They are sensations of difference-tones; they correspond, as their name indicates, to the difference between the number of vibrations in two primary tones. They may have a twofold origin, either from the interference of the vibrations in the outer ear, especially in the tympanum or chain of ossicles (Helmholtz's combi- [p. 99] nation-tones), or from the interference of the vibrations in the auditory nerve-fibres (Koenig's beat-tones). The first are, from the very character of their origin, weak tones; especially in comparison with the original tones, they are always relatively very weak. The second class, on the other hand, are generally stronger and may even surpass the original tones in intensity. It is probable that the first appear onlv in the case of harmonious compound clangs, while the second appear also in dissonant compound clangs. The fusion of difference-tones with the chief tones of the compound is the more complete the less intense the former are, and the more they tend to form a simple harmonious tonal series with the original components of the clang. As a result of these attributes, the difference-tones are to compound clangs what the overtones are to single clangs. They are, however, almost entirely independent of the clang-color of the components of the compound, but vary greatly with the relation in which the principal tones of these components stand to one another. This explains the relative uniformity in the character of a given compound clang even when the clang-colors of its components vary.
7. A compound clang may pass through all possible intermediate stages into a third form of intensive auditory ideas, that of noises. When two tones are no longer included within a series of harmonious tones and when at the same time the difference between the number of their vibrations does not exceed certain limits, for higher tones about sixty vibrations and for lower thirty or even fewer, there arise disturbances in the compound clang, which correspond in number to the difference between the number of vibrations in the primary tones, and are due to the alternating coincidence of like and opposite phases of vibration.
These disturbances are either interruptions of the clang-sensation, [p.
99] beats, or, especially in the case of deep tones, intermittent sensations of a difference-tone, tonal beats. If the differences in the number of vibrations exceed the numbers mentioned, the tones at first sound continuous, for the interruptions disappear, but they are harsh.
Later the harshness disappears and we have pure dissonance. Ordinary dissonance is made up of a mixture of beats or harshness and pure dissonance.
The first two are due to perceptible or just disappearing interruptions of the sensation, the latter to the entire absence of the unity of the clang, that is, of the consonance that would have arisen if a complete or partial fusion had taken place. This lack of accord in tones, due to the relation of their pure qualities, may be designated bissonance .
If through the simultaneous sounding of a great number of non-accordant tones the various conditions for an ordinary dissonance, beats, tonal beats, harshness, and bissonance, are all added together, a noise is the result. On the psychological side this means that the predominating tonal elements disappear entirely or become mere modifying elements in the total idea. For our apprehension of noises, in the case of those which last a short interval only, the general pitch of the most intense elements is determinative, in the case of those which last longer, the form of the disturbance resulting from the rapidity of the beats, from the accompanying tonal beats, etc., also has an influence.