The Mathematics of Music

Simply from the words we use when we talk about music, it should be apparent that there is an intimate relationship between number and both tone and rhythm. "Start playing sixteenth notes there" we might say, or "play a diminished 7th." We count things off, "One - Two - Three - Four", and even Zuckerkandl uses numbers to describe his elusive "dynamic qualities." But what exactly is that relationship, and how significant is it to our understanding and appreciation of music?

If we understand music as a series of tones related through time, number seems to have direct connections to two distinct spheres of that understanding - both the generation of those tones, and the way in which they unfold through time. What we are concerned with, in understanding music in relation to number, is not the mere presence of such a relationship - because it is apparent that this relationship exists - but rather the degree to which this relationship affects our listening.

First, we will discuss the relation of number to tone - the sounds we hear unfolding through time. The Pythagoreans understood this relationship to be the primary one in the consideration of music - and, at least chronologically, it is - for the tones that we hear could not be produced without the vibration of strings (or things acting like strings, in the case of columns of air) in ratios to each other. We know, for example, that we hear an octave when two strings with a 2:1 relationship of length are plucked. Such relationships are present in all combinations of tones, with a tendency of these combinations to become more dissonant as these ratios become more complicated, veering away from low whole numbers. But when we hear an octave, are we hearing this "two-to-oneness"? In a sense, of course, we are, because this relationship is inherent in the octave itself. But things grow far more complicated when we put this octave into the context of a melody. Then, we have many such ratios all in ratio to each other, and our simple of understanding of pleasure coming from small whole numbers goes right out the window. In essence, each tone we hear in a melody has within it a conflict between two ratios - the ratio of the preceding tone to the current one, and the ratio of the current tone to the root. Could this tension be responsible for what Zuckerkandl refers to as "dynamic qualities?" At any rate, we see that that what we previously understood as a simple correlation between number and melody becomes something both more complicated and more artful when put in such a context - and, while the basic relationship of vibration to sound doesn't disappear, it becomes far less significant in comparison to the relationship of one sound to another that is built upon it.

But while the pleasure in musical tones comes out of an artful complication of number, we see something quite different in the case of rhythm. Rather, the art of musical rhythm is born of an artful subversion of such a relationship - we build a structure of beats and note-lengths only to destroy and escape such a structure. At first glance, the presence of number in rhythm seems simple, but incidental - in order to keep track of the tones unfolding through time, we create a unit of time - the beat - and base our tonal movements on that unit. The problem is, however, that music built entirely on such a unit simply isn't pleasurable. In fact, it's pretty boring. If we take the oneness of the beat as absolute, we are left with few options in the creation of a piece. Zuckerkandl talks about the monotony of the metronome in this fashion - it becomes a cage in which we are unable to move freely through time. So, in order to create the interest that we require in music, we subvert the beat in two ways. First, we make the beat inconstant. In a waltz, for example, the third beat of a measure will often come a bit late, for both emphasis of this beat and to facilitate the sliding-into-the-next-measure that allows the piece to be danced to. Secondly, we deny the oneness of the beat - we divide it into smaller parts, which allows us syncopation. We expect notes to fall directly on the beat, but when they fall between the beats, and are still sounding during the beat, that expectations is attacked - a tension is generated between what we hear and what we expect to hear. Similar to the tension between the ratios of present tone to preceding tone and present tone to root tone, this tension creates interest in a piece.

It's clear that number has a very direct place in our understanding of music as tones sounding through time - but it's not a simple one. We can't reduce it to simple ratios or beat-units. At least, we can't do so if we hope to understanding any sort of interesting music. This relationship has a strange place in our understanding - straddling the line between being purely incidental and dramatically crucial. It's a thing to be both compounded and subverted - to be at once moved with and against. And it is this duality and tension that allows us to hear beauty in something as simple as sound.