Let us consider the strings of a piano. (If possible, look at the strings in some instrument.) The range of the piano is 7-1/3 octaves. Its lowest note, A_{4}, has about 27 vibrations per second. Its highest, C⁴, about 4176. This great range in vibration rate is secured by varying the length, the tension, and the diameter of the strings. =343. The Laws of Vibrating Strings.=--The relations between the vibration rate, the length, the tension and the diameter, of vibrating strings have been carefully studied with an instrument called a _sonometer_ (Fig. 335). By this device it is found that the pitch of a vibrating string is raised one octave when its vibrating length is reduced to one-half. By determining the vibration rate of many lengths, the following law has been derived: (Law I) _The rate of vibration of a string is inversely proportional to its length._ [Illustration: FIG. 335.--A sonometer.] Careful tests upon the change of vibration rate produced by a change of _tension_ or pull upon the strings show that if the pull is increased four times its vibrations rate is _doubled_, and if it is increased nine times its rate is tripled, that is: (Law II) _The vibration rates of strings are directly proportional to the square roots of their tensions._ Tests of the effects of diameter are made by taking wires of equal length and tension and of the same material but of different diameter. Suppose one is twice as thick as the other. This string has a tone an octave lower or vibrates one-half as fast as the first. Therefore: (Law III) _The vibration rates of strings are inversely proportional to the diameters._ These laws may be expressed by a formula _n_ ∝√(_t_)/_dl_. The vibration of a string is rarely a simple matter. It usually vibrates in parts at the same time that it is vibrating as a whole. The tone produced by a string vibrating as a whole is called its _fundamental_. The vibrating parts of a string are called _loops_ or _segments_ (see Fig. 336), while the points of least or no vibration are _nodes_. Segments are often called _antinodes_. [Illustration: FIG. 336.--A string yielding its fundamental and its first overtone.]