 1900–1930: A Romance of Many Dimensions, Linda Dalrymple Henderson explores the deep influences that physics and notions of higher spatial dimensions have had on art. In 1991, Henderson’s ideas were extended to music in Art and Physics: Parallel Visions in Space, Time, and Light, by Leonard Shlain, who argued that painters such as Picasso and composers such as Schoenberg were influenced by the spirit of the time, whether consciously or not.

As Music by the Numbers moves forward, Maor reaches further back in history, and onto the stage strides Pythagoras of Samos, the legendary 6th-century Ionian philosopher credited with discovering his eponymous theorem (although it was known to the Babylonians and Indians much earlier). Maor treats us to a variation of an oft-told tale about how Pythagoras discovered the correspondence between integer ratios and intervals of musical notes. As the philosopher strolled past a blacksmith’s shop, he heard various notes produced by hammers striking metal. Looking in, he realized that the pitches he was hearing were directly proportional to the weights of the hammers in the hands of the blacksmiths.

Too bad physicists say this is impossible, spoiling a popular story. It’s a safer bet that Pythagoras experimented with a monochord—a wooden box with a single string stretched across the top—and discovered that when two strings of the same thickness and tension but different lengths are plucked simultaneously, the sounds combine most agreeably when the ratio of their lengths is among certain integer ratios, such as 1:2, 2:3, and 3:4.

For Pythagoras, who regarded numbers as alive and music as divine, the discovery must have seemed like a window into the minds of the gods. It also presented him with a tricky problem: How does one choose a finite number of integer ratios from among infinitely many possible so that the corresponding intervals form a pleasant musical scale? Maor’s explanation of Pythagoras’ solution, today called the Pythagorean scale, is clear and concise.

As any musician knows, producing a note is one thing, but making it sensuous is another. When we hear middle C played on a piano, we are also hearing many other notes, or overtones, each with less volume, but all of them combining to produce the quality, the timbre, of the sound that reaches our ear.

Much different from a piano is the humble tuning fork, which the author informs us was invented in 1711 by the English trumpet and lute player John Shore. Strike it and bring it close to your ear. The sound that you hear has almost no overtones. The resulting compression and rarefaction of air causes our eardrums to move in and out. A graph of the movement with respect to time would be a sinusoidal wave. With an array of the correct tuning forks, struck simultaneously, each with the correct force, one could faithfully reproduce the piano’s middle C.

The problem of decomposing sound into pure waves has a long history, one to which Maor devotes the bulk of the book. The author conducts his quick tour of harmonic analysis with little expectation of mathematical or physics background on the part of the reader. Maor demonstrates his ability to explain difficult ideas with ease.

Music by the Numbers allows the reader to relax from time to time by providing short, amusing interludes. We learn about the lowest known note in the universe, a note originating in the distant Perseus cluster, or Abell 426. The galaxy cluster, which is 250 million light-years away, bathes in hot plasma, sending concentric ripples—acoustic waves—into space. From the distances between the ripples we know that the note produced is B-flat, 57 octaves below middle C. According to a report in the astronomy magazine Sky and Telescope, “You’d need to add 635 keys to the left end of your piano keyboard to produce that note!”

Mood Music

Confronted by so many clues about the relationship between mathematics and music, the reader might now think that the book before us is a mystery, possibly with no resolution. The preface did contain a warning:

In the end, though, the attempts to relate mathematics to music are inherently limited by the contradictory goals of the two disciplines: mathematics—and science in general—aims at our intellect, our capacity to analyze abstract patterns and relations in an objective, logical manner, while music strives to touch our hearts, to awaken our emotions to its sounds, its rhythms, and its temporal aural patterns.

Generally speaking, music does strive to awaken our emotions. But to discuss the relation between mathematics and music, we need a much more subtle, expansive definition of emotion than that of 19th-century Romantics, who largely focused on intense feelings. There is emotion in a Johann Sebastian Bach fugue or an Indian raga or Japanese koto music or Thelonious Monk’s jazz compositions, but it is not the sort of dramatic emotion that composers such as Franz Liszt or Hector Berlioz hoped to convey.

There is also emotion felt when mathematical comprehension dawns, especially when it comes after prolonged effort. It might begin with a sudden burst of laughter, a release of tension. The feeling that follows is comparable to the emotion described in the poem “July Mountain,” by Wallace Stevens:

The way, when we climb a mountain,​
Vermont throws itself together.

We should not give up in our efforts to understand the relationship between mathematics and music because we underestimate the emotional power of either subject.

Painters and composers at the turn of the 20th century were influenced by the spirit of the time, whether consciously or not.

