In cords composed of the same material, and of uniform thickness, the time of a plete musical vibration, or double oscillation, is where / is the length of the cord, p its weight, P the force with which it is stretched, and feet.
In order to apply this formula to the vocal ligaments, let a be their depth, b their breadth, and d their specific gravity ; then p will be equal to and equation (1.) becomes i=21/ "be - - - - (2.) 2g1) and the number of such vibrations in 1" will be / 2gP 21Nla63 ) We observe, in the first place, that if all other things remain the same, the number of vibrations Varies inversely as the length of the cord ; hence, if the vocal ligaments were divided by nodes into a ventral segments, each segment might be considered a separate vibrating ligament, whose length would be -1.th of the vocal cord, and consequently the number of its vibrations in a given time would be ea times as many as that of the whole cord.
Owing to the elasticity of the thyro-aryte noid ligaments, their lengths, when in a state of repose, differ considerably from those which they present under the greatest tension. They differ also in the two sexes. In a series of experiments by Midler, the differences of length were observed to be as represented in the following table, the figures of which arc in inches and decimals of an inch. From these experiments it appears that the lengths of the male and female vocal cords in repose are nearly as 7 to 5, and in tension as 3 to 2. In boys at the age of fourteen, the length is to that of females after puberty as 6'25 to 7, so that the pitch of the voice is nearly the same. These experiments afford an idea, although an imperfect one, of the elasticity of the vocal ligaments. It has always been a subject of surprise, if the thyro-arytenoid ligaments obey the laws of strings, how such short and narrow laminm should produce such very grave tones as many bass singers are capable of uttering ; and this struck M. Blot as one of the circum stances which in his opinion prove their mode of vibration to be unlike that of strings. He asks, " OA pourroit-on trouver la place ne cessaire pour donner A cette corde la longucur qu'exigent les sons les plus graves P" The author is acquainted with some bass singers who can produce the note C which results from sixty - four musical vibrations.
Let us now investigate this phenomenon more closely, and endeavour to explain how such grave tones are produced by such extremely short membranes. Muller has contrived several ingenious pieces of mechanism, seen in figs. 891, 892, and 893, by means of which he was enabled to estimate the amount of tension, lateral compression, and atmospheric pressure on the vocal cords, during the pro duction of sound. In order to find the variations in the amount of condensation of air in the vocal organs in the production of sounds differing in pitch and intensity, the apparatus was furnished with a manometer (fig. 893, v). From that portion of the ex periment which was confined to the investiga tion of the effects produced by tension of the vocal, compared with that of musical cords, he obtained results which are recorded in the following table.
In the above table Cl answers to 256 vibra tions. Muller states that the numbers of vibra tions in these experiments are not exactly in the direct ratio of the square roots of the stretch ing forces, and that the weights 4, 16, 64, did not produce the octaves, but generally from a semitone to two or three tones lower. Now this result should have been anticipated ; but it does not seem to have occurred to him that whilst he increased the tension he at the same time increased the length, and we know (eq. 3.) that the number of vibrations in this ?tension case varies as , and conse of cord fluently the numbers actually produced by the weights above mentioned ought (agreeably to Miller's experiments) to be less than those which correspond with octaves. We see by the first experiment in the above table that the tension sufficient to produce 818 musical vibrations is 64 loths, or very nearly 33 ozs. If, therefore, we take the mean length of the vocal ligaments under the greatest tension at •91 of an inch, and substitute in equation (3.) their values for all known quantities, remem bering that P represents the tension of one vocal cord, we shall find the weight of each ligament, viz.