Therefore D is equal to one-half of 594, or 297. We start Proportion (3) with c, of 528 vibrations, the octave above C, and we obtain the numbers of vibrations per second which give the sounds A and F. We here write in their proper order these notes of the gamut, and place under them their numbers of vibrations. The notes of the gamut are also designated as 1st, 2d, 3d, 4th, etc., so as to indicate what are called intervals. Thus the G forms to the C the interval of the 5th. The E is the interval of the 3d to C. 264 297 330 352 396 440 495 528 An examination of these numbers will show that each may be divided by eleven. Doing this, we obtain the following series of numbers, which gives the relative numbers of the vibrations for the notes of the gamut in any octave of the musical scale : C: DE: F: G: A: B: c. 24: 27: 30: 32 : 36: 40: 45: 48. EXPERIMENT 86.-Of the correctness of the above mode of forming the gamut, you may convince yourself by cutting another disk for the siren having eight instead of four circles of holes, each circle having, in order, these numbers of holes, viz.: 24, 27, 30, 32, 36, 40, 45, 48. Turning the disk, by giving to the crank a uniform motion of 22 revolutions in 10 seconds, while you successively blow into the circles, you will hear in succession the eight notes of the gamut of the octave of C, of 264 vibrations. EXPERIMENT 87.-Even the disk with four circles of holes may be made to give all the notes of the gamut, but only four notes in each experiment. You will find on making the calculation that, if you turn the handle of the rotator 22 times in 10 seconds, you will make the C of Proportion (1); 33 turns in 10 seconds will give the G of Proportion (2); while 29 turns in 10 seconds will give the F of Proportion (3). Hence, if you blow into the four circles of holes, while the disk has in succession these three different velocities, you will successively get the numbers of vibrations making the sounds of the gamut given in Proportions (1), (2), and (3). CHAPTER XII. EXPERIMENTS WITH THE SONOMETER, GIVING THE SOUNDS OF THE GAMUT AND THE HARMONICS. FIG. 46 represents a wooden box 59 inches (150 centimetres) long, 4 inches (12 centimetres) wide, and 4 inches (12 centimetres) deep. The sides are made of oak inch (12 millimetres) thick, and the two ends of oak 1 inch (25 millimetres) thick. These are carefully dovetailed together. In the side-pieces are cut three holes, as shown in the figure. There is no bottom to the box, and the top is made of a single piece of clear pine inch (3 millimetres) thick, and glued on. Two triangular pieces, inch (2 centimetres) high, and glued down to the cover of the box, just 474 inches (120 centimetres) apart, form bridges over which the wires are stretched. There is also, as shown at E, a loose piece of pine 24 inches (6.35 centimetres) wide, inch (2 centimetres) thick, and about 4 inches (12 centimetres) long. At a, b are two screweyes set firmly upright at one end of the box in the oak. At c, d are two piano-string pegs. From these to the screw-eyes are stretched two pieces of piano-forte wire (No. 14, Poehlemann's patent, Nuremberg). In putting on these wires, the ends must be annealed, by making them red-hot in a stove, before they are wound round the screw-eyes or pegs. Such an instrument is called a sonometer, and will make a useful and entertaining instrument for our experiments. When it is finished, the wires may be drawn up tight by means of a wrench or pianotuner's key, and then we shall find, on pulling the wire one side and letting it go, that it gives a clear tone that lasts some time. EXPERIMENTS WITH THE SONOMETER, GIVING THE SOUNDS OF THE GAMUT. EXPERIMENT 88.-Place the sonometer (Fig. 46) in front of you, and with a metric measure lay off distances from the left-hand bridge to the right, of 60 and 30 centimetres. Tighten the wire till it gives, when plucked, a clear musical sound, not too high in pitch. Then place the block E (Fig. 46) under the wire, with its edge on the line marked 60 centimetres, and place the end of the finger on the wire at this edge of the block. Pluck the wire at the middle of this length of 60 centimetres, and listen attentively to the pitch of the sound. Then at once remove the block and pluck the wire in its middle so that the whole wire vibrates. You will perceive that the sound now given is like the one given when the half-wire vibrated, only it differs in this, it is the octave below it. With the block placed at 30 centimetres, vibrate one-quarter of the length of the wire, and you will find that we have the sound of the first octave above that made by half the wire, and the second octave above the sound given by the whole wire. Our siren has proved that by doubling the number of vibrations the sound rises an octave. Therefore, when a wire is shortened one-half it vibrates twice as often, and when shortened one-quarter it vibrates four times as often, as when its whole length vibrates. This then is the rule, or law, which governs the vibrating wire. The force stretching the wire remaining the same, the numbers of vibrations of the wire become more frequent directly as its length is shortened. Thus, if the wire be shortened, 1, 3, or 1, the number of its vibrations per second will increase 2, 4, 3, or 9 times. EXPERIMENT 89.-Knowing this law we can readily stretch a wire on the sonometer till it gives say the C of 264 vibrations per second, and then determine the various lengths of this wire which when vibrated will give all the notes of the gamut. We have seen that the relative numbers of vibrations which give the sounds of the gamut are as follows: Notes..... C D EFGA B с Relative number of vibrations... 24 27 30 32 36 40 45 48 We have seen that, if the whole length of 120 centimetres of wire gives C, then 60 centimetres must give c of the octave above, and, as the relative numbers of vibrations of G and C are to each other as 36 is to 24, it follows that the length of the C-wire must be longer than the G-wire |