Introduction Music is an important component of the information society. It is known that music can render both positive and negative influence on human being and society. However mechanisms of this influence in many respects remain unclear. Network science application to the analysis of pieces of music is an effective approach of modern cognitive technologies and can play an important role in understanding not only problems of music and other kind of information impact on a person, but also other global problems facing of the present society. The reductionism as research method dominating in modern science, assumes that the studied system can be understood if properties of its elements are described. Music belongs to number of so-called complex systems which don't manage to be described and understood formally by means of such approach. Methods Since the end of the last century for studying of complex systems the new effective instrument of research - the theory of complex networks [1] have been developed. Nodes in such networks represent elements of these complex systems, and links between nodes – interactions between elements. Such networks form a peculiar backbones of the relevant complex systems that allows to model such systems in general as a whole and to overcome some shortcomings inherent to a reductionism. Results and Discussion The purpose of this presentation is to show that works of music can be described as multilayer networks, which structural and dynamic properties can throw new light on the nature of music as complex system. A musical melody can be easily converted into a network structure if we take the musical notes of all possible durations as its nodes. It can be easily calculated that the number of nodes in one voce in such network shall not exceed 1800. Indeed, the number of piano keys equals 88; if we multiply it by 20 – the number of all possible time durations of notes (halves, quarters, eights e.t.c.) - we get the number of 1760. Connections between nodes (notes) in the network are established according to chronological principle: if note I starts to sound at the moment in time T, when note J at this particular moment finished to sound, there is a connection between the respective nodes of the network [2]. In our approach the same notes in different octaves belong to different network layers. On an example of " Prelude in A major " by Chopin will be described relationship between melodic and harmonic structures of music. Each of these structures can be represented as networks, and music - view as multilayer network. We constructed directed network structure for melody of F.Chopin’s “Prelude A-Dur”. Figure 1 shows melody network of the piece. Figure 1. Melody network structure for the Frederic Chopin’s prelude A-Dur N7 . The thickness of links corresponds to time of repetitions of appropriate musical intervals. (See PDF version for the Figure). Harmonic structure of a musical work has qualitatively different structure of relations between notes and should be described as a separate layer. We have created a network of harmonic structure of the Frederic Chopin’s prelude A-Dur N7 as the second layer. If a piece of music has polyphonic nature, it is easy to describe this musical work as multilayer network too. Multilayer structure of musical works is the consequence of multilayer organization of the human brain networks. It is assumed to briefly discuss the possible mechanism of emotional influence of music from network science standpoint. Conclusions Our understanding of complex systems is always associated with incompleteness of information on their structure and properties. A quantitative measure of incompleteness of information on system is its entropy. Recently in the theory of complex networks methods of calculation of entropy both simple monolayer, and multilayer networks on the basis of generalization of the most fundamental concepts and methods of statistical physics are developed [3, 4]. We are developing entropic approach for investigation of music as complex system now. Acknowledgments This work is performed with assistance of the Russian Humanitarian Scientific Fund (grant N 14-04-00369) References and Notes Caldarelli, G.; Scale-Free Networks: Complex Webs in Nature and Technology. Cambridge University Press: Cambridge, England, 2007 Liu, X., Tse, C., Small, M. Complex network structure of musical compositions: Algorithmic generation of appealing music. Physica A 389, 2010, 126–132 Anand K., Bianconi G, Severini S. Shannon and von Newman entropy of random networks with heterogeneous expected degree. Physical Review E, 83, 2011, 036109 . Bianconi G. Statistical mechanics of multiplex networks: Entropy and overlap. Physical Review E, 87, 2014, 062806

