The Contribution of Music to Mathematical Discovery
Figure 1 shows some of the ways in which a string can vibrate: A produces a certain note; B produces the first harmonic, an octave higher; and C fifth higher.
In 1715, the English mathematician Brook Taylor gave this a mathematical form. If we choose a suitable unit of time and a unit of length that makes the string have length , the displacement of the string at position x and time t is given by
y = sin
x cos t for vibration A
So far, these vibrations are separate possibilities. However the French composer,
Rameau, said that when a note was played he heard many of these harmonics at one and the same time. In 1726 he
published a theory of harmony (which I believe is still regarded as valid) based on this idea.
His series contains an infinite number of constants an which can be chosen at will. Bernouilli expressed the
belief that this series represented all the vibrations possible when a stretched string was put in some position
and released from rest. The initial shape of the string is found by putting t = 0. Then all the cosines have the
value 1 and the series becomes
Violent opposition to this view came from the French mathematician d'Alembert and the famous Swiss mathematician Euler. Indeed mathematicians argued for the next fifty years without getting any further forward. Euler and d'Alembert pointed out that the simplest way to set a string vibrating was to pluck its middle point and give it the shape in Figure 2.
This shape consists of two lines with different equations. They maintained that
it was impossible for the single formula (1) to start off representing one line and then halfway through, change
its mind and decide to represent another. If it started off representing part of one line, then it had to go on
for ever showing that line.
This series converges in much the same way as the geometrical progression with a common ratio of 1/2. Now f(x) shows a break at every point at which one of the terms has a break - that is to say, at every place where is a fraction of which the denominator is a power of 2. These points are so close together that it is impossible to draw the graph; there are no two points joined by a continuous piece of curve. Yet there is a Fourier series for f(x).
This is a simplified account of an argument given by Bernhard Riemann in 1854 in a dissertation 'On the representability of a function by a trigonometric series'. With regard to functions like f(x) that cannot be drawn, he says, 'Such functions have never been considered before'.
Coefficients in the Fourier series can be calculated by integrating the function. Since f(x) has a Fourier series, we naturally consider whether it can be integrated. Riemann showed that it could. This remarkable result means that although the graph of f(x) cannot be drawn, there is a definite area under it.
It is also remarkable that one can combine the curves of sine graphs to produce straight lines, and that one can combine sine curves, which are continuous, in such a way as to produce the gap in Figure 3, let alone the infinity of jumps that are seen in the graph of f(x).
The Germans have a rather good term, 'Anschauliche Mathematik' - mathematics that you can understand by looking at it. The corresponding English word is 'intuitive' - mathematics that you feel should be true. In the 17th and 18th centuries, mathematics was essentially intuitive; mathematicins used procedures that looked right and seemed reasonable. It was in this spirit that calculus was discovered and it is still in this way that learners should be introduced to the beginnings of calculus. Indeed intuition is still important at the stage of advanced work; it is essential to have a feeling for what you are doing.
The 19th century was the century of rigour. The free and easy style of the earlier
mathematicians had been immensely creative, but at times it had led to errors and absurdities. Intuition had to
be supplemented by careful analysis and logical proof. One reason was the increased complexity of the material
being studied. It was all very well, in the early stages, to explain that integration gave the area under a curve,
and everyone had an idea of what area meant. It is not the same when you are dealing with a perforated graph like
that of Riemann's f(x). Does it indeed determine an area? If so, we have to define exactly what is meant by that
area and how to measure it. This is characteristic of 19th century mathematical works. It is no longer felt that
everyday ideas are sufficient: we have to make sure that each concept is accurately defined, and that we proceed
logically from this definition when we use it.
The explanation lies in the meaning of the sum of an infinite series, about which early mathematicians were rather vague, and which was clarified in the 19th century. The objection would be perfectly valid if we were talking about the sum of a finite number of terms. However many terms of the series (1) you may take, its graph will always be free from breaks. But y in equation (1) is not equal to the sum of any finite number of terms; it is something that is approached more and more closely, but never reached as you take larger and larger numbers of terms.
The series for f(x) is
Figure 5 shows the graph for the sum of the first 10 terms of this series. It does not consist of straight lines nor does it have a break in it, but it is doing its best to resemble the graph of F(x) in Figure 4a.
This article first appeared in the November 1990 issue of Mathematical Review.
Copyright © W. W. Sawyer & Mark Alder 2001
Version: 22nd March 2001