Chapter 11

The vibration rates per second of the vibrating bodies, strings, steel rods, etc., which produce those musical tones which are consonant, are in definite and small mathematical ratios to each other. Thus the rates of C-C' are as 1:2; of C-G, C-E, as 2:3, 4:5. In general, the simpler the fraction, the greater the consonance.

But no sonorous body vibrates in one single rate; a taut string vibrates as a whole, which gives its fundamental tone, but also in halves, in fourths, etc., each giving out a weaker partial tone, in harmony with the fundamental. And according to the different ways in which a sonorous body divides, that is, according to the different combination of partial tones peculiar to it, is its especial quality of tone, or timbre. The whole complex of fundamental and partial tones is what we popularly speak of as a tone,--more technically a clang. These physical agitations or vibrations are transmitted to the air. Omitting the account of the anatomical path by which they reach the inner ear, we find them at last setting up vibrations in a many-fibred membrane, the basilar membrane, which is in direct connection with the ends of the auditory nerve. It is supposed that to every possible rate of vibration, that is, every possible tone, or partial tone, there corresponds a fibre of the basilar membrane fitted by its length to vibrate synchronously with the original wave-elements. The complex wave is thus analyzed into its constituents. Now when two tones, which we will for clearness suppose to be simple, unaccompanied by partial tones, sounding together, have vibration rates in simple ratios to each other, the air- waves set in motion do not interfere with each other, but combine into a complex but homogeneous wave. If they have to each other a complicated ratio, such as 500:504, the air- waves will not only not coalesce, but four times in the second the through of one wave will meet the crest of the other, thus making the algebraic sum zero, and producing the sensation of a momentary stoppage of the sound. When these stoppages, or beats, as they are called, are too numerous to be heard separately, as in the interval, say, 500:547, the effect is of a disagreeable roughness of tone, and this we call discord. In other words, any tones which do not produce beats are harmonious, or harmony is the absence of discord. In the words of Helmholtz,<1> consonance is a continuous, dissonance an intermittent, tone-sensation.

<1> _Lehre v.d. Tonempfindungen_, p. 370, in 4th edition.

Aside from the fact that consonance, as a psychological fact, seems positive, while this determination is negative, two very important facts can be set up in opposition. As a result of experimental investigation, we know that the impression of consonance can accompany the intermittent or rough sound- sensations we know as beating tones; and, conversely, tones can be dissonant when the possibility of beats is removed. Briefly, it is possible to make beats without dissonance, and dissonance without beats.

The other explanation makes consonance due to the identity of partial tones. When two tones have one or more partial tones in common they are said to be related; the amount of identity gives the degree of relationship. Physiologically, one or more basilar membrane fibres are excited by both, and this fact gives the positive feeling of relationship or consonance. Of course the obvious objection to this view is that the two tones should be felt as differently consonant when struck on instruments which give different partial tones, such as organ and piano, while in fact they are not so felt.

But it is not after all essential to the aesthetics of music that the physiological basis of harmony should be fully understood. The point is that certain tones do indeed seem to be "preordained to congruity," preordained either in their physical constitution or their physiological relations, and not to have achieved congruity by use or custom. Consonance is an immediate and fundamental impression,--psychologically an ultimate fact. That it is ultimate is emphasized by Stumpf<1> in his theory of Fusion. Consonance is fusion, that is, unitary impression. Fusion is not identical with inability to distinguish two tones from each other in a chord, although this may be used as a measure of fusion. Consonance is the feeling of unity, and fusion is the mutual relation of tones which gives that feeling.

<1> _Beitrage zur Akustik u. Musikwissenschaft_, Heft I, Konsonanz u. Dissonanz, 1898.

The striking fact of modern music is the principle of tonality. Tonality is said to be present in a piece of music when every element in it is referred to, gets its significance from its relation to, a fundamental tone, the tonic. The tonic is the beginning and lowest note in the scale in question, and all notes and chords are understood according to their place in that scale. But the conception of the scale of course does not cover the ground, it merely furnishes the point of departure,-- the essential is in the reference of every element to the fundamental tone. The tonic is the centre of gravity of a melody.

The feeling of tonality grew up as follows. Every one was referred to a fundamental, whether or not it made with it an harmonious interval. The fundamental was imaged TOGETHER WITH every other note, and when a group of such references often appeared together, the feelings bound up with the single reference (interval-feelings) fused into a single feeling,-- the tonality-feeling. When this point is once reached, it is clear that every tone is heard not as itself alone, but in its relations; it is not that we judge of tonality, it is a direct impression, based on a psychological principle that we have already touched on in the theory of rhythm. The tonality- feeling is a feeling of form, or motor image, just as the shape of objects is a motor image. We do not now need to go through all possible experiences in relation to these objects, we POSSESS their form in a system of motor images, which are themselves only motor cues for coordinated movements. So every tone is felt as something at a certain distance from, with a certain relation to, another tone which is dimly imagined. In following a melody, the notes are able to belong together for us by virtue of the background of the tone to which they are related, and in terms of which they are heard. The tonality is indeed literally a "funded content,"--that is, a funded capital of relation.

