A great place to begin your exploration of the connection between waveform and timbre is the study of classic waveshapes and spectra such as the sine, sawtooth, triangle, and pulse, and noise. However, for simplicity's sake, let's take a look at waveforms associated with noise first. The waveform graph in Fig. 1 contains 4 seconds of white noise followed by 4 seconds of pink noise. Examine the waveform of each type of noise. Click the playback button to the right of Fig. 1. What do you notice about the timbre of these two types of noise? White noise is completely random. There is equal energy throughout the entire frequency spectrum of white noise. As a matter of fact, the white noise in Fig. 1 was produced by asking the computer to choose random numbers. Contrastingly, pink noise has equal energy per octave, though it is also essentially random within those frequency bands. We associate noise with random variation patterns.

Fig. 1. A waveform graph (a vs. t) generated by Bias' PeakVST containing
4 seconds of white noise follwed by 4 seconds of pink noise shown

Playback:

Fig. 2. A graph of a sinusoidal waveform.

Playback:

Now lets take let's take a look at the completely predictable and repetitious waveform shown in Fig. 2. This waveshape is called of sine wave. It is the product of sinusoidal motion and thus is often referred to as a sinusoid. First, notice how its waveform repeats. It has been determined experimentally that number of repetitions of the waveform per second is associated with the pitch we perceive. A waveform that repeats its waveshape is said to be periodic. Periodic and nearly-periodic waveforms generate a sense of pitch, and our sensation of pitch is associated with the number of times per second that the aveform repeats itself. For example, an A-string on a violin vibrates 440 times per second, and we perceive its pitch to be A4 - 440 Hz, or A above middle-C.

Simple Harmonic Motion

The waveform of a sine wave may be derived from circular motion a type of motion physicists refer to as simple harmonic motion. As a tuning fork ossilates back and forth it produces a single tone that most would describe as "pure." The term pure describes our perception of the absense of overtones in the sound. Using a computer, it is possible to produce a sound that is very nearly pure. Physicists call this type of oscillation simple harmonic motion. This type of regular or periodic oscillation pattern may be found throughout nature. However, the most common example cited is a pendulum. A pendulum swings back and forth under the influence of gravity (and friction, of course, but that is conveniently ignored) in a manner similar to the tuning fork. In 1851, Foucault dramatically used a pendulum to demonstrate that the earth rotates on its own axis. He suspended a pendulum (now called a foucault pendulum) from the dome of the Pantheon in Paris. As the pendulum swung back and forth, it traced out a straight line in the sand that changed its orientation as the day progressed. A very dramatic and useful example of simple harmonic motion indeed. Obviously, a pendulum's regular motion could be used to mark the passage of time as, say, in a grandfather clock.

Fig. 3. The Tacoma Narrows bridge set into oscillation/sympathetic vibration by strong winds.

To get a better picture of the type of motion we are talking about, lets graph the motion of the tuning fork over time. To do this, attach a pencil to the tuning fork. Now turn the tuning fork upside down and press the tip of the pencil gently against a piece of paper. The oscillating tuning fork draws a straight line. Try it. Now, to simulate, how the pattern changes through time, drag the paper to the right as the pencil draws. The pattern drawn is roughly what mathematicians refer to sinusoidal motion or a sine wave.

Because a sine wave is a product of simple harmonic motion, it produces a sound containing a single frequency. Thus, unlike musical tones, it produces no overtones. Just as a laser produces light at a single frequency because everything is in phase It is used in the acoustical laboratory to explore the properties of sound. A tone consiting of two or more sine waves is called a complex tone. A periodic variation in the amplitude of a complex tone that is produced when waveforms of two pure tones is slightly out of phase. In 1966, Steve Reich composer a piece called pendulum music for 2-4 microphones suspended over their respective speakers which produced 2-4 varing tones.