FUND AMm\TAt., PARTIAL, AND COM RINATIONA.L V I IIIIATIoNS. Musieal instruments may be di vided roughly into two classes, wind and string, instruments. Lt the former class are ineluded organ-pipes, horns, flutes, ete.; in the latter, pianos, violins. harps. Pte. in all wind instru ments a column of air inclosed in a metal or wooden tube is set in vibration by suitable means, and this vibrating mass primitives the waves in the surrounding air. In string in struments, flexible strings are stretched between pegs fastened to a solid frame—in general a wooden board—anti are set in transverse vibration by bowing. plucking, or striking. As a result of the vibration of the string, the frame holding the pegs is itself set in vibrations of the same frequency, and it, as well as the string itself, produces the waves. The importance of the so-called sounding-board is at once evident.
A stretched flexible string, A B, can vibrate in many waves: as a whole, with its middle point its point of greatest amplitude, as in 1 (fig. 3); in two parts, with its middle point, h, at rest, and the two halves vibrating like separate strings in opposite phases, as in 2 (rig. :3); in three parts, with two points, c and tt, at rest, di viding the string into three equal vibrating seg ments, as in 3 (fig. :3), etc. Time frequencies of these different modes of vibration are in the ratios of 1 : 2: 3: 4, etc. The vibration of the string as a whole is called the "fundamental:" the others, the "upper partials." The frequency of the transverse vibrations of a stretched flex ible string is given by the formula "=rall , where 7' is the stretching force or tension, as is the mass of each unit length of the string. h is the length of the vibrating segment. Thus, in the fundamental, L is the length of the string; in the first upper partial it is one-half the length of the string, etc. When I Ile string is set vi brating by a random blow or bowing, it will make eomplex vibrations, resulting from the emnhination of the fundamental and some of the tipper partials, the number anti relative in tensities of these depending largely on the point where the blow is struck, or the bow applied, and on the character of the impulse. So, when
ever a musical tone is produced by a string instrument, the ear can recognize in the complex sound simple tones due to the fundamental and the upper partials; and differences in the qual ity of sounds caused by different string instru ments, which have fundamentals of the same frequency, are due to differences in the number and character of the upper partials, which de pend in turn on the material of the string, the point where the impulse is applied to set the string in motion, and the character of this im pulse. Similarly, the vibrating column of air in organ-pipes, horns, etc., can vibrate in different ways; and in a complex vibration there is a fundamental and upper partials whose frequen cies are in the ratios of 1 : 2 : 3 : 4, etc. The frequency of the vibrations of the fundamental in an open organ-pipe is given by the formula: N 2 L where T' is the velocity of waves in the gas which fills the pipe, and L is the length of the pipe approximately. The similar formula for a "stopped" pipe is: V L (in stopped organ-pipes the vibrations are in the ratios 1 :3 :5 : 7, etc.) In other in struments than wind and string ones, such as drums, cymbals, etc., there are upper partials besides the fundamental; but there is no simple mathematical relation between their frequencies. When two organ-pipes on the same wind-chest are "sounded" loudly, the resulting waves in the air are not due simply to each fundamental and its upper partials, but also to certain extra vi brations clue to the combined action of the two vibrating columns of air on the surrounding air. Thus, if the fundamentals of the two pipes have frequencies 1000 and GOO, there will be present waves showing the existence of vibrations whose frequencies are 1000 -I- 600 and 1000-600. The sounds heard owing to these vibrations are called "summational" and "differential" tones, or, in general, "combinational" tones; they are always difficult to hear. The existence of both partial and combinational vibrations may, however, be established by means of resonators (q.v.).