Suppose a point moving back and forth along AC, in simple harmonic motion corresponding to uniform motion around the body sounding c vibrates six times, and therefore makes one sixth of a vibration while the body sounding c makes one eighth of a vibration. Lay off the circumference DEF into six equal parts, and the other circumference into eight. From the points of division draw lines perpendicular, respectively, to AB and AC, and prolong them; then their intersections will give the points for the required curve. In order that the curve connecting the points shall be smooth, intermediate points must be determined. The curve representing the combination of any other two tones can be constructed in the same way. 236. The Pendulum Method traces the curve. Demonstration. On the opposite sides of a baseboard, about 40 cm. square, fasten two uprights 102 cm. long above the upper surface of the base. Fix a crosspiece to the top of these. Bore a hole in the middle of this and fit a handle so that it will turn snugly. Make a lead disk 10 cm. in diameter and 2 cm. thick, and through the middle drill a hole 5 mm. in diameter. Suspend this by three cords as in Fig. 210, and at the point A tie these three cords to two others which run through the holes B and C in the crosspiece and H B K R then through a hole in the handle H. Wind a ring of copper wire R about the two cords, so that it can be slipped up or down, and unite the two into one, as RA. Place a glass plate on the baseboard, and sift sand upon it from a tin flour dredge. Select a glass rod or tube that will slip easily through the hole in the disk, and make one end small and rounded in a Bunsen flame. Put the rod through the disk; then draw the disk back and release it so that it will vibrate across the base in the direction DE. The disk swings as a pendulum from the points C and B, and the rod traces a straight line in the sand. Vibrate again in a direction GF, at right angles to DE. The rod will again trace a straight line, swinging from the point R. Now draw the disk aside midway between these directions, and when it is released the rod will trace a curve which will be the result of combining the two motions, and the form of which will depend upon the relative lengths of the two pendulums, i.e. of the points K and R from the middle of the lead disk. FIG. 210 The distance of K from the middle of the disk can be kept at 1 m. by turning the handle H; and by making the distance of R from the middle of the disk such that the short pendulum vibrates three times while the long one vibrates twice, the curve corresponding to the combination of the tones sol and do is obtained. If the times of vibration are as 2: 1, the curve will represent the combination of a tone and its octave. By applying the law of the pendulum for length and time of vibration, the length of the short pendulum can be easily found for most musical intervals. FIG. 211 Figure 211 shows some of the simpler figures that can be obtained if the ratios of vibrations are such as 1:2, 2:3, 3:4. If the ratios are of large numbers, the figures become more complicated. E 237. Manometric Flames. The optical method devised by König, to which he has given the name of manometric flames, consists of bringing the condensations and rarefactions of sound waves to act upon a gas flame and regulate its height, and of observing the effect in a revolving mirror. The prin H M B FIG. 212 ciple of the apparatus is shown in Fig. 212, and the complete form in Fig. 213. A wooden or metal box is divided into two chambers, A and B, by an elastic diaphragm D. FIG. 213 Two pipes open into A and one into B. The pipe C brings in gas, which is burned as a small, round flame at the top of the tube E. The pipe H opens into B and conveys the sound waves made before its open end at M. When D is struck by a condensation, it bends toward A, making that chamber smaller, increasing the pressure, and making the flame burn higher at E. When a rarefaction comes to D, the chamber A is made larger, the pressure is decreased, and the flame drops down to a shorter one. These changes follow one another so rapidly that the eye cannot detect them unless the image of each flame is separated from the others. This can be done in two ways: first, by turning the eye quickly and throwing the line of sight across the flame, when the images will be separated in the eye; and second, by the use of a revolving mirror. If the mirror is turned while the flame is burning steadily, the reflection of the flame seen in the mirror will be a plain band of light; but if a simple tion will have twice the number of points. If the tone sung is accompanied by overtones, the reflection will show a compound form in which smaller waves are impressed upon the fundamental as in Fig. 214, B. NOTE. Experiments with vibrating flames and rotating mirror will not give satisfaction unless carried on in a dark room. Sing the tones of the scale before the mouth Demonstration. piece, calling each tone O. peat with do, re, mi, etc. pound one? Notice the change for each pitch. ReCan you tell a simple tone from a com 238. Helmholtz Resonators are spherical shells, of various sizes, each having at one side a short tube to receive the sound and directly opposite a smaller tube which is held Each resonator will increase the loudness of a to the ear. tone of some particular pitch only, whether that tone is a fundamental or an overtone. These instruments were devised by Helmholtz, and by their use he discovered just which overtones are present in the sounds of various musical instruments. FIG. 215 A modified form of Helmholtz resonator is shown in Fig. 215. This consists of two tubes, one of which slides within the other so that the instrument can be adjusted to tones of varying pitches. The flexible tube is connected to the manometric flame apparatus, and by this means the character of the vibrations present is determined. By combining sounds giving all the different vibrations observed, it is possible to reproduce a sound having the same quality as the original. Demonstration. Procure a glass tube about 8 mm. in diameter and about 20 cm. long and draw it down to a small jet. Bend this tube at right angles and fasten it to a small board with a wire staple. Place this under a tripod covered with wire gauze, as shown in Fig. 216. Turn on the gas and light it above the gauze. Regulate the position of the glass tube and the pressure of the gas until you have a flickering blue flame, broad at the base and pointed at the top. Place over this a tube 5 cm. in diameter and of almost any FIG. 216 |