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length from each end. The wave length in this case is evidently one-third of that in the former, and the frequency of vibration three times as great.

Rods of other metals or of wood or glass may be caused to vibrate in this way and the velocities of sound in them may be compared by their frequencies of vibration as shown by the tones which they give out.

343. Longitudinal Vibration of Wires.-Longitudinal vibrations may also be set up in wires firmly clamped at both ends by

rubbing them lengthwise with a bit of rosined cloth.

FIG. 196.-Transverse vibration of rod.

The clamped ends of the wire are nodes in this case and the middle is a loop. The pitch depends only on the velocity of sound along the wire and on its length and is quite independent of its tension except in so far as the tension affects the elasticity of the wire.

344. Transverse Vibrations of Bars.-The transverse vibrations of bars are determined by their mass and stiffness, and

FIG. 197.

-N

hence depend on Young's modulus of elasticity, since it is this coefficient which determines a bar's resistance to bending. If a uniform free bar is struck at the middle point it tends to vibrate as shown in the figure, with a node near each end, and it may be supported at the nodes on wooden bridges without materially affecting its vibration.

In the xylophone or kaleidophone the bars of wood or metal vibrate transversely and are supported at their nodes.

345. Tuning-forks.-A tuning-fork may be considered a bent bar vibrating in the mode shown in figure 197. For there are two nodal points, one on each leg of the fork near the bottom. The prongs swing alternately toward and away from each other, while the stem of the fork, being attached to the vibrating segment between the nodes, vibrates up and down. This is made apparent by the loud tone given out when the stem of a vibrating fork is touched to a wooden table top or sounding board.

Tuning-forks are often mounted on wooden resonators, boxes enclosing an air chamber capable of responding to the vibrations of the fork.

346. Law of Similar Systems.-When two vibrating systems are made of the same material and are exactly similar in dimensions, though not of the same size, their periods of vibration are proportional to their linear dimensions. This law is shown by mathematical reasoning to be a consequence of mechanical principles, and is illustrated in many familiar instances.

For example, if two stopped organ pipes are constructed with cubical resonating chambers, but one having half the dimensions of the other, the smaller will vibrate with twice the frequency of the larger. Two tuning-forks of equally stiff steel and exactly similar in shape will be an octave apart if one is twice as large as the other. And so, also, if we take two straight steel bars, one of which has half the dimensions of the other in each direction, the smaller will make twice as many vibrations per second as the larger when vibrating in the same

manner.

347. Vibration of Plates.-The vibration of flat plates of various shapes was studied by Chladni who scattered sand on

FIG. 198.-Chladni's figures.

the plates and observed the figures formed by the nodal lines in which the sand gathered when the plates were bowed. Some of these forms known as Chladni's figures are shown in figure 198. The upper row shows different modes of vibration that may be set up in a square plate supported at its centre and bowed at some point on the edge. The slowest mode of vibration is the first, in which the vibrating segments are the four corners. Segments separated by a nodal line must always be opposite in phase, one vibrating up, while the other swings down. This opposition of phase is indicated by marking them alternately plus and minus.

If a resonator or wide-mouthed bottle which can respond to the vibrations of the plate is held with its mouth over any vibrating segment it will respond strongly, but if moved over a nodal line so that it is simultaneously acted on by two adjoining segments it is silent because the segments are in opposite phases.

So also when the plate is vibrating as shown in the first or second diagram in figure 198, if the hands are held just above two similarly vibrating segments so as to quench the sound waves coming off from them, the sound from the plate will be heard louder than before.

348. Bells.-The blow of its tongue on a bell causes the circular rim to spring out into slightly elliptical shape, from which it springs back passing through the

circular form into an ellipse with its greater axis at right angles to the first; thus it oscillates in four segments with four intermediate nodes as shown in the figure. Making the rim of the bell thicker causes it to oscillate more quickly by reason of its increased stiffness and thus raises its pitch.

FIG. 199.-Vibration of bell.

The above is its fundamental or slowest mode of vibration, but simultaneously with this the blow of the hammer sets up higher modes of vibration in which the rim may vibrate in 6, 8, or 10 segments with intermediate nodes. These higher tones are not in the harmonic series of the fundamental and hence the tones of bells are unsuitable for music. When the bell is first struck the higher tones are more prominent than the fundamental, but as the tone dies away the fundamental tone persists the longest.

The beating or throbbing heard as the tone of a bell dies away is due to want of uniformity in the rim, in consequence of which there are two fundamental tones of slightly different pitch. One or the other of these is excited according to the point struck by the hammer, though in general both are simultaneously set up.

MUSICAL RELATIONS OF PITCH.

349. Musical Intervals Depend on Ratios.-The musical effect of two tones when sounded together depends upon the ratio of their frequencies. This is well shown by means of the

siren (§304). If four rows of holes in the siren are simultaneously used, in which the numbers of holes are proportional to 4, 5, 6, and 8, respectively, a combination of tones will be produced which will be recognized as the major chord,-do mi sol do. And this musical relationship holds whatever may be the speed of the siren, showing that whatever the pitch may be it is the ratio of the frequencies of two tones which determines their musical relationship.

350. Harmonious

Ratios.-Tones are harmonious whose frequencies are proportional to any two of the simple numbers 1, 2, 3, 4, 5, 6. The most important harmonious ratios and their musical names are here given:

1:1 unison

12 octave

13 twelfth

2:3 fifth

34 fourth

45 major third

56 minor third

The names are derived from the ordinary musical scale; thus the octave is the relation of the first and eighth tones of the scale; the fifth, that of the first and fifth; the fourth, that of the first and fourth, etc.

351. Major Scale.-Three tones whose frequencies are in the ratio 4 5 6 form what is known as a major triad.

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The major scale is a sequence of tones so related that the first, third, and fifth tones form a major triad; also the fourth, sixth, and eighth, and the fifth, seventh, and ninth. The first note. of the sequence is called the key-note and the triad starting with. the key-note is the triad of the tonic. The fifth tone is known as the dominant and the fourth as the subdominant, and their triads are, respectively, known as the triads of the dominant and of the subdominant.

If the tones of the scale are represented by letters as in ordinary musical notation, their ratios for the key of C will be as follows:

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352. Tones and Half Tones.-If the ratio of the vibration frequency of each tone to that of the one immediately preceding

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These ratios determine the musical character of the intervals. When the ratio of the frequencies of two tones is or, they are said to differ by a whole tone, while those whose ratio is 19 are said to be a half tone apart.

353. Minor Triad.—In the major triad, three tones whose ratios are 4:56, the interval between the first and second tone is a major third, while that between the second and third is a minor third. If we had three tones in the ratio 10 12 15, the interval between the first and second would be a minor third (56) while the interval between the second and third would be a major third (4:5). Such a combination of tones is known as a minor triad.

354. Minor Scale.-A scale based on minor triads in the same way that the major scale is based on major triads is known as the minor scale. In the key of C the tones C, D, F, G are the same on both scales, while E, A, B each differs from the corresponding note of the major scale by the interval, the minor tone being lower in each case. These tones of the minor scale may be designated E flat, A flat, B flat.

355. Temperament.-Since there are two kinds of wholetone intervals ( and 10) and also two kinds of half-tone intervals (1 and 2), and since a note a half tone higher than D, for example, which is called D sharp, would not be the same as E

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