DIMENSIONS AND OVERALL FORM
Let us first of
all attempt to conjure up what the Sonata for Two Pianos and Percussion
must have meant to an educated bourgeois listener, for example at the Basle
premiere of the work in January 1938. 76)
MOVEMENT I (NEGATIVE WORLD)
For to describe the basis which holds steady
the entire universe is no child's play
Perhaps never before had there been heard such alien and disturbing sounds, shuddering tremolo and sul ponte effects, ghostly harmonics and magical reverberations on the gong, "atavistic" and almost nightmare-like timpani glissandos, in a word such tendencies that when taken together, could signify a symbolic departure into a primitive, animistic time, a "totem and tabu" society among ghosts, shadows and demons - if permitted we could say to the border of waking and sleeping, or perhaps an irrational subconscious world, where the basic feeling is of fear and immeasurable anxiety, where a touch is enough to give a fatal shock, or where levels burst open releasing superstitious, evil and dark powers to the surface. So became music once more a shaman ceremony, a magical means of evocation, of summoning forth, of controlling the gods, a technique for wizardry, and so for the bourgeois West became the whole adventure a celebration, in so far as, in the words of Freud: "celebrations represent the ceremonial breaching of what is forbidden".
When the first movement was being examined - in contrast to the gleeful third movement - it was the dream-like psychological layering that seemed the most obvious, the continuous expansion and contraction, the covering of layers by each other (e.g. bs. 133 and 161), great openings up (bs. 6 and 10), and penetration through dream-walls (bs. 17 and 330-331), along steps (bs. 208-17) and across strange bridges (bs. 80-83) - among scenery and signs which, even though they are unknown, arouse the impression of being already experienced, occasionally with a quite startling effect, as when lengthened figurations appear in the closing theme 77)(bs. 105, 135, and 200!) like primitive phallic symbols, or in the opening bars of the work, where we descend to such depth that contact is made with the completely general and suprapersonal, the world of the collective. These are nameless motions, with reason given the impersonal pronoun "Es" in psychology (Cocytus symbol). During the course of the first movement we often encounter a technique familiar from the field of hypnosis and magic where through the unbroken repetition and intensification of a single motive, and the concentration of the attention on a single point, it explodes in an electric discharge, a piercing flash, and a short circuit (bs. 195-217). For in the first movement the themes are being continually forced into a ceaselessly contracting and expanding "rotation", a returning into themselves with wheels driven by a timeless law and destiny. Among these ambulatory systems we are led along as in Dante's Inferno, progressing from circle to circle, and if we pass over from one to another, all the earlier layers act on us unconsciously, turning in their circles without aim or relief. The suspicion grows within us (readily confirmed by our studies of other works) that some kind of deep and basic connection exists between the rotations and the axis system, and that the two things merge naturally into one another. For example, how typical it is that Moussorgsky creates the purest axis system in the "clock scene" of Boris Godunov: "...blood rises to the head and returns to the heart: everything rotates...". The circle continually bending back on itself, allowing no release from its prison, is the closest relative of madness and obsession. A great many times does this motif appear in the psychology of trecento culture (literature and painting). Perhaps it is after all no coincidence that the techniques found throughout the first movement: organum, fauxbourdon and mixtures, stem from medieval music. This makes even more relevant the fact that (as mentioned in the introduction to the analysis of the finale) the system underlying the basis of the first movement (golden section) applies to the "oldest layer" of the music.
All things considered,
it would appear that behind this archaic psychological intent lies a general
disgust of civilization, caused by a continual craving for the banished
laws of nature, and the burdens of an artificial way of life: the desire
for an uncritical and uninhibited, so to say impartially naive way of life,
and for a cultic renewal of life. It would be good to return to the sun,
the wind, the elements; to have celebratory occasions tied to the soil
- Walpurgis night and spring sacrifices - in the words of Baudelaire "how
pleasant to cling to the memories of bare primeval times." To return our
pathologically undermined world to something wholesome, intensely vital
and very energetic: this perhaps is the reason why in the years 1936 and
1937 the Music for Strings, Percussion and Celesta and the
for Two Pianos and Percussion were written for percussion instruments
(of course, just for the Basle listener referred to above) for the first
movement is in its essence a negative world, desiring a resolution,
a resolution it will find in the gleeful third movement.
