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3. Find the number of vibrations per second of a fork which produces resonance in a closed pipe 1 ft. long; in an open pipe 1 ft. long. (Take the speed of sound as 1120 ft. per second.)

4. A gunner hears an echo 51⁄2 sec. after he fires. How far away was the reflecting surface, the temperature of the air being 20° C.?

5. The shortest closed air column that gave resonance with a tuning fork was 32 cm. Find the rate of the fork if the velocity of sound was 340 meters per second.

6. A tuning fork gives strong resonance when held on its flat side or on its edge, but when held cornerwise over the air column the resonance ceases. Explain.

7. What is meant by the phenomenon of beats in sound? How may it be produced, and what is its cause?

8. What is the length of the shortest closed tube that will act as a resonator to a fork whose rate is 427 per second? (Temperature = 20° C.)

9. A fork making 500 vibrations per second is found to produce resonance in an air column like that shown in Fig. 349, first when the water is a certain distance from the top, and again when it is 34 cm. lower. Find the velocity of sound.

10. Show why an open pipe needs to be twice as long as a closed pipe if it is to respond to the same note.




398. Physical basis of musical intervals. Let a metal or cardboard disk 10 or 12 inches in diameter be provided with four concentric rows of equidistant holes, the successive rows containing respectively 24, 30, 36, and 48 holes (Fig. 357). The holes should be about inch in diameter, and the rows should be about

inch apart. Let this disk (a siren) be placed in the rotating apparatus and a constant speed imparted. Then let a jet of air be directed, as in § 385, against each row of holes in succession. It will be found that the musical sequence do, mi, sol, do' results. If the speed of rotation is increased, each note will rise in pitch, but the sequence will remain unchanged.

FIG. 357. Siren for producing musical sequence do, mi, sol, do'

We learn, therefore, that the musical sequence do, mi, sol, do' consists of notes whose vibration numbers have the ratios of 24, 30, 36, and 48, that is, 4, 5, 6, 8, and that this sequence is independent of the absolute vibration numbers of the tones.

Furthermore, when two notes an octave apart are sounded together, they form the most harmonious combination which it is possible to obtain. These characteristics of notes an octave apart were recognized in the earliest times, long before anything whatever was known about the ratio of their vibration numbers. The preceding experiment showed that this ratio is the simplest possible, namely, 24 to 48, or 1 to 2. Again, the next easiest musical interval to produce, and the next


most harmonious combination which can be found, corresponds to the two notes commonly designated as do, sol. Our experiment showed that this interval corresponds to the next simplest possible vibration ratio, namely, 24 to 36, or 2 to 3. When sol is sounded with do', the vibration ratio is seen to be 36 to 48, or 3 to 4. We see, therefore, that the three simplest possible ratios of vibration numbers, namely, 1 to 2, 2 to 3, and 3 to 4, are used in the production of the three notes do, sol, do'. Again, our experiment shows that another harmonious musical interval, do, mi, corresponds to the vibration ratio 24 to 30, or 4 to 5. We learn, therefore, that harmonious musical intervals correspond to very simple vibration ratios.

399. The major diatonic scale. When the three notes do, mi, sol, which, as seen above, have the vibration ratios 4, 5, 6, are all sounded together, they form a remarkably pleasing combination of tones. This combination was picked out and used very early in the musical development of the race. It is now known as the major chord. The major diatonic scale is built up of three major chords in the manner shown in the following table, where the first major chord is denoted by 1, the second by 2, and the third by 3.

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The chords do-mi-sol (the tonic), sol-si-re (the dominant), and fa-la-do (the subdominant) occur frequently in all music. Standard middle C forks made for physical laboratories all have the vibration number 256, which makes A in the physical scale 4263. In the so-called international pitch A has 435 vibrations, and in the widely adopted American Federation of Musicians' pitch, 440.

400. The even-tempered scale. If G is taken as do, and a scale built up as above, it will be found that six of the above notes in each octave can be used in this new key, but that two additional ones are required (see table below). Similarly, to build up scales, as above, in all the keys demanded by modern. music would require about fifty notes in each octave. Hence a compromise is made by dividing the octave into twelve equal intervals represented by the eight white and five black keys of a piano. How much this so-called even-tempered scale differs from the ideal, or diatonic, scale is shown below.


Diatonic key of G


C D E F G A B C' D' E' F G'
256 288 320 341 384
4263 480
512 576 640 682.2 768
432 480
512 576 640 720 768
256 287.4 322.7 341.7 383.8 430.7 483.5 512 574.8 645.4 683.4 767.6


401. Laws of vibrating strings. Let two piano wires be stretched over a box or a board with pulleys attached so as to form a sonometer (Fig. 358). Let the weights A and B be adjusted until the two wires emit exactly the same note. The phenomenon of beats will make it possible to do this with great accuracy. Then let the bridge D be inserted


FIG. 358. The sonometer

exactly at the middle of one of the wires, and the two wires plucked in succession. The interval will be recognized at once as do, do'. Next let the bridge be inserted so as to make one wire two thirds as long as the other, and let the two be plucked again. The interval will be recognized as do, sol.

Now it was shown in § 398 that do' has twice as many vibrations per second as do, and sol has three halves as many. Hence, since the length corresponding to do' is one half as great as the first length, and that corresponding to sol two thirds

*This discussion should be followed by a laboratory experiment on the laws of vibrating strings. See, for example, Experiment 41 of the authors' Manual.

as great, we conclude from this experiment that, other things being equal, the vibration numbers of strings are inversely proportional to their lengths.

Again, let the two wires be tuned to unison, and then let the weight A be increased until the pull which it exerts on the wire is exactly four times as great as that exerted by B. The note given out by the A wire will again be found to be an octave above that given out by the B wire.

We learn, then, that the vibration numbers of similar strings of equal length are proportional to the square roots of their tensions.

In stringed instruments, for example the piano, the different pitches are obtained by using strings of different length, tension, and mass per unit length.

402. Nodes and loops in vibrating strings. Let a string a meter long be attached to one of the prongs of a large tuning fork which makes in the neighborhood of

FIG. 359. String vibrating as a whole

100 vibrations per second. Let the other end be attached as in the figure and the fork set into vibration. If the fork is not electrically driven, which is much to be preferred, it may be bowed with a violin bow or struck with a soft mallet. By making the tension of the thread, for example, proportional to the numbers 9, 4, and 1 it will be found possible to make it vibrate either as a whole, as in Fig. 359, or in two or three parts

(Fig. 360).

FIG. 360. String vibrating in three

This effect is due, as explained in § 397, to the interference of the direct and reflected waves sent down the string from the vibrating fork. But we shall show in the next paragraph that in considering the effects of the vibrating string on the surrounding air we shall make no mistake if we think of it as clamped at each node, and as actually vibrating in two or three or four separate parts, as the case may be.

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