The Pythagorean Concept of Measure

ERNST LEVY The Pythagorean ci Concept of Measure MEANINGS TO BE FOUND IN AN OLD | Since THE TIME RENAISSANCE MAN turned from the reading of. the. Book Diye + > 033575 yine to that of the Book Natural, an astounding development ‘of science and PERCEPTION THAT QUANTITY technology has taken place. Their spectacular successes induced, little by - little, an intellectual inebriation culminating, about the turn of the century, (MEASURE) AND QUALITY (VALUE) . © GoExIST AT THE ROOT OF NATURE in a sort of a religious faith in reason believed to permeate man’s nature, thus making him a being molded as well as moved by reason. The year 1914 brought a fearful awakening from the absurd dream. The subsequent events disclosed even to those who had inclined toward an interpretation of the catastrophe as a momentary setback, the existence of a frightening gulf between the state of science and the state of man. It appeared that man's moral being had remained untouched by man's scientific activities. It appeared that modern science, being concerned with only one value, that of “finding out," had in fact become a specialty within the circle of human endeavors—hence, had become humanistically irresponsible. Today's universal anguish is not simply a result of the fear of destruc- I tion. If doomsday came to be announced for tomorrow, our fear thereof LEV>»,E. Wi 1q65 J6A5-F TPYCOMHEANSACESPFUAPRNET. JAONUORUNCEAMELNT 2151-575 ENSLISH JAOURNAL LERVNYSsTMCURAPEINTNS» Languages: lTyeoc: DMeEPsTYATcMPHQArEGYAUOSPISeAECURrSLN’RoISErMTSsY: would have a quality quite different from the anguish we presently experience, because our anguish is coupled with a feeling of senselessness. Science has given us marvellous insights into nature. Technology has made our life more comfortable in many ways. However, both science and technology have failed to make our life meaningful. The destruction which threatens us at this hour assumes the character of a stupid accident, showing none of the moral implications of doomsday, except perhaps to one who chose to detect the workings of a diabolic principle in the very fact that humanity did run away on the road of intellectual arrogance. The loud accent we are putting on the scientific attitude does not imply an unawareness of the existence of other values. People still profess to be religious, to respect philosophy and art. It is apparent, however, that these values have been largely dissociated from “real” life. They have been pushed unto the fringe of life—they have become “Sunday values.” In fact, they have been “museumized,” just as beauty in our cities finds itself “parked away” at safe distance from a prevailing and overwhelming ugliness, Necessarily, the inner dichotomy of our being is mirrored in our creations which, in a thousand different ways, point to the rift existing within us. In this our present situation, help cannot be expected from the outside, P

| We must heal our inner being, and we might well begin ‘by trying to bridge the gap in us, which in the last analysis is one that separates the realm of quantity from that bration-form, frequency. In general, he will define tone ag ‘ something produced by periodical vibrations of the air striking the ear-drum. These are all quantitive terms, of quality. The physicist mathematizes the musical impression. The Is it necessary to say that whereas the postulate of a change of attitude, of a reversal of values, can be clearly stated, there exists no automatic cure, no panacea for the evil that besets us? Is it necessary to explain that when a goal is passionately striven for, there will be no want of ways to approach it? But also, is it not evident that a chosen method should be pursued till the goal is reduction of qualities to quantities, to that which can be measured, is characteristic of the scientist's approach. Whitehead says: “All science as it grows toward perfec. tion becomes mathematical in its ideas.” And further: . reached? It is in this spirit that we wish to direct the attention of the reader to a trend of thought endeavoring to bridge the gap from which we are suffering. That trend, originated, according to tradition, by Pythagoras (6th cen‘tury 3.c.), founded anew and developed in our time by Hans Kayser, takes as its starting point the musical tone, proceeding toward an acoustical philosophy, an understanding and evaluation of things through the sense of description which ignores me and you and him, and also ignores sight and touch and taste and smell.” (1) Thus, the qualities of phenomena disappear in the scientific processing. I expose the physicist to a tone, and he hands me back a figure. He really does not explain the tone, but describes some measurable counterpart in the physical world. That abstraction from quality, that voluntary —one is tempted to say, heroic—self-limitation has made hearing (akroásis). It might at first seem an odd idea to choose the musical tone for an epistemological tool. Let us say at once that philosophy, like art or mathematics or religion, presupposes an individual gift, a talent. Any