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Music by the Numbers is a very personal account, reflecting the author’s preference for Classical and Romantic composers such as Wolfgang Amadeus Mozart and Ludwig van Beethoven, as well as his skepticism about other, later styles of music. Maor devotes two chapters to Schoenberg, an influential Austrian-American composer of the 20th century. Much of his music is too difficult for audiences, and hence it is not often performed. We read about Maor’s personal difficulties listening to Schoenberg’s compositions, and he recounts locating only a single recording of Schoenberg’s work in the CD section of a local shop. The author concludes, “Judged by this admittedly nonscientific survey, I think that it is fair to say that Schoenberg’s music has failed to meet its creator’s high expectations.”

Maor wishes to make the point that Schoenberg intended to subjugate music to mathematical rules. Like others before him who have tried to impose mathematics on music, Maor argues, Schoenberg did not succeed.

We must appreciate, however, the crisis unfolding in music at the turn of the 20th century. As traditional melody, harmony, and rhythm were being questioned, composers such as Schoenberg were trying to bring back structure. As in mathematics, music without rules—axioms, if you will—is impossible.

On Time and Cognition

One might not care for Schoenberg, but fortunately the new musical universe is large and always expanding. The important 20th-century Greek composer Iannis Xenakis, mentioned only briefly in Music by the Numbers, encouraged efforts to bring mathematical tools to music yet opposed effort to impose them. Inspired by mathematics, especially stochastic processes, Xenakis composed challenging but rewarding music.

Maor writes that whether mathematically directed music becomes part of the mainstream repertoire “remains to be seen.” Indeed, appreciating unfamiliar music or mathematics requires time and patience. The mathematician John von Neumann was approached once by a young physicist who complained about his trouble understanding a technique for solving partial differential equations. Von Neumann replied, “Young man, in mathematics you don’t understand things. You just get used to them.” The same could be said about music.

Bach’s music, for example, was neglected for more than 70 years until the composer Felix Mendelssohn began to champion it. And, happily, Xenakis’ works are being appreciated and performed today by a new generation of musicians.

Mathematical work that was unappreciated but later found important includes that of Évariste Galois, a teenager in 19th-century France who laid the foundations of group theory and what is today called Galois theory, which helps us understand the structure of the set of roots of a polynomial equation.

Ultimately, mathematics and music might be related simply because of the way we think. If there is a relationship, then it will be found by cognitive psychologists.

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It is well known that mathematical theorems admit reinterpretation and application in ways that were unforeseen at the time of their discovery. It is impossible to imagine, say, that in the 17th century either Isaac Newton or Gottfried Wilhelm Leibniz anticipated all of the consequences of their Fundamental Theorem of Calculus, including Stokes’ theorem on manifolds and differential topology and geometry.

In a similar way, musical compositions can reveal new ideas when they are performed. The Spanish cellist, conductor, and composer Pablo Casals brought Bach’s cello suites into the concert hall and public consciousness, but he declined to publish his own edition of the six suites because he realized that they could be interpreted in so many different, musically valid ways. In a recent interview with National Public Radio, cellist Yo-Yo Ma testified to the wide range of emotions that can be drawn from just the sarabande movement of the 6th suite: “I’ve played this piece at . . . friends’ weddings and, unfortunately, also at their memorial services.”

Cultural critic Edward Rothstein’s 1995 book, Emblems of Mind, explained this phenomenon in psychological terms. Musical phrases, he says, are abstraction “units” drawn from daily life:

These units may not even be objects in the ways that we usually think of them—for the boundaries between musical units of meaning are generally ambiguous or malleable. Even a single phrase can have multiple layers, with melodic, rhythmic, and harmonic ideas intertwined. But the significance of these articulated elements comes from the connections we make upon listening, as we recognize similarities and differences among the objects we have discerned.

Rothstein introduced the word mapping to describe what our minds are doing when we listen to music. We discern the interconnections among the series of abstract musical units, appreciating their economy but also their surprises. The collection of units is mapped to the network of our thoughts, memories, and emotions.

Pianist Alon Goldstein once demonstrated musical mapping for an audience by introducing Maurice Ravel’s Une Barque sur l’Ocèan with a story about a father and child riding together in a boat on the sea. Listen for this image as you listen to me play the piece, he instructed. Everyone listened and afterward agreed that they had seen the images. Then Goldstein confessed that he had merely made up the story. Such is the nature of profound music and its performance that different mappings are achievable.

In his book, Rothstein reminds us that the objects of mathematics are also abstract units drawn from daily life. Numbers, spaces, and functions have no physical existence. As we understand a mathematical proof, we imagine interconnections and appreciate the economy of the argument as well as its surprises. We build mental models to help us see what is going on, similar to the mapping that happens when we listen to music.

Ultimately, mathematics and music might be related simply because of the way we think. We are restless symbol manipulators and makers of metaphors, automatically trying to reconcile the two activities. If there is a complete explanation of the relationship between mathematics and music, then it will be found by cognitive psychologists.

Daniel S. Silver is an emeritus professor of mathematics at the University of South Alabama. His research areas are geometric topology and history of mathematics. He is the programming vice president of Mobile Chamber Music, a concert series now in its 58th year.