Motion appears in many aspects—but there are two obvious kinds, one which appears in astronomy and another which is the echo of that. As the eyes are made for astronomy so are the ears made for the motion which produces harmony: and thus we have two sister sciences, as the Pythagoreans teach, and we assent.
PLATO
In this chapter we intend to show how trigonometric functions have given man his first real insight into the nature of musical sounds, and how this knowledge is utilized in the design of such devices as the telephone, the phonograph, the radio, and sound films.
The mathematical study of musical sounds did not start with the application of trigonometric functions. Indeed, it goes back to the very first emergence of any real mathematics and science, namely the beginning of the classical Greek period. For example, the Pythagoreans discovered that the lengths of two equally taut plucked strings whose sounds harmonize are related by simple numerical ratios such as 2 to 1, 4 to 3, and 3 to 2. The lower note in each case originates with the longer string. They also designed musical scales whose notes, as measured quantitatively by the lengths of the vibrating strings, possessed precise numerical values. From Pythagorean times onward, mathematicians and scientists were convinced that musical sounds had important mathematical properties, and music, along with arithmetic, geometry, and astronomy, became part of the quadrivium. These four subjects were studied together right through the medieval period. Although Greek, Arab, and medieval mathematicians continued to investigate musical sounds and wrote books on music, their work was essentially limited to the construction of new systems of scales for instrumental and vocal music.
It was the mathematicians and scientists of the seventeenth century who initiated other investigations and made the next series of important discoveries. Familiar names, such as Galileo, his French pupil and colleague Father Marin Mersenne (1588–1648), Hooke, Halley, Huygens, and Newton, obtained significant new results. Whereas the Pythagoreans had studied strings of different length but equal tension, Mersenne studied the effect of changing tension and mass of a string and found that an increase in mass and a decrease in tension produce lower notes in a string of given length. This discovery was very important for stringed instruments such as the violin and the piano; to secure the range of pitch which these instruments possess by variations in length only would require exceedingly long strings. Galileo and Hooke demonstrated experimentally that each musical sound is characterized by a definite number of air vibrations per second, a statement which will mean more to us in a few moments. The determination of the velocity of sound (about 1100 feet per second in air) was another achievement. It is of interest that the clocks which some of these men designed and constructed were essential to the progress made in the study of sound because, as we can see from the results cited, the ability to measure small intervals of time was an indispensable condition for any work in this field.
The best mathematicians of the eighteenth century, Leonhard Euler, Daniel Bernoulli (1700–1782), Jean le Rond d’Alembert (1717–1783), and Joseph Louis Lagrange, studied vibrating strings, such as the violin string, and vigorously disputed whether trigonometric functions were adequate to represent the vibrations. The mathematical analysis of sound waves soon followed and proved to be the chief tool in the theoretical mastery of musical sounds. We can readily see why mathematics was invaluable in these investigations, for observation of the air, even of air in the process of propagating sound, reveals nothing.
Before we undertake to study just what the nineteenth-century mathematicians and scientists learned, we must make some distinctions. The first is a matter of terminology. We shall be interested in the analysis of musical sounds as opposed to noise. However, in the present context, the term “musical sound” is used in a technical sense and includes not only those sounds commonly understood to be music, but also the sounds of ordinary speech. As a matter of fact, the physicist’s meaning might be more appropriately represented by the term intelligible sound. Just what is meant by either phrase will be clear in a few moments.
The second distinction one must make is between sound as a motion of air and sound as a sensation which human beings experience. The former is a physical phenomenon which takes place in space and whose physical and mathematical properties are fixed. On the other hand, the sensations which human beings may receive because moving air strikes their ears and stimulates certain nerves depend upon their auditory mechanism and may vary from one person to another. There are, for example, physical sounds which humans cannot hear at all. Though we shall have something to say about the perception of sound, our first and main concern will be to understand the physical phenomenon.
