The object of music being to please the ear, or the mind through the ear, there is no test of excellence nor criterion of fitness, in any one detail, except the opinion of the best judges. This seems to assume the question, for the best judges can only be described as those who beet know what is good music. This circle cannot be avoided, either in speaking of music or any other of the fine arts; to taste we must appeal, bet not to the ta.sto of every ono. All we have here to do with this is to remark, that the mathematical considerations employed in an article like the present are not to be considered as placing the musical scale upon a mathematical basis, but simply as showing that there is something like an explanation of those rules which derive their authority, not from the mathematical system which embodies them, but from the sanction of the majority of cultivated ears. These things which are agreeable in practice are found to be in certain mathema tical relations to one another which make the theory of the musical male sample and interesting : but had it been otherwise, we should have left mathematical simplicity, and preferred a more pleasing complexity.
The sounds which are agreeable to the ear are found to be those which are the consequence of vibrations of equal duration following one another : and the pitch of the note depends on the rapidity of vibration only. [Aeoeseics.] The note called A, for instance, sounded at the same time on a harp, a flute, and a honi,presents three different characters, three different intensities, but only one species of vibration as to the time of lasting. If the first instrument communicate 430 vibrations in a second to the air, so does the second inetrunicut, and also the third. With the difference of intensity or loudness, and with the difference of character, tho twang of the harp, or the tone of the horn, we have nothing to do in considering the place of the noto they sound in the scale; a cultivated car discovers that they sound the same note, and a mathematician knowa that they severally conunimi cate to the air the same number of vibrations per second.
Let us then suppose a string to be mounted, and stretched at both ends, or, better still perhaps, suspended vertically • by one cud, and bearing a weight at the other. If this string be then test in vibration by the finger or by the bow of a violin, a musical (that is, a ploasaut) sound is produced, if the string be not too long, nor stretched by too small a weight. With the phenomena of vibration, as connected with the length, material, and stretching weight of the string, we have here nothing to do [Come] except to remark,—1, That the ear observes that, material and tension remaining the same, the longer the oaring the lower the tone, and eke rersd. 2, That the mathematician knows that, arteries paribus, the longer the string the fewer the number of vibrations in a given time, in inverse proportion to the length. Thus if a certain string, stretched by a certain weight, give 100 vibrations per second, a string of half the length, stretched by the same weight, will give 200 vibrations per second. If a vibration mean a double motion of the string, once backwards and once forwards, the effects begin to be musical soon after the string is short enough, or stretched enough, to give 30 vibrations per second.
The number of musical tones is, theoretically, infinite : that is, between any two tones as many different tones as we please can be interposed, no one of which is so high as the higher, nor so low as the lower. Highness and lowness of tone are terms which are purely relative, and refer to an effect upon the ear which does not admit of definition ; common terms usually distinguish only extreme cases; thus, a tone disagreeably high is a squeak, and one disagreeably low is a growl. There is no absolute reason why we should all the former
high and the latter low, rather than the contrary ; and in fact the earlier Greeks (naming them after the parts of the throat in which they thought they were produced) called the squeaking sounds low, and the growling ones high. But while we endeavour to separate names from things, we must not forget that there is much which all men acknowledge of real connexion between the associations which accompany sounds and those derived from other sensible phenomena. For instance, it would be impossible to persuade any one, that if light and darkness were to be imitated by musical tones, the light ought to be represented by low notes, and the darkness by high notes : and a composer who should accompany words expressive of transition from darkness to light by a marked descent from the higher part of the scale to the lower, would be thought to mean irony or burlesque. No satisfactory explanation has ever come to our knowledge as to what associations are awakened by the lower notes of the scale which con nect them with darkness; but that this connexion does exist is certain.
Taking such a string or mouochord (single string) as above described, it is immediately found that any alteration of its length produces seine alteration of the tone. If the change be very slight, a dull or unpractised ear may not readily perceive it ; but let the alteration lie carried a little further, and there can be no difficulty. Such tunes, near to ono another, when sounded together, have a disagreeable and jarring effect, accompanied by beats [Acousries]; but when the second string has been considerably shortened (say that this is done gradually), the disagreeable effect ceases almost at once, and at the moment when the shortened string is to the longer one as five to six. Two sounds are then heard which harmonise together, and on their joint effect the ear dwells with pleasure until it becomes monotonous this very common word is itself derived, as to its common significa tion there used, from the wearying effect of the same tone, or set of tones, long continued). In the mean while, and during the shortening of the string, the joint effect, theugli always disagreeable, is not equally eo throughout ; and there is one place iu particular where the effect, though not agreeable to the beginner, is bearable for a little while, and highly agreeable to the practised ear, which knows that fall compensation is at hand in what is called the resolution of the discord, or transition to a more harmonious combination in a manner which seems peculiarly natural. This intermediate and more tolerable phase of sound takes place when the shortened string is to the other as eight to nine. Moreover, it may be observed that this last com bination, hardly bearable, is rendered perfectly so if the two tones. instead of being eounded together, are made to follow each other in succession, no matter how rapidly. In both these cases the student will observe that the proportion of the lengths of the strings is that of some small numbers, five to six, and eight to nine. And it is matter of experiment, that the more simple the proportions of the lengths of two strings (stretched by the same weight), the more useful the com bination in music—it is usual to say, the inure agreeable sounded by itself ; but to this we cannot subscribe, as we believe that to most cars the more complicated combination of a third (presently to be described) is more agreeable than the less complicated one of a fifth.