MUSICAL INTERVALS-THE GAMUT. 101 commensurable with those numbers, or with each other; so that all notes thus chosen must be discordant. But the required object is best attained by inserting between a note and its octave, or double vibration, six other notes which are respectively due to 1, 14, 13, 11, 13, and 111⁄2 more vibrations per second than the lower note, which is called c, or Do, while the others represent the remaining notes of the gamut, namely, Hence, supposing that whatever be the number of vibrations per second necessary to produce the note c, we agree to represent it by unity or 1, then the numbers necessary to produce the other seven notes of the octave above will be as follows: D = 1, E = {, F = 3, G = 3, A = 5, B, C = 2. It will be observed, that the seven intervals or tones are by no means equal, for these numbers do not form a geometrical progression. Thus the ratio between c and D is that of 8: 9, while that between D and E is rather less, namely, as 9 : 10; the next, or that between E and F is only as 15: 16; the next, or that between F and G, as 8 : 9 ; that between G and A, as 9:10; that between A and B, as 8 :9; and the last, or that between в and c, is only as 15: 16. Thus we see that the intervals between E and F, and between B and c hardly exceed the half of each of the other intervals. Hence the reason that these two intervals do not, like the others, admit of subdivision into semi tones. 68. A very remarkable proof of the vibratory nature of sound is heard when two notes very nearly, but not quite unisonant, are sounded together. A periodical interruption of the sound called a bent, occurs at intervals which are longer, the nearer the two notes approach to perfect identity, and may often be as long as half a second or more. To understand this, we must remember that each pulse or vibration of air consists of two contrary motions to and fro. Now, if one source of sound tend to produce the forward motion exactly when the other source would excite a backward motion, and vice versâ, the two, if equal, will annihilate each other, and two sounds will produce silence. But with two notes, very slightly differing in pitch, this must occur at certain regular periods, between which the contrary effect takes place, and the two vibrations coincide so as to reinforce each other. The effect is precisely similar to that which is seen when two very regular sets of parallel lines, one having rather wider spaces than the other, are superposed. Thus, two iron gratings or railings, Fig. 24, which, Fig. 24. though really of equal intervals, are made to appear slightly different from the effect of perspective, produce the appearance of broad beats, or alternations, in one of which the bars coincide and conceal each other, while in the next they fall into each other's intervals. Or to take a closer analogy from the sense of hearing instead of that of sight: if we listen to a train drawn by two locomotives whose driving wheels differ slightly in size, their beats are heard distinctly for a few seconds, they are then lost in confusion for a short time ; they are again heard distinctly, and again blended PROPAGATION OF SOUND. 103 together alternately. Now, suppose any vibrating body to strike the air 100 times in a second, and another 104 times in the same space, the latter will produce a sound about 3rd of a tone sharper than the former; the waves of sound will coincide and reinforce each other 4 times every second, and will oppose, and, if equal, will destroy each other 4 times in the same period, thus producing an audible beat about as rapid as that of a watch. 69. In our introductory treatise on Natural Philosophy, we have shown the method by which the velocity of sound in air has been ascertained. At the temperature of 62° sound travels at the rate of 1125 feet per second. It has been stated (61) that, when a given note is sounded in air, the sound is propagated in waves similar in character to those which may be so beautifully studied when the wind is blowing over a field of standing corn. (See Fig. 25.) Now, when it is said that sound travels at the rate of 1125 feet per second, it is not meant that the particles of air move through that distance any more than the ears of corn travel from one end of the field to the other; it is only the form of the wave which so travels. It is the same with the particles of air; their individual movement is confined within narrow limits, but the effect of this movement is propagated from particle to particle with the rapidity of 1125 feet per second. As soon as the particles first disturbed have moved to such a distance as their elasticity will permit, they return to their former position, and acquire in so doing a momentum sufficient to carry them a certain distance in the opposite direction, and by this means an oscillating or vibrating movement is established. Each particle is disturbed a little later than the one preceding it, and thus the particles are in different states of motion, some moving onwards while others are moving backwards, the two sets being separated by particles at rest, or in the act of turning from the completion of one movement to commence the next. Now these turning sets of particles are alternately more condensed and rarefied than in their natural condition. (See Fig. 20.) Those which have just commenced their backward motion are rarefied as the ears of corn at RR R, Fig. 25, and those which are beginning their Fig. 25. R forward motion are condensed as at c c c. This has given rise to the term wave of sound, a wave being understood to include particles in all the various states of vibration, each wave being exactly similar to all the others for any given note. The length of a wave is the distance between any two particles which are in precisely the same stage of vibration. It matters not whether we measure it from the most forward to the most forward set of particles, or from the most backward to the most backward set; or from one place of greatest rarefaction to the next, or from one place of greatest condensation to the next; just as it is a matter of indifference, in estimating the length of a wave of water, whether it be taken from one elevation to the next, or from one depression to the next. Waves of water do not all travel with the same speed, their speed being proportional to the square root of their length, so that, the slower or less frequent the oscillation, the faster does it travel; those of the ocean, for instance, travel faster than those of the English Channel, and these than the waves of a river. LENGTH OF WAVES OF SOUND. 105 Waves of air, however, whatever their frequency, all travel with the uniform speed of about 1100 feet in a second. Hence, when we know their frequency, or how many of them arrive at and pass a fixed point in a given time (such as a second), we may at once find their length, which is 1100 feet, divided by their number per second. Thus, the c above the tenor c is produced by 512 vibrations per second; 512 waves fall on the ear in that time, during which each would travel 1100 feet, so that this distance contains 512 waves, each of which must accordingly occupy about 2 feet 1 inch. Consequently, every time this note is sounded on any instrument, the air which conveys this note to our ears is thrown into waves, each of which measures about 2 feet. All the innumerable particles of air between us and the instrument form a series of little pendulums, the amplitude of whose oscillations depends on the loudness of the sound, and is, in all cases, very minute; but the distance from each particle to the next, which is in precisely the same part of its oscillation as from c to c or R to R, Fig. 25, depends entirely on the pitch, and is, in this case, about 2 feet. In the same way we find the waves of the gravest note to be about 64 feet long, and those of very shrill sounds to be less than an inch*. * A very curious effect of the rapid motion of the observer on sound has just been brought before the notice of the British Association by Mr. Scott Russell. He found that the sound of a whistle on an engine, stationary on a railway, was heard by a passenger travelling in a train in rapid motion to give a different note, in a different key from that in which it was heard by the person standing beside it. The same remark applies to all sounds. The passenger in rapid motion heard them in a different key, which might be either louder or lower in pitch than the true or stationary sound. Now, as the pitch of a musical sound is determined by the number of vibrations which reach the ear in a second of time, if an observer in a railway train move at the rate of 56 miles an hour towards a sounding body, he will meet a greater number of undulations in a |