The variety of sounds given off by musical instruments, the human voice, phonographs, radios, and whirring machinery, for example, is so great that one cannot hope to study all of them in one swoop. Hence it would seem wise to start one’s investigation with simple sounds. But which sounds are simple? If we rely upon our ears to decide this question, then the sounds given off by tuning forks seem to be simple. The ear may, indeed, be deceived here, but let us follow up this suggestion.
If either prong of a tuning fork is struck, both prongs will move inward and then outward very rapidly and will repeat this motion for a long time. Let us consider one prong, say the right one shown in Fig. 19–1. Before the prong is struck, it occupies what we might call the rest position. After being struck, the tip is displaced some distance to the right. It then moves to the left, to a position somewhat to the left of the rest position, and then moves to the right. The sequence then repeats itself many times. The displacement of the tip varies with time, and the first question one might raise is, What is the relationship between displacement and time? There are two considerations which suggest that the formula is sinusoidal: First of all, the prong resembles a spring-and-bob arrangement. The spring is the prong itself, though the motion is a sidewise oscillation rather than an expansion and contraction. The mass which corresponds to the bob is the mass of the prong itself, though admittedly this mass is not concentrated in one place as it is in the case of a bob on a spring. The second consideration is that as the tip of the prong moves farther and farther out from the rest position, the force which the prong exerts to return to the rest position may be expected to increase with the displacement. The simplest assumption one might make in this case is that the force increases directly with the displacement. From formula (17) of the preceding chapter we can see that this is indeed the mathematical law which underlies and determines the sinusoidal motion of the bob. Hence it seems reasonable to expect that the relation between displacement of the tip of either prong and time is sinusoidal. The amplitude of this relation is the maximum displacement of the tip, and the frequency is the frequency per second with which the prong oscillates.
Fig. 19–1.
A vibrating tuning fork.
Of course, we are not so much interested in the motion of the tuning fork as we are in the sound it creates. Hence what matters next is, How does the air respond to the vibration of the tuning fork? The fundamental fact about the behavior of air which is of importance in this connection is that air pressure seeks to become uniform everywhere. This means that if the air pressure for any reason should become high in one place, the air will spread out from that place into neighboring regions where the pressure is lower and so try to equalize the pressure in the entire region under consideration. With this physical fact in mind, let us see what happens when the right prong of the tuning fork moves, say to the right. The prong pushes the molecules of air near it to the right and thus crowds them into a place occupied by other molecules. The pressure becomes high in this place, and since the molecules of air cannot move to the left because the prong is there, they will move off farther to the right (and in other directions) in order to equalize the pressure. But this motion means that the crowding now occurs a little farther away from the tuning fork and again, to equalize the pressure, the molecules move farther to the right. The process continues, and the crowding, or condensation, as it is usually called, moves off to the right.
Fig. 19–2.
Motion of air molecules under pressure of a vibrating tuning fork.
The prong, having moved as far to the right as it can, will now move back not only to its rest position but farther to the left. This motion leaves an empty region—the place that the prong had occupied—and so the molecules of air on the right rush into this empty space. Molecules still farther to the right also move to the left because the pressure has become less to their left. Thus a state of low pressure, or rarefaction, as it is called, moves to the right as molecules move to the left to equalize the pressure in their neighborhood. With each successive vibration of the prong, a condensation and rarefaction move off to the right (Fig. 19–2). The successive condensations and rarefactions also move out in other directions, but it is sufficient for our purposes to follow what happens in one direction.
The action of the air is somewhat complicated because it consists of billions of molecules, and they do not all behave in exactly the same way. However, there is an average effect. It is convenient to speak of a series of typical molecules to the right of the prong which represent the average behavior of the entire collection. If we consider the action of any one typical molecule, say one near the prong, then what it does is to move to the right when the prong moves to the right. When the prong moves to the left, the typical molecule will also move to the left because the air pressure has been lowered. Like the prong, it will move past its rest position, and continue to the left. Then, as the prong moves to the right, the molecule will be pushed to the right, will pass its rest position, go farther to the right, and, from this time on, it will continue to oscillate.