"Think of Infinity; shudder at it and
bow your head before it"
Artists in the West found in the inscrutable ancient culture of Asia something tuned in every respect to the contemplation of nature, in which the greatest desire of man was to open itself to nature, and immerse its individuality in it. For these artists the architecture (art) of the East from the start sets out to be symbolic of the world of the cosmos. It is just these two motives, the cosmic architectural principle and the idea of the symbolic expression, which we tend to find in the structural relationships and harmonic organization of the Sonata. Are we justified in suggesting that the basic architectural forms (pyramid, stupa, ziggurat) deriving from the sun and the constellations display, with their enclosed centralized layout facing the four points of the compass, some inner mental kinship with the forms we are dealing with, and the symbols of the axis system? We must, however, remove one misunderstanding that surrounds Bartók's art.
Thomas Mann in his novel Doktor Faustus, which portrays the great crisis of Occidental art, mentions as one of the decisive characteristics of cosmopolitan music its Pythagorean propensities. In the works of the hero of the novel, the musician Leverkühn (in reality Schönberg), we hear the music of the cosmic spheres with its icy, clear, glacially transparent, wryly dissonant, but at the same time unfathomably unearthly and alien sounds. Leverkühn seems to handle the teeming world of the stars, spiralling mists, cosmic clouds and galaxies as though his acquaintance with them derived not from books, but from personal experience, and due to some "intimate" initiation regarding the universe, he has expanded the concept of citizenship to immeasurable proportions. These attempts all form part of a tendency perceptively described somewhere as the "dehumanization of art". The artist, who was not able to partake of "citizenship" in the society of his own time, now sought shelter in the security of the "star-lined crystal spheres" (Baudelaire). From this sprang the need to discover the Pythagorean propensities of music: the attraction of music towards the spheres, binding the two together with pure numbers and proportions.
It would be rash to claim that Bartók's music has much in common with such esoteric "music of the stars"; scholarship however cannot conceal that in many respects it did join these tendencies, not through pride or a lack of humanness, but on the contrary, from the purest humanity. For it is one thing to desire homelessness, and another to suffer it. One thing to deny something and another to emphatically affirm it through a double denial. It cannot be coincidence that it is precisely in these movements that the lonely voice of Bartók is heard at its most intense.
While studying the proportions of the Sonata the cosmic workshop of Dürer's Melancholia continuously hovered involuntarily before our eyes, with its astral diagrams, geometrical devices, the crystal form, and the mathematical formulas. As the work progressed, so this comparison became all the stronger, because (as is numerically supported at the end of this chapter) this laboratory gradually grew and modernized to include wave theory from atomic physics, and the diagrams of hyperspheres and curved spaces, with which Bartók, on the evidence of letters and biographical data, was closely familiar. All the signs point to the fact that the Sonata acquired its measurements and final form under the influence of Eddington's book "The New Pathways of Science" (1934), Bartók's thoughts turning to the idea of writing a musical Macrocosmos - curiously enough exactly at the time (1937) when he finished composing Microcosmos. Later, when we study the organization of this Macrocosmos, we shall see why it is through the medium of this "nature music" that the outer two movements, two polar opposite worlds, must meet and become reconciled.
There is an essential difference, however, between Bartók's cosmos and that of "Leverkühn-Schönberg". Schönberg's cosmos is "no-man's-land" (Pierrot Lunaire), an imaginary area above and below reality - Laputa land; Bartók's cosmos, on the other hand, can be compared to Dante's, in which, as in the 3 parts of the Divine Comedy, each with its 33 Cantos in terza rime and terzine, everything has its place in the line, but this world is a real world, containing experiences and characters like those which the artist encountered daily in real life. "One feels in Dante’s Inferno" writes a critic of the Divine Comedy "as if one were visiting an Italian town of the period." It is possible that in a few hundred years time somebody will write: "One feels in Bartók's cosmos is if one were visiting the Europe of the period."
