MATHEMATICS AND MUSIC
It is possible that the old heading “Arts and Sciences” has been responsible for some of the barrenness which is so conspicuous a feature of æsthetic theory. For the heading seems usually to have suggested, not only that there is a difference between the arts and the sciences, but that the difference is of a fundamental kind. For the purposes of æsthetic theory the various arts are assumed to have more in common than any one of them has with any of the sciences. We find the writer on æsthetics expounding his principles in chapters headed Painting, Sculpture, Poetry, Music; but it is rare indeed to find the argument extended to mathematics and physics. Yet there is no evidence that such omissions are due to deliberate reflection; the philosopher has not decided, after examination, that the sciences are unæsthetic objects; we must assume that accidents of taste and education have prevented him from paying attention to what may conceivably be useful data for the formulation of a theory of æsthetic. Within the last two or three generations scientific men have been thinking and writing a good deal about the philosophic basis and implications of their study, and it is significant that this inquiry has led many of them to insist on the æsthetic character of the satisfactions that science affords. The late Henri Poincaré, in particular, has shown that scientific theories are akin to works of art, and in this country, Dr. Norman Campbell has asserted his belief that great men of science are essentially great artists. The point of view is an interesting one, and suggests that fresh light may be thrown upon æsthetic problems by a new grouping of their subject-matter. Instead of putting the arts and the sciences on opposite sides of the fence, it may be helpful to see whether certain members of these two groups have not a natural affinity with one another, and so gain hints for a different and more comprehensive classification.
It is noteworthy, in this respect, that music has always occupied an exceptional position among the arts. Pater tried to relate it to other arts by saying it was the art to which all others aspire:
The arts may be represented as continually struggling after the law or principle of music, to a condition which music alone completely realises; and one of the chief functions of æsthetic criticism, dealing with the products of art, new or old, is to estimate the degree in which each of these products approaches, in this sense, to musical law.
It is characteristic of Pater’s criticism, and of much of the criticism of his school, that it exists, as it were, within a world of its own. The meanings to be attached to his most important terms are always suggested or insinuated; they are never defined. The method is useful, perhaps even necessary, in dealing with a complex and elusive object, and where appeal is made to perceptions which lie on the fringe of consciousness. But it runs the grave danger of becoming altogether too tenuous to be intelligible when we make direct reference to the object it is supposed to illuminate. When, for instance, Pater says of the best music, “the end is not distinct from the means, the form from the matter, the subject from the expression,” we become acutely aware of the absence of definition in each of these primary terms directly we think of any actual composition. We feel, indeed, that the terminology is not natural; in contemplating a poem the mind may be naturally impelled to distinguish between subject and expression as a kind of first effort in analysis; it is doubtful whether, in listening to music, this direction for analysis ever presents itself. So that to say that in music subject and expression are identical is not to say anything useful about music, but merely to declare that that kind of analysis is irrelevant. It is very probable that nothing is to be gained by first making distinctions which have a meaning for other arts and then bringing music into the scheme by saying that for music such distinctions become meaningless.
But if we are to maintain that this kind of criticism is irrelevant, then music becomes not only an isolated art but the art of which we know least. If it cannot be accommodated as an example within the general body of æsthetic criticism, the criticism that uses such terms as Pater uses, then whatever general conclusions the multifarious writings of the last two centuries on the “beautiful” may be considered to have reached are not applicable to music. In this extremity it is natural, nowadays, to become “scientific.” Comparative studies are undertaken: the music of Java is compared with the music of Bach: the evolution of musical devices is made clear; the psychological condition of the patient under music is examined: the time taken for the right degree of hypnosis to be induced is determined. That such methods may one day stumble upon important facts it would be rash to deny, but nothing has yet been reached which illuminates the particular problem that music presents. We are frankly of the opinion that, so far, the difficult utterances of certain mystical or semi-mystical writers throw more light on the real nature of music than do those of common sense.