Fig. 19–3.
Motion of a typical molecule.
Typical molecules farther to the right will behave like the typical molecule near the prong; however, their reaction will be slightly delayed since condensations and rarefactions reach them a little later. Figure 19–3 illustrates the motion of a typical molecule reacting to a series of prong oscillations.
Two important facts emerge from the above discussion. The first is that the average, or typical, molecule follows, in effect, the motion of the prong. Any one molecule acts as if it were attached to the prong by a spring. When the prong moves to the right, it contracts the spring. The latter seeks to restore its length and so pushes the molecule to the right. While the molecule moves to the right, the prong moves to the left, and hence the spring is extended. It now seeks to contract and so pulls the molecule to the left. The molecule moves to the left, and the spring contracts. But now the prong is ready to move to the right, and consequently the motion of prong and molecule repeats itself. The action of air pressure is indeed like the action of the spring. In fact, Hooke used the phrase, “the spring of the air,” to describe the effect of air pressure.
The second fact is that the sound wave which moves from the prong to some person’s ear, say, consists of the series of condensations and rarefactions induced by the prong’s motion. Each molecule merely oscillates about its rest position, but in doing so it produces the increase and reduction of pressure which cause the neighboring molecules to oscillate.
The nature of the sound wave may perhaps be made clearer by comparing it with a water wave. If the end of a stick is quickly moved back and forth in still water, a series of waves will spread out from the end of the stick. However, the individual water molecules do not move out. Each oscillates about its original position, but the increase and decrease in pressure which the stick creates cause the molecules farther away to duplicate the motion of the molecules near the stick.
Since the motion of any typical molecule whether near or far from the prong is the same, let us study the motion of any one of these molecules. Specifically, let us seek the relationship between the displacement from rest position and the time that the molecule is in motion. What formula relates displacement and time? We have already produced two crude physical arguments suggesting that for the prong, displacement and time are related by a sinusoidal formula. Since the motion of any typical air molecule duplicates the action of the prong, the formula which relates displacement and time for the typical air molecule should also be sinusoidal. Actually these physical arguments do not really prove that the formula is sinusoidal. However, this fact can be established either by a rather complex mathematical analysis of air motion or, experimentally, by converting the air pressure to electric current (by means of a microphone, for example) and by then displaying the current on a cathode-ray tube (television tube).
We shall take for granted that for a typical air molecule, the formula relating displacement and time is sinusoidal. Hence, if y is the displacement and t is the time, then, in view of what we have learned in the preceding chapter, the formula is
where D is the amplitude, or maximum displacement, and f is the number of oscillations, or the frequency of cycles per second. We wish to emphasize that the formula applies to the sounds produced by tuning forks or to what we have reason to believe are simple sounds.
To use formula (1), we must know D and f. The value of f is the frequency with which the tuning fork oscillates. A frequency commonly used to standardize the pitch of sounds is 440 per second. This then is a typical value of f. The value of D, the amplitude of the motion of a typical air molecule, is not the amplitude of the prong’s motion, but depends upon the medium in which sound spreads out or is propagated. It depends, so to speak, on the “springiness” of the medium. In air, 0.001 inch can be considered to be a reasonable value for D. Hence a typical formula for a simple sound is
Thus, a typical air molecule oscillating in accordance with formula (2) shuttles back and forth about its mean, or rest, position 440 times per second or, as we say, it goes through 440 complete cycles in one second. The farthest distance from the mean position that it reaches, that is, the amplitude of its motion, is 0.001 inch.
Fig. 19–4.
Graph of displacement versus time of a typical molecule executing 400 oscillations per second.
Figure 19–4 illustrates the relationship between displacement and time for a simple sound such as formula (2) represents. Of course the typical molecule shuttles backward and forward, but on the graph its displacements are plotted as ordinates and the time elapsed is shown by the corresponding abscissas.
