quality of every musical sound is determined by the number, orders, and relative intensities of the upper partial tones which accompany the fundamental whenever any ordinary instrument is sounded. Every such compound tone can be graphically represented by its own curve, the form of which may be varied not only by varying the elements just mentioned, but also by varying the phases in which the separate components are joined together. Helmholtz endeavored to test the influence of change of phase in using his apparatus for acoustic analysis, but the results were negative, and his conclusion was, that variation in phase has no physiological effect. Koenig has since attacked this problem, employing wave-sirens of his own invention, by which he has established quite conclusively the existence of this fourth element of musical quality. . . . The wave-siren may be briefly described as an apparatus in which a blast of air is forced through a narrow cleft against the edge of a moving plate or disk on which a series of determinate wave-forms have been cut. Each sinuosity, as it passes the cleft, interrupts the egress of air, so that a series of compound pulses are propagated whose grouping is determined by the form of the curved edge. The pitch is determined by the speed of rotation and the wave-length cut in the metal, through either the convex surface of a cylinder which rotates on its axis, or the edge of a disk which rotates about its center. A number of such wave-forms, each with its own wind-cleft, may be operated at the same time, with the same speed and with the same pressure of air at each cleft. They may be arranged to either coincide or differ in phase to any required extent. By the use of this new instrument Koenig has found that the complex sound obtained by the composition of a series of harmonics, of even as well as odd orders, quite independently of their relative intensity, has always its maximum of strength and its greatest acuteness of quality for a difference of phase of a fourth of a wave-length; the minimum of strength, and the softest quality, for a difference of phase of three fourths of a wave-length. It may be said that, if changes in the number and relative intensity of the harmonics produce differences of quality, such as are observed in instruments belonging to different families, or such as the human voice shows in the different vowels, the changes due to difference of phase between the same harmonics are yet capable of producing differences of quality at least as sensible as those which are noticeable in instruments of the same kind, or in the same vowels sung by different voices.
All musicians are able to perceive the general smoothness or roughness of a combination of sounds; but the analysis of the combination requires exquisite sensitiveness of ear for the detection of variation in both pitch and harmony. In the tuning of the standard forks which are issued from Koenig's laboratory, his