And if someone
were now to ask what we hear at the end in this music, then the reply must
be that it is this movement, and in Bartók’s oeuvre the cosmic adagios
generally, that most markedly conceals within itself the voice of humanity,
even of love. Not for nothing did Bartók position this movement
in the centre of the Sonata for Two Pianos and Percussion, the centre of
MOVEMENT III (POSITIVE WORLD)
Coloured like gold, translucent, glittering,
I saw a Ladder reaching up so high,
that to my sight it was past following.
At this point the reader is justified in interrupting with the question - what does all this have to do with the gleeful finale of the Sonata? The overall form of the Sonata rests on the polar unity of the outer movements: in this connection the "gleeful" third movement forms a natural contrast and resolution of the "demonic" first movement (the two worlds linked by the "principle of nature" - the nature music of the second movement). In Western art of the thirties, however, the complementary opposite of the demonic was not the gleeful, but intellectual humour: the ironic, which in extreme cases showed itself in the above described cynical and frivolous forms. The sureness of Bartók's morality found expression in the fact that he chose the more optimistic path of the two (it is probably not just coincidence that in this Sonata the Mephisto tone is found in the secondnd half of the firstst movement: in the fugue, i.e. in the negative movement of the work). On the other hand, Bartók often made use of this distorted mocking tone exactly in order to express the crisis of European culture. It is true that in his hands, this language sounds bitter and vitriolic, using grotesque features of Voltairean, Rabelaisean - indeed Swiftean proportions, the unsparing irony of which left their mark on the whole of Bartók's activity. There is nothing, however, binding the content of the finale of the Sonata to this bankrupt music, on the contrary, it is characterized by an unbridled and uncontrollable good humour which at most can only be compared to the finale of the Music for Strings, Percussion and Celesta - yet still the only way of giving some explanation for the effect of this explosive movement is to say that Bartók sharpened the blades of his humour to such a point with these grotesque and burlesque pieces that they "kill not with anger, but with laughter". Bartók really does teach us here to "laugh with the laugh that sparks creation - to answer the spirit of hardship with dances and songs of mockery - to both laugh and be sublime" - and this unending, limitless yes, in this movement "dances out of itself: all good things are laughing."
A characteristic point in the third movement, however, must be dealt with separately. We might call it a kind of "wallflower", its justification stemming from the fact that it jumps to the fore out of the context of the final liberation at the last minute as a restraining force, in the hurdy-gurdy style solos of the coda (bs. 379-394), whose banal drumming (from b. 387) together with the hiccupping timpani strokes 78) here cloak not just would-be primitiveness, but the full irony of the evocatory side-drum and glissandos on the timpani in the first movement, and similarly in the texture, the deliberately banal use of the "theme and inversion" technique; in other words a pretended affectation and false naivety, in essence parodying and mocking his own style, using the "Dorfmusikanten" style, but somehow beyond anything dilettantish: with a strange grimace the style "falls on its knees before itself." (We do not know whether the mentioned hurdy-gurdy theme's reminiscence of the ironic ending of Haydn's Oxford Symphony is intentional or not.) It is interesting that this restraining force is found just before the final ending not only in the Sonata for Two Pianos, but appears with a similar significance at the end of the Fifth String Quartet (meccanico!) and the Divertimento (pizzicato theme).