particular way of philosophizing also presupposes a talent, not only because the talent will determine the subjects about which to philosophize, but also because it will incline one towards the use of some philosophical tools rather than others. Now the Pythagorean viewpoint presupposes the gift of musical sensitiveness. Those who do not possess a ‘musical instinct will neither understand nor admit Pythagorism. A philosopher to whom the world would be the same were music absent from it, will never be a Pythagorean. But there will be Pythagoreans as long as there are musical philosophers, or philosophical musicians. “For example, it might be asked at dinner: ‘What was it which underlay my sensation of sight, yours of touch, and his of taste and smell’? the answer being ‘an apple. But in its final analysis, science seeks to describe an apple in terms of the positions and motions of molecules, a i li; The essence‘of: the Pythagorean doctrine may be ex: plained with the aid of a string stretched between two fixed points, and. some ‘device for the division of the science great. It has also made it one-sided, and its inherent disregard for quality, that is to say for value, proves increasingly dangerous to man. So far the gap between the two worlds, as exemplified in the difference between the musical and the physical terms, has not been bridged, but rather the sharpness of the contrast has been more nicely stated. We are now going to produce tones of “higher” pitch (physically, of increased frequency) by shortening the string, while avoiding the alteration of its tension. This is done simply by applying the bridge to the string, thereby dividing it into a sounding and a silent portion. By moving the bridge along the string, we may obtain as many different divisions as we wish. We proceed systematically, and one of two methods may be chosen. According to the first method, we divide the string successively into two, three, or more parts, corresponding to the series A of integers, and we then observe the resulting intervals! According to the second method, we divide the string by . ear, seeking to obtain certain intervals arranged in 2. string, as for instance a bridge. Such an apparatus is hierarchical succession, and we then observe the result- ‘called a monochord, and has been used from time immemorial for acoustical experiments. ing ratios. It will be noticed that a musical principle of order can only be a principle of value, whereas a scientif- . The stretched string, when plucked, will emit a tone, ic principle of order can only be one of size, Let us proceed according to the second method. The _ which for convenience we shall call “C,” its absolute pitch being of no concern to us. The plucking of a single first two intervals, in order of their musical importance note on a string is certainly a most primitive experiment, are octave and fifth. Allowing our ear to be the judg® and yet it provides food for thought. We set the string and continually testing the string while moving the into motion, and hear a tone. What is a tone? It is, first bridge, we shall notice that the sound of the octave of all, an impression received through a sense, the ear. By occur when the string-length is exactly one half. Conti?” - analysing our impression, we shall be able to distinguish uing to push the bridge in the same direction, we next several attributes: loudness, timbre, pitch—specific qualities, like bitter, hot, red, or soft. The physicist, however, string length. Now, knowing that one-half the string” will define those qualities in his own terms: amplitude, vi. 52 W! obtain the fifth above the octave at one-third of the to length corresponds to the octave, we can predict tha

oath will all correspond to octaves of the fundamental ‘sone “C.” In short, we may say that all powers of 2 prooctave? What our musician says about the octave holds, « e multiply the fig‘ duce octaves. Similarly, if we dividor course, for any interval, any chord; in each case we ma 3 ‘obtain the tone “G,” in various octaves. On the other cal phenomenon. Thus we introduce quality into quai tity, value into size. We are musicalizing mathematics, “ ‘pre 1/8 corresponding to the fifth “G,” by powers of 2 (2/9, 2/8, 413 ~~ + 118» 1/6, 1/22, 1/24...),we shall always “hand, it is easy to see that multiplying and dividing by =" owers of 3 will correspond to a progression by fifths: - ‘1/8 C 1/3: G 1/3? D... 3°/1C 81/1F 92/ıBb....In other =. words, we may predict the tone corresponding to any constituted by powers of two, three, and their . ascribe to a figure the value of the corresponding mus to speak. The figure, from a mere symbol of quantit becomes a symbol of both quantity and quality. It b comes what should fitly be called a number in counte distinction to a figure. Specifically, we call such an “e pressive number” a musical number. It is a bridge b ‘* combinations. The tone of any following prime number, tween quality and quantity, between the worlds of valu: ! however, will have to be determined by experiment, or at © Jeast its presumed hierarchical position will