Mirror symmetry in music.
S.S. Goncharenko
Novosibirsk: State Conservatory,1993 [in Russian],
372 p. ISBN 5-7196-0467-7

Order and symmetry in musical art: logical and historical aspect.
L.V. Aleksandrova
Novosibirsk: Conservatory, 1995 [in Russian]
ISBN 5-7196-0481-2

The structural analysis of musical works is a long established tradition. One of the fundamental directions in the frames of this analysis is the detection of the laws of symmetry. These laws form the basis of not only musical thought but also of human mentality. Clearly, this is why in recent years the scientists from the Novosibirsk conservatory have paid close attention to the problem of interrelation between symmetry and musical thinking.

This review considers two researches characterized by different approaches to the given problem: the book about mirror symmetry by S.S.Goncharenko and the book that tells about the diversity of forms of symmetry in musical works (written by L.V.Aleksandrova).

S.S.Goncharenko considers mirror symmetry in wide culturological context and derives from it unrealizable processes of thought active in creative work. She calls this peculiarity of thinking the reversibility (recurrence, progressive development that repeats the stages already covered in the reverse order) and reveals its manifestations in the very different layers and spheres of culture and art: in fairy-tales, exorcisms, rites, literature, music, and so on. The reversibility of thinking processes took roots in diverse forms of the mirror symmetry and its various manifestations are found by S.S.Goncharenko also in such simplest musical structures as consonances, phrases, tunes, palindromes in literature and in the most complicated evocative and compositional constructions, for example, in operas of R.Wagner and C.Debussy. This widespread use of the mirror symmetry and its capability to exist at different levels of musical and literary works is explained by the author by presence of the diverse sources from which the ethnocultural models of reversibility penetrate into the musical art.

Another explanation of such all-penetrating nature of symmetrical structures is the examination of peculiarities of the artistic thinking proper: its existence at the boundary of binary contrapositions, formation of thinking processes in a way similar to that of natural phenomena, their similarity to the processes of internal speech, and participation of synesthetic processes in artistic thinking. The author's erudition, the theoretical validity of her approach, the diverse gamut of her arguments made it possible to draw a number of thorough aesthetic conclusions. One of these conclusions is as follows: the reversibility mechanism of artistic thinking manifests itself in mirror symmetry. This mechanism aims at correcting binary contraposition in thought. Therefore mirror symmetry reflects the origination of specific boundary states of thinking in the contrapositions of material and spiritual things. In the author's opinion, such boundary states fix, hold, and protect the integrity of human mentality, since they lie in its near non-reflectable, deepest layers.

If S.Goncharenko considers the problem of symmetry in its wide culturological aspect, L.Aleksandrova uses a quite different approach; she analyzes in detail the very different forms and ways of ordering formation in the music alone. Her research is an attempt to formalize methods of creating musical structures. She puts forward an axiom: musical constructions are generated by universal laws of thought. The author employs this principle in the attempt to project a mathematical and logical apparatus onto the dynamic development of music; that is, she makes use of the fields of knowledge where the laws of cognition are realized in the most rational and abstract way. Such formal structural analysis allowed her to reveal in musical material the relations of orders, equivalence, tolerance, and also their components, that is, transitivity, symmetry, antisymmetry, asymmetry, reflectivity, and others (see p.5 of the book). As a transition to the mathematical and logical relations, she analyses the mechanisms of couplings that provide another image of musical order: disjunctive, conjunctive, implicative, and equivalent couplings.

In her most comprehensive effort the author applies the idea of complex systems of interrelations between musical order and symmetry to the diatonic system (see book, essay no.5). The resultant systemic evolution formed one of the most important features of present-day music: the so-called tolerant spaces .The author does not confine herself to revelation of formal symptoms of order and symmetry in music.

She reveals also the stabililty of "figures of musical logic" fixed in the process of their repetition and determines their semantic load . She calls this layer of the musical material the pre-compositional layer of music (see p.7 of the book) and completes her study by trying to determine the laws of dynamics in music. The basis of these laws forms the most intricate system relating retention of and variation in musical structures. These two basic phenomena are considered through the prism of the symmetry and the antisymmetry of musical works. This approach is most sequentially represented in her analysis of Bartock's compositions. At different levels of his works the author reveals a certain combination of stable, unchangeable modules (constructions) with episodes based on free variance .

The analysis of various forms of order and symmetry aims at the creation of a complementary model of creative thinking which, on the one hand, strives for irrefutability, and, on the other hand, is caused by ambiguity. This is why the author attempts to use two complementary approaches: the probability method and the theory of fuzzy sets. Both approaches lean, in their turn, upon a systematic structural approach which makes it possible to establish the hierarchy of couplings at different levels of musical arrangement and to describe the process as a change of stable and variable constructions.

The high degree of formalization makes the author's observations most valuable and useful for construction of different models of ordering of musical material (for example, in computer-aided simulation of music). At the same time, this rational/logical approach leads to a loss of the sensual, emotional, and spontaneous nature of music. This fact is connected, obviously, with disregard for a large number of pairings between musical and mathematical thinking. In connection with this it is notable that semantic functions of various images of order in music are studied very poorly (this aspect is also the most weak point in the previous book of S.S.Goncharenko). The semantic study of symmetric and other models could be of primary importance when substantiating the causes of formation of deep couplings within musical thinking with similar phenomena in other kinds of art. Both studies of the Novosibirsk scientists contain a large stock of problems unsolved. They are very important not only from the standpoint of the derivation of certain universal laws of cognition, but could reveal new possibilities in the present-day evolution of musical dynamics.

Combination of thorough formalization with well substantiated aesthetic and culturological approaches makes it possible to avoid as much as possible the shortcomings inherent in each of these analyses independently. At the same time, the profound study of music viewed as moving forms and constructions may result, in my opinion, in interesting computer-assisted ideas as applied to the simulation of thought processes in music.

Ye. Sintsov

Published in Leonardo Music Journal, 1998, v.8, pp.78- 79; in Leonardo Digital Reviews, October, 1997, v.5, N9.