ЭкспериментИскусство_2011

how to arrange them in an appropriate manner, i.e., to create such a composition which would form the integrity?

When building such a composition, the most acute problem is to organize links between rather remote elements of the structure which pretends to be integral. In those cases when the object is perceived as a sequence of events (e.g., when the object is a literary work), the most important problem is to establish links between the initial element of the structure – and its final element. The simplest version of such a link is the so-called ‘ring structure’: at the end of the work those elements are used, which were used at its beginning. [So, here we again deal with the phenomenon of repetition, either full or partial.] Due to such a device, the work can be perceived as something ‘closed,’ and hence, completed. That is why ring composition became typical for most lyrical poems (Zhirmunsky, 1975). As well in music there exists such a device as ‘reprise,’ i.e., the repetition of the exposition (initial musical theme in a sonata).

[Apropos, women feel the need for such ring structures very sharply. For instance, they prefer to use identical (or similar) colors for their foot-wear and a hat, or a scarf. Due to such a ‘rhyme’ between the top and the bottom, all the intermediate components, including parts of the body, are perceived as a unity, a kind of a ‘poetic image’. As well, women resort to the help of many other means capable of generating the integrity of their images, as far as this task is very important for them.]

In addition, the requirements to the structure of the work, are not restricted by coincidence (full or partial) of its beginning and the end. To become the integrity, the internal structure of the work should be ‘centralized’, i.e., it should possess a certain ‘core element’ through which most other elements would be connected with each other. Such centralization provides maximal economy of recipient’s efforts when dealing with any object, irrespective of its destination. That is why the phenomenon of centralization is inherent in objects belonging to various spheres: techniques, commerce, science, religion, etc. (see Golitsyn & Petrov, 1995).

Fig. 5 presents two versions of a certain system (e.g., such a system belonging to the sphere of techniques, as the telephone net):

a) before centralization;

b) after centralization, when a central ‘core element’ is introduced.

204

Fig. 5. To the phenomenon of centralization: a system of the telephone net at its early stage (a) and after introducing ‘central exchange’ (b)

It is easily seen that the first version (a) is not very economical: in order to connect with each other n elements (e.g., n telephone subscribers), it is needed to use

links [e.g., 8 subscribers require 8 (8–1) / 2 = 28 links]. The second version (b) occurs much more advantageous: to connect n elements, only N = n links are needed [e.g., for 8 subscribers, we need only 8 links connecting each subscriber with the ‘central exchange’, i.e., 28 / 8 = 3.5 times less!]. Due to such economy, centralized structures occur very advantageous for various systems, including works of art.

Of course, centralization assists to perceive any work as an integrity. That is why to be perfect, any work should possess a ‘central element’ around which are grouped most other elements. It is especially evident in works of fine arts (see, e.g., Arnheim, 1986; Bouleau, 1963), where in most cases a painting possesses a certain compositional center. All other elements of the painting are connected with this ‘core’ element. As well it is known that the color structure of each painting should be also ‘centralized,’though the form of this centralization is rather specific: all color elements of the painting are referred to the so-called ‘color-and-light standard,’which is nothing else but the white element (representing the sunlight – see Gribkov & Petrov, 1996; Petrov & Gribkov, 2001).

Quite similar are the evidences of this phenomenon in literature, and first of all in the plot of prosaic works. Thus, it is simply advantageous to have a certain personage as the central one (e.g.,

205

D’Artagnan in several novels byAlexander Dumas), so that all other personages would be interconnected through this ‘central exchange’. As well, the role of such central element can be played by a principal line of narration (e.g., the development of a war).

However, beside the link between the beginning and the end, there exists another problem dealing with the general contours of the entire structure of the work. In particular, this problem seems to be important in the case when the central element of the structure is evident. For instance, usually in a painting there exists an evident compositional center; so the task is to find the best place for this central point. Another problem relates to the ‘frame’ of the work, meaning its proportions (e.g., the ratio between the height of an architectural construction and its width). For most cases of such a kind, a rather general regularity is valid: appropriate relationships are subdued to the so-called ‘Golden Section’ proportion. [Information approach to the advantages of the Golden Section, together with considerations concerning the limits superimposed upon this regularity, see in: Petrov, 1997. A review of empirical data about the Golden Section, its pro et contra, see in: Hoege, 1997.]

