Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Previously in our discussion, we have seen how the tones generated by different instruments are really mixtures of some fundamental vibration, or oscillation, and whole number multiples of that frequency, called overtones. The various combinations of fundamental tones and overtones are what give instruments their characteristic sounds. This understanding began with the Pythagorean observation that strings with commensurable lengths sound harmonious when plucked together. We've progressed from understanding the relations of string lengths to understanding how waves work and how the frequencies of waves are what we perceive as pitch. We've also seen how we can express simple sine and cosine waves as periodic functions of time via a connection to trigonometry. In essence, we have learned that musical tones are complicated mixtures of waves, and we now know how to express simple waves mathematically. We are now ready to use our mathematical tools to tackle complicated waves, such as the tones that real instruments make. To do this, we need some concepts and tools from an area of study that, when it began, had nothing to do with music, but rather heat: Fourier analysis.
Joseph Fourier was an associate of Napoleon, accompanying the great general on his conquest of Egypt. In return for his loyalty, Fourier was made governor of southern Egypt, where he became obsessed with the properties of heat. He studied heat flow and, in particular, the temporal and spatial variation in temperature on the earth. He realized that the rotation of the earth about its axis meant that its surface was heated in some uneven, but periodic way. In reconciling the different cycles involved in the heating of our planet, Fourier hit upon the idea that combinations of cycles could be used to describe all kinds of phenomena.
Fourier said that any function can be represented mathematically as a combination of basic periodic functions, sine waves and cosine waves. To create any complicated function, one need only add together basic waves of differing frequency, amplitude, and phase. In music, this means that we can theoretically make any tone of any timbre if we know which waves to use and in which relative amounts to use them. It's not unlike making a meal from a recipe—you need a list of ingredients, you need to know how much of each ingredient to use, and you need to know how and in what order to combine them.
The ingredients used in Fourier analysis are simply sine and cosine waves. Of course, these simple waves can come in different frequencies. For sounds that we consider pleasing and musical, the sine wave mostly will come in frequencies that are whole number multiples of a fundamental frequency. For sounds that are "noisy," such as white noise, the sine-wave ingredient frequencies can be anything.
NOTE: In the following discussion, we'll be using the shorthand terms "sin" and "cos" to represent "sine" and "cosine," respectively.
To begin, let's look at a simple example, sin t:
Now, consider a modified sine function, sin 2t:
Combining these two functions gives us a new waveform, f(t) = sin t + sin 2t.
This waveform is comprised of equal parts sin t and sin 2t. It has features of both but is a new waveform. We don't have to combine the two simple waves in equal parts, however. Let's look at what happens when we use only "half as much" sin 2t:
Just as we find when cooking, using different proportions of the same ingredients yields a different result. This waveform is different than the one we obtained previously, illustrating the effect that altering the coefficient of a function can have on the graph, or wave. The coefficient corresponds to the amplitude of a wave, and, in our combined function, essentially determines how much each sine term contributes to the final waveform.
Now let's see what happens when one of the terms is offset in phase.
This produces yet another waveform, illustrating the effect of each component wave's phase. Notice that the graphs of sin (t+) and cos t are identical. This shows us the natural phase relation between sine and cosine functions. Now that we've seen how simple sine waves can be combined to create somewhat more complex waves, let's see how to make a more complicated wave, such as a sawtooth wave.
First, let's just look at the sawtooth waveform.
Notice that the graph has a series of "ramps" that indicate that the function increases at some constant rate, then instantaneously drops to its minimum value as soon as it reaches its maximum value. Each of the ramps looks like the function y = x, which we can express as f(t) = t, given that we have been talking about values relative to time. So, this sawtooth wave can be made by some sort of function that periodically looks like f(t) = t. It has a period of 2π, so we can say that this function is f(t) = t for –π to π. According to Fourier, even a function such as this can be written as the sum of sines and cosines.
To see this, let's start with a sine wave of period 2π, a period equivalent to that of the sawtooth wave above.
Now, let's subtract another sine wave of twice the original frequency.
The equation that represents the function we've built so far is:
f(t) = 2sin t – sin 2t
Let's add a third sine wave of three times the original frequency.
With the addition of the third term, our Fourier expansion is now:
2sin t – sin 2t + sin 3t
At this point we are just guessing which frequencies and amplitudes, or coefficients, to use. Fourier's great contribution was in establishing a general method, using the techniques of integral calculus, to find both the coefficients, and by extension, the component frequencies of the expansion of any function. This, as we shall soon see, has given mathematicians a greater range of manipulative capabilities with functions that are difficult to deal with in their standard form. Fourier's specific method is beyond our scope in this text, but the idea that certain functions can be represented as specific mixtures of sine and cosine waves, is an important one.
Returning to our sawtooth exercise, we can see that as we add more terms, the resultant wave begins to take on the sawtooth shape.
