the table on page 381, where brackets show the various intervals which are taken as equal on the equally tempered scale.

291. Tones - Harmonics - Overtones. When a single note is sounded on many kinds of musical instruments, a practised ear can detect that, in addition to the note the frequency of which corresponds to the note、sounded, there are present notes corresponding to other frequencies, though these are very much less intense than the principal note. A note which the ear cannot break up in this manner is called a tonc. Thus musical notes are in general composed of tones, the pitch of the note being that corresponding to the lowest tone it contains.

If n is the frequency of a tone, then the tones of which the frequencies are 2n, 31, 4n, &c., are called the harmonics of the tone n, and this tone is called the fundamental.

The tones which go to build up a note are not necessarily the harmonics of the lowest tone, so, for distinction, they are called the overtones or upper partials of the fundamental.

In the case of tones the vibrations of the sounding body, as well as the waves produced in the air, are simple harmonic vibrations; and it is from this fact, first discovered by Ohm, that the name harmonic vibration is derived.

291A. The Major and Minor Chords.-Three notes which when sounded together form a consonant combination, are called a chord or triad. Three notes, of which the frequencies are as 4:5:6, constitute what is called a major chord or triad. Thus the notes C,E,G, G,B,d, and F,A,c each form a major chord. Any one of the notes in a major chord may be replaced by its octave, or accompanied by its octave, without destroying the characteristic consonant character of the combination.

If the frequencies of three notes are as 10:12:15, the consonance is not quite so complete as with the major chord, and the combination is called a minor chord. The notes E,G,B form a minor chord. As in the case of the major chord, any note may be replaced or accompanied by its octave.

The scale with which we have dealt in § 289 is called the major scale, and the following scheme shows the major and minor chords which can be formed with the tones which constitute the scale :

The sequence of notes shown in the following scheme is called the

CHAPTER IV

REFLECTION, REFRACTION, AND INTERFERENCE

292. Reflection of Sound. We have a familiar case of the reflection of sound in the echo, which is due to the reflection of the sound-waves by some large vertical surface, such as a

FIG. 234.

B

cliff or the side of a house.

The reflection of sound may also be shown by means of what is called a sensitive flame, which consists of a gas flame produced by a pin-hole burner, in which the pressure of the gas has been increased till the flame is on the point of flaring. Such a flame forms a very sensitive detector of sound-waves, particularly those of a very high pitch. The form of the flame when unaffected is shown at B (Fig. 234), while, when a sound of suitable pitch is produced, the flame flares and shortens into the shape shown at A. The sensitive flame is placed at A (Fig. 235), in front of a tube CD, and a whistle B is placed opposite the end of another tube EF, while a screen is placed at CH, so as to screen off the direct action of the whistle on the flame. The pressure of the gas is adjusted till the flame just does not flare when no reflector is placed at 1; it will then be found that, on placing a reflecting surface at I in such a way as to be equally inclined to the axes of the two tubes, the flame immediately begins to flare, and continues to do so as long as the reflecting surface remains in position.

The direction in which the sound-wave proceeds after reflection can be obtained in exactly the same way as that adopted in

$ 274; and, as there, it will be found that the angle of incidence is always equal to the angle of reflection. This fact is also proved by the experiment described above, for it is only when the normal to the

reflector I bisects the angle between the axes of the tubes that the flame flares; and when the normal bisects this angle, the angle of reflection is equal to the angle of incidence.

ments.

Use is made of the reflection of sound in several well-known instruThus in the ear-trumpet, the sound-waves that are caught by the bell-shaped mouth are reflected from the sides of the trumpet, and the cross section of the

another application of the reflection of sound. It consists of a reflecting surface placed so as to reflect those sound-waves that strike it down towards the audience. Hence the waves, that would otherwise spread up to the roof and be irregularly reflected there, are directed downwards, and assist in making the speaker audible.

In the case of speaking-tubes, the sound-waves, instead of spreading out in spheres, as they would do in the open air, are, by reflection at the sides of the tube, confined within the tube, so that they travel forward with comparatively small decrease in amplitude, the wave-front remaining of the same cross section throughout. A similar effect is produced when a watch is held against one end of a long wooden rod, and the ear is held against the other end. The ticking of the watch can be heard almost as clearly as if it were held close to the ear. The reason is that the sound-waves in the wood, when they reach the bounding surface between wood and air, are almost entirely reflected, and thus the wave proceeds down the rod without much of the energy escaping into the surrounding air.