have to be ‘ tested. The musical meaning of the prime number is the pirth of a new tone. This shows that the value (tone) and of sizes. It represents the Pythagorean concept ı measure as one both haptical and normative, the non being more particularly defined as musical. ‘cannot be derived from the measure (string-length) any Thus the antithetical concept-pairs, soul-intellect, fee ing-thinking, intuition-reflection, are synthesized in tl “more than the measure may be derived from the value, musical number as a psycho-physical symbol. Its tw \ In fact, we did not derive one from the other. We only sides—musical value and haptical measure—are connec : put the elements of two orders into a one-to-one correCES ed by a relation which is exactly and spontaneously aj prehensible. Those who find it: difficult to accept il ‘We are now ready to bridge the gap. To accomplish Pythagorean viewpoint might ponder the followir this, a mental attitude is required which many among us quotation from Viktor Goldschmidt: “Our capacity ı i” will be reluctant to ‘assume, because it is contrary to the apprehend the outside world may be explained thus, th: scientifically determined mind. The definition of that attitude is simple enough. It consists in this, that we there are processes in our mind (microcosm) which a: analogous to those in nature. These psychological pro must be willing to ascribe equal reality and equal importance to quality and quantity. On the physical side, the esses we call natural laws.” (2) Now, there does not see: to be any good reason why this epistemological postula tone is represented by a figure. It is a symbol for a mashould be monopolized by the haptical outlook. If I a terial, or “haptical” measure (from “haptikos” in Aristotle, meaning “able to Jay hold of”). On the musical side, tone is a qualitative experience. We have heard the physicist give his explanation of the tone. Let us now lismit quantity in the outside world and a psycho-physic. parallelism which makes that quantity apprehensible, stand on equally firm (or equally uncertain) groun when projecting into the outside world the value appr ten to the musician with equal seriousness. His argument might run as follows: words, and once more taking the octave as an exampl hended by the same psycho-physical process. In oth: Take the octave as the object of our discussion. Here Why should there be more truth in the numerical rat we have two tones which are distinct from one another, “one to two” than in the musical “octaveness"? WI different from one another, and yet in a way identical— should I not claim that the psycho-physical parallelis so much so that indeed we call them by the same name. is equally valid for quality? If an affirmative answer : That particular identity of the octave, that sameness and these questions is granted, it still remains to be prove yet otherness—truly a remarkable and unique phenomethat our value-concepts are not merely an emanation : our soul. If they are, then they would stand only in tern of an aesthetic outlook. Their value would remain stric non—how is it possible to apprehend it except by the musical ear? If 1 asked you to name something which is the same and not the same, would you not be tempted to ly humanistic, and could not be paralleled in the out scorn the answer, and accuse me of mysticism? If I then world. If, however, it could be shown that the acous: had you listen to an octave, and ‘asked you to describe it, ratio one to two, does it not seem perfectly legitimate to cal value-scale manifests itself in a corresponding valu scale in nature, then the postulate of a psycho-physic parallelism would stand not only in respect to quantit but also in regard of quality. One of the tasks of hz monics is to find out whether such a corresponden exists. Results obtained so far permit us to affirm that identify that ratio with the octave, to let it partake of does. would you not say that the two tones composing it were alike and different at the same time? Now, having discovered that this remarkable interval corresponds to the ratio which comes right after that of identity, namely the that specific octave-quality? Wouldn't it, then, appear One might object that an optical apprehension of tl that the octave is an “explanation” of the ratio one to world could be postulated with the same right as we pc