Although formula (2) represents only simple sounds—we have yet to discuss the formulas describing more complicated sounds—it enables us to understand what we meant earlier by the phrase “intelligible sounds.” We see that a simple sound has a regularity or periodicity. The motion of the air molecules repeats itself a number of times a second. When the ear receives many cycles of this motion, it can identify the sound. If, on the other hand, the motion of the air molecule is not regular but varies irregularly with time, the ear still hears sound, but sound that does not convey any meaning, i.e., noise.
1. What is the basic mathematical formula which represents simple sounds? State the physical meaning of the various letters in the formula.
2. State the formula which describes the relationship between displacement and time for a simple sound whose frequency is 300/sec and whose amplitude is 0.0005 in.
3. If y = 0.002 sin 2π · 540t is the mathematical description of a sound, what are the frequency and amplitude of this sound?
4. If a sound has a frequency of 400 cycles/sec, how many cycles would the ear receive in 1/20 sec?
We have now a good mathematical representation of simple sounds. But interesting musical sounds, whether vocal or instrumental are, as a rule, not simple, and the really significant contribution of mathematics to the understanding of musical sounds lies in the analysis of more complex sounds. To comprehend this contribution, we must first examine a relevant mathematical idea. Instead of considering simple sinusoidal functions such as (2), let us take the function
What sort of relationship between y and t does formula (3) represent?
A good way to investigate this question is to draw a graph of the above function. Since we wish to obtain merely some general idea of how y varies with t, we shall seek only a sketch rather than a very accurate graph. We could proceed by selecting values of t, calculating the corresponding values of y, and then plotting the points whose coordinates have thus been determined. However, there is a quicker method which is also more perspicuous. Let us consider the two functions:
and
We have used the notation y1 and y2 to distinguish the dependent variables in (4) and (5) from the y in formula (3). Formulas (4) and (5) are easily graphed. Formula (4) is the ordinary sine function which goes through the regular cycle of sine values in each unit of t. Formula (5) has a frequency of 2 in each unit of t; that is, the y-values go through the complete cycle of sine values twice in each unit of t. Let us sketch both functions on the same set of axes (Fig. 19–5).
Fig. 19–5.
The graph of y = sin 2πt + sin 4πt obtained by addition of ordinates.
Now the y of formula (3) is clearly the sum of y1 and y2. Hence adding the values of y1 and y2 at various values of t will yield y. Since we are interested only in a sketch, let us perform the addition by using Fig. 19–5 to obtain the values of y1 and y2. Thus for t = 0, the graphs show that y1 and y2 are both zero. Hence y, the sum of y1 and y2, also is zero. At 
, we see from the graph that y1 is about 0.7 and y2 is 1. Hence y = 1.7 when . At 
, we find that y1 = 1 and y2 = 0. Hence y = 1 when . At 
, y1 is about 0.85 and y2 is about −0.85. In adding the last two values for y1 and y2, we must take into account that one is positive, the other negative, and their sum zero. Hence at , y = 0. By selecting a few more values of t and estimating the corresponding y1- and y2-values, we can obtain more y-values. Finally, we join the various points which belong to the graph of formula (3) by a smooth curve. The result is the heavy-lined curve shown in Fig. 19–5. The method just described for graphing y as a function of t provides a rough sketch. If one wishes to obtain a more accurate graph, he can calculate the value of y for each value of t.
How far need we carry this process of determining y-values corresponding to various t-values? We note that the function y1 = sin 2πt repeats itself when t becomes larger than 1. The function y2 = sin 4πt goes through two full cycles in the interval from t = 0 to t = 1 and begins its third cycle of sine values as soon as t increases beyond 1. Thus at t = 1 both functions begin to repeat the values which they had taken on at t = 0 and, in the interval from t = 1 to t = 2, both functions will repeat the behavior exhibited in the interval from t = 0 to t = 1. Since y1 and y2 repeat their former behavior, it follows that y, which is the sum of y1 and y2, will also repeat its former behavior. In other words, in the interval from t = 1 to t = 2, y will behave precisely as it did in the interval from t = 0 to t = 1. And in each succeeding unit interval of t-values, the function will repeat the behavior exhibited in the interval from t = 0 to t = 1. If we therefore determine the behavior of y in the interval from 0 to 1, we know how it behaves for all larger values of t.
