Fourier, part 2.4

Fourier Approximations and Music

Part 2: Musical notes

2.4 Harmonics and pure tones

To understand why the graphs of pressure functions for the same notes might have different shapes, we will find it helpful to study the vibration of a single stretched string -- for example a violin or guitar string. The pressure function associated with the vibration of a single string is determined by the motion of the string. If we stretch the string so that its profile resembles a sine graph and then release it, the resulting vibration will produce a pure tone.

The following applet shows the motion of the string when its initial profile resembles one arch of a sine function.

First harmonic

This particular vibration is called the fundamental or first harmonic. Another example of a vibration that will produce a pure tone occurs if the initial profile of the string contains two arches of the sine curve, i.e., one complete period of a sine graph. The next applet depicts this vibration.

Second harmonic

This vibration is called the second harmonic or first overtone.

8. What other possible initial profiles for the string will produce pure tones? Give three examples by drawing the initial profiles of the string that will produce the 3rd, 4th, and 5th harmonics.

The next applet displays a number of possible pure tone vibrations of the single string.

Harmonics

You may have noticed that the more sinusoidal oscillations present in the initial profile of the string, the faster it vibrates. In fact, if the fundamental frequency of the string is w hertz, the frequency of the second harmonic will be 2w hertz, and, in general, the frequency of the nth harmonic will be nw. If you did not notice the frequency increasing for the higher harmonics, return to the last applet and pay careful attention to the period of the oscillations.

Now we turn to consideration of the pressure functions associated with pure tone vibrations of the string. Such a function will have the same frequency as that of the vibration of the string. If the fundamental vibration of the string has frequency w, then the associated pressure function p₁can be described by

p₁ = c₁ sin(wt - s₁).

We need to say a word about the phase shift s₁. If we are measuring the pressure with say a CBL attached to a calculator, it is unlikely that the zero time for the measuring device and the "zero" of the pressure function occur at the same time. The s₁ term allows us to model the data without changing the time scale. However, for single pure tone vibrations, we could rescale time so that s₁ = 0.

Write an expression that represents the pressure function associated with the third harmonic of a vibrating string. (Assume that the first harmonic has frequency w.)
Write an expression that represents the pressure function associated with the nth harmonic of a vibrating string. (Assume that the first harmonic has frequency w.)

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