” tulate an acoustical one. That is true. However; the ear tones and string-lengths, between that which can be - possesses certain advantages over the eye which seem to evaluated and that which can. be measured. However, make it a better and more readily available tool for the his attitude toward the results of the experiment was | building of a bridge between quality and quantity. We significantly different from that which is characteristic * have remarked a moment ago that the relation between of the modern scientist. Whereas the modern scientist : tone and number is apprehended “exactly and spontadiscards the value (tone) to retain but the haptical meas. neously.” We may, exactly and spontaneously, apprehend ure (string-length), Pythagoras’ enthusiasm was aroused by ear a mathematical ratio. We notice, for instance, rather by the fact that mathematics could be heard. The whether an octave is “perfect,” as we say, or even slightly ear confers value-meaning on the numbers; thus the pro- “out of tune.” We are thus able to divide a string acportions varied in music will also be varied, not only for cording to certain ratios, by the sense of hearing. The eye the eye, but for the cosmos of our soul as well as for the is unable to perform a similar operation in respect to cosmos of the universe, both being subjected to the same color. Indeed, the ratio between two color frequencies laws. “The same numbers that appeal to the ear appeal cannot be recognized by the eye. Another, though less in a marvelous way to the eye and to the soul.” This senimportant, observation to be made concerns the range tence, ascribed to Pythagoras, very well sums up his poof frequencies, which for the eye is restricted to about one sition. octave. Thus the ear appears more immediately useful The influence of Pythagorean thought on Greek art for our purpose, and is even likely to provide a basis for and philosophy was considerable; it accounts to a great ‘the study of optical value-scales. Attempts in this direcextent for the special consideration enjoyed by music, tion, from Newton on, have been rather crude and a bit which was believed to contain the secret of the world. fanciful. It is hoped that the application of the more Now it should not be supposed that the Pythagoreans, strict methods of harmonics will yield better results in confining themselves to general ideas, had built their the future. cosmology upon the simple hint provided by the string Aristotle says of the Pythagoreans, “They assumed experiment. On the contrary, they had worked out a systhat the elements of numbers were the elements of all tem which was based on a table obtained through an things.” The assumption of number as the first principle, interpolation of the division and multiplication series of or “world-stuff,” constituted an immense advance over the string—an operation, by the way, strikingly similar to the hypotheses of the earlier philosophers, Thales had the setting up of the Periodic Table. That Pythagorean thought that the world-stuff was water. Anaximenes had Table has been reconstructed by Albert von Thimus said that it was air. It is at once apparent that number (1806 - 1878) from a diagram and indications found in Iamblichus. Thimus’ monumental work, entitled Harhas two advantages over all other assumed first principles: a great variety of things can be reduced to number (possibly all of them) and a system may be built. Modern science has borne out the Pythagorean hypothesis, more than any Pythagorean would ever have dared to monikale Symbolik des Altertums, (DuMont-Schauberg, Köln, 1868) served Kayser as a starting point for his own work. Whereas Thimus had been interested mainly in the historical aspects of Pythagorism, Hans Kayser (1890 dream. Burnet says: “It is impossible not to be struck by -1963) recognized at once the intrinsic significance of the resemblance between this doctrine and Dalton’s thethe tabula pythagorica for modern thought. ory of chemical combinations... the diagrams of modern The Pythagoreans themselves had been led to remark ©. “i stereo-chemistry have. also a curiously Pythagorean apable concepts by the Table symbolism, which among»; many other views suggested to them that of helioce® — pearance. We sometimes feel tempted to say that Pythagoras had really hit upon the secret of the world when he said “Things are numbers.’ ” (3) To modern science indeed, “all things are numbers.” trism, two thousand years before Copernicus. All through *: the Middle Ages, Pythagorean thought, transmitted by . such men as Aristotle, Vitruvius, Boethius, may be tra in various forms. Number symbolism as embodied in the _ But from these numbers, the soul of things has been excluded. They do not represent the whole, but only its proportions of Gothic cathedrals belongs to the Pyth28 _ quantitative, haptical side. They provide only the sizeorean tradition as well as to St. Augustine's fundamet* . measure of things, disregarding the value-measure. By introducing musical significance into the haptical number, we reestablish measure in its Pythagorean meaning as both size-measure and value-measure. The day Pythagoras hit upon the discovery of the relation between pitch and string-length was the birthday of Western science. His experiment was indeed a scientific one. He succeeded in proving the existence of an exact mathematical relationship between the pitch of DI 54 tal views on music and poetry. In Kepler, (1571-1630) harmonical thinking produced its last achievement _ Kepler's Third Law was indeed found through a pr. cally harmonical method—a fact which modern scientisis _ unfortunately shrink from recognizing. With the advance of experimental science, the emancipation of num be: from value progressed rapidly, and by the end of seventeenth century the haptification of science had come pretty general.

