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II. PSYCHICAL COMPOUNDS
§9. INTENSIVE IDEAS.
1. A combination of sensations in which every element is connected with any second element in exactly the same way as with any other, is called an intensive idea. Thus, for example, a compound clang made up of the tones d f a is such an intensive idea. For the immediate apprehension, each of the partial combinations into which this compound clang can be resolved, as df, da, fd, fa, ad, af, are all entirely equivalent, in whatever order they are thought of. This is obvious at once if we compare the compound clang with any succession of the same tones, where df, da, fd, fa, etc., are essentially different ideas. We may define intensive ideas, accordingly, as combinations of sensational elements, in which the order of the elements may be infinitely varied.
It follows from their nature, that intensive ideas do not have, arising from the way in which their elements axe united any characteristics, by means of which they can be resolved into separate parts. Such a resolution is possible only through the differences in the constituent elements themselves. Thus, we discriminate the elements of the compound clang d f a, only because we hear in it the qualitatively different tones d, f, and a. Still, the separate components in such a unitary idea are less clearly distinguishable than in their isolated state. This fact, that the elements are pushed into the 'background by the impression of the whole, is of great im- [p. 94] portance for all forms of ideational combination. We call it the fusion of sensations, and in particular, for intensive ideas, intensive fusion. If the connection of one element with others is so close that it can be perceived as a part of the whole only through unusual concentration of the attention aided by experimental variation of the conditions, we call the fusion complete. If, on the other hand, the elements are immediately recognized in their proper qualities, and merely recede somewhat into the background in comparison with the impression of the whole, we call the fusion incomplete. If certain particular elements are more prominent in their characteristic qualitites thin others, we call them the predominating elements. The concept of fusion as here defined as a psychological concept. It presupposes that the fused elements of the idea are really subjectively distinguishable. It must not be confounded with the entirely different and purely physiological concept of the fusion of external impressions into a single resultant stimulation. For example, when complementary colors unite and give white, the fusion is, of course, not psychological.
In reality, every intensive idea always enters into certain spacial and temporal combinations. Thus, for example, a compound clang is always a process having a certain duration, and is at the same time localized by us in some direction or other, though often only very indefinitely. But since these temporal and spacial attributes can be indefinitely varied, while the intensive character of the ideas remain the same, we may abstract from the former in investigating the intensive attributes.
2. Among ideas of the general sense we have intensive fusions in the form of combinations of sensations of pressure with those of hot or cold, or combinations of pain-sensations with those of temperature or pressure. All these fusions [p. 95] are incomplete, and very often there is no decidedly predominating element. The combination of certain sensations of smell and taste are more intimate. This is obviously favored on the physiological side by the proximity of the sense-organs, on the physical side by the regular connection between certain stimulations of the two senses. In such cases the more intense sensations are generally the predominating elements, and when these are the sensation of taste, the composite impression is usually regarded as a taste-quality only. Thus, most of the impressions known in ordinary life as "tastes", are in reality combinations of tastes and smells.
The greatest variety of intensive ideas, in all possible gradations of complexity, are presented by the sense of hearing. The relatively most simple of these ideas and those which are most closely related to simple tones, are the single clangs. As more complex forms, we have compound clangs. Complex noises may arise from the latter when they are united with sensations of simple noises, and under certain other circumstances.
3. A single clang is an intensive idea which is made up of a series of tonal sensations regularly graded in quality. These elements, the partial tones of the clang, form a complete fusion, in which the sensation of the lowest partial tone becomes the predominating element. The pitch of the tone is determined by this principal tone. The other elements are higher and are, accordingly, called overtones. The overtones are all grouped together under the name clang-color as a second determinant of the clang, added to the predominating tone. All the partial tones that go to determine the clang color are placed along the tonal line at certain regular intervals from the principal tone. 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 simple 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. 
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 eb. 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. 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.
Human articulations are characteristic examples of different forms of noise. The vowels are intermediate between clangs and noises with predominantly clang character; the resonants are noises of long duration, and the proper consonants noises of short duration. In whispers the vowels become simply noises. The circumstance that the differences in vowels are perfectly distinct in whispers, goes to prove that the character of vowels depends essentially on their noise-elements. It is [p. 101] probable that simple sensations of noise (p. 49) enter into all noises together with the numerous tonal elements that no to make them up. The irregular air-vibrations arising from the disturbances in the tonal waves, excite both the nervous elements in the vestibule of the labyrinth, which are sensitive to such stimulations, and the auditory nerve fibres themselves.
7a. Helmholtz's resonance hypothesis has aided us materially in understanding the physiological substratum of intensive auditory ideas, especially those of clangs (p. 51). It is assumed that certain parts of the auditory organ are so tuned that tonal waves of a given rate always set in sympathetic vibration only the part ,correspondingly tuned. This explains in a general way the analyzing ability of the auditory sense, as a result of which we can distinguish the elements not only in a compound clang, but to some extent even in a single clang. The resonance hypothesis, however, accounts physiologically for only one side of tonal fusion. the persistence of the single sensation in the total intensive idea, not for the other side, the more or less intimate combination of the elements. The assumption of an imaginary "organ of fusion" in the brain for this purpose, is one of those fictions that are more harmful than helpful, in which the attempt is made to satisfy a demand for explanation with an empty word. The tonal elements that produce an intensive clang-idea persist as real sensations and still give up their independence more or less in the total idea. Tonal fusion is, then, a psychical process and requires a psychological explanation. But since this fusion is very different under different objective conditions, as, for example, when the impressions are due to the combined vibrations from a single source or to vibrations from several distinct sources; these differences must have some physiological and physical grounds for their explanation. The most natural way to attempt such an explanation is properly to supplement the resonance hypothesis. If we assume that besides the analysing parts of the auditory organ, the resonant membrane, still others exist which are effected by the total, unresolved clang, we have a sufficient physiological substratum for the different effects of the various conditions. The [p. 102] observations (p. 41) on birds deprived of their labyrinths make it possible to infer that the auditory nerve-fibres in the canals of the labyrinth may be such organs. Then, too, the existence of beat-tones (p. 99), which sometimes surpass the primary tones in intensity, and the observation that the interruptions of a single tone may unite to form a second sensation when sufficiently rapid, both seem to require a similar supplementation of the resonance hypothesis.
 The case is different when the fundamental itself contains overtones of noticeable intensity, which are also repeated as independent clangs in the compound tone. The single clangs of such a series arrange themselves in the same phases as these overtones, and the compound clang has the character of a single clang with very strong overtones. Helmholtz concluded from experiments in which he combined in various ways simple clangs from tuning-forks, that differences in phase have no influence on the clang-color. But as the idea of a single clang can not be produced in this way, it is probable that an entirely constant relation of the phases of different tonal vibrations from independent sources can never be brought about with this method. Experiments by R. Koenig tell for the influence on the clang-color, of the form of the clang as determined by the relation of the vibration-phases