Introduction

A detailed discussion of the electronic circuits used in conjunction with radiation detectors would be out of place in the present monograph. The design of these circuits has become so specialized that they are no longer ‘home–constructed’, certainly not by the crystallographer to whom this text is primarily addressed. The aims of the present chapter are much less ambitious. They are

1. (1) to indicate what ‘black boxes’ are required in a diffractometer installation;

2. (2) to assist the crystallographer, who is contemplating the assembly of a complete installation from individual commercial units, in making his selection from the large range which is available;

3. (3) to describe the precautions necessary to secure a satisfactory performance from these units; and

4. (4) to indicate the amount of trouble–shooting which the user of the diffractometer may have to undertake himself.

It must be remembered that an automatic diffractometer is largely an electronic instrument and the user must have some acquaintance with electronic technology to ensure the correct functioning of his equipment. The brief notes offered in this chapter cannot supply the knowledge which can only be derived from general texts on electronics. The diffractometer user should also study more detailed discussions of detector circuitry (for example, chapter 11 of the monograph by Price, 1964).

Diffractometer input and output

The automatic diffractometer, whatever its type, can be regarded as a black box which accepts numerical input information of one kind and produces numerical information of another kind. The input information consists of the settings of the various crystal and detector shafts and all the instructions necessary to make valid measurements of the intensities of Bragg reflexions. The output data consist of the results of those measurements. The generation of the input information and the processing of the diffractometer output must inevitably be carried out by means of a computer: the rate at which measurements are made by an automatic diffractometer is such as to make manual computation with a desk calculator quite impracticable.

It is important, therefore, that the input and output medium of an automatic diffractometer be one which is readily generated and read by a computer. (In the case of analogue diffractometers, discussed in §3.3, only the output medium needs to be considered.) The most common media are punched cards or punched paper tape: the choice between these two is generally dictated by the computer installation which is used in conjunction with the diffractometer. It may be noted in passing that punched cards offer a greater flexibility in that the sequence in which reflexions are to be measured and the sequence in which experimental results are listed can be varied by simply resorting a stack of cards.

We began our study of sound-curves by examining the curve produced by a tuning-fork. A tuning-fork was chosen, because it emits a perfectly pure tone. But, as every musician knows, its sound is not only perfectly pure, but is also perfectly uninteresting to a musical ear—just because it is so pure.

The artistic eye does not find pleasure in the simple figures of the geometer—the straight line, the triangle or the circle—but rather in a subtle blend of these in which the separate ingredients can hardly be distinguished. In the same way, the painter finds but little interest in the pure colours of his paint-box; his real interest lies in creating subtle, rich or delicate blends of these. It is the same in music; our ears do not find pleasure in the simple tones we have so far been studying but in intricate blends of these. The various musical instruments provide us with readymade blends, which we can combine still further at our discretion.

In the present chapter we shall consider the sounds which are emitted by stretched strings—such as, for instance, are employed in the piano, violin, harp, zither and guitar—and we shall find how to interpret these as blends of the pure tones we have already had under consideration.

Experiments with the Monochord

Our source of sound will no longer be a tuning-fork but an instrument which was known to the ancient Greek mathematicians, Pythagoras in particular, and is still to be found in every acoustical laboratory—the monochord.

The last two chapters have been concerned with the vibrations of tuning-forks and of strings. The vibrations of tuningforks proved to be mainly of theoretical interest, helping us to understand vibrations and sound-curves in general. On the other hand, the discussion of the vibrations of strings established direct contact with practical musical problems, the sounds produced by the violin, piano, harp, etc. In the present chapter we shall consider a further class of musical instruments in which the vibrating structure is a column of air—organ-pipes, flutes, whistles, oboes, fifes, etc.

The Spring of Air

For our first experiment we need only very simple apparatus—an ordinary bicycle or motor-tyre pump with a reasonably close-fitting piston. Let us cork up the tube at the outlet end, and stand the pump vertically, with the piston near the top, as in fig. 39.

The piston does not immediately fall to the bottom, because the pressure of the air in the tube holds it up. We can push it down by pressing hard on the handle, but the moment we take the pressure off, it bounces up again, just as though the air inside the tube formed a spring. Indeed we have discovered what Robert Boyle called “the spring of air”.

We shall understand the mechanism of this “spring of air” if we bear in mind that a gas consists of an immense number of molecules which dart about to-and-fro at very high speeds, each moving in a straight path until it either collides with another molecule or runs into some solid object.

