He wrote: "Before setting out to make my way in the music business, I was in training to become a 'pure' mathematician. Such esoteric subjects as algebraic topology, measure theory, and nonstandard analysis were my preoccupations. I would stay up nights trying to solve knotty mathematical problems, playing with abstract phrases and structures.
"But at the same time, I would be lured away from these constructions by another activity. With an enthusiasm that could come only when critical faculties are in happy slumber, I would listen to or play a musical composition again and again, imprinting my ear and mind and hands with its logic and sense. Music and math together satisfied a sort of abstract 'appetite,' a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical."
Rothstein went on to say that such an experience is by no means unique to him. He noted that music and mathematucs have been associated throughout history.
Pythagoras and his followers saw numbers as models of everything in the physical world, and they identified music with number, noting its scales, tempos, and other regularities. Johannes Kepler envisioned planetary motions as the "music of the spheres." Galileo Galilei speculated on the numerical reasons why some combinations of tones are more pleasing than others. Leonhard Euler considered the same problem in a treatise on consonance and whole numbers.
On the music side, Johann Sebastian Bach sometimes treated the composition of canons and other types of music as exercises akin to solving mathematical puzzles. Frederic Chopin described the fugue as "pure logic in music." And twentieth-century composers have applied sophisticated mathematical theory in their works.
In his 1993 book The Music of the Spheres:Music, Science and the Natural Order of the Universe, music critic Jamie James examined and pondered the history of the concept of a musical universe—a cosmos envisioned as a stately, ordered mechanism both mathematical and musical. Music and science were once intimately intertwined and united by a grand vision, he pointed out. But music, like much of the most fundamental art and literature of our culture, has now been relegated to the obscure margins of the curriculum.
On the music side, Johann Sebastian Bach sometimes treated the composition of canons and other types of music as exercises akin to solving mathematical puzzles. Frederic Chopin described the fugue as "pure logic in music." And twentieth-century composers have applied sophisticated mathematical theory in their works.
In his 1993 book The Music of the Spheres:Music, Science and the Natural Order of the Universe, music critic Jamie James examined and pondered the history of the concept of a musical universe—a cosmos envisioned as a stately, ordered mechanism both mathematical and musical. Music and science were once intimately intertwined and united by a grand vision, he pointed out. But music, like much of the most fundamental art and literature of our culture, has now been relegated to the obscure margins of the curriculum.
"All art, including music, was a much more serious matter before the self-conscious aestheticism of the late nineteenth century took root," James argued. "It is a recent notion that music is a divertissement to be enjoyed in comfortable surroundings at the end of the day, far removed from the hurly-burly of life's business."
The intriguing interplay between mathematics and music, which goes back to antiquity and possibly much earlier in human history, hints at a deep and fundamental connection.
In concluding his book, Rothstein commented that music and mathematics share not only the clarity of their expression but also their beauty and mystery.
He wrote: "Our attempt to comprehend music and mathematics, to understand their workings and their purposes, is ... a model for our coming to know at all—a model for our education, for the ways we make distinctions and connections.
"We begin with objects that look dissimilar. We compare, find patterns, analogies with what we already know. We step back and create abstractions, laws, systems, using transformations, mappings, and metaphors. This is how mathematics grows increasingly abstract and powerful; it is how music obtains much of its power, with grand structures growing out of small details."
American Mathematical Society links on mathematics and music, including the PBS video "The Majesty of Music and Math."
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