The difficulty of hearing sounds at a distance on certain days is supposed to be due to the fact that on such acoustically opaque days there exist columns and layers of air at different temperatures, and that the sound-waves get partly reflected at each passage from air at one temperature to air at another. Such reflection must occur, for the velocity with which the sound-waves travel will be different if the temperature of

the air is different, and whenever a wave passes from one medium to another, in which it moves with a different velocity, it is partly reflected at the boundary between the two media.

293. Refraction of Sound.-When a sound-wave passes from one medium to another, the direction in which the sound-wave is travelling is in general altered, and is said to be refracted. The laws which govern the refraction of sound are the same as those in the case of light (see $ 341). In the case of sound, however, it is difficult to obtain quantitative results. The fact that sound can be refracted may, however, be shown by having a lens-shaped india-rubber bag filled with a gas other than air. If the lens is convex (§ 348) and is filled with carbon dioxide, a gas in which sound travels with a smaller velocity than in air, the sound-waves will be brought to a focus, so that by placing a whistle at one side and a sensitive flame at the other, it will be possible to arrange matters so that the flame does not flare when the lens is not interposed, but does when the lens focuses the waves on the flame. If the lens is filled with hydrogen, a gas in which sound travels more rapidly than in air, then the convex lens will act as a diverging lens (§ 348), and a somewhat similar experiment can be performed in which the flame flares without the lens, but, owing to the spreading of the sound-waves by the lens, does not do so when the lens is interposed.

Sound-waves are often refracted on account of the motion of the wind. Thus suppose we have a plane sound-wave, in which the wavefront is a vertical plane, moving against the wind. Near the surface of the earth the velocity of the wind is in general less than at some distance above the surface. Now the sound-wave will travel at the same speed through the air, but, owing to the contrary motion of the air, the distance moved through by the wave-front relative to the earth will be greater near the surface of the earth than higher up. The wave-front will therefore become inclined, the top lagging behind the bottom, and, since the motion of the wave is at right angles to the wave-front, the direction of motion of the wave, instead of being parallel to the surface, will be inclined upwards. Thus the sound-waves may pass over the head of an observer who is on the windward side of the place where the sounds originate. When the sound is travelling with the wind the opposite effect is produced, the waves being refracted downwards. This effect partly accounts for the greater distance sounds can be heard when the sound is moving with the wind, than when the sound is moving against the wind.

Another cause of the refraction of sound-waves is the unequal heating of the various strata of air. In general during the day the air near the ground is hottest. As sound travels quicker in hot than in cold air, the result is that the waves get refracted upwards.

294. Interference of Sound-Waves.-The fact that sound-waves can interfere may be easily shown, and the wave-length of the note used

measured (only roughly it is true) by means of the instrument shown in section in Fig. 236. A whistle or reed is sounded at A, and the sound-waves

the path which the sound has to traverse can be made longer in this tube than the other. If the tube is gradually pulled out, a position such as D' can be obtained, for which the waves that travel by the two paths are in opposite phase when they reach C, and hence they interfere, so that the flame is unaffected although the whistle is sounding as strongly as before. When this occurs the path AD'C traversed by one wave is longer than the path ABC traversed by the other by half a wave-length, so that when a crest reaches C via B, the preceding trough which has travelled via D' has only just reached C, and these two neutralise one another, so that the air at C is undisturbed.

M

/M'

G

If the tube D is pulled further out, the paths differ by more than half a wave-length, and hence the two sets of waves commence to strengthen one another again, till when the tube is pulled out to D", and the difference between the paths amounts to a whole wave-length, the flame is almost as strongly affected as it was at first, when the paths were of equal length. The difference between the lengths of the two tubes when

interference occurs is equal to half the wave- EA

length of the note used.

B F

The interference of sound-waves is also very clearly shown in the case of the waves produced in air by a tuning-fork. As we shall see later, the extremities of the two prongs shown in section at A and B (Fig. 237) vibrate in such a way that they move alternately away from and towards each other. As a result, while N they produce a condensation in the air between, they produce a rarefaction in the air towards E and F, and vice versa. Hence each prong starts two sets of waves, and

H

FIG. 237.

N'