| It is interesting to follow that process of haptification, ppal transition from the medieval to the modern attitude, d to consider some of its consequences. SE The theologian-philosophers of the Middle Ages had È grected a grandiose cosmic monarchy in which, toward «he end, the world became a petrified system of symbols. We may Say, in Pythagorean language, that value was works of the Creator ....In fine, 1 am inclined to believe some general laws of the Creator prevailed with respect to the agreeable or unpleasing affections of all our senses; at least the supposition does not derogate from the wisdom or power of God, and seems highly consonant to the macrocosm in general.” (6) Little by little, the archetypal form-concepts, which «© over-emphasized, the phenomenon neglected. Such an _ gttitude is characteristic of exclusively deductive thinkmay be regarded as a sort of grand-style hypothesis, were replaced by so-called working hypotheses. In former ology. The world is an organism ordered by God. times, the accent had been on the form-concept. The phenomenon was expected to conform. If it did not, “suo danno” as Torricelli says—too bad for the phenom- . One of its outstanding.qualities is a sense for mor- Knowledge starts with God-given, that is to say archetypal form-concepts, proceeding thence down to the phenomenon. The obvious danger of such a speculation “from © above” lies in a trend to blindness toward the evidence ‘of facts, or in a temptation to distort the facts to make them fit the preconceived ideas, the religious-aesthetic enon! Now, however, the stress was shifted to the phenomenon, which was asked to speak through the experiment. True, some a priori assumption was still needed. But if it proved false, “suo danno”—too bad for the hypothesis, which then had to be changed to fit the newly acquired "archetypes. On the other hand, the danger of inductive + 8Stzence lies in disregard for morphological value. There The failure of science to tackle successfully questions is a trend to uniformize, to make everything appear of of morphology—a failure intimately tied to its very methequal importance, lastly, to become purely descriptive, od—is one of the more obvious consequences of the a position of renunciation, or of resignation, which many modern scientists have deliberately taken. It is a logical purely haptical position. It is clear that a mechanistic or utilitarian view will never do when faced with the task outcome of the haptical view. In 1928, the embryologist of explaining, say, the forms of living organisms, Cell- Joseph Needham wrote, “Materialism is the only philosophy upon which science can get to work...and if I do not believe in it for a moment as a man, I recognize division alone will not provide an explanation. The its fundamental importance as an experimentalist.” (4) It is impossible better to express the dichotomy of modern man, that “double life” mentioned before. The founders of modern science, however, did not take such an extreme stand. They were still aware of the necessity stones do not explain the architecture. The tones do not explain the symphony. We read that “they say that Pythagoras had been the first to call the universe Cosmos— that is: order, harmonical structure.” (7) Now order means diversification of uniformity and unification or harmonization of diversity. Of the three terms—uniformity, diversity, harmonization—we disregard the last to unite the deductive and inductive attitudes. In a letone, which, on the cosmic level, raises questions which ter addressed to Michelangelo Ricci, on February 10, 1646, the physicist Torricelli, disciple of Galileo, writes: “Whether the principles of the doctrine de motu be true may be forever unanswerable. It may be noted that someor false is of very little concern to me. For, if they are not fied with the last one true, one has to imagine that they are true according to what we had assumed, and then one has to take all the other speculations derived from those principles...in a ‘to uniformity. The scientific progression from effects to times the extreme terms are mixed up, in the sense that the first term (construction unit) is erroneously identi(supreme morphological principle). The main business of science is to reduce diversity causes is descending, not ascending. Morphology, howpurely geometrical way. I imagine or assume that some ever, has to show the flowering of diversity from uniforbody or point is moving up or down with the known proportion, and horizontally with equal speed. This mity. As Warrain puts it, “... for every scientific problem whose object is to reduce the diversity of the real assumed, I then say that all Galileo and myself have said world to a unity of origin there exists another scientific will follow. Now if the balls of lead, iron, stone do not question whose object is to report on the actual selection observe those assumed proportions, that’s just too bad for them; we shall say that we simply won't talk about them of the realities among the possibilities allowed by the generality of the law.” (8) In