Is this mockery really directed at the first movement? Let us compare the codas of the outer movements. The first movement ends with an apotheosis-like climax with the first subject in lengthened-out lines, while the coda of the finale, more accurately the burlesque coda, ends with hurdy-gurdy versions of the first subject. A striking feature of the instrumentation of the last movement is the complete absence of the magic timbre of the gong; also absent are the expanding and contracting processes which provided the psychological character of the first movement, these being replaced by constant shapes. Where the demonic element does even so appear, parody serves as the excuse, as though by doing this he wanted to counterbalance or "disparage" all the dangers conjured up in the first movement. Hence in bs. 117-136 with their approaching outbursts of drumming, the listener understandably thinks he is hearing the subterranean sounds of the first movement, and during the general pause filled with lightning flashes of tension he waits with baited breath for the crash of a thunderbolt; the resolution however is inverted, not just thematically, but with an inversion of mood as well: the music is light, sticks out its tongue, has Till Eulenspiegel-like phrases, is serenade-like and divertimento flavoured. The whole closing theme of the finale is simply the preparation for the great joke that takes place in bs. 134-140. But is not the whole thematic material of the finale, with its spark-flashing runaway character, an express denial of the undestructibly revolving melodies of the first movement? It is as though the rockets of the third movement (first subject), or for example the firework themes in bs. 207-228 whirling on the spot 79), then spattering everywhere, were harvesting triumph over these karmic melodies.And is it not striking that genuine diatonic melodies, Western harmony 80) and construction using 8+8 bar periods are only employed in the third movement, while the first movement in its melodic and harmonic structure follows the chromatic system and the GS, and morphologically follows asymmetry? We can amplify this by saying that the characteristic minor third symbolism of the first movement takes place in pursuance of the chromatic system; the conspicuous fourth-based motivic construction of the third movement on the other hand, in pursuance of the diatonic system. The first subject of the third movement is evidence, along with other works, that for Bartók, the most perfect diatonicism was expressed in the form of the complete acoustic scale. 81) This seems to be symbolized by the fact that at the beginning of the first movement there is a chromatically "closed" melody, and at the beginning of the third movement an "open-ended" acoustic melody. It can further be established that true polyphony, where each voice is independent and sensitively written, permeates only the first movement, in contrast to which in the third movement, in addition to the purely harmonic technique, there is often pseudo-polyphony - i.e. a succession of schematically composed canons and quasi-entries (e.g. at bs. 18, 160 and 229, timpani and xylophone). All of this of course chimes in with the intellectual programme of the work, for in this context the chromaticism and the GS, the narrow minor third, the essentially dissonant polyphony and the asymmetry form a "demonic" world, while the diatonicism, the motivic texture based on wide fourths, the homophony and the division into symmetrical periods are all used to serve a "gleeful" world.
The most important observation has been left to the end, namely the contrast between the opening and closing bars of the whole Sonata. In bs. 1-5 of the first movement we have already discovered the motive of "sinking" and "descending", immersion in a dark and helpless element. Using analogy, we saw in it the Inferno, sinking "below the earth", into the catacombs (Moussorgsky), immersion into the depths of the well, which "we shall find is deep; should we not say it is unfathomable?" The silent tremors of the timpani indeed detach themselves from the negative pole of life, from "pre-consciousness, the mythical maternal womb", and from perfect night, which already has a key ("f#", cf. The Night's Music and Bluebeard's Castle). In bs. 405-420 of the third movement, all these omens are exchanged: above the movement and the whole work dance unrestrainedly the ostinatos of the "flipped" cymbal (according to the instruction, played with the finger nails) and the thin drumsticks ("very light and thin sticks on the edge of the side-drum"), flitting down the sunlit paths with lithe dainty feet and gentle caresses (C major, the opposite pole of f#) almost weightless and intangible, dissolving into light and air: "children of wind and light".82)
We must therefore regard the "ends" of the Sonata - its opening and closing bars - as negative and positive poles, which we can describe as pairs of opposites - depth and height, heaviness and lightness, passive immobility and unrestrained activity, inflamed darkness and cold light, the unconscious and sparkling wit - or as the attraction of two poles, almost suggesting the analogy with a magnet between whose poles is created a bouncing current of energy.
