Fourier Approximations and Music

Part 2: Musical notes

2.3 Pure tones

Now we begin our investigation of timbre. Even though they have the same frequency, the same note played on two different musical instruments sounds different because the pressure function associated with each instrument has a different shape. A guitar string that vibrates the air at 440 hertz creates a different pressure function than the reed on an oboe playing the same note. When we look at the pressure functions associated with different musical sounds we see that their shapes vary considerably and are usually not "simple."

Here are graphs of the pressure functions created by three different instruments playing the same note.

Trumpet	Bassoon	Tuning fork

The simplest pressure function is produced by a tuning fork. With its elementary back and forth oscillation, a tuning fork produces a pressure function with a graph resembling a sine curve. The corresponding sound is called a pure tone; the associated pressure function may be described by

6. Describe the roles of c , w, and s in the description of the function p.

7. Use an appropriate trigonometric formula to show that pure tones can also be represented by pressure functions of the form
   p(t) = a cos(wt) + b sin(wt).