Among such writers on music Schopenhauer is notorious; and it is worth while to dwell a little on his speculations, fantastic as they may seem, since they contain an element common to all such interpretations, which does serve to isolate the essential problem of music. In Schopenhauer’s æsthetic the object of all arts, except music, is to lead, by the description of objects, to the recognition of the Ideas (Platonic) whose appearance in multiplicity constitutes the world. All arts, therefore, have a transcendental function; their aim is to reveal to us the Platonic world of eternal essences or Ideas. But they have to raise us to this region via the objects of experience; in that sense they are also, therefore, concerned with the world of appearance and are dependent upon it. The case is different with music. Music is not concerned with the external world either as a symbol or as a reality. It is not even, in Schopenhauer’s language, concerned with the Ideas, but refers directly to that “Will” which, in Schopenhauer’s philosophy, underlies the Ideas themselves. The essence of his theory is given in the following passage:
... so ist die Musik, da sie die Ideen übergeht, auch von der erscheinenden Welt ganz unabhängig, ignoriert sie schlechthin, könnte gewissermaassen, auch wenn die Welt gar nicht wäre, doch bestehen: was von den anderen Künsten sich nicht sagen lässt. Die Musik ist nämlich eine so unmittelbare Objektivation und Abbild des ganzen Willens, wie die Welt selbst es ist, ja wie die Ideen es sind, deren vervielfältigte Erscheinung die Welt der einzelnen Dinge ausmacht.
Or, as he says a little later on, the world may be regarded as embodied music.
It is not likely that anyone will take Schopenhauer’s philosophy of music seriously; and even those who are sympathetic to his general view are not likely to find their sense of the ludicrous undisturbed by his identification of bass notes with the planets, tenor notes with the vegetable world, and so on. The intensity of his response to music and his humourless courage have led him to what are perhaps the most fantastic statements in all his writings. But what is worth noting is that so imaginative and fertile a speculator, because he was genuinely sensitive to music, had to give it a profoundly isolated position in his æsthetic. In so doing we think he recognised one very important difference between music and the other arts. It is true that music is independent of the world of experience in a way that other arts are not. It is true that there is a sense in which Schopenhauer is right when he says that music would exist even if the world did not. We can see what is meant if we compare the development of a “dramatic” piece of music, such as the first movement of Beethoven’s C minor Symphony, with a great tragedy. The tragedy, as a condition of success, must make reference to our experience of life. The ostensible matter of the tragedy, the characters and incidents, must not violate our conception of reality if they are to be accepted. The tragedy must be plausible. Such considerations obviously do not apply to music. It is meaningless to say that the development of a composition must satisfy our sense of probability. Yet there is a meaning in saying that its development seems either arbitrary or inevitable. The analogy that immediately presents itself is a chain of logical reasoning, as in the sustained development of a mathematical theorem. Such development is independent of all experience; the mind is obeying none but its own laws, and is paying no attention to any alien elements. And it is this characteristic of mathematics which seems responsible for the fascination the study possesses for its devotees.
Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world,
says Mr. Bertrand Russell in a typical passage. A strain of romantic eloquence seems, indeed, to be inseparable from the writings of mathematicians on their subject. But the analogy can be pressed more closely. There are elegant and inelegant mathematical demonstrations, those which merely “command assent,” as Lord Rayleigh said, and those which provide a very high degree of æsthetic satisfaction. In these latter demonstrations the mind seems to be moving with more swiftness and freedom; the whole demonstration seems to flower in a natural and spontaneous way; we have the impression of inevitability. Mathematical elegance, as Poincaré has put it, “n’est autre chose que la satisfaction due à je ne sais quelle adaptation entre la solution que l’on vient de découvrir et les besoins de notre esprit.” It is as if there were a mode of living natural to the human spirit, an unadapted life, a life free from the necessity of accommodating itself to the elements, so largely alien, of the actual world. Mathematics is the expression of this life so far as the intellect is concerned. Is it too much to say that music is a fuller embodiment of this free life?
If we are to say this we must acknowledge that more than the intellect is capable of this free life, that there is a logic of the emotions as well as of the mind. This assumption is not difficult to make; indeed, if we reflect on our experience of some compositions, such as, to take the same example, the first movement of the C minor Symphony, it is difficult to avoid making it. And, in considering the matter from this point of view, we may gain some results useful for musical criticism in general. The theme of the movement in question is characteristic of many of Beethoven’s themes in that it does not serve merely as a kind of structural skeleton on which a composition is to be built. In this respect it differs from, for instance, many of Bach’s themes. The theme immediately, in its ominous and arresting quality, throws the mind into a certain state of expectancy, a state where a large number of happenings, but only the happenings belonging to a certain class, can logically follow. As an analogous vague yet restricted initial preparation we may instance the entry of the witches at the beginning of Macbeth. As the music proceeds this rich, but more or less definite, state in the hearer becomes more and more precise, more and more subtle. It is, as it were, explored and shown in all its height and depth. What was pregnant in the theme is exhibited to us in all its extent, its definiteness, and its force. The theme was the entrance to a world. And we have the consciousness of logic, of inevitability, because at no point are we constrained. We exult because we are free; this is how we, too, would move but for our fetters, our alien, arbitrary fetters from which, for this time, we have been freed. And in none of this, unless we have incurably literary minds, are we ever reminded of experience. This life is no life that we have lived or that, on this planet, we could live. Music is as independent of the world as mathematics, but it cannot, like a system of geometry, even be applied to the real world as an hypothesis. It is even doubtful how far the emotions it expresses, when it is merely expressing emotion, correspond to those of real life. The sorrow of the bereaved father is not the same thing as the sorrow of the bereaved lover, but music can express sorrow with thousands of nuances. It is customary to say that the emotions of music are generalised emotions; that its sorrow, for instance, is a kind of common denominator of all sorrows. But the exact opposite seems to be the case. The situations of real life, like the resources of language, are probably too limited to afford correspondences to the immense variety of emotions expressible in music. The musician is as free as the non-Euclidean geometer to create worlds which have no objective counterpart.