We must, however, begin further back. With Einstein's theory of relativity the universe became enclosed, the closed nature of space being due to its curvature; according to this the universe resembles an enormous globe, more accurately the surface of a globe (hypersphere). A straight line starting on the surface of the globe eventually meets itself; if we look through a telescope pointing into infinity, we ought to see ourselves. And I would add here another thought. According to Eddington, if we were to initiate a debate competing to construct a new universe, even then we would have to prescribe the same numbers, as "constants". The constant factors of nature are not the results of an arbitrary choice, creation increasingly having the appearance of a purely mathematical operation. One of these constants regarded by science is the number 256, whose basic element can be reduced to the most simple number, number 2:
256 = 162 = 44 = 28
The component parts of this number coincide in many respects with the time construction of classical music; the constructional basis of classical melody is after all the periods consisting of 4, 8, or 16 bars each, without doubt because of the connection with the number 2, whose multiplied powers - as we saw above - allow the greatest possible number of alternatives and opportunities for symmetry.
Let us now return to the Sonata. Ideally the listener should bring the final bar of the work into an associative connection with the opening of the work: the positive and negative poles make contact, "close" and from the aspect of form an electric circuit is set up. Both "ends" of the Sonata are shoreless, open-ended (!), and this open-endedness practically provides the enclosing, an enclosure in which the ends meet in opposite directions, i.e. in a "circle", or some similar shape. The dynamic plan of the whole work also confirms this idea:
Where polarity is so much supported by the open-endedness of the work, where the negative and positive poles point so much into boundlessness, there is no alternative but the example of Einstein's curved space: "those marching in one direction must meet those marching in the opposite direction." That this is how Bartók conceived the structure of such a Macrocosmos can be proved mathematically. The value (time-value) of all the notes in the whole Sonata amounts to 804 whole notes. If we stick to the symbol of curved space, then the above number corresponds to a hypersphere whose radius is the number 8, well-known from classical formal construction, and whose diameter is the number 16:
4r2p= 4x82p= 256p = 804
The core of the formula is the number 256, which has been mentioned before, and from which a scientific hypothesis (Fürth) has deduced the number of particles in the universe.83)
With this discovery we have found the key to the proportions of the first movement. When we were studying the symmetries (v. chapter entitled The Proportions of the Sonata) we observed in the structure of the first movement units of 56 (57) and 28 (28+28=56) bars. Now 28 bars falls within the boundary of 256 (!) eighth notes (because there are 9 eighth notes in one bar; 28x9=252). The dimensions of the second movement are 256+256+256 eighth notes. The "nature-sounds" (!) marked on the sketch (v. analysis) appear in the places where the + signs are (bs. 31 and 62). The exposition of the third movement (bs. 5-133) and the section extending to the recapitulation of the first subject (bs. 5-260) also seem to be connected with the number 256.
The Sonata's measurements therefore are not arbitrary and subjective, but are determined by inner necessity in the material. With the addition of the 804th whole note the open-endedness of the work comes to a close and the (804th+1) unit - to use Eddington's words - is left stuck in "nowhere".