There are several major facts to be learned from this example. First of all, since the function (3) repeats its behavior in every unit of t-values, it is periodic. Moreover, because the term sin 4πt goes through two cycles of sine values in exactly the t-interval in which sin 2πt goes through one cycle, the entire function repeats itself with the frequency with which y = sin 2πt repeats itself. Hence the frequency of formula (3) is one cycle per unit of t. Thirdly, the shape of the graph of formula (3) shows that the formula is not sinusoidal even though it is periodic. In other words, the sum of two sine functions can yield a function whose shape is quite different from that of a sine function, but the sum can nevertheless repeat itself.
We might expect that functions built up of three or more sine functions could have quite strange shapes and yet be periodic if the summands all began to repeat at some value of t, say t = 1, the values they had taken on at t = 0.
1. By following the method described in the text sketch the graph of
a) y = sin 2πt + sin 6πt,
b) y = sin 2πt + 
sin 4πt,
c) y = sin 2πt + sin 3πt.
2. What is the frequency, in one unit of t, of the function
a) y = sin 2πt + sin 8πt,
b) y = 2 sin 2πt + sin 4πt,
c) y = sin 2πt + sin 4πt + sin 6πt,
d) y = sin 2π · 100t + sin 2π · 200t + sin 2π · 300t?
We have already mentioned that the sounds given off by almost all musical instruments and by the human voice are not simple sounds; that is, they are not representable by functions of the form (1). Yet these sounds are intelligible, which means that they must be periodic or that the pattern of displacement versus time must repeat itself. The shapes of the curves which represent such sounds are, however, quite varied. In fact, to each sound there corresponds a characteristic shape. For example, Fig. 19–6 shows the shape corresponding to the sound of a piano note C. To obtain this graph, the sound is converted to electric current and the vibration of the current is made visible by means of a cathode-ray tube. In view of the variety of musical sounds it may seem that we have reached an impasse in our attempt to analyse all such sounds mathematically. But by a stroke of good luck mathematics provided the very theorem which gives us remarkable insight into all complex sounds. The stroke of good luck was the mathematician Joseph Fourier (1768–1830).
Fig. 19–6.
Displacement versus time of a typical molecule for the note C on a piano.
Fourier was the son of a French tailor. While attending a military school he became intrigued with mathematics. Since he realized that his low birth would not permit him to become an army officer, he let himself be persuaded by members of the Church to study for the priesthood. However, he abandoned the priesthood to accept a professorship of mathematics at the military school that he had attended. Later he became a professor at the École Normale and at the École Polytechnique, universities founded by Napoleon.
Fourier’s main interest was mathematical physics, and his most important work in that domain concerned the conduction of heat; for example, he studied how heat travels along metals. His chief contribution, a book entitled The Analytical Theory of Heat (1822), is one of the great classics of mathematics. In the development of the theory of heat Fourier established a mathematical theorem whose value extends far beyond the physical application for which it was intended. Our interest in the theorem lies in what it does to analyze complex musical sounds.
Fourier’s celebrated theorem says that any periodic function is a sum of simple sine functions of the form D sin 2πft. Moreover, the frequencies of these component functions are all integral multiples of one frequency. To illustrate the significance of this theorem, let us suppose that y is a periodic function of t. Then the formula which relates y and t must be of the form
The numbers in this formula depend, of course, on the choice of the initial periodic function, but let us suppose that they are correct and see what they stand for. The numbers 1, 0.5, 0.3 are the amplitudes of the respective sinusoidal components of the entire periodic function. The lowest frequency per second, that of the first term, is 100. The second term has frequency 200, or twice the lowest frequency. The third term has frequency 300, or three times the lowest frequency, and so on. The dots at the end of formula (6) imply that we might need additional terms like the ones shown, to represent any given periodic function. In accordance with the theorem, all frequencies occurring in such additional terms must be multiples of 100.