It is natural, therefore, in comparing the arts, that we should class mathematics and music together, since they resemble one another by their most intimate characteristics and differ, in these respects, from all other arts. It is worth noting, in this connection, that it is only in mathematics and music that we have the creative infant prodigy. Experience and learning, compared with what we vaguely call “instinct” or “gift,” play a comparatively insignificant rôle; the boy mathematician or musician, unlike other artists, is not utilising a store of impressions, emotional or other, drawn from experience or learning; he is utilising inner resources so obviously independent of experience that, like Plato’s slave, he seems to have brought them with him from some anterior life. And the artistic progress of a musician, if it be a true progress, means primarily that he is making ever more accessible the riches of this inner life. It is difficult to avoid mysticism, or at least Platonism, at this point. But here again it seems to us that Schopenhauer understood something essential. When he says that music, like the Platonic ideas, is an embodiment of the “Will” that underlies all things, he does at least say that what is revealed to us by a composition is something other than the “personality” of the composer. The function of music is not, like that of literature, to illuminate this world, but there is a world it illuminates—a world at least as vast and independent of this one as that mathematical “cosmos” described by Mr. Bertrand Russell.
There is much music, of course, which suggests no such mystical fancies. With most of Wagner’s music, for instance, there is no hint of other worlds, but rather a gorgeous colouring of this one—or of those aspects of this one which excite romantic poets with strong bodily appetites who can assume the background of the vigorous material prosperity of the nineteenth century. Such music is fully comparable with a certain kind of literature; all it lacks is the definiteness of statement, and hence the intellectual clarity, which the use of language affords. It may be even more powerful and subtle than literature can be—Tristan und Isolde expresses certain emotions with immense adequacy. But it is not doing something which music alone can do; and, for that reason, it throws very little light on the peculiar problem of music. For the peculiar problem of music consists in its independence, in its power of transporting us to a world which is not otherwise revealed. To Schopenhauer, to whom both the world and music were embodiments of the same Will, there was a musical equivalent for every experience; and, it would seem to follow, for every musical utterance there is a corresponding experience. The two worlds are independent, but there exists between them, as a mathematician would say, a one-to-one correspondence. Yet he very strangely goes on to accept the theory that a musical utterance is a kind of generalisation of a number of distinct experiences. He points out that the musical setting of a poem, for instance, will serve for a number of similar poems. It is the “kernal” of all these poems which is given directly by the music. But it is equally true that the same poem will serve for several musical settings. When Beethoven, as one of sixty-three composers, composed his setting of Carpani’s poem “In questa tomba oscura,” he probably composed the best setting, since it is the only one that has survived; but among the other sixty-two there must have been many which, in Schopenhauer’s phrase, were expressions of the “kernal” of the poem. The fact seems to be that, unless music is deliberately illustrative, it is not concerned with what is otherwise expressible. That is why musicians are always dissatisfied with “literary” descriptions of music. However good in their own kind they may be, they are always felt to be irrelevant and even, in some way, a degradation of the actual musical utterance. It is felt that they exhibit a certain insensitiveness or lack of taste, as in that curiously popular image which likens twin hills to a woman’s breasts.