Nevertheless we are not for a moment left in any doubt that this closed sphere conceals negative tendencies - negative in the Nietzschean sense, for in it "all reality is curved, a circle biting itself", in fact more than that, existence is itself a circular wheel - "the will of the ring coils within it". And when we at the same time discovered the complementary pair of this negative power and its adversary, the active force directed at the positive, which continually forces these closed systems - "circles" - to open up, to "straighten up", then in this endeavour we recognized what is probably one of the deepest mysteries of Bartók's music (cf. all that has been said concerning the closedness and openness of the tonality, interval symbolism, rhythm, proportions and formal construction). In the course of this particular symbolism, the motivic texture at the beginning of the Sonata depicts a circle, while at the end (and in the first subject of the finale as well) it depicts a straight line: 84)
Incidentally, a good analogy suggests itself here. The symbol of Dante's Inferno is the "circle", of the Paradiso an "arrow". The concentric circles of the Inferno narrow downwards to the lowest point, Cocytus, those of the Paradiso, by contrast, open out into the heights of the infinite Emphyreum. 85)
Here for the first time it becomes clear that we have in fact to consider two different interpretations of the work, which diametrically oppose one another. The first (the hypersphere) presented the work from its architectural aspects, as a taxonomic whole - as if projecting the form into space - which led us to the concepts of Einstein; the second is the "musical" analysis which - emphasising the formal sections as they take place in time - described the path of the Sonata "from darkness towards the light"along a straight line of development. This analysis took the lead over the first, as in its closing bs. the work left it behind; to be exact, it "swept" out of the circle with newfound lightness, and sparkling with laughter aspired towards an ultimate freedom, losing all encumbrances which bound it to the enclosed sphere. In other words, the first framework signified a geometrically and ideologically circumscribed world for Bartók, while the second contained an endeavour directed specifically against being enclosed, providing the ultimate meaning of the work: to break out of the "cosmic prison", and become completely free of it. This ultimate liberation and joy in this way gains a double emphasis: on the one hand the opening up of the prison cell, on the other, following the programmatic thinking of the Sonata, the triumph over the "darkness" - having grown out of it.
* * *
Let us finally narrow down the circle of our investigations, and put the results of the numerous completed operations into a single framework of logical equations. The "law of balance" in an equation, as we know, counterbalances factors which have opposing tensions: addition and subtraction, multiplication and division and so on. "Without the principle of contrast" writes Nyestyev, "realistic, life-like, in particular monumental and comprehensive compositions, cannot be created". Where there is no positive and negative, there is no tension, and where there is no tension, we are deprived of a living musical form. Thus artistic form demands tension, and tension demands effect and countereffect. Homogeneous factors are in themselves not viable, and have no significance. When factors with contrasting values clash and combine, liberating energies in turn are changed into a form-building agency. The starting point for an aesthetic based on economical principles might be this: life's processes are of necessity energetic in nature. Energy, however, can only originate from the tension of opposites. The process of compensation between tensions is the resulting energy itself. To have no tension or stimulus is to be dead. The mechanical foundation of musical form is polarity, the creation of some differences of tension, and not by chance is form creation called a "power game". Enjoyment of a work (quantitative) is none other than the resolution of tension in the direction of the slope of energy. The energy itself, however, is at the same time a quantitative factor, its quality depending on the manner of its presentation. In this sense the tonal and other formulas used in this analysis bear the same relationship to Bartók's music as the formulas of classical harmony do to the art of Haydn, Mozart and Beethoven. These too acquire their quality, or "content" from the way in which they are used.
If we accept the
conjecture that the Sonata for Two Pianos was intended as the culmination
of Microcosmos, as a "Macrocosmos", then we have to conjecture as well
that in the conscious organization of its material, the work competes with
its younger partner, Microcosmos, which in detail and in miniature embraces
the whole repertoire of Bartók's technique, and which was only waiting
for a summing up that would gather its elements into a single conception.
Consider for example the connections that exist between the outer movements
the Sonata regarding tonality, use of intervals, rhythm, motivic texture,
structure, and not least intellectual content:
Let us consider further how consistent Bartók is throughout in his opposition of "pole and counterpole" (elsewhere "main branch and sub-branch"), in his use of "theme and inversion" and of complementary rhythms ("basic figures"), of proportions and dynamics with positive and negative values, of this continuous double-visaged "yin-yang" technique - as it is prescribed in the canons of oriental art. Heraclitus could hardly have imagined a more perfect Macrocosmos, when he thought he had recognized its secret in the system of regulating opposites.