In the present book, we are dealing with subjects which lie partly within the province of science and partly within that of art, and the boundary between the two provinces is not always perfectly clear. If the question is debated as to whether the music of John Sebastian Bach is superior to that of his son Philipp Emanuel, science can bring nothing to the discussion. The question is purely one for artists, and it is quite conceivable, although perhaps rather improbable, that they may not be able to agree as to the answer. On the other hand, if the question is whether the music of either Bach is superior to that produced by a chorus of cats singing on the roof, there will be little doubt as to the answer. The artists will all agree, and science is able to explain to a large extent why they agree.

To say the same thing in another way, the aim of music is to weave the elementary sounds we have been discussing into combinations and sequences which give pleasure to the brain through the ear. As between two pieces of music both of which give pleasure in a high degree, only the artist can decide which gives more, but the scientist can explain why some give no pleasure at all. He cannot explain why we find Bach specially pleasurable, but he can explain why we find the cat music specially painful.

The Coming of Music

The lantern of science, throwing its light down the long corridors of time, enables us to trace out the gradual evolution of terrestrial life. Far away in the dim distances of the remote past we see it emerging from lowly beginnings—possibly single-cell organisms on the sea shore—and gradually increasing in complexity until it culminates in the higher mammals of to-day, and in man, the most complicated form of life which has so far emerged from the workshop of nature. And as living beings become more complex, they acquire an ever more intricate battery of sense-organs which help them to find their way about the world, to escape danger, to capture their food and avoid being themselves captured as food.

One of these is of special interest to musicians, for out of it has developed our present organ of hearing. Sunk into the skin of a fish, and running the whole length of its body, from head to tail on either side, there is a line of pits or depressions. Under these lies an organ known as the “lateral-line” organ. This is believed to register differences of pressure in the water, which will acquaint the fish with the currents and eddies in which he is swimming, and may also warn him of the proximity of other fish, especially of large fish of hostile intentions.

We have now considered the generation of sound and its transmission through the air to the ear; we must finally consider its reception by the ear, and transmission to the brain.

When the air is being traversed by sound-waves, we have seen that the pressure at every point changes rhythmically, being now above and now below the average steady pressure of the atmosphere—just as, when ripples pass over the surface of a pond, the height of water in the pond changes rhythmically at every point, being now above and now below the average steady height when the water is at rest. The same is of course true of the small layer of air which lies in contact with the ear-drum, and it is changes of pressure in this layer which cause the sensation of hearing. The greater the changes of pressure, the more intense the sound, for we have seen that the energy of a sound-wave is proportional to the square of the range through which the pressure varies.

The pressure changes with which we are most familiar are those shewn on our barometers—half an inch of mercury, for instance. The pressure changes which enter into the propagation of sound are far smaller; indeed they are so much smaller that a new unit is needed for measuring them—the “bar”. For exact scientific purposes, this is defined as a pressure of a dyne per square centimetre, but for our present purpose it is enough to know that a bar is very approximately a millionth part of the whole pressure of the atmosphere.

Much has been added to our scientific knowledge of musical sound, since Helmholtz published his great work Tonempfindungen in 1862. The new knowledge has been often and well described, but mostly by scientists writing for scientists in the technical language of science.

In the present book I have tried to describe the main outlines of such parts of science, both old and new, as are specially related to the questions and problems of music, assuming no previous knowledge either of science or of mathematics on the part of the reader. My aim has been to convey precise information in a simple non-technical way, and I hope the subject-matter I have selected may interest the amateur, as well as the serious student, of music.

I need hardly say that I am indebted to many friends and books. A considerable fraction of my book is merely Helmholtz modernised and rewritten in simple language. Another considerable fraction is drawn from the wealth of material provided in the notes added to Helmholtz's book by his English translator, A. J. Ellis. On the less technical side, I have borrowed largely from Dayton C. Miller's book The Science of Musical Sounds (The Macmillan Company, 1934), and am especially indebted to the author for permission to reproduce eleven excellent photographs of sound-curves.

We have now considered the way in which musical sounds are produced by various instruments, and the way in which the quality of those sounds depends on the proportion of the various harmonics of which they are the blend. We have further considered the combining of single sounds into chords, and the choice of a musical scale which shall give as much pleasure as possible—or, perhaps, more accurately, as little pain as possible—to the sensitive ear. But our problem does not end with the production of musical tone; it ends only with the perception of this tone by the brain. After a musical tone has been produced in the orchestra, it has to go through two further stages—transmission from the instrument to the ear-drum of the listener, and transmission from his ear-drum to his brain. We consider these two stages in this and the following chapter.

The Transmission of Sound-Waves

Imagine that we are listening to an orchestra out in interstellar space. As we have seen that some material medium is necessary to transmit the sound from the instruments to our ear-drums, let us give free rein to our fancy, and imagine interstellar space filled with ordinary air.

Sound leaves each instrument in the form of waves which spread in all directions. If there is nothing to absorb their energy, they travel on for ever, but their intensity naturally diminishes as they spread out.