other words, after having reduced reality The statement is delightfully humorous—yet its essential seriousness cannot be doubted. On May goth, 1698, Newton writes to John Harington, to some primary “world-stuff’ of the ancient building-stone—the Greek philosophers, the “energy-quantum” of modern physics—it remains to be “I have, in former considerations, examined these things, explained why the world-stuff agglomerates into such and such phenomena. We might even ask: If the world and wish my other employments would permit my furis but a set of “variations” on a “theme,” why did the ther noticing thereon, as it deserves much our strict scrutiny, and tends to exemplify the simplicity in all the Creator go at all into the trouble of creation? Why did the world become phenomenalized? These are questions

. which cannot be answered “from below.” To investigate those last spiritual strongholds. If we don’t take heed, ‘them, we need morphological concepts, which are valuewe might, in the end, resemble the man ‘trying to propel concepts. the boat in which he sits, by blowing into the sail. One such concept in music is, for instance, that of the so-called perfect chord, or triad, contained within the first six numbers, or senarius (prime: 1-2-4, fifth: 3-6, third: 5). Octaves being considered identities, the senaric values extend, first, over the numbers 8-16-32 ... 12-24-48 Our postulate, then, must be the reunion of philosophy and science, a reunion in which neither. partner will be the handmaiden of the other. Such a reunion should eventuate in the establishment of a philosophic-scientific body of knowledge fitly termed “Normative Science,” . 10-20-40 . . . , and further, over products between a science where philosophy tests its findings by the meth- “senaric values, e.g., 9 (3.3), 15 (3.5), etc. The senaric ods of science, and where science acknowledges the selecprinciple is widely and variously embodied in nature, for tion-principles of philosophy. instance in whorled flowers. A. particularly striking and beautiful example is the passion-flower (passiflora coerulea), where a tripartite pistil is combined with a quinquepartite stamen, the octave of which is shown in the petals. Such a form-tendency can only be understood when paralleled with the corresponding musical phenomenon. Another consequence of the haptification of modern science is its separation from philosophy. Newton is still speaking, not of “natural science,” but of “natural philosophy.” With the disappearance of the value-concept from haptified science, the “Universal Harmony,” as it was still called in the seventeenth century, came to be regarded as of no more concern to science, and was handed over to the philosophers. Thus a break between There is nothing utopic in such a postulate. On the contrary, it is the separation that ought to be looked upon as artificial. All creative processes of the mind, from mathematics to art, are characterized by a peculiar interplay between deduction and induction, the idea and the experiment, the method “from above” and that “from below.” Now if we imagine on one hand a valueconcept foreign to science, and on the other hand a haptical concept, foreign to philosophy, what we should look for is a means of translating one into the other. Such a translation should be based on phenomena in which the attributes of quality and quantity are united * in such a way as to afford an exact and spontaneous apprehensibility of their relationship. At the present time, we know only one such phenomenon, though science and philosophy occurred. One might be tempted to classify it under the heading of “division of labor,” others may exist: the musical tone, as symbolized in the and it may perhaps be considered an occurrence parallel to that brought about by the Industrial Revolution. Its real significance, however, lies in this, that it is a break It would seem that for some time there has been a musical number. within man himself, a division of his innermost being— growing awareness of the necessity for a change of mind. Strangely enough, it has been facilitated, while probably the dichotomy to which we have pointed before. The not prompted, by the recent revolutions in physics—as effect of the separation on science has been described: much, perhaps, as it has been promoted by the anguish of our times. The symptoms are numerous. Whitehead’s science has become “amoral.” The effect on philosophy is just as disastrous. Squeezed between religion whose source of divine inspiration she doesn’t share, and sci- _ ence whose methods she cannot use, philosophy has in Science and the Modern Man is easily among the most significant ones. In a speech delivered July goth, 1928, on the occasion of the unveiling of a monument at main turned critical and historical, abandoning all Kepler's birthplace, Sir Arthur S. Eddington said, “It is: “¿claim to the royal position she once held. That is the only in the latest years that we have gone back to some situation brought about by the tremendous development thing like Kepler’s outlook, so that the music of