Perhaps the reader will forgive here an interjection. The above pairs of opposites, like the left and right sides of an equation, are of equal value to one another, and like in an equation, if one side is cancelled, the other also disappears, with the result that they cannot exist without each other. (It is interesting that the concept "left" and "right" cannot be defined absolutely.) They postulate and preclude one another, forming a unity and a contrast. They deny the specific, the relative and the variable (i.e. they aspire towards the universal, the absolute, and the constant: they are not concerned with exceptions, they do not depend on subjective caprice, they cannot change their make-up), their verification stemming from the equals sign that is wedged between their elements. Absolute order, which thus finds its centre within itself "in the absolute unity of its nature", extends to all the elements of the music, making it impossible to remove a single detail; the difference between a part and the whole is only a question of magnitude (changing any note would bring about a collapse of the whole system). I do not know if the reader can also feel how exacting is Bartók's music in dealing with these questions, but the analyst cannot even so avoid drawing the decisive line that divides Bartók and the formalists. For example for Schönberg a logical formula always meant a "magic square", an order fixed by the stars, which "allows no notes, not one single note, which do not have a formal function in the structure of the work: free notes are no more!" - so say the followers of Schönberg. For Schönberg, form is an end (an aim); for Bartók, a means (a medium) that serves expression and communication. Bartók not only knew that f# and c are counterpoles, but also what it means to travel from darkness to light; he teaches us not only to see dialectical antithesis in chromaticism and diatonicism, but how to get from the Inferno to the Paradiso. All we have to do is translate in sequence the structural formulas of the Sonata into the language of thoughts and experience, and the work immediately outgrows the framework of formal analysis. It turns into such a dramatic picture of the philosophical contradictions and tensions of its time, its programme presenting such a deep - following the spirit of the work we might say "light-craving" - desire and need of the "path leading to the good" (Gorky), that the most an analysis can do is give a pale reflection of it.
79) It is highly likely that the major second series of bs. 207-226 are to be interpreted as the diatonic variants and counter images of the minor second series in movement I (cf. from bs.42 and 174). (See next paragraph.)
81) End of Cantata Profana: "only from a cool spring"; movement IV of Music, metamorphosis of the chromatic fugue theme (b.204); 4th String Quartet, movement IV as the diatonic variant of the chromatic theme of movement II, etc.
82) It is perhaps not too far-fetched to state that in the starting bar of the work (timpani), only the spatial-material element finds an outlet, as it is not articulated metrically - while the last bars of the work only have a temporal, metric dimension (side-drum ostinato).
83) Using circularity as a symbol, it is noteworthy that 804 is identical with the circumference of a circle whose diameter is 256 (!), or taking area, 804 is the area of a circle with a radius of 16. Let us quote Jung's observation at this point: from time immemorial, humanity has used the symbol of the circle whenever it has wished to express unity, fullness, completeness.
Similar symbols can be discerned in folklore, myths, linguistic creativity, sayings. The circle and circulation in general is the symbol of life for the Chinese. In Gothic architecture (the Gothic arch) being "squeezed erect" is materialized. - In Freud's study of dreams, anything that is closed (box, wardrobe, pit, room, mouth) obtains a peculiar meaning, and so does everything that stretches long (whip, stick, snake, sharp dagger, tower). It is not surprising that Bartók had such a great liking for Greco, that he admired his elongated figures almost transfigured into flames. On the "Mount of Olives", closedness suggests dreaming, openness suggests wakefulness (the duality of the unconscious and the conscious). The motivic circle and line have the same meaning in the tenor aria of Bach's Matthew Passion ("wachen" - "schlafen"). In a physical sense, circular motion required constant tension due to the permanent change of direction, while linear motion is tensionless due to inertia. It is no accident that one has long been associated with the idea of life, that other with that of death. The circle organizes, the line disorganizes. By the same token, the closedness of the GS system and the openness of the acoustic system also belong here: the golden section only applies to organic materials, while the acoustic (overtone) system derives from the movement of inorganic matter (e.g. string): one has been verified physiologically, the other physically. An intriguing analogy may be cited: Th. Mann opined that the intellect, death, cognition and irony are Apollonic qualities. - These associations closely correspond to what is summarized in the tables about movements I and III below.