Before we consider the significance of Fourier’s theorem for the study of musical sounds, we should satisfy ourselves that formulas such as (6) do represent periodic functions. In this connection, two results of our work in Section 19–3 should be helpful. We learned there that the sum of two sine functions can produce a rather peculiarly shaped but nevertheless periodic graph. Moreover, because the second term in formula (3) had twice the frequency of the first one, the frequency of the entire function was the lower of the two frequencies. The situation in (6) is very much the same. It is a sum of sine terms, and the graph of this sum may indeed have a peculiar or irregular shape. But the shape will repeat itself because during the time that the first term goes through one cycle, namely the interval t = 0 to 
, the second term will go through two cycles, and the third term through three, so that the entire function will repeat itself as soon as the first term does. Since the frequency of the first term is 100 in one unit of t, the entire function has the frequency of the first term.
And now what does Fourier’s theorem have to do with the analysis of musical sounds? The application of this theorem to music was made by a German, Georg S. Ohm, a teacher of mathematics and physics, who lived in the first half of the nineteenth century. As pointed out earlier in this section, every musical sound is a periodic function; that is, the relation between displacement and time of a typical air molecule oscillating under the pressure exerted originally by the source of the sound is a periodic function of t. But Fourier’s theorem says that every such function is a sum of simple sine functions of the type illustrated in (6). Each simple sine function corresponds to a simple sound such as is given off by a tuning fork. Hence one arrives at the important conclusion that every musical sound is a sum of simple sounds. Moreover, the frequencies per second of these simple sounds are all multiples of one lowest frequency. To put the matter differently, every musical sound can be duplicated by a combination of tuning forks, each vibrating with the proper frequency and amplitude.
The musical sound whose graph is shown in Fig. 19–6, for example, is a sum of five simple sounds. The frequencies of these sounds and their respective amplitudes are tabulated below. The amplitudes are expressed in terms of the first one entered, which is chosen to be 1.
We should note that the frequencies are all multiples of the lowest one, which is 512. The formula representing this sound is then
The assertion that every musical sound is no more than a combination of simple sounds is so surprising that, although it is backed by unassailable mathematics, one wishes to see it confirmed by experimental evidence. Such evidence is available. First of all, a trained ear can recognize the simple sounds present in a complex sound. Secondly, if one releases the dampers on the strings of a piano and then strikes a note, a number of other strings will also begin to vibrate, namely those whose basic frequencies are the same as the component frequencies present in the note struck. The physical explanation is that the note struck gives off several frequencies—the frequencies of its component simple sounds. Each of these frequencies sets off air vibrations which in turn force into vibration all other strings whose basic frequencies are the same as those of the simple sounds.
Perhaps the best experimental evidence is furnished by some specially designed instruments. The distinguished nineteenth-century physician, physicist, and mathematician Hermann von Helmholtz (1821–1894) gave two kinds of demonstrations. In the first one he designed special pipes, called resonators, each of which selected and rendered audible only that frequency which was suited to the dimensions of the pipe. A resonator in the neighborhood of a complex sound will pick up and render audible any component of the sound whose frequency excites the resonator. By using resonators of different sizes Helmholtz was able to show that the frequencies present in the complex sound were just those called for by Fourier’s theorem. Then Helmholtz demonstrated the reverse. He set up electrically driven tuning forks of the proper frequency and amplitude such that the combination of simple sounds duplicated a given complex sound. A modern version of this latter device is the electronic music synthesizer.
There is no question, then, that any musical sound is no more than a sum of simple or sinusoidal sounds. The simple sound of lowest frequency is called the fundamental, or first partial, or first harmonic. The simple sound whose frequency is twice that of the lowest one is called the second partial or second harmonic; and so on. The frequency of the entire complex sound is the frequency of the first harmonic for the reason already given in our discussion of Fourier’s theorem. The amplitudes of the individual sine terms are the amplitudes or strengths of the harmonics present.
