


2.6 Synthesis of periodic functions
The guitar experiment reveals that a plucked string vibrates in a number of different modes simultaneously. The frequencies of these modes are the same as those of the harmonic vibrations of the string. So, when the string was lightly touched at its midpoint, the fundamental vibration was prevented from sounding. The resulting sound was primarily that of the second harmonic, and, hence, its tone was one octave higher. When the string was touched onethird of the way from the end, the fundamental and the 2nd harmonic were prevented from sounding. The resulting sound was made up primarily by the 3rd harmonic. So while the vibration associated with a pure tone has only one natural frequency, a string that is plucked or bowed will generally vibrate with many different natural frequencies at the same time.
What we have seen is that the string vibrates at only one frequency exactly when the shape of the string at any instant of time is the graph of one or more arches of a sine function. In this case, the sound produced is a pure tone, and the corresponding pressure function is a sinusoidal function of time. Since the string is tied down at both ends, the possible "puretone" configurations of the string are the fundamental one corresponding to a single arch, and then those corresponding to two, three, and more arches. The tone frequency corresponding to the double arch configuration is twice that of the single arch, and, in general, the tone frequency for an narch configuration is the nth harmonic of the fundamental frequency.
However, if the string is plucked or bowed, the string vibrates with many frequencies at the same time. However, this complicated vibration pattern appears to be just a weighted combination of the fundamental frequency and various harmonics. Thus, we may expect that the pressure function corresponding to a complicated vibration is itself a combination of a terms corresponding to the fundamental frequency.
and terms corresponding to the harmonics
for n = 2, 3, .... (Here the a_{n}s and the b_{n}s are the weights.)
Now we have come to the central point in this discussion. We might hope that the pressure function for any musical tone can be represented in the same way. Indeed, we will see that this is true for a wide range of periodic functions  whether or not they represent the pressure function for a musical tone. We illustrate this with the following applet. Here, you have a given "general" periodic function of period 2*pi, i.e. w = 1. Use the sliders to adjust the coefficients so that the combination of sine and cosine functions produces a graph that matches the given graph.
In general, the pressure function associated with the sound of any musical instrument has a complex form that can be represented as a combination of a fundamental frequency and various harmonics. If the pressure function p has frequency h hertz, then it can be approximated by an expression of the form
p(t) =  a_{0} + a_{1} cos(wt) + b_{1} sin(wt) + a_{2}cos(2wt) + b_{2} sin(2wt) 
+ a_{3} cos(3wt) + b_{3} sin(3wt) + ... + a_{n} cos(nwt) + b_{n} sin(nwt), 
Notice that we have added an a_{0} term to the expression. This term represents the average or normal air pressure. In other words, a_{0} is determined by our choice of 0 on the pressure scale.
Two different musical instruments playing the same note have distinctive sounds because of the different strengths of the harmonics produced by the different instruments. In terms of our representation, the pressure functions have the same w but different a_{n}'s and b_{n}'s. These different coefficients determine the graphs of the pressure functions and, so, reflect the different timbres of the instruments.
As you saw in the previous exercises, it is a rather easy task to construct the corresponding pressure function for a musical sound when we are given the amplitudes of the individual harmonics. On the other hand, the reverse problem of determining the harmonic structure of a musical sound, when we have the graph of the pressure function, is a more interesting and more challenging problem. In parts three and four of this module you will develop the necessary mathematical tools to accomplish this task. In fact, you will be able to use these tools to represent almost any given periodic function as the sum of sine and cosine functions.



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