This Mathematical Song of the Emotions
Four note chords
by Dmitri Tymoczko
Two hundred years ago, there were no CDs or MP3s and the primary way to preserve your music was to write it down. Not surprisingly, notated composition was a culturally central activity: Roughly 20,000 people are said to have attended Beethoven’s funeral, a non-negligible fraction of Vienna’s population. These days music is largely transmitted through recordings rather than pieces of paper, a change that has altered listeners’ expectations while opening doors for talented but illiterate musicians.
Of course, notated music, like archery, continues to exist. Hollywood has an almost endless appetite for swooping orchestral passages that evoke the requisite degree of mythic solemnity. (Come to think of it, Hollywood probably hires its share of archers as well). But there is also the smaller world of the modern composer, the underpaid and somewhat self-conscious inheritor of Beethoven, Debussy, and Schoenberg. Notated art music has in effect become a subculture, a niche activity largely disconnected from the surrounding world. And like many subcultures, it tends to police its boundaries. If you are influenced by Pygmy music, free improvisation, or outsider art, you will encounter few objections from fellow composers. If your big thing is Brahms, on the other hand, then you may have a harder row to hoe.
The world of modern composition has thus been shaped by a twofold process of abandonment. On the one hand, the listening public (including intellectuals such as philosophers, novelists, and visual artists) has largely gravitated toward popular music, with its simpler structures and aural-tradition roots. At the same time, aficionados of classical music (including any of number of music historians, theorists, and even performers) have gravitated toward the past, deciding, at least implicitly, that the classical tradition is rich enough to sustain their interest—and depriving contemporary composers of a good part of their potential audience.
My book A Geometry of Music grew out of dissatisfaction with this situation. Unable to imagine a life either as a popular musician or an avant gardist, I wanted to write a kind of music that was artistically serious, yet still grappled with the harmonic and melodic techniques of earlier centuries—music like Shostakovich’s, which is recognizably of our time and yet also rich in something like the way that Bach’s music is. A music analogous to the prose of Thomas Pynchon or David Foster Wallace, accessible but modern and overflowing with intensities of both thought and feeling.
The book therefore pursues a twofold agenda. On the one hand, I wanted to take a step back and consider some very general questions about (broadly) “classical” music. Why is this music so interesting? What gives listeners the sense of a rich and multidimensional coherence, like that of a beautiful abstract tapestry? And on a more technical level, what makes for a “good” melody, scale, or chord?
A basic conclusion here is that traditional tonal techniques are “overdetermined” in a very profound sense. Rather than there being an infinity of possible musical syntaxes, each unrelated and incompatible with the others, it turns out that there are just a few ways to combine very basic ingredients of musical coherence. As a result, different tonal styles turn out to be fundamentally similar when you consider their deepest structure.
At the same time, I wanted to sketch an alternative history of twentieth-century composition, one in which atonality and the avant garde were less central than they are sometimes taken to be. The book thus proposes the existence of a shared “common practice” with its roots extending back before the beginnings of the classical tradition, and its fruits still visible in very recent music. Central to this narrative is the idea that composers like Debussy, Ravel, and Stravinsky are in the same basic line of work as sophisticated improvisers like Art Tatum, Miles Davis, and Bill Evans, with these musicians jointly resolving technical problems inherited from nineteenth-century Romanticism. (Jazz is of course substantially non-notated, but it is a very sophisticated tradition involving extensive training). From this point of view, the history of twentieth-century tonality runs straight through jazz, as improvisers developed and transmitted the breakthroughs of earlier decades.
I haven’t yet mentioned mathematics, or given any reason to think that geometry has anything to do with this story. But it turns out that even the simplest musical questions can have interesting mathematical answers. In particular, there is a natural “geometry of chords”—that is, a collection of remarkable geometrical spaces representing any possible chord in any conceivable scale. In these spaces, “distance” corresponds to the aggregate physical effort it takes to move from one chord to another on an instrument like the piano. (For instance, C major and E major are close because one can transform the first into the second by moving C down one step to B and G up a step to G-sharp). These spaces have a beautiful and twisted geometry, a consequence of the way in which musicians are consider different collections of notes to be “functionally equivalent.”
What this gives us, I argue, is a new and systematic way to think about particular pieces and even entire genres. We can, for example, understand how Shostakovich or Steve Reich move between “nearby” scales, thus generalizing the earlier procedures of Bach and Mozart. Or we can see how the unusual chord progressions in Wagner and Chopin trace systematic paths through a higher-dimensional space of harmonic possibilities. More generally, I argue that these new tools can suggest new compositional possibilities, allowing us to imagine musics that can, hopefully, compete with Bach, Mozart, and Brahms.
I don’t mean to suggest that this is the only music worth pursuing. Music can be many different things at many different times. Sometimes musicians are like athletes, coolly focused on the challenge of getting their bodies to make the right collection of physical motions. Sometimes they are like actors, screaming theatricalized rage into microphones. But music can also be, as Leibniz said, the “unconscious exercise of our mathematical powers”—an uncanny mixture of logic and emotion, exemplified for me by Bach and Bill Evans. It is this sophisticated beauty that I am trying to support, a kind of beauty that is sometimes lost amid the noisier sounds of popular and avant garde musics (each wonderful in its way). I’d be happy to think that A Geometry of Music had helped make some more space for this fragile, mathematical language of the emotions.
About the Author:
Dmitri Tymoczko is a composer and music theorist who teaches at PrincetonUniversity. His book A Geometry of Music is available from Oxford University Press, and his CD, Beat Therapy (which sounds like jazz/funk until you listen more carefully) is available from Bridge Records.