the - of one branch of human pursuit, and the gradual etiospheres is no longer drowned by machinery.” The introduction of the Gestalt concept not only into psychology, - lation of all other branches. In these days of collective ‘man-made catastrophes, no one dare say that the probbut into other fields as well, is also significant: “People ‘ lem is an academic one. And if it were? The disparaging - banded together in an economic community are more term "academic”—to wit, purely theoretical—does it not than a simple sum of individual beings. As ‘members of depict our state of mind, a haptical mind, valuing only one another’ they form a biological organism whose that which is “practical” in an immediate and utilitarwhole is always greater than the sum of its parts.” (Dewtan sense? Does it not, in substance, mean that ideas are ey and Dakin) (9) In general, publications disclosing 4 nothing, haptical successes everything? True, we despermore or less conscious or avowed Pythagorean viewpoint are no longer a rarity today. Future generations, howately cling to a few "ideas”—liberty, democracy, justice— because we obscurely feel that in the last analysis it is ever, will recognize Kayser as the first modern pioneer Precisely such “academic” things that keep us going and of harmonics, in both an historical and a qualitative acting. Yet we should be warned: a civilization that sense, To date he is the only author to have treated stakes everything on haptics may finally abandon even harmonics methodically and comprehensively. His first

(Orpheus) was published in 1924, while his first =ic book appeared in print in 1932. (10) co there is a good chance that once more we are continual contact with values—though we learn the story but little by little, and may never learn it whole—is so highly satisfying that our patience actually grows as the randing on the threshold of a renewal in man’s thinkwork advances. We begin to understand that we do not renewal which would be characterized by the genhave to erect at once a comprehensive system to soothe jdea of the “Pythagorean Measure.” The road will se long, but we have learned to bridle the impatience hat in former times prompted the erection of great deguctive systems. The lesson of haptical science has not lost. Normative science will be built up gradually. And it appears, surprisingly but naturally enough, that our impatience. It is enough that each stage of the slow advance confirms the existence of immutable values. But those are thimgs everybody has to experience for himself. “What we need is not to ‘know’ the truth, but to experience it.” (ung) (11) Thus we transform facts into values. References ae A.N. Whitehead, An Introduction to Melia | 8. . Warrain, Francis, Essai sur P'Harmonices Mu nds es Victor Goldschmidt, Ueber Harmonie und Compli- | = © Johann Kepler, Paris, Hermann & Cie,-194858: 9. E. R. Dewey and E. F. Dakin, Cycles, N. Y., Holt, cation, Berlin, 1901. 8 John Burnet, Greek Philosophy, London, Macmil- - de lan, 1932. Joseph Needham, Man a Machine, N.Y., Norton, 1928. 1947. 10. akustischen Weltbildes, Berlin, among which should be mentioned Grundriss eines Systems der harmonikalen Wertformen, Abhandlungen zur Ektypik harmonikaler Wertformen, 1938 Wissenschaft. 7. Quoted from Hawkins, General History of the Science and Practice of Music, London, 1776. Plutarch, quoted from Kayser, Abhandlungen (see and 1946, and Lehrbuch der Harmonik, 1950, all 11. ref. 10 below). Ernst Levy, who is at present Professor of Music at Brooklyn College, began his career at the age of five in the Conservatory of his native town, Basel, where he studied with a number of eminent musicians. Before coming to the United States permanently in 1941, Professor Levy served as head of the Piano Department at Basel Conservatory, concertized extensively in Europe, and founded the Choeur Philharmonique de Paris, one of the outstanding Lambert Schneider, 1932. Kayser wrote some twelve principal works, 5 Quoted from Hugo Dingler, Zusammenbruch der 6. Hans Kayser, Der horende Mensch]Elemente eines published by Occident Verlag, Zurich. C. G. Jung, Seelenprobleme der Genenwart, Leipzig und Stuttgart, Rascher, 1931. peared as soloist with the Boston Symphony, and has given piano recitals in Boston, New York, and other cities. Professor Levy has written extensively, principally on musical subjects. At present, in collaboration with Dr. Siegmund Levarie, he has in preparation a textbook of acoustics. He has also written fourteen symphonies {of which three have been performed in this country), in addition to a cello concerto and works for orchestra, for: at Bennington College, the University of Chicago, and the the organ and the piano, choral works, and chamber music. His piano music has been published by Boosey and Hawkes and Associated Music Publishers, and Mercury has published his choral setting of Four Folk Songs from Massachusetts Institute of Technology; he has as well ap- Switzerland. choral organizations of the continent. In this country he has taught at the New England Conservatory in Boston,