Fourier Approximations and Music

Part 1: The nature of a musical sound

Some sounds we perceive as musical and others as just noise. Our goals in this module are to explore the differences between musical sounds and noise, as well as the differences in sounds produced by various musical instruments. Our analysis will lead us to represent complex musical sounds as combinations of quite simple ones. By the time we have finished, we will have developed a very general tool for analyzing a wide range of periodic behaviors.

We will concentrate on sounds as they move through the air from the source -- a violin or a car horn -- to our ears. So in this module, we will think of sound as varying air pressure near our ear drums. (This ignores all the physiological effects within the ear and the nervous system.)

Sound moves as a pressure wave -- compressing and then rarefying the air. Since we cannot see this wave in the air, it may help to use an analogy. Think of a long, soft spring lying on its side on a smooth surface. If you suddenly push on one end of the spring, a wave of compression and then expansion moves along the spring. In a similar way, the source of sound generates compression and expansion of the nearby air. The regions of compression have slightly higher air pressure than average and the regions of expansion have lower pressure. When this pressure wave reaches our ear, it alternately pushes in on the ear drum and allows it to relax.

Thus, we will describe a sound by a function that relates air pressure (at some particular location) to time. Now we may restate our first question from this point of view. What distinguishes the pressure function corresponding to a musical note from a pressure function corresponding to random noise?

We consider two particular sounds. Pictured below are the graphs of air pressure as functions of time for these two. One graph represents a random noise and the other a note played on a clarinet. (A note is the simplest musical sound --e.g., the sound made by striking a single piano key.)

Graph 1		Graph 2

1. Which graph represents the sound of the clarinet?

2. How did you determine this?