These are the general facts of tonality. But what is its meaning for the nature of music? Why should all notes be referred to one? Is this, too, an ultimate psychological fact? In answer there may be pointed out the original basic quality of certain tones, and the desire we have to return to them. Of two successive tones, it is always the one which is, in the ratio of their vibration rates, a power of two, with which we wish to end.<1> When neither of two successive tones contains a power of two, we have no preference as to the ending. Thus denoting any tone by 1, it is always to 1 or 2, or 2n that we wish to return, from any other possible tone; while 3 and 5, 5 and 7, leave us indifferent as to their succession. In general, when two tones are related, as 2n:3, 5, 7, 9, 15--in which 2n denotes every power of two, including 2o=1, with the progression from the first to the second, there is bound up a tendency to return to the first. Thus the fundamental fact of melodic sequence may be said to be the primacy of 2 in vibration rates. But 2n, in a scale containing 3, 5, etc., is always what we know as the tonic. The tonic, then, gives a sense of equilibrium, of rest, of finality, while to end on another tone gives a feeling of restlessness or striving.

Now tone-relationship alone, it is clear, would not of itself involve this immediate impulse to end a sequence of notes on one rather than on another. Nor is tonality, in the all- pervasive sense in which we understand it, a characteristic of ancient, or of mediaeval music, while the tendency to end on a certain tone, which we should to-day call the tonic, was always felt. Thus, since complete tonality was developed late in the history of music, while the closing on the tonic was certainly prior to it, the finality of the tonic would seem to be the primary fact, out of which the other has been developed.

We speak to-day, for instance, of dissonant chords, which call for a resolution--and are inclined to interpret them as dissonant just because they do so call. But the desire for resolution is historically much later than the distinction between consonance and dissonance.... "What we call resolution is not change from dissonant to consonant IN GENERAL, but the transition of definite tones of a dissonant interval into DEFINITE TONES of a consonant."<1> The dissonance comes from the device of getting variety, in polyphonic music, by letting some parts lag behind, and the discords which arose while they were catching up were resolved in the final coming together; but the STEPS were all PREDETERMINED.<2> Resolution was inevitably implied by the very principle on which the device is founded. That is, the understanding of a chord as something TO BE RESOLVED, is indeed part of the feeling of tonality; but the ending on the tonic was that out of which this resolution- feeling grew.

<1> Stumpf, op. Cit., p. 33. <2> Grove, _Dict. Of Music and Musicians_. Art. "Resolution."

Must we, then, say that the finality of the tonic is a unique, inexplicable phenomenon? giving up the nature of melody as a problem if not insoluble, at least unsolved?

The feeling of finality in the return to 2n is explained by Lipps and his followers, from the fact that the two-division is most natural, and so tones of 2n vibrations would have the character of rest and equilibrium. This explanation might hold if we were ever conscious of the two-division as such, in tones --which we are not; so that it would seem to depend on the restful character of a perception which by hypothesis is never present to the mind at all.

The experience is, on the contrary, immediate,--an impression, not a perception; and this immediacy points to the one ultimate fact in musical feeling we have so far discovered. The whole development of the scale, and the complex feeling of tonality, is an expression of the desire for consonance. Every change and correction in the scale has gone to make every note more consonant with its neighbors. And naturally the tonic is the tone with which all other tones have the most unity. Now this "return" phenomenon is a simpler case of the desire for the feeling of unity. The tonic is the epitome of all the most perfect feelings of consonance or unity which are possible in any particular sequence of tones, and is therefore the goal or resting-place after an excursion. The undoubted feeling of equilibrium or repose which we have in ending on the tonic is thus explained. Not that consonance itself, the feeling of unity, is explained. But at any rate consonance is the root of the "return," and of its development into complete tonality.

The history of music is then the explicit development of acoustic laws implicit in every stage of musical feeling. That feeling covers an ever wider field. When Mr. Hadow says that the terms concord and discord are wholly relative to the ear of the listener,<1> and that the distinction between them is not to be explained on any mathematical basis, or by any a priori law of acoustics,--that it is not because a minor second is ugly that we dislike it, for it will be a concord some day,--he is only partly right. The minor second may be a "concord," that is, we may like it, some day; but that will be because w have extended our feeling of tonality to include the minor second. When that day comes the minor second will be so closely linked with other fully consonant combinations that we shall hear it in terms of them, just as to-day we hear the chord of the dominant seventh in terms of its resolution. But the basis will not be convention or custom, except in so far as custom is the unfolding of natural law. The course of music, like that of every other art, is away from arbitrary--though simple--convention, to a complexity which satisfies the natural demands of the organism. The "natural persuasion" of the ear is omnipotent.

