Another and more complete analogy to the condition existing within the tube of Fig. 360 may be had by simply vibrating one end of a two-meter or three-meter rope, as in Fig. 362. The trains of advancing and re flected waves which continuously travel through each other up and down the rope FIG. 362. Nodes and loops in a cord will unite so as to form a series of nodes and loops. The nodes at c and e are the points at which the advancing and reflected waves are always urging the cord equally in opposite directions. The distance between them is one half the wave length of the train sent down the rope by the hand. SUMMARY. The length of the shortest resonant closed pipe is one-fourth wave length. (There is also resonance at any odd multiple of this length.) The length of the shortest resonant open pipe is one-half wave length. (There is also resonance at any multiple of this length.) Musical beats are caused by the alternate reënforcement and interference of two sets of wave trains differing in wave length, the number of beats per second being the difference in the two frequencies. QUESTIONS AND PROBLEMS* 1. Why do the echoes which are prominent in empty halls often disappear when the hall is full of people? 2. Four seconds after a cannon was fired an echo of the report was heard from a distant iceberg. At what distance from the cannon was the iceberg, the temperature being 0° C.? 3. A gunner hears an echo 5 sec. after he fires. How far away was the reflecting surface, the temperature of the air being 20° C.? 4. Account for the sound produced by blowing across the mouth of an empty bottle. The bottle may be tuned to different pitches by adding more or less water. Explain. 5. Explain the roaring sound heard when a sea shell, a tumbler, or an empty tin can is held to the ear. * Supplementary questions and problems for Chapter XVI are given in the Appendix. 6. State clearly the meaning of resonance and the meaning of forced vibration, and point out the difference between them. 7. 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.) 8. The shortest closed air column that gave resonance with a tuning fork was 32 cm. Find the rate of the fork if the speed of sound was 340 m. per second. 9. 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 1130 ft. per second.) 10. What change, if any, is produced in the tone of an organ pipe by a rise in temperature? Give reason for your answer. 11. Two tuning forks, one of which has a frequency of 256 per second, emit 5 beats per second when sounded simultaneously. What are the possible rates of vibration of the other fork? 12. A standard tuning fork of frequency 256 gives 4 beats per second when sounded with another fork. When a piece of wax is attached to the standard fork the number of beats is reduced to 3 per second. What is the frequency of the other fork? CHAPTER XVII PROPERTIES OF MUSICAL SOUNDS MUSICAL SCALES 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. 363). 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 § 386, 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. 363. 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, 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. 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 shown above, it will be found that six of the notes in each octave in the table above 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. So, to compromise, the octave is divided into twelve equal intervals represented by the eight white and five black keys of a piano. How this so-called even-tempered scale differs from the ideal, or diatonic, scale is shown below. Note G C D E F C D 512 576 512 576 VIBRATING STRINGS* 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. 364). 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 CHAGFED C FIG. 364. The sonometer accuracy. Then let the bridge D be inserted 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 * This discussion should be followed by a laboratory experiment on the laws of vibrating strings. See, for example, Experiment 53 of "Exercises in Laboratory Physics," by Millikan, Gale, and Davis. |