As the present authors will readily attest, introducing oneself as a mathematician is generally not an effective way to start a social conversation. But, as Cambridge mathematician Tim Gowers explains, there is a "miracle cure": just explain that you, as well as many other mathematicians, are also a musician or at least are deeply interested in music.
The present authors are not the best examples of this, because neither is very good at musical performance, although both have an abiding interest in listening to music. One of us listens to an eclectic collection of mostly modern music while he works (rock, jazz, classical, singer-song-writer, show tunes, alternative, folk, country, adult contemporary and world music to name a few -- plus an Apple music subscription). The other one of us has a large collection of classical music, including the entire works of Bach (nearly 200 hours total), on his iPhone, all of which he has listened to many times.
Einstein and music
Perhaps the best real-life example of a mathematician-musician was Albert Einstein, who, as many who knew him personally would attest, was also an accomplished pianist and violinist. His second wife Elsa told of how Albert, while during deep concentration on a mathematical problem, would sit down at the piano and play for a while; after one two-week period, interspersed with random piano playing, Einstein emerged the first working draft of general relativity. He once said, "If ... I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music."
So why is it that a remarkable number of professional mathematicians are also into music? Are the two disciplines so similar? Or is there a genetic link? Or is it simply that both mathematicians and musicians are likely to have been raised in households where mathematics, music and other scholarly, artsy subjects were valued, and where the mathematicians and/or musicians were encouraged on by eager parents? Good questions.
Mathematics and music
There does seem to be a credible connection between the sort of mental gymnastics done by a mathematician and by a musician. Additionally, current work on synesthesia supports the notion that creativity is enhanced by a mixing of the senses.
To begin with, there are well-known mathematical relationships between the pitch of various notes on the musical keyboard. An octave is separated by a factor of two; a fifth interval (say C to G) by the ratio 3/2, and two adjacent notes on the keyboard are separated by the twelfth root of two = 1.059463...
But beyond mere analysis of pitches, it is clear that the deeper world of musical syntax and structure is akin to the sorts of sophisticated structures, syntax and regularities that are part and parcel of mathematical thinking. Mozart is well-known to have written music that is both beautiful and structurally impeccable. As Salieri explained in the musical Amadeus, "Displace one note and there would be diminishment. Displace one phrase and the structure would fall."
Arguably the most "mathematical" composer of all was Johann Sebastian Bach, who was a master of counterpoint and polyphony. His work typically starts with a fairly simple theme (in the case of the monumental Brandenburg Concerto #5, it is merely a simple four-note pattern), then combines the theme with offsets, much as a chorus does when singing "rounds," in ever-more-elaborate ways, thus producing an often stunning result.
One example, which can be visualized using a very nice online tool, is Bach's "Great" Fugue in G minor (BWV 542). The theme is introduced immediately, and then developed into countless polyphonic variations. An even better example is Bach's Fugue in E minor (BWV 548), known as the "Wedge" fugue, so named for a strong theme that develops as an expanding sequence of notes in the shape of a wedge, quite obvious in the printed score (see the upper clef of the second and third measures shown in the excerpt below), and then is repeated in countless polyphonic variations, all connected in a sophisticated high-level structure.
With these connections between mathematics and music, it was perhaps inevitable that both mathematicians and musicians would turn to computers. Thus the field of computer music was born. Numerous tools have been developed to assist in this task; indeed, many modern-day musicians, covering a wide range of specialties, now utilize computers in their work.
Some of the more interesting work in this area is to program computers to actually compose music. David Cope, for instance, has written computer programs that can analyze a corpus of music, say by a particular composer, and then create new works in a similar style. He was most successful in replicating and producing variations of the music of Bach and Mozart, which is perhaps not surprising given the highly mathematical structures used by these composers. One of Cope's programs, known as "Experiments in Musical Intelligence," created what he dubbed "Mozart's 42nd Symphony."
Is there a link between mathematical and musical talent?
So to return to one of our original questions, is there a link between mathematical talent and musical talent? There are certainly many musicians who didn't make it past algebra in school, and there are certainly many mathematicians who cannot carry a tune. Furthermore, it is not as easy as one might think to scientifically test such a proposition. As Tim Gowers observes:
[I]f you want to show that professional mathematicians are on average better at music than other people, then you have to decide quite carefully who those "other people" are. You might expect that the kind of person who becomes a professional mathematician is much more likely than average to come from the kind of family that would consider music to be an important part of a child's education, so for that reason alone one would expect at least some "background correlation" between the two. ... Identifying and controlling for these kinds of effects is difficult, and as far as I know, ... there has been no truly convincing study that has shown that musical ability enhances mathematical ability or vice versa.
In short, while it is problematic to claim any kind of innate link between mathematical ability and musical ability, it is clear that the two disciplines have a deep commonality.
One of us (Borwein) has taught many students who were vacillating between musical, medical, and academic careers. And at many mathematical conferences, entertainment is provided by international-level pianists or violinists whose day job is mathematics.
What's more, in the era of modern high-performance computing, the future may bring the two disciplines together in ways that we can scarcely imagine at the present time. May your mathematical future also be a musical one!