According to this regularity, exactly proportion .62 is optimal (i.e. carries maximal information) when dividing a certain fragment into two parts. Proceeding from the classical standpoint, this proportion is optimal because it responds to perceptual integrity of the object, and hence, it carries ‘harmony.’ For instance, in the case of a rectangular form, its lesser side (a) should relate to the larger one (b) as the larger one to their sum:

Because of such ‘harmonizing’ impact, pictures often contain this proportion in its various ‘faces’ which was shown in numerous empirical investigations (see, e.g., Cline & Konecni, 2000). For instance, in many paintings their compositional center is slightly shifted from the middle, so the painting occurs divided in the ratio close to .38 / .62. [Besides, this shift, as well as the phenomenon of Golden Section in general, can be explained also by such a property, as the above mentioned threshold of perception, which, in turn, is caused by the volume of human operative memory; for the last one the information approach gives the value 8 units (Golitsyn & Petrov, 1990; Petrov & Mazhul, 2008; Petrov, 2007, 2008): T = 1 / 8 = .125. Hence, in order to provide minimal asymmetry, i.e., distinction from the middle, the form should possess a proportion either .5

206

– .125 = .375 or .5 + .125 = .625, both values being very close to the Golden Section (in fact, the proportion .375 is nothing else than the proportion 1 – .375 = .625). So, the information approach evidences in favor of high self-consistency of our mental systems, quite analogous to the self-consistency of the Universe treated by famous Antropial Principle!]

Many other visual objects, especially works of architecture, possess contours of the Golden Section. As well, some literary works and musical ones possess their culmination point which divides the work in proportion .38 / .62 (see, e.g., Voloshinov, 2000). Recently an attempt was undertaken to unite several aspects of the Golden Section: a group of artists organized in California a theatrical action in a hall with ‘golden’ proportions, scene with exactly such proportions, play divided (in time) in accordance with golden ratio, etc.; the performance was a great success (Havill, 1976). So the Golden Section may be considered as one of important structural properties providing perceptual integrity of works of art.

Meanwhile, the Golden Section is not the only proportion advantageous for aesthetic perception. There exist the whole set of rather advantageous relationships; this set can be obtained on the basis of the so-called Fibonacci series: in this series each next item is a sum of the two previous items: 1; 1; 2; 3; 5; 8; 13; 21… Each time, the relation of two consequent items is advantageous for perception. The Golden Section is simply a partial case, which is realized when

These relations are also used in many cases, in dependence of concrete ‘boundary conditions’ of perception. For instance, ‘minilocal’ harmony can be achieved when using the simplest of these relations which correspond to the consonances of the musical scale, namely:

1 / 1 = tonic; 2 / 1 = octave; 3 / 2 = fifth; 5 / 3 = large sixth; 8 / 5 = little sixth.

Nevertheless, the Golden Section is of most importance for numerous applications in artistic creativity: it responds to a certain ‘ideal’limit of the Fibonacci series. This situation is realized in such conditions of perception when its time is not restricted, which is typical for visual objects. [On the contrary, when acoustic perception, the time of perception is principally restricted, so in these conditions

207

other ratios are usually realized – see Golitsyn & Petrov, 1995. This thesis is in contradiction with the above examples of the Golden Section in music and literature.]

3. Global integrity: a struggle against a giant number of elements

At last, very often we deal with rather specific problem which is caused by the multiplicity of elements (their ‘numerical strength’). In fact, sometimes it is desirable to organize the perception in the regime of the operative memory capable of keeping up to 8 elements (Golitsyn & Petrov, 1990). Meanwhile, usually much more elements are to be integrated. Sometimes the number of elements is gigantic (e.g., words in a novel), and it is desirable that they would form a ‘harmonious whole.’ How to overcome this difficulty? – Two ways seem to be possible here.

The first way is to ‘aggregate’ primary elements, i.e., to come to certain ‘secondary elements’ each of them consisting of a group of several primary ones. In general, this way is realized in many phenomena of mental life, especially in language (because it requires rather high speed of communication and hence, its procedures should be based on the recipient’s operative memory). Thus, a group of phonemes form a morpheme, a group of morphemes being a word, a group of words forming a phrase. Due to such aggregating, rather large amount of primary elements can be perceived almost instantly.

In the arts, exactly this phenomenon is used, especially when it is needed to provide instant perception, i.e., mainly in visual objects. For instance, if an architectural construction consists of many identical elements (e.g., many rows of windows of a sky-scraper), it is desirable to divide this massif into several groups of elements, in order these groups would form a certain integrity. In large prosaic works, a flux of the narration is usually divided into paragraphs, then into chapters, etc., and due to such ‘aggregating division’ the work can keep some features of integrity.

Meanwhile, there exists the second way, which is prospective when the amount of elements is giant, and the potentialities of their aggregating are exhausted. This way is to arrange the elements in such a natural manner which would not lay obstacles to the natural perception of the work. How to do this?

