inward impulse must be communicated to the free end of the spring during the time which elapses while a pulse traverses four times the length of the pipe. Reverting to the actual conditions of our problem, we have the resonance of the air-column in place of the alternate lengthening and shortening of the spring. The to-and-fro impulses at A are impressed by a vibrating fork. The Sound-pulse traverses four times the length of the pipe while the fork is performing one complete vibration. We know, however [§ 15 p. 32], that during this latter period the Sound-pulse produced by the fork's action traverses precisely one wave-length corresponding to the pitch of the note produced by the fork. Hence, for maximum resonance in the case of a pipe closed at one end, the wave-length corresponding to the note sounded must be four times as great as the length of the air-column, or the length of the column one quarter of the wave-length.

41. These principles give us the explanation of a valuable appliance for intensifying the sound of a tuning-fork. Such a fork, when held in the hand after being struck, communicates but little of its vibrations to the surrounding air; when, however, its handle is screwed into one side of a wooden box of suitable dimensions, in the way shown in Fig. 24, the tone becomes much louder. The vibrations of

the fork pass from its handle to the wood of the box,

Fig.24

and thence to the air-column within, which is of appropriate length for maximum resonance to the fork's note. This convenient adjunct to a tuningfork goes by the name of a 'resonance-box.'

42. When a number of musical sounds are simultaneously sustained it is generally difficult, and often impossible, for the unaided ear to decide whether an individual note is, or is not, present in the mass of sound heard. If, however, we had an instrument which intensified the note of which we were in

search, without similarly reinforcing others which there was any risk of our mistaking for it, our power

of recognising the note in question would be proportionately increased. Such an instrument has been invented by Helmholtz. It consists of a hollow ball of brass with two apertures at opposite ends of a diameter, as shown in Fig. 25.

The larger aperture allows the vibrations of the external air to be communicated to that within the ball; the smaller aperture terminates in a nipple of convenient form for insertion in the ear of the observer. The air contained in the ball resounds very powerfully to a single note of definite pitch, whence the instrument has been named by its inventor a resonator. The best way of using it is, first to stop one ear closely, and then to insert the nipple of the instrument into the other. As often as the resonator's own note is sounded in the external air, the instrument will sing it into the ear of the observer with extraordinary emphasis, and thus at once enable him to single out that note from among a crowd of others differing from it in pitch. A series of such resonators, tuned to particular previously selected notes, constitutes an invaluable apparatus for analysing a composite sound into the simple tones of which it is made up.

CHAPTER IV.

ON QUALITY.

43. THE laws of resonance enable us to establish a remarkable, and by most persons utterly unsuspected fact, viz. that the notes of nearly every regular musical instrument with which we are familiar, are not, as they are ordinarily taken to be, single tones of one determinate pitch, but composite sounds containing an assemblage of such tones. These are always members of a regular series, forming with each other fixed intervals which may be thus stated: if we number the separate single tones of which any given sound is made up, 1, 2, 3, &c., beginning with the lowest, we have

(1) The deepest, or 'fundamental,' tone, whose

pitch is ordinarily regarded as that of the whole sound.

(2) A tone one Octave above (1).

(3) A tone a Fifth above (2), i.e. a Twelfth above (1).

(4) A tone a Fourth above (3), i.e. two Octaves above (1).

(5) A tone a Major Third above (4), i.e. two

Octaves and a Major Third above (1).

(6) A tone a Minor Third above (5), i.e. two Octaves and a Fifth above (1).

These are the most important members of the series. Their vibration-numbers are connected by a simple law, which is easily deduced from the above relations. If the fundamental tone makes 100 vibrations per second, (2) will make twice as many, i.e. 200; (3), being a Fifth above (2), will have for its vibration-number

3

2

× 200, or 300. For (4),

which is a Fourth above (3), we get similarly

4

3

6

5

5

× 300, or 400; for (5), × 400, or 500; for (6),

4

× 500, or 600. Thus the numbers come out 100,

200, 300 and so on; or generally, whatever be the vibration-number of (1), those of (2), (3), (4), &c., are respectively twice, three times, four times, &c. as large. If C in the bass clef be selected as the fundamental tone, the series, complete up to the tenth tone, is shown in musical notation as follows: