60. Sound is caused by the vibrations of sounding bodies, which communicate similar vibrations to the air, which last affect our organs of hearing. When the vibrations of the sounding body. and consequently those of the air, are of vnilarat velocity. that is, when the numbers of vibrations in equal times are equal, our organs of hearing are umform/y affected, which we express by saying that the sound conti nues at the sante pitch. If by any cause the velocity of the vibrations be increased, the sound is perceived to be what we call more acute or sharp; and if by any cause the ve locity of the vibrations be retarded, the sound is perceived to be what we call graver or lower. -When the accelerations or retardations arc gradual, so that the alterations of acute ness or gravity take place by insensible degrees, the sound is continuous, and is not properly musical; (such for exam ple, as is produced by sliding the finger along the string of a violin, while it is acted on by thc bow.) When, on the other hand, the pitch is the same for some ascertainabk time, and the alterations of acuteness or gravity take place by certain ascertained degrees or intervals, the sounds are musical; and one principal object of musical science is to ascertain those relations in the pitcli of different sounds, which are proper for producing music.
That we may be able to reason concerning the diffe rent relations of musical sounds, it is necessary that w-e possessed of some unequivocal method of delermining or de fining them; and the most convenient is found to be the division of a musical string. There are three circumstances which determine the number of vibrations made by a string in a given time, or, in other words, the pitch of its S01111C1, N iZ. the length of the vibrating string, the weight of a given portion of it, and the force with which it is stretched. By supposing these two la.st circumstances always the same, the pitch of musical sounds can be very conveniently expressed by the length of the strings which respectively produce them. It has been demonstrated that the numbers of vibrations are inversely proportional to the lengths; consequently unisons are of the same length: the string of the more acute sound is shorter, and that of the graver sound longer.
61. We shall here show the relative proportion of the strings, which give the simplest scale of musical sounds. When the sound of a string is compared with that of one half its leng-th, the latter is octave higher than the former; so called because such is the relation of the eighth sound of the scale. The relation between a note and its octave is, next after that of unison, the simplest and most perfect in nature; and when the two notes are sounded at the same time, they almost entirely unite. Indeed so striking is the
resemblance of the one sound to the other, that it often re quires care and attention to distinguish them from unisons. It is for this reason, that by whatever name or letter any sound in a scale is denoted, its octave is denoted by the same name or letter; and should we divide the string which gave us the octave to the former, again by 2, so as to obtain its octave, this new sound, whose string is of the first, is called its double octave, and is still denoted by the sante letter ; and so of -A-, or the triple octave, &c. In like man ner, should we take a string double in length of the first, it would give a sound octave below it, that again doubled, its double octave below, and so its triple octave below, &c. And all these sounds, given by strings in the proportion of 8, 4, 2, 1, &c. are denotedly the same letter, and are considered as repetitions of eaMother. Hence, when we fill up one octave with its proper intermediate sounds, we describe the whole system of musical intervals. It is plain from this statement, that if we should use no other proportions than that of equality, and those obtained by bisection, we could have no other musical intervals than uni sons, octaves, double octaves, &c. But if we take of the primary string., we obtain that sound which is fifth in the scale, and if we take of the primary strin7, we have the sound which is third in the scale. By combining these ra tios, the whole system of musical relations is obtained.
62. Thus, if we take a fifth to the fifth of the scale, we take of that is t of the primary string. As 4 is less than ?_;., it is plain that this sound is more acute than the oc tave; but by doubling it, that is, taking its octave below, we have 4 of. the primary string, and as this is greater than 1, is an Intermediate sound between that of the primary string, and the third of the scale. It is the second of the scale. If we take a stt ing, such that the primary string shall be fifth to it, its length will be expressed bv the improper frac tion q, and taking its half in order to have its octave, we have 3I of the printar , which it is plain will be an interme diate sound between the third and fifth ; it is the fourth of the scale. If we take a fourth to the third of the scale, or a third to the fourth of the scale, it AS ill be 31 xl=.1.4=; this is the sixth of the scale, sume%s hat more aeute than the fifth. Lastly, if we take a fifth to the third, or a third to the fifth, it nil] be xst-=-18,, which is the seventh of the scale, some what graver than the °eta.% e. See ACOUSTICS, APOTOME, CO.II31A, &C. &C. FAREY'S NOtati0I1, INTF:RVAL, nu.vr tox.