Considering the lowest level of organization of most works, we do see there numerous ‘micro-elements’: color spots on the plane of the picture surface, words constituting the speech flux of a novel,

208

etc. Surely, these micro-elements should serve the above structural purposes, i.e., they should be used to realize ring composition, centralization, and due proportions of the structure of the work.

Sometimes these micro-elements can be divided into several classes, and the number of such classes is also giant. For instance, in a color structure of a painting we can single out the following color elements: white, grey, black, red, orange, and so forth, as well as their numerous hues. But in a literary work, the number of elements coincides with the number of different words used (namely words, or more strictly, morphemes are those minimal elements which are capable of carrying sense of any speech communication). Their nomenclature may vary from 40-80 units in a sonnet to many thousands in a novel. What might be the relationships within this world of classes of micro-elements?

Again we shall split the problem into two sub-problems.

*) Sub-problem of proportions: what should be the number of micro-elements belonging to different classes? The answer is well known: in order to function optimally, the structure of classes constituting the system, should be subdued to the so-called Zipf’s law. [Its different versions are also known under names of Pareto, Lotka, Mandelbrot, etc. – see, e.g., Martindale, 1995. It relates to the so-called ‘hyperbolic distributions,’ belonging to the class of ‘stable non-Gaussian distributions’ – see Petrov & Yablonsky, 1980.] Its essence is in the following: if we order all the classes in accordance with the number of their micro-elements, so that the rank r = 1 would be ascribed to the most numerous class, the rank r = 2 to the less numerous, and so forth, then the number of microelements nr belonging to the class with r-th rank, is determined by this rank:

C and γ being constants characterizing the system considered. Turning to logarithmic co-ordinates, we come to

i.e., the linear dependence with the slope (–γ).

Exactly such were empirical dependences built for many perfect works of art, belonging to its different kinds. Fig. 6 presents appropriate dependence for a novel ‘The Ulysses’ by James Joyce: the number of words, in function of their ranks (see: Zipf, 1949; Petrov & Yablonsky, 1980).

209

Quite analogous dependence for painting is presented by Fig. 7: areas of different color elements used in the picture ‘Over the eternal peace’ by Isaac Levitan (1894), in function of the ranks of these colors (Orlov, 1980). About 95% of color elements (out of the totality of 341 different colors) are described by a linear dependence (6).

Both examples presented show very good agreement with theoretical prediction. Such linearity was observed in many other works of different kinds of art, including musical and literary ones (see, e.g., Orlov, 1980). In literary works there exists a giant number of words (sometimes up to several millions ones!), so it is easy to realize appropriate statistical investigations. And they did confirm the Zipf’s law, which occurred valid for many outstanding works of art, each of them being an artistic integrity. However, if the work is not an integrity, the Zipf’s law occurs to be not valid, and we cannot observe linear dependences. (Usually a break, or fracture is observed in the dependence discussed.) For instance, each of 4 volumes of the novel ‘War and Peace’ by Lev Tolstoy, showed absolutely perfect linearity, evidencing in favor of the integrity of each volume; nevertheless, the novel as a totality occurred to be not an integrity – according to the character of its dependence. So, subduing to Zipf’law may be treated as a criterion of the integrity of the work.

Meanwhile, evidently, Zipf’s law should be valid not only for the entire set of those micro-elements which constitute the struc-

210

ture of the work. It may include some substructures which can be more or less autonomous, if they respond to more or less independent aspects of the work’s functioning. Hence, we may suppose that each such substructure possesses its own distribution of its microelements which would be subdued to Zipf’s law. [This situation reminds of the structure of a crystal, where beside the ‘genuine’lattice of atoms, some additional ‘sub-lattices’may exist, for instance, sublattices responding to different admixtures present in the given crystal, though sometimes in quite negligible quantities. Nevertheless, usually atoms of these admixtures form regular structures which are named sub-lattices.]

In fact, some empirical investigations confirm existence of such substructures. Thus, a literary work may contain several such substructures, one of them relating to the so-called ‘color life,’ i.e., perception of spectral properties of the surroundings. So, the distribution of appropriate micro-elements constituting this ‘color substructure,’ is supposed to be subdued to Zipf’s law. Figure 8 presents such a distribution of colors in the novel ‘The Demons’ by Fyodor Dostoyevsky. Though the rank dependence presented is not clearly linear, its slope is close to the value ( 1.0) typical for most linguistic systems. Moreover, three the most frequent colors (red, white, and green) coincide with Russian national color triad dominating Russian painting (Gribkov & Petrov, 1996). Hence, we undoubtedly observe the evidences of ‘color substructure’ in this novel. (Other Dostoyevsky’s novels show similar regularities.) Possibly this substructure not only reflects the creative process of this author, but influences upon the perception of his novel as a certain integrity.

