We may, therefore, assume that if the separate fibres of the cochlea were to be irritated in any other manner than by sound, as, for instance, by mechanical or electric irritation, their several appropriate tones would be heard. In fact, tones and noises are heard when an electric current is conducted through the head, which are produced by the irritation of the entire auditory nerve. It has been observed, further, that persons suffering from an affection of the ear often experience a constant subjective sensation of a certain tone in the ear, which has been explained as arising from one of the nerve-fibres in the organs of Corti being irritated from some cause due to disease. Indeed, in some diseases, a deafness to a certain series of tones has been noticed which, in such cases, is due to the destruction of certain of the organs of Corti.

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CHAPTER VII.

Notes or Compound Tones-The production of Harmonics-The perception of the latter by means of Resonators-Graphical representation of Quality or Colour.

IT is a fact well known from experience, that the tones of different instruments and of the human voice are distinguished from each other by their note or quality. If we sing a tone of a certain pitch, for instance a', strike the same note upon the piano, or if it is sounded upon a violin, a flute, trumpet, or organ, we shall always obtain a tone of the same pitch, making 440 vibrations in a second, and yet these tones differ very widely from each other in their quality.

Into the cause of this we have now to enquire. We must return to the process of vibration to find the principle of this fact. Now, since the tones in the abovementioned instruments, when they possess the same pitch, invariably possess the same number of vibrations, it is impossible for the number of vibrations to decide. the quality of the tone, and we are forced to seek the cause of the difference in quality in some other property of the several vibrations.

Now it is a fact which has long been known to musicians, that a certain number of higher tones are produced

with every tone which is sounded upon our instruments. In acoustics these higher tones have been called the Harmonics of the given fundamental tone. If, for instance, the note c is struck upon the piano, an ear musically trained will recognise in this note the c above it. This may be still better accomplished by means of apparatus, of which we shall presently speak more particularly.

The production of these harmonics, which has been investigated by Helmholtz in a very ingenious manner, may be most easily understood from a vibrating string. Let us suppose a stretched string to be thrown into vibration, we shall observe that the essential part of the action consists in the string bending first in one direction and then in the other. The action, however, is in reality not quite so simple as this, but other movements take part in it also. A vibrating string has always a strong inclination to divide into two parts, each of which will perform its own vibrations, as shown in fig. 80 b. Thus we can imagine that, whilst the whole string is vibrating, each half is at the same time vibrating independently; and, therefore, in addition to the fundamental tone, a softer tone, the first harmonic, is produced. This must, of course, be the octave of the fundamental tone, since it is produced by the vibrations of the half of the string, the number of which must have been exactly double that of the fundamental note. Therefore, in the same time that the whole string completes one vibration, the two halves will have completed two.

This action is even still further complicated. Not only is the string divided into two halves, but it is at the same time further divided into three equal parts

though in a less degree, as shown in fig. 80 c. Each of these three parts, again, vibrates independently, with three times the velocity of the whole string, and by this vibration the second harmonic is produced. If we now imagine these three different vibrations to be combined, we shall have, for every point of the string, a very complicated motion, which, however, we can compose from the separate motions.

But the action has not even reached its limits with the division into three parts, being still further divided into four, five, six, and more parts, and as each of these several parts,,, etc., of the entire length, performs its own vibrations, an entire series of harmonics is produced in accordance with a fixed law. They become weaker and less perceptible as their pitch increases, so that the fundamental tone predominates; but still they give to the fundamental tone a peculiar character: its quality or colour.

We may easily convince ourselves of the existence of the harmonics in a string, and also determine their pitch, by means of a few experiments. For this purpose we may very well make use of the string of a monochord, but the string of a piano or of a stringed instrument will serve equally well. In the first place we must measure the exact half of the string, which we will suppose to give the tone, c. This we shall do most easily by pressing the finger upon the string, and moving it about till the half of the string, when struck, gives the octave, c'. Now hold the left forefinger a little distance above the centre, draw the string aside to some distance forcibly with the right hand, and immediately touch the string lightly with the left forefinger. By this means the fundamental

tone, and also a series of harmonics, will be damped, and the first harmonic alone will sound distinctly, because the vibrations of the two halves have not been disturbed. This tone, however, must have existed before the string was touched.

In the same manner we may make the higher harmonics sound distinctly, by laying the finger which acts as the damper upon,, etc., of the string. The higher they are, the weaker they become.

It is the fundamental tone which determines the pitch of all harmonics. For whilst the fundamental tone makes one vibration, the first harmonic makes two, the second three, the third four, the fourth five, and so on. Thus if the fundamental tone is c, the series of harmonics will be: c' g' c" e" g" b"" c'", for while c makes one vibration, c' performs two, g' three, c' four, e" five, g'"' six, b" seven, c'"' eight.

That it is possible for a string to vibrate in such a manner that the two halves, or each third and fourth, will vibrate independently, may be demonstrated by a singular experiment, which every violin player knows from practice. If the finger is laid lightly upon the centre of the string, without pressing it, and the string is then struck, the octave will be heard, with a very delicate quality, called flageolet. If a third part of the string is measured off by the touching finger, the fifth of the octave is heard. In this case, not only does the third part of the string vibrate when struck, but each other third vibrates independently, as may be seen in fig. 80 c. Now between each of these two thirds there is a point which is at rest; this may easily be proved in the following manner. If a small piece of paper is placed