Four terms: f(t) = 2sin t – sin 2t + sin 3t – sin4t
Five terms: f(t) = 2sin t – sin 2t + sin 3t – sin4t + sin5t
Six terms: f(t) = 2sin t – sin 2t + sin 3t – sin 4t + sin 5t – sin 6t
Seven terms: f(t) = 2sin t – sin 2t + sin 3t – sin 4t + sin 5t – sin 6t + sin 7t
As you can see, the sum of the sine series is starting to look like a sawtooth wave. In order for it to look exactly like one, however, will require an infinite number of terms. To suggest an infinite sum, we often use the "dots" convention, as in this equation:
F(t) = 2sin t – sin 2t + sin 3t -… + bnsin nt
The dots indicate that the established pattern goes on and on. However, there is a more precise way to represent this sum (or more confusing, depending on your point of view!). This is called the "summation notation:"
This representation encodes the fact that the index "n" starts at 1 and keeps on going, and that for every index n there is a coefficient bn that is the "weight" on the mode sin nt (of frequency 2πn). So, the bn's are the amplitudes of the component frequencies, and in the case of the sawtooth wave, we can express them by the formula . We find this by using Fourier's technique for finding expansion coefficients (i.e., by computing an integral). The details of this, although outside the scope of this text, can be found in most standard calculus textbooks.
The final Fourier expansion of the sawtooth wave is then:
In the Fourier series for this sawtooth wave, note that there are no cosine terms. That's because all of the coefficients that would correspond to cosines are zero. In general, a Fourier series expansion is composed of contributions from sine terms, sin nt (with amplitudes bn), cosine terms, cos nt (with amplitudes an), and a constant offset, or bias, a0. So, in summation notation the general formula for a Fourier expansion of a function, f(t), is:
Notice in the progression that we constructed earlier that as the number of component waves increases, the overall waveform increasingly approaches the look of the ideal sawtooth. Each additional term has a higher frequency than the preceding term and, thus, provides more detail than the term before it. We can get as close as we want to the form of the ideal sawtooth by adding as many high-frequency components as we choose. This is analogous to a sculptor roughing out a general shape and then refining details after multiple passes.
Being able to take any function and express it in terms of these fundamental pieces is an extremely useful tool. In mathematics, functions that may otherwise seem impenetrable may give up their secrets when transformed into a Fourier series. In the realm of music, Fourier analysis gives musicians and sound engineers extraordinary control over sound. They can choose to augment or attenuate specific frequencies in order to make their instruments sound perfect. Also, with today's synthesizers, musicians can build up fantastic sounds from scratch by playing with different combinations of sines and cosines.
As we have seen, Fourier analysis can be used to represent a sound, or any signal, in the frequency domain. This view of a wave in terms of the specific mixture of fundamental frequencies that are present is often called a signal's spectrum. Analyzing the spectra of different signals can yield some surprising information about the source of the signals. For example, by looking at the light from stars and identifying the presence or absence of specific frequencies, astronomers can make extremely detailed predictions about the chemical composition of the visible layers of the star. In audio engineering, technicians can monitor the frequencies present in a sound and then amplify or attenuate specific frequency bands in order to control the makeup and quality of the output sound.
Each sine or cosine term in a Fourier expansion represents a specific frequency component. We can graph these frequencies in a histogram in which each band represents a range of frequencies. The height of each band corresponds to the amplitude of the contribution of those frequencies to the overall signal. This visual representation of sound may be familiar to you if you've ever used a graphic equalizer.
Using the "sliders" of a graphic equalizer, one can adjust the amplitude of the contribution of each frequency range to the overall sound. This makes it possible to change the "color" of the sound coming out of the system. Boosting low frequencies increases the bass tones and "richness" but can make the sound "muddy." Boosting higher frequencies improves the clarity but can make the sound seem "thin." The more sliders you have, the more precisely you can sculpt the sound produced.
Taking a natural sound and breaking it up into its component frequencies may seem like a daunting task. Computers are quite good at it, but they are by no means the only way of accomplishing the feat. In fact, the human ear does something like this to help us distinguish one kind of sound from another.
The basilar membrane in your ear is formed in such a way that sounds of different frequencies cause different areas to vibrate, more or less going from low to high as you progress from one end of the membrane to the other. Tiny hairs on this membrane, corresponding roughly to frequency bands, "pick up" the relative amplitudes of the components of the tones you hear and relay this information to the brain. The auditory processing part of your brain translates the information into what we perceive as tones. Our ears and brains naturally do a Fourier analysis of all incoming sounds!
In addition to helping us to distinguish the sounds of music, Fourier analysis has broad application in many other fields, as well. Its signal-processing capabilities are of use to scientists studying earthquakes, electronics, wireless communication, and a whole host of other applications. Any field that involves looking at or using signals to convey information, which covers a pretty broad swath of modern endeavors in science and business, uses Fourier analysis in some way or another.
Up until this point, we have been concerned with simple, one-dimensional waves, such as those evident in a cross-section of the ripples on a pond. However, a more realistic, complete analysis would have to involve the vibrations of the entire surface of the water—in three dimensions. In the realm of sound, we're now moving from the vibration of a string to a musical surface—such as a drum!