We have now to consider the subject of the quality_ of musical sounds; and one of its most important branches, what constitutes the distinction between the various vowel sounds. It had long been recognized that the only possible cause of this distinction between sounds musically identical mast lie in the nature of the fundamental noise, or, to express it differently, the nature of the periodic motion of each particle of air. But it appears that Helmholtz was the first to enter upon a complete examination of the point, both mathematically and experimentally, and the results he has arrived at form by no means the least remarkable of the contents of his excellent work, Die Lehre von den Tonempfindungen, recently published.
It was established by Fourier that any periodic expression whatever may be resolved into the sum of a number of simple harmonic terms, whose periods are, respectively, that of the original expression, its half, its third part, etc. Hence any periodic motion of air (i.e., any musical sound) may be resolved into a series of simple pendulum vibrations (i.e., pure musical sounds, such as those of turning-forks), the first vibrating once in the given period, the second twice, and so on. These notes are, as we have seen, the several harmonics of the lowest. Hence the quality of a musical sound depends upon the num ber and loudness of the harmonics by which it is accompanied.
Two experimental methods were employed by Helmholtz, one analytical, the other synthetical. In the first he made use of resonance-cavities fitted to the ear, and giving scarcely any indication of external sounds until one is produced which exactly corre sponds in pitch with the note which the cavity itself would yield. With a series of such cavities, tuned to the several harmonics of some definite note, the note was examined when played on various instruments, and when sung to different vowel-sounds. It was thus ascertained which harmonics were in each case present, and to what extent, pro ducing the particular quality of the sound analyzed. The second method was founded on the fact, already noticed, that a tuning-fork gives an almost pure musical sound (i.e., free from harmonics). A series of tuning-forks, giving a note and its harmonics, were so arranged as to be kept constantly in vibration by an electromagnetic apparatus.
Opposite to each was fixed a resonance-cavity exactly tuned to it, and capable • of being opened more or less at pleasure. When all the cavities were shut, the sound was scarcely audible; so that by opening them in various ways, any combination of harmonics might be made to accompany the fundamental note. These combinations were varied by trial, until the quality of the resultant sound was brought to represent as nearly as possible that of some vowel. The results of this second series of experiments coincided with those of the first. It appears from these investigations that the German U is the quality of a simple sound, though it is improved by adding faintly the two lowest harmonics; that 0 depends mainly on the presence of the third harmonic; and so on with the other sounds. it also appears, and it is well known by experience, that different vowel sounds, to be sung with accuracy, require to be sung to different notes. the proper note being that for which the cavity of the mouth is adapted for the production of the accom panying harmonics which determine the quality of the particular vowel.
In strings and pipes, as we have seen, the higher notes are strictly harmonics of the fundamental note, and therefore the sounds of instruments which depend on these sim ple elements are peculiarly adapted for music. On the other hand, when, as in masses ^ of metal. etc., the higher notes are not harmonics of the fundamental note, the mixed sound is always more or less jarring and discordant. Such is the ease with bells, trum pets, cymbals, triangles, etc.; and, in fact, these sounds are commonly characterized as "metallic." To produce from such instruments a sound as pleasing as possible, they must be so struck that as few as possible of the higher notes are produced, and these as feebly as possible. Thus, for instance, to get the most pleasing sound from a piano-forte ' wire, it should not be struck at the middle, as in such a case the first, third, fifth, etc., harmonics of the fundamental note will be wanting. If, however, it be struck at about of its length from one end, the harmonics produced will be mainly the first five; and these all belong to the chord of the fundamental note. A valuable recent wofk is lord Rayleigh's Theory of Sound (2 vole. 1878).