June 19, 2004
Decyphering the Grammar of Mind, Music and Math
magine a locked room in which a person sits alone staring into space. There is nothing to look at. Nothing to touch or taste or smell. Most of the world is stripped away. Except for sounds.
But these sounds resemble nothing heard before. They lack all similarity to experience and any reference to surroundings. Now imagine that those sounds — heard for the first time — are the sounds of a Beethoven Symphony. Or an Indian raga. What would that disembodied ear and mind make of them? How much would be understood?
In recent decades such a situation would have been considered artificial, abstract and irrelevant. What kind of musical understanding can grow out of this kind of isolation, lacking the resonance of a cultural framework, lacking the expectations provided by the knowledge of a style and lacking some sense of historical and political context? To understand music, we have been taught, that room has to be unlocked, the windows opened and the world fully engaged.
But now the emphasis may be changing. The appeal of a more abstract way of thinking about music may be growing. There is a search for timeless laws and principles; it may be that something can be learned from the listener in the locked room.
A new field, for example, sometimes called biomusicology, is preoccupied with how music affects the brain. What regions of the brain respond to changes in harmony or melody? Is there a single region that makes sense of music? Is there a difference between the way neurons react to frequency differences in speech (intonation) and frequency differences in pitch (melody)? In such research the contingencies of culture and history are often stripped away. The foundations of musical perception are sought, as are the biological laws that make music a human universal.
This change in emphasis may also be contributing to a renewed interest in the relationship between music and mathematics. More than 20 years ago when I wrote an article about the subject for The New York Times and later when I explored it in a book, I argued that there was more to musical meaning than was evident in influential political and programmatic interpretations. For all of music's deep connections to human experience and social life, music was also similar to mathematics in important respects, as ancient philosophical and musical texts insist.
Harmony and counterpoint, after all, are sonic reflections of ratio and number. Musical languages seem to possess their own premises and laws. And a coherent and elegantly phrased composition can display the beauty and inevitability of a mathematical proof. Mathematicians and musicians have long had reciprocal interests. For a time, though, such musical idealism became something of an eccentricity.
But more recently seminars in music and mathematics have been proliferating at universities. Last fall Oxford University Press published an anthology, "Music and Mathematics: From Pythagoras to Fractals." On Thursday night in Weill Recital Hall, the Siemens Foundation also presented four recent winners of the Siemens Westinghouse Competition, a high school science research competition, playing various instruments. The event featured a discussion by scientists interested in music, including Dr. Mark Jude Tramo, the director of the Institute for Music & Brain Science at Harvard University, who has studied the neural basis of musical perception for over a decade.
What sort of picture of musical understanding is taking shape with this renewed interest? Much of the brain research is teasingly inconclusive. Every effort to examine the effects of single musical variables — pitch, meter, harmony — inadvertently shows just how much more music is than the sum of its parts. Despite attempts to identify a particular musical region of the brain, for example, Dr. Tramo has shown that many regions are active when music is heard; even motor areas of the brain can become active though the body might be at rest.
The relationship between music and language is also complex. The Russian composer Vissarion Shebalin continued to write music for a decade after a stroke in 1953 damaged his speech and language understanding. In one classic study, brain-damaged patients could identify instruments and wrong notes but could not recognize melodies. Some acoustic phenomena — say variations in pitch — are interpreted differently when heard in music and in speech. A recent paper on the tone-deaf by researchers who included Aniruddh D. Patel at the Neurosciences Institute in San Diego shows that while such individuals may have no trouble discriminating intonations in speech, they are stymied by sliding musical imitations of those intonations.
Sonic events may be experienced differently when they become part of music. This is where that locked room becomes so suggestive. We have all been in the position of that abstract listener, particularly when hearing music of an alien culture for the first time or hearing new music that can at first seem like little more than random sounds.
But music has a power unique among forms of human communication: it can teach itself. Gradually over repeated hearings, without the use of a dictionary or any reference to the world outside, music shows how it is to be understood. The listener begins to hear patterns, repeated motifs and changes in meter and realizes that something is happening, that sounds have punctuation, that phrases are being manipulated, transformed and recombined.
Gradually, the listener gains a form of knowledge without ever referring to anything outside the music. Sounds create their own context. They begin to make sense. Similar processes with varying richness and power take place in all forms of music, which is why it is much easier to understand another culture's music than another culture's language.
Nothing else is quite like this self-contained, self-teaching world. Music may be the ultimate self-revealing code; it can be comprehended in a locked room. This is one reason that connections with mathematics are so profound. Though math requires reference to the world, it too proceeds by noting similarities and variations in patterns, in contemplating the structure of abstract systems, in finding the ways its elements are manipulated, connected and transformed. Mathematics is done the way music is understood.
Of course, this does not exhaust music's possibilities. Open the locked door and meanings proliferate. A composition is related to others in the same musical style; it has a place in history; it has an impact on the body and emotions; it speaks of things outside itself. Its abstract patterns take on other meanings, just as abstract mathematical reasoning can find unexpected applications. Music is a metaphor; the world provides its analogies.
This means that music can be fully understood only by maintaining access between the room and world; neither can be closed off. And in its ability to combine opposing realms that are both timeless and timely, absolute and relative, mental and physical, abstract and concrete, music might become a teacher of more than just sound.