1. State Fourier’s theorem.
2. Suppose that a complex sound is representable by the function
y = 0.001 sin 2π · 240t + 0.003 sin 2π · 480t + 0.01 sin 2π · 720t.
What is the frequency of the complex sound? What is the amplitude of the third harmonic?
3. Write the formula for a musical sound whose frequency is 500/sec and whose first, second, and third harmonics have amplitudes of 0.01, 0.002, and 0.005, respectively.
4. If the relationship between displacement and time for the fundamental of a musical sound is y = 3 sin 2π · 720t, what is the frequency of the third harmonic?
5. Explain why the frequency of a complex musical sound is always that of the first harmonic.
Musical sounds as received by the ear seem to possess three essential properties; that is, the ear recognizes what are commonly called the pitch, the loudness, and the quality of a sound. One of the major values of the mathematical analysis of musical sounds is that it clarifies and makes precise just what we mean by these properties. We shall consider them in turn.
In our subjective judgment, sounds vary from low or deep tones to high or piercing ones. Verbal descriptions of the pitch of sounds are, of course, qualitative and vague. If one experiments with tuning forks of different pitch, he readily discovers that high pitch means high frequency of fork vibration and therefore high frequency of oscillation of the air molecules. Correspondingly, low pitch means that the fork and the air molecules vibrate with low frequencies. Prior to the availability of the analysis examined in the preceding section, the notion of pitch was not clear for complex sounds. But we now know that all musical sounds have a definite frequency, namely the frequency of the fundamental. Thus, although complex sounds contain other frequencies, that is, the frequencies of the higher harmonics, it is the over-all frequency of the composite sound which determines whether it appears high- or low-pitched to the ear. For example, as one strikes the notes on a piano going from left to right, the fundamental frequency steadily rises.
The loudness of a musical sound is determined by the amplitude of the corresponding molecular motion, but the relationship between loudness and amplitude is not quite so simple as that between pitch and frequency. Let us note, first of all, that amplitude means the maximum displacement of the typical air molecule or the largest y-value of the corresponding graph. Physicists call the square of this amplitude the intensity of the sound. Thus intensity is still a physical or objective property of a musical sound. Among sounds of a given frequency, the more intense sound will seem louder to the ear. However, this is no longer true if the frequencies of the sounds differ. The average ear is most sensitive to a frequency of about 3500 per second and less so to frequencies above and below this value. Hence a very intense sound at a frequency of 1000 per second may sound softer to the ear than a less intense one at a frequency of 3500 per second. As a matter of fact the average human ear does not hear at all sounds above about 16,000 vibrations per second, no matter how intense they are. Loudness depends not only on the intensity and the frequency of the sound but also on the shape of the graph within any one period. Two sounds may possess the same frequency and the same amplitude, but may have differently shaped graphs. Such sounds will, in general, not sound equally loud to the ear.
The most interesting and from an aesthetic standpoint the most important aspect of musical sounds is their quality. It is this property which determines whether or not a sound is pleasing. The quality of a sound depends upon which harmonics are present in the sound and the amplitudes of these harmonics. Thus a sound emitted by a piano and a sound of the same frequency emitted by a violin create different effects on the ear because they differ in the harmonics present and in the amplitudes of these harmonics. Since the harmonics and their amplitudes determine the shape of the graph, it follows that, mathematically, the quality of a sound is the shape of the graph within any one period.
Sounds or tones vary greatly with respect to harmonics and their amplitudes. Some sounds, for example, the sounds of tuning forks, some notes on the flute, and sounds produced by wide-stopped organ pipes, possess only a few harmonics or, in effect, merely the first. On the other hand, most instruments give off sounds containing many harmonics, but some of these may have small or almost zero amplitude. For example, the sounds of organ pipes are, in general, weak in the higher harmonics. The sounds of a violin possess a great number of harmonics and are usually strong in the first six harmonics. The relative amplitudes of the harmonics present in violin sounds are about the same for all notes; however, there are enough differences for the ear to distinguish, say the A- from the D-string, even though both are sounded at the same frequency. The uniformness of quality may explain why the sounds of a violin are so pleasing. The sounds of a piano also contain many harmonics, but the relative amplitudes of the harmonics in any one sound depend upon the velocity with which the hammer strikes the string.
