Thus the pitch of the tonic absolutely fixes the pitch of every note, in the scale of which it is the starting-point.

Before we proceed to investigate the mechanical equivalent of the third element [§ 24] of a musical sound, its quality, it will be convenient briefly to examine a subject possessing an important bearing on that enquiry. This we shall do in the next chapter.

CHAPTER III.

ON RESONANCE.

37. When a sounding body causes another body to emit sound, we have an instance of a very remarkable phenomenon called resonance. The German term for it, 'co-vibration' (Mitschwingung), possesses the merit of at once indicating its essential meaning, namely, the setting up of vibrations in an instrument, not by a blow or other immediate action upon it, but indirectly as the result of the vibrations of another instrument. In order to produce the effect, we have only to press down very gently one of the keys of a pianoforte, so as to raise the damper, without making any sound, and then sing loudly, into the instrument, the corresponding note. When the voice ceases, the instrument will continue to sustain the note, which will then gradually fade away. If the key is allowed to rise again before the sound is extinct, it will abruptly cease. A similar experiment may be tried, as follows, on any horizontal pianoforte which allows the wires to be uncovered. Each note is, it is well

known, produced by two, or by three, wires. Having, as in the previous case, raised one of the dampers without striking the note, twitch one of the corresponding wires sharply with the finger-nail, and then wait a few seconds. The vibrations will, in this interval, have communicated themselves to the other string, or strings, belonging to the note pressed down: if, now, the first wire be stopped by applying the tip of the finger to the point where it was at first twitched, the same note, produced by these transmitted vibrations, will continue to be sustained by the remaining wire or wires.

A more instructive method of studying resonance is to take two unison tuning-forks, strike one of them, and hold it near the other, but without touching it. The second fork will then commence sounding by resonance, and will continue to produce its note though the first fork be brought to silence. It is essential to the success of this experiment that the two forks should be rigorously in unison. If the pitch of one of them be lowered by causing a small pellet of wax to adhere to the end of one of its prongs, the effect of resonance will no longer be produced, even though the alteration of pitch be too small to be recognized by the ear. Further, the phenomenon requires a certain appreciable length of time to develope itself; for, if the silent fork be only momentarily ex

posed to the influence of its vocal fellow, no result ensues. The resonance, when produced, is at first extremely feeble, and gradually increases in intensity under the continued action of the originally-excited fork. Some seconds must elapse before the maximumresonance is attained. The conditions of our experiment show, directly, that the resonance of the second fork was due to the transmission, by the air, of the vibrations of the first, the successive air-impulses falling in such a manner on the fork as to produce a cumulative effect. If we bear in mind the disproportionate mass of the body set in motion compared to that of the air acting upon it,-steel being more than six thousand times as heavy as atmospheric air, for equal bulks,—we cannot fail to regard this as a very surprising fact.

Let us examine the mechanical causes to which it is due. Suppose a heavy weight to be suspended from a fixed support by a flexible string, so as to form a pendulum of the simplest kind. In order to cause it to perform oscillations of considerable extent by the application of a number of small impulses, we proceed as follows. As soon as, by the first impulse, the weight has been set vibrating through a small distance, we take care that every succeeding impulse is impressed in the direction in which the weight is moving at the time. Each impulse, thus applied, will

cause the pendulum to oscillate through a larger angle than before, and, the effects of many impulses being in this way added together, an extensive swing of the pendulum is the result.

When the distance through which the weight travels to and fro, though in itself considerable, is small compared to the length of the supporting string, the time of oscillation is the same for any extent of swing within this limit, and depends only on the length of the string. My readers will find this important principle illustrated in any Manual of Elementary Mechanics, and I must ask them to take it for granted here. For the sake of simplicity, let us suppose that we are dealing with a second's pendulum, i.e. one of such a length as to perform one complete oscillation in each second, and therefore to make a single forward or backward swing in each half second, It will be clear, from what has been said above, that the most rapid effect will be produced on the motion of the pendulum, by applying a forward and a backward impulse respectively during each alternate half second, or, which is the same thing, administering a pair of to and fro impulses during each complete oscillation of the pendulum. We have a simple instance of such a proceeding in the way in which a couple of boys set a heavily laden swing in violent motion. They stand facing each other, and each boy,