**) Sub-problem of interactions is the second worry of our consideration devoted to micro-elements: what might be consequences of co-existence of constituents meaning their impact in the integrity of the work? One may suppose that a certain kind of joint appearance, being repeated regularly, would ‘tint’ the work, adding a definite ‘hue’ to its mood. But to do this, such situations are to be frequently met, which in turn requires that:

–the number of appropriate micro-elements should be rather large, they would be numerous;

–a certain ground should exist for dividing the micro-elements into different categories, so various particular combinations of members of different categories, would form invariants possibly

creating due ‘common mood’ of the work.

211

Both requirements are satisfied in works of prose. Really, prosaic texts usually consist of many micro-elements: words. As well, such elements as words, can be divided into several grammatical categories substantial for perception: nouns, adjectives, verbs, prepositions, etc. That is why exactly in literary works, such interactions would be expected. And they were really observed, at least in several cases. We shall present three examples of such a kind, appropriate phenomena being based on the concept of the so-called ‘transparency’ of

prose (see in detail: Petrov, 1994, 2007a; Koptsik et al., 2004). The basis of the model is the contraposition of two stylistic poles:

–‘non-transparent’style, oriented on concrete images, i.e., sensual associations generated by the text perceived;

–‘transparent’style, having nothing (or almost nothing) in com-

mon with concrete images or associations; it is oriented on a ‘game’ of abstract symbols, creation of a definite mood, etc.

To realize each of the above styles, there exist at least three methods, i.e., devices of creating appropriate type of perception.

The first device deals with controlling concentration of microelements belonging to definite parts of speech.As it was shown theoretically and proved empirically, the inclination to ‘non-transparent’ pole can be realized by increasing the share of nouns and adjectives, to the detriment of the share of verbs. On the contrary, to be oriented on ‘transparent’ pole, it is desirable to increase the share of verbs to the detriment of nouns and adjectives. So, the degree of stylistic orientation of each prosaic text (at the level of words used) can be estimated by the ‘coefficient of transparency’

v, n, and a being the number of verbs, nouns, and adjectives, respectively. This coefficient was calculated in texts written by 6 most

212

eminent Russian prose-writers of the 19th – 20th centuries. These calculations permitted to build the evolutionary trajectories of ‘two lines’ of Russian prose: ‘transparent’ line and not-transparent one (Petrov, 2007a, pp. 418-420).

The second device deals with controlling concentration of verbs at the ends of phrases. Really, in a recipient’s memory mostly that word remains (after perceiving a phrase), which was placed at the end of a phrase. So the transparency of the phrase is strongly dependent on the transparency of its last word. As a result, the share of phrases with verbs at the ends, can also characterize the transparency of the text (at the level of phrases). The usage of this device was proved by calculations of this share in prosaic works by Alexander Pushkin and Mikhail Lermontov, both writers being contemporaries, but presenting, respectively, ‘transparent’ and ‘non-transparent’ lines of Russian prose. In agreement with theoretical predictions, this number was about 27-28% in Pushkin’s texts, and only 17-20% in texts by Lermontov, the difference being statistically significant at the 1%-level (Chi-square criterion). Besides, the absolute values of these frequencies also occurred in good agreement with theoretical predictions (see Section 1): this device was used by Pushkin not very often, not crossing the threshold of awareness (A ≈ .30 – .35), so its usage was not perceived as importunate (Petrov, 2007a, pp. 420-421). Due to such difference, appropriate ‘coloring’ of each prosaic work as an integrity, occurs possible.

[Once upon a time, the author of this text, made an experiment using this device in his own literary creativity (see Petrov, 2002a). One of my short stories, ‘Nita Zho’ (about medieval life) written together with painter Alex Melamid in 1965, seemed to be too dreary, monotonous. Then we changed the word order in most phrases – keeping all the words of the text, but putting verbs at the ends of phrases. And suddenly all the story occurred transfigured, and even became quite mysterious!]

At last, the third device controls the order of episodes of prosaic text. For instance, if a writer organizes his/her text in such a manner that the episodes of the narration would occur intermixed, fully disordered – then the narration would be perceived as almost ‘nonrealistic’; at least, the text cannot generate concrete associations, it seems to be absolutely ‘transparent,’ remaining (after reading) only a certain ‘general mood.’ Exactly such was the situation in the short story ‘Light Breathing’ by Ivan Bunin. (The psychological analysis

213