<1> W.H. Hadow, _Studies in Modern Music_, 1893.

V

It has been said that the feeling of tonality is a motor image or "form-quality" and that the image of the tonic persists throughout every sequence of tones in a melody. Now these are not only felt as having a certain relation to the tonic; that relation is an active one. It was said that we had a positive desire to end on a certain tone, and that a tendency to pass to that tone was bound up with the hearing of another tone. The degree of this tendency is determined by their relation. The key, the tonality, is determined by the consensus of intervals which have been felt as more or less consonant. Then steps in this scale which come near to the great salient points--that is, the points of greatest consonance, which is unity, which is rest--are felt as suggesting them. This is the reason why a semitone progression is felt as so compelling. In taking the scale upward, C to C', that element in the tone- Space already clearly foreshadowed by the previous tones is C'; B is so near that it is almost C'--it seems to cry aloud to be completed by C'. Then the tendency to move from B to C' is especially strong. In the same way a chromatic note suggests most strongly the salient point in the scheme to which it is nearest--and "tends" to it as to a point of comparative rest. The difference between the major and minor scales may be found in the lesser definiteness<1> with which the tendency to progression, in the latter, is felt--"a condition of hovering, a kind of ambiguity, of doubt, to which side the movement shall proceed." We may then understand a melody as ever tending with various degrees of urgency, of strain, to its centre of gravity, the tonic.

<1> F. Weinmann, _Zeitschr. f. Psychol._, Bd. 35, p. 360.

It is from this point of view that we can see the cogency of Gurney's remark, that when music seems to be yearning for unutterable things, it is really yearning only for the next note. "In this step from the state of rest into movement and return, the coming again to rest; on what circuitous ways, with what reluctances and hesitations; whether quick and decisively or gradually and unnoticed--therein consists the nature of melody."<1>

<1> Weinmann, op. cit.

Or in Gurney's more eloquent description, "The melody may begin by pressing its way through a sweetly yielding resistance to a gradually foreseen climax; whence again fresh expectation is bred, perhaps for another excursion, as it were, round the same centre but with a bolder and freer sweep,...to a point where again the motive is suspended on another temporary goal; till after a certain number of such involutions and evolutions, and of delicately poised leanings and reluctances and yieldings, the forces so accurately measured just suffice to bring it home, and the sense of potential and coming integration which has underlain all our provisional adjustments of expectation is triumphantly justified."<1>

<1> Op. cit., p. 165.

This should not be taken as a more or less poetical account under the metaphor of motion. These "leanings" are literal in the sense that one note does imply another as its natural complement and satisfaction and we seek to reach or make it. The striving is an intrinsic element, not a by-product for our understanding.

There is another point to note. The "sense of potential and coming integration" is a strong factor of melody. If it cannot be said that the first note implies the last, it is at least true that from point to point the next step is dimly foreseen, and this effect is cumulative. If melody is an ever-hindered striving for the goal, at least the hindrances themselves are stations on the way, each one as overcome adding to the final momentum with which the goal is reached. It is like an accumulation of evidence, a constellation of associations. AB foretells C; but ABCDEF rushes yet more strongly upon G. So it is that the irresistibleness, the "unalterable rightness" of a piece of music increases from beginning to end.

The significance of this essential internal necessity of progression cannot be overestimated. The unalterable rightness of music is founded on natural acoustic laws, and this "rightness" is fundamental. A melody is not right because it is beautiful, it is beautiful because it is right. The natural tendencies point out different paths to the goal; and thus different ways of being beautiful; but the nature of the relation between point and point, the nature of the progression, that is, the nature of melody, is the same.

Up to this point we have consistently abstracted from the element of rhythm in melody. Strictly speaking, however, it is impossible to do so. The individuality of a melody is absolutely dependent on its rhythm, that is, on the relative time-value of its tones. Gurney has devoted some amusing pages to showing the trivial, dragging, lustreless tunes that result from ever so slight a change in the rhythm of noble themes, or even in the distribution of rhythmical elements within the bar. The reason for this is evident. The nature of melody in the sense of sequence consists in the varied answers to the demands of the ear as felt at each successive point. Now it is clear that such "answer" can be emphasized, given indifferently, held in suspense, in short, subjected to all kinds of variation as well by the rhythmical form into which it is cast, as by the different choice of possibilities for the tone itself. The rhythm helps out the melody not only by adding to it an independently pleasing element, but, and this is indeed the essential, by reinforcing the intrinsic relations of the notes themselves. Thus it is in the highest degree true that in melody and rhythm we do not have content and form, but that, strictly speaking, the melody is tone-sequence in rhythm.