As compared with literature, music is abstract. It is independent, as literature is not, of the facts of life. But just as there is some music which approaches to the condition of literature, so there is some literature which approaches to the condition of music. Such literature, while it is concerned with the world of experience, as literature must be, is concerned with that world as symbol and not as reality. Such literature, we might say, is not concerned to illuminate the world of what we here call experience, but to reveal something about the soul of man itself—or, if we prefer scientific jargon to mystical, to deal with the normally subconscious rather than with the normally conscious. Both kinds of literature have been called realistic, but they are realistic from entirely different points of view. Dostoevsky, for instance, regarded the realism of such writers as Zola as trivial. And can Macbeth be regarded as a realistic work, on the basis of the French conception of realism? Macbeth is, indeed, a striking example of the extent to which literature can approach the condition of music. The whole apparatus of the play, the witches, the characters, the incidents, are so obviously not presented for their own sakes, but as symbols through which an overwhelming perception was to be conveyed. Here the fact that the literary artist must accommodate himself to the laws of the real world, that he must satisfy our sense of probability, seems hardly a hindrance. Our sense of probability is, indeed, purposely lulled by the entrance of the witches at the beginning. We are made aware that not the real world alone is concerned. In this respect the supernatural “machinery” of Macbeth performs an altogether different function from that in Hamlet. The whole of Hamlet is perfectly realistic in the tight sense. But the fact that literature must always use symbols differentiates it utterly from music. And just as we have seen that real life may present no analogies to what is revealed in music, so it may happen that the literary artist who has access to a wide and deep inner life may find no symbols, such as are essential to literary art, to convey his perceptions. Mr. T. S. Eliot has stated that Hamlet is an artistic failure because the whole play, considered as presentation in terms of symbols, does not adequately convey the emotions or perceptions we confusedly feel Shakespeare is trying to express. Whether or not Mr. Eliot is correct in his instance, his general thesis is perfectly sound. Even if Hamlet could be re-written so as to satisfy Mr. Eliot, it is still true that there are some perceptions, states of mind, emotions, or whatever one likes to call them, which it is very difficult to believe are expressible in literature at all. Santayana gives a neat but somewhat trivial instance in one of his essays where he says that there is no human incident or group of incidents which can serve as a fitting symbol for pure radiant joy—a sort of prolonged, exultant, celestial state of joy. A shadow, to the mature mind, lies over the brightest and most delightful of life’s happenings. He suggests that a poet who should try to imitate music in this respect could do little but write the word “Joy!” with exclamation marks. He could write nothing else that was unambiguous. And, indeed, a symbol is always ambiguous unless, like the symbols of the mathematician, its meaning is completely exhausted by its symbolic intent. The symbol distracts; it brings with it a crowd of irrelevant associations, and for that reason, even when the symbols are most superbly handled, as in Macbeth, the resultant communication is less definite than with music. But the very great, the immense, importance of literature lies in the fact that it can, partially at least, shape the facts of life so as to make them consonant with the nature of man. If experience can furnish symbols which express the deepest needs and aspirations of the soul, then life can be, at least partially, illuminated. For man can understand nothing which is not consonant with his own nature. The literature which truly illuminates life is the literature which interprets life most fully in terms of our own emotions and aspirations. In this sense not only all literature, but all science, is anthropomorphic. Science is only possible in so far as it is logical. That is to say, the universe can only be understood in so far as its happenings are obedient to the laws of man’s own mind. In its relation to mathematics, where the mind pays no attention to the arbitrary conditions of experience, physics plays something of the same part as is played by literature in its relation to music. Both physics and literature, in their universal function, are concerned with a world which need not obey the laws native to the spirit of man. Such illumination as they can give is dependent upon, as it were, what correspondences they can find. The revelation of life afforded by The Brothers Karamazov, for instance, consists in relating the phenomena of life to the deepest impulses of the spirit of man. Only so does life become in any measure truly comprehended; and it is in this respect that such works differ from those reports on life where we may recognise and assent to everything, but where our comprehension of anything is not deepened. Such works as The Brothers Karamazov may be called philosophic, if we use the word to include something other than purely intellectual understanding.
We have suggested that, if mathematics may be taken as the intellectual analogue of music, then it is not perhaps too far-fetched to say that such a science as physics may be taken as the intellectual analogue of literature, since both are concerned to interpret what we call the real in terms of what we call the ideal, while the two former arts are not concerned with the real. And the question arises whether the arts, mathematics, and music, which are not concerned to illuminate experience, are worthy of serious attention. In the case of mathematics the answer is not doubtful, since it has repeatedly shown itself applicable to real happenings, however little notions of utility may have played a part, or need have played a part, in its creation. Even the most remote mathematical theorems are not certainly immune from practical application. But no such claim can be made for music, and it is for that reason that to some philosophers music is a pleasing but essentially trivial art. To such philosophers music, while it may suggest spiritual profundities, is, after all, saying nothing of any possible significance. The adventures of the soul that it depicts are less significant even than a stage fight. Its one justification is the pleasure it affords; it takes us out of ourselves in a way no other art can do, and after this refreshing interregnum we return to the things that matter. It may be so; we can give no proof that it is not so; we can only say we find the point of view incredible. On this point, again, we certainly find the mystical view of Schopenhauer, if less intelligible, at least more convincing than that of common sense.