The complete series of possible overtones in a clang consists of the first octave of the principal tone, the fifth of this octave, the second octave of the principal tone, and the major third and the fifth of this [p. 96] second octave, etc. This series corresponds to the following proportions between the number of objective tonal waves: 1 (principal tone), 2, 3, 4, 5, 6, 7, 8, . . . . (overtones). When the pitch of the principal tone remains constant, only the second determinant of the tonal quality, the clang-color, can vary according to the number, position, and relative intensity of the overtones. In this way we can explain the great variety of clang-colors in musical instruments, as well as the fact that for every instrument the clang-color changes somewhat with the pitch; for in the case of low tones the overtones are generally relatively strong, in that of high relatively weak, while they disappear entirely when they are too high to be audible. Even the slight differences in clang-color in single instruments of the same kind, are to be explained in the same way.
From a psychological point of view the chief condition for the rise of a single clang, is the complete, or approximately complete, fusion of several tonal sensations with only one predominating element. As a rule, it is impossible to distinguish with the unaied ear the overtones in a clang. They can be made perceptible by the use of resonators (resonator-tubes tuned to the overtones sought), and after they have been isolated in this experimental way, the stronger ones can be successively heard in the clang, even without the aid of the resonators, if the attention is directed to them.
4. There are three conditions necessary if there is to be only one predominating element in a tonal fusion.
First, one tone must be relatively more intense. Secondly, in its qualitative relations to the other partial tones, the principal tone must be the fundamental of a series whose members are all harmonious. Thirdly, all the partial tones must be uniformly coincident. This coincidence is objectively guaranteed by deriving the clang from a unitary source, (that is, producing [p. 97] the clang through the vibrations of one string, one reed-pipe, etc.) The result is that the objective vibrations of the partial tones always stand in the same relation to one another -- a result which can not be secured when clangs from several sources are united. The first two of these conditions relate to the elements, the third to the form of their combinations. The first is the least essential to the idea of a single clang. If the second is not fulfilled, the combination becomes a compound clang when the predominating fundamental is wanting, or a noise when the series of tones is not harmonious, or a mixed form between a clang and a noise when both parts of the condition are unfulfilled.
If the third condition, of constancy in the phases of the partial tones, is not met, the clang becomes compound even when the first two conditions are complied with. A series of ****** clangs from a number of tuning-forks which should unite to a single clang so far as intensity and quality are concerned, always produces in reality the idea of a compound clang. [ 1 ]
5. A compound clang is an intensive combination of single clangs. It is in general an incomplete fusion with several predominating elements. There are, as a rule, all possible [p. 98] grades of fusion in a compound clang, especially when it is made up of single clangs of composite quality. In such a case, not only does every single clang form a complete fusion in itself, but these single clangs fuse the more completely with one another the more their fundamentals approach the relation of elements of a single clang. So it comes that in a compound clang made up of single clangs rich in overtones, those components whose fundamentals correspond to the overtones of some other single clang in the compound, fuse more completely with this related clang than with others.
The other clangs, in turn, fuse the more completely the more their relation approaches that of the first members of a series of overtones. Thus, in the compound clang c e g c' the clangs c and c' form a nearly complete fusion, while the fusions of the clangs c and g, c and e, are incomplete. Still less complete is the fusion between c and e b . A measure for the degree of fusion may be obtained in all these cases by allowing an observer to hear the compound clang for a very brief interval, after which he is to decide whether he perceived only one clang or several. This experiment is repeated many times, and the relative number of judgments in favor of the unity of the clang is a measure for the degree of fusion.
6. Besides the elements contained in the single clangs of a compound, there are always, arising from the combination of vibrations in the auditory organ, additional elements which cause new tonal sensations, characteristic for the different kinds of compound clangs. These may also fuse more or less completely with the original clang.