The vowel sounds of the human voice are rich in harmonics. For example, the sound of “oo” as in tool, expressed at a fundamental frequency of 125 vibrations per second, has as many as 30 detectable harmonics. The relative amplitudes of the first six are 0.4, 0.7, 1, 0.2, 0.2, and 0.2, respectively. The higher harmonics, though present, have lower amplitudes. However, not only do the number of harmonics present and their relative amplitudes vary considerably from one vocal sound to another, but even the same sound issued at two different pitches will have different harmonics and amplitudes.
The physical reason for the differences in quality among the many types of musical instruments is, of course, the nature of the device itself. The piano and violin both use vibrating strings, but piano strings are struck whereas violin strings are bowed. The clarinet, oboe, and bassoon are operated by forcing air against vibrating reeds. Air is forced past the edge of an opening in the organ pipe also, but here the edge or lip is rigid. In addition, each instrument possesses a resonance device which emphasizes certain harmonics. The sounding board of the piano, the hollow box of the violin, and the pipes of an organ are resonance devices.
Although two people may not quite agree about their reactions to sounds, it is on the whole true that the qualities of sounds which we describe by such words as soft, piercing, rich, dull, braying, hollow, bright, and the like, are due to the harmonics and their relative amplitudes. Sounds which contain only the first harmonic are soft but dull. For brightness and acuteness the higher harmonics are essential. Sounds which possess the first six harmonics are grand and sonorous. If harmonics beyond the sixth or seventh are present and have appreciable amplitudes, the tones are piercing and rough. In general, the amplitudes of harmonics decrease as the frequencies increase. However, if the amplitudes of higher harmonics are too large compared with that of the fundamental, the tone is described as poor rather than rich.
1. Suppose that two sounds are represented respectively by y = 0.06 sin 2π · 200t and y = 0.03 sin 2π · 250t. Which one is louder? Which one is higher pitched?
2. Explain in mathematical terms the meaning of a simple sound.
3. What is the mathematical criterion of a musical sound as opposed to noise?
4. Which mathematical properties of the formula for a complex musical sound represent the pitch of the sound and the quality of the sound?
5. Discuss the assertion that music is basically just mathematics.
1. The construction of musical scales. Include, in particular, the work of J. S. Bach on the equal-tempered scale.
2. The human voice as a source of musical sounds.
3. The functioning of the human ear.
BENADE, ARTHUR H.: Horns, Strings and Harmony, Doubleday and Co., New York, 1960.
FLETCHER, HARVEY: Speech and Hearing, D. Van Nostrand Co., Princeton, 1929.
HELMHOLTZ, HERMANN VON: On the Sensations of Tone, Dover Publications, Inc., New York, 1954.
JEANS, SIR JAMES H.: Science and Music, Cambridge University Press, London, 1937.
MILLER, DAYTON C.: The Science of Musical Sounds, 2nd ed., The Macmillan Co., New York, 1926.
OLSON, HARRY F.: Musical Engineering, McGraw-Hill Book Co., Inc., New York, 1952.
REDFIELD, JOHN: Music, A Science and an Art, A. A. Knopf, Inc., New York, 1926.
SEARS, FRANCIS W. and MARK W. ZEMANSKY: University Physics,3rd ed., Chaps. 21 to 23, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1964.
TAYLOR, LLOYD WM.: Physics, The Pioneer Science, Chaps. 24 to 28, Dover Publications, Inc., New York, 1959.
VON BERGEIJK, WILLEM A., JOHN R. PIERCE and EDWARD E. DAVID, JR.: Waves and the Ear, Doubleday and Co., New York, 1960.
WOOD, ALEXANDER: The Physics of Music,6th ed., Dover Publications, Inc., New York, 1961.