The intimate bondage of tone-sequence and rhythm is grounded in the identity of their inner nature; both are varieties of the objective conditions of embodied expectation. It is not of the essence of music to satisfy explicit and conscious expectation--to satisfy the understanding. It meets on the contrary a subconscious, automatic need which becomes conscious only in the moment of its contenting. Every moment of progress in a beautiful melody is hailed like an instinctive action performed for the first time. Rhythm is the ideal satisfaction of attention in general with all its bodily concomitants and expressions. Tone-sequence is the satisfaction of attention directed to auditory demands. But the form-quality of rhythm, the form-quality of tonality, is an all but subconscious possession. Together, reinforcing each other in melody, they furnish the ideal arrangement of the most poignant of sense- stimulations.

VI

It is strange that those who would accept the general facts of musical logic as outlined above do not perceive that they have thereby cut away the ground from under the feet of the "natural language" argument. If the principle of choice in the progress of a melody is tone-relationship, the principle of choice cannot also be the cadences of the speaking voice. That musical intervals often RECALL the speaking voice is another matter, as we have said, and to this it may be added that they much more often do not. The question here is only of the primacy of the principle. Thus it would seem that the facts of musical structure constitute in themselves a refutation of the view we have disputed. To say that music arose in "heightened speech" is irrelevant; for the occasion of an aesthetic phenomenon is never its cause. It might as well be said that music arose in economic conditions,-- as indeed Grosse, in his "Anfange der Kunst," conclusively shows, without attempting to make this social occasion intrude into the nature of the phenomenon. Primitive decorative art arose in the imitation of the totemic or clan symbols, mostly animal forms; but we have seen that the aesthetic quality of the decoration is due to the demands of the eye, and appears fully only in the comparative degradation of the representative form. In exactly the same way might we consider the "degradation" of speech cadences into real music,--supposing this were really the origin of music. As a matter of fact, however, the best authorities seem to be agreed that the primitive "dance-song" was rather a monotonous, meaningless chant, and that the original pitch- elements were mechanically supplied by the first musical instruments; these being at first merely for noise, and becoming truly vibrating, sonorous bodies because they were more easily struck if they were hard or taut. The musical tones which these hard vibrating bodies gave out were the first determinations of pitch, and of the elements of the scale, which correspond to the natural partial vibrations of such bodies. "The human voice," Wallaschek<1> tells us, "equally admits of any pentatonic or heptatonic intervals, and very likely we should never have got regular scales if we had depended upon the ear and voice only. The first unique cause to settle the type of a regular scale is the instrument." To this material we have to apply only that "natural persuasion of the ear" which we have already explained, to account for the full development of music.

<1> _Primitive Music_, 1893, p. 156.

The beauty of music, in so far as beauty is identical with pleasantness, consists in its satisfaction of the demands of the ear, and of the whole psychophysical organism as connected with the ear. It is now time to return to a thread dropped at the beginning. It was said that a common way of settling the musical experience was to make musical beauty the object of perception, and musical expression the object, or source, of emotion. This view seems to attach itself to all shades of theory. Hanslick always contrasts intellectual activity as attaching to the form, and emotion as attaching to the sensuous material (that is, the physical effects of motion, loud or soft sound, tempo, etc.). He speaks of the aesthetic criterion of INTELLIGENT gratification. "The truly musical listener" has "his attention absorbed by the particular form and character of the composition," "the unique position which the INTELLECTUAL ELEMENT in music occupies in relation to FORMS and SUBSTANCE (subject)." M. Dauriac in the same way separates the emotion of music<1> as a product of nervous excitations, from the appreciation of it as beautiful. "It is probably that the pleasure caused by rhythm and color prevails with a pretty large number, with the greatest number, over the pleasure in the musical form, pleasure too exclusively PSYCHOLOGICAL for one to be content with it alone....The musical sense implies the intelligence....The theory...applies to a great number of sonorous sensations, and not at all to any musical perceptions." Mr. W.H. Hadow<2> tells us that it is the duty of the musician not to flatter the sense with an empty compliment of sound, but to reach through sensation to the mental faculties within. And again we read "the art of the composer is in a sense the discovery and exposition of the INTELLIGIBLE relations in the multifarious material at his command."<3>

<1> "Le Plaisir et l'Emotion Musicale," _Rev. Philos._, Tome 42, No. 7. <2> Op. cit., p. 47. <3> Grove's _Dict._ Art. "Relationship."