Introduction is an important component of the information society. It is known that music can render both positive and negative influence on human being and society. However mechanisms of this influence in many respects remain unclear. Network science application to the analysis of pieces of music is an effective approach of modern cognitive technologies and can play an important role in understanding not only problems of music and other kind of information impact on a person, but also other global problems facing of the present society.The reductionism as research method dominating in modern science, assumes that the studied system can be understood if properties of its elements are described. Music belongs to number of so-called complex systems which donu0027t manage to be described and understood formally by means of such approach.Methods Since the end of the last century for studying of complex systems the new effective instrument of research - the theory of complex networks [1] have been developed. Nodes in such networks represent elements of these complex systems, and links between nodes – interactions between elements. Such networks form a peculiar backbones of the relevant complex systems that allows to model such systems in general as a whole and to overcome some shortcomings inherent to a reductionism.Results and Discussion purpose of this presentation is to show that works of music can be described as multilayer networks, which structural and dynamic properties can throw new light on the nature of music as complex system.A musical melody can be easily converted into a network structure if we take the musical notes of all possible durations as its nodes. It can be easily calculated that the number of nodes in one voce in such network shall not exceed 1800. Indeed, the number of piano keys equals 88; if we multiply it by 20 – the number of all possible time durations of notes (halves, quarters, eights e.t.c.) - we get the number of 1760. Connections between nodes (notes) in the network are established according to chronological principle: if note I starts to sound at the moment in time T, when note J at this particular moment finished to sound, there is a connection between the respective nodes of the network [2].In our approach the same notes in different octaves belong to different network layers. On an example of Prelude in A major by Chopin will be described relationship between melodic and harmonic structures of music. Each of these structures can be represented as networks, and music - view as multilayer network.We constructed directed network structure for melody of F.Chopin’s “Prelude A-Dur”. Figure 1 shows melody network of the piece.Figure 1. Melody network structure for the Frederic Chopin’s prelude A-Dur N7 . The thickness of links corresponds to time of repetitions of appropriate musical intervals.(See PDF version for the Figure).Harmonic structure of a musical work has qualitatively different structure of relations between notes and should be described as a separate layer. We have created a network of harmonic structure of the Frederic Chopin’s prelude A-Dur N7 as the second layer. If a piece of music has polyphonic nature, it is easy to describe this musical work as multilayer network too. Multilayer structure of musical works is the consequence of multilayer organization of the human brain networks. It is assumed to briefly discuss the possible mechanism of emotional influence of music from network science standpoint.ConclusionsOur understanding of complex systems is always associated with incompleteness of information on their structure and properties. A quantitative measure of incompleteness of information on system is its entropy. Recently in the theory of complex networks methods of calculation of entropy both simple monolayer, and multilayer networks on the basis of generalization of the most fundamental concepts and methods of statistical physics are developed [3, 4]. We are developing entropic approach for investigation of music as complex system now.AcknowledgmentsThis work is performed with assistance of the Russian Humanitarian Scientific Fund (grant N 14-04-00369)References and NotesCaldarelli, G.; Scale-Free Networks: Complex Webs in Nature and Technology. Cambridge University Press: Cambridge, England, 2007   Liu, X., Tse, C., Small, M. Complex network structure of musical compositions: Algorithmic generation of appealing music. Physica A 389, 2010, 126–132Anand K., Bianconi G, Severini S. Shannon and von Newman entropy of random networks with heterogeneous expected degree. Physical Review E, 83, 2011, 036109 .Bianconi G. Statistical mechanics of multiplex networks: Entropy and overlap. Physical Review E, 87, 2014, 062806

In "Ambiguity and Art" (Visual Mathematics, 2 (2000), 1), it has been shown that structural and semantic ambiguities in some artworks can be considered as symmetry breaking in their perception (H.Haken, Haken-Krell, Erfolggeheimnisse der Wahrnemung, Ullstein, 1992). In this presentation, we showed that structural ambiguity of art of sculpture and acting are sometimes used for symmetry breaking in plots development of some literary works and movies. Sculpturing involves an ability to depict representatives of living nature (most often humans and animals) with materials of inanimate nature. The plots of such literary works as Copper Horseman, Stone Guest by Alexander Pushkin, plot of opera Don Giovanni by W.Mozart, ancient legend about the sculptor Pigmalion, and others are based on the idea of animated statue, that is symmetry breaking "inanimate-animate". In the movie Kane 18 (Russia, 1963) sculpture of girl portrays a real living person. One can see a large number of such "living sculptures" on the Rambla in Barcelone. Just as the ambiguity of art of sculpture gives rise to plots about animated statue, ambiguity of art of acting an "actot-role" makes possible to use symmetry breaking called "character invasion" for plot development. The main hero of the film A Double Life is the theater actor who plays the role of Othello for such long time that it begins to affect him psychologically, making him more and more jealous of his beloved, and like the stage character, he strangles her and kills himself. In the film Jesus of Montreral, an actor playing the role of Jesus Christ becomes transformed into a Christ-like figure. We will demonstrate also ambiguities in plot development for other kinds of art.

The model of patterns recognition or attractor network model of associative memory, offered by J.Hopfield 1982, is the most known model in theoretical neuroscience. This paper aims to show, that such well-known laws of art perception as the Wundt curve, perception of visual ambiguity in art, and also the model perception of musical tonalities are nothing else than special cases of the Hopfield’s model of patterns recognition.

The ability of our brain to respond to small extrinsic or intrinsic perturbations points out that the brain as a complex system is operating close to instability, or criticality, because any system at the critical state has a very high sensitivity to tiny perturbations [Haken 1996]. Per Bak gives another reason why the brain should be critical: the input signal must be able to access everything that is stored in the brain. The brain cannot be in subcritical state. In this case input signal would be access to only a limited part of information. But the brain cannot be supercritical either: in this case any input would cause an explosive process in the brain, and connect the input with everything that is stored in the brain [Bak 1996]. Hence, the waking brain must operate strongly at the critical state, where a neural network reveals Weber-Fechner logarithmic law and Steves power law [Kinouchi 2006]. The critical point maximizes information transmission within a neural network.

A non-linear theory proposed different models of perception of ambiguous patterns, describing different aspects of multistable behaviour of the brain. This paper aims to review the phenomenon of ambiguity in art and to show that the mathematical models of the perception of ambiguous patterns should be regarded as one of the basic models of artistic perception. The following type of ambiguity in art will be considered. Visual ambiguity in painting, semantic (meaning) ambiguity in literature (for instance, ambiguity which V. Shklovsky called ‘the man who is out of his proper place’), ambiguity in puns, jokes and anecdotes, and mixed (visual and semantic) ambiguity in acting and sculpture. The complexity theory of the brain revealed that the human brain as a complex system operates close to the point of instability and ambiguity in art must be regarded as an important tool for supporting the brain near this critical point that gives human beings possibilities for better adaptation.

Traditional theory of art usually tries to explain every concrete artwork—its unique features—that differentiates this artwork from others. Complexity theory of art reveals universal features of art, that make this artwork genuine art. This might be compared with the study of biology. Before the XX century, biology studied mostly phenotype features, describing unique traits of living organisms. But the biology of the XX century studies common genotype features, inherited to the whole animated matter, as genetic code, protein-folding, etc. Traditional theory of art might be compared to phenotype studies in biology, whereas complexity theory of art, to genotype studies.

The attractor network model embodies the properties of an associative content-addressable memory. Every stored pattern (meaning or note) corresponds appropriate minimum on the potential function. Imprinting a pattern lowers its energy and the energy of all pattern in the vicinity. This creates a basin of attraction. Such system would be able to recognize inputting pattern which is pulled into one of the closest keypattern. Almost all familiar melodies are built around a central tone toward which the other tones gravitate and on which the melody usually ends. This central tone is the keynote, or tonic. Three stable steps of tonality: tonic, median, and dominant are keynotes or attractors of neural network model. Others steps of tonality play the role of recognizable patterns, gravitating to some or other keynote. Some recent experiments indicate that music possibly represents control of chaos in the brain and one may suggest that stable steps of tonalities are namely these tiny perturbations by which this control is realized.