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The role of symmetry in music work
As we observe the symmetry structure of many musical pieces, the logical question about its function is posed. Some of the psychological implications of symmetry have previously been explained from the aspect of the "law of pregnancy": every unresolved situation shows a great amount of tension directed towards its resolution which in turn leads to the decrease of tension. In connection to this we will discuss the following questions:
Predictability is the form of symmetry in time, a possibility of the listener anticipating the further course. In the chapter Perceiving Music Intellectually of his book A Composer's World, P. Hindemith describes perception of the musical structure in a way that allows a comparison with language. "While listening to a musical structure unfolding in his ears, the listener mentally constructs, in a manner parallel and simultaneous to the structure, an image in a mirror. Registering the components of the piece as they come to him, he tries to fit them to the corresponding parts of his own mental construction, or he simply predicts the possible further course of the music and compares it to the image of the musical structure he has preserved in his memory after previous experiences." Anthropologist C. Levi-Strauss elaborates on the same subject: "Musical emotion stems from the fact that at every moment the composer conceals or reveals more or less of what the listener anticipates based on the patterns he thinks he can predict, but is not wholly capable of predicting because of his own subjection to a two-fold periodicity: the rhythm of breathing, determined by his individual nature, and the periodicity of the scale the experiencing of which depends on the level to which he has mastered it. If the composer conceals more than what we are anticipating, we experience a wondrous sensation of falling. We feel as though we have been thrown off a stable point of support of the musical scale into an empty space, just because the point of support we expected as not there. If the composer conceals less, the opposite occurs: he compels us to perform gymnastic exercises considerably more complex than our own. At times he is the one who moves us and at other times he incites us to move by ourselves, always transcending that which we would be able to do with only our own resources. The esthetic experience is grounded in a variety of stimuli and moments of rest, in a fulfilled or unfulfilled expectation of our anticipation" (Gombrich, 1979). Performing an extrapolation in time, the listener is constantly testing his own hypotheses whose correctness naturally depends on the individual ability and musical knowledge of the listener, and also on the level of organization of the musical piece itself. In an aleatory composition based on the stochastic principle, prediction as a heuristic component of a musical piece loses its meaning. On the other hand, different forms of symmetric organization offer the listener a more solid base for building hypotheses and constructing that which P. Hindermith calls a "mental model." If we accept C. Burt's hierarchical model of the structure of human capabilities, according to which the highest level in the hierarchy of human capability belongs to the perception of relations, their combinations and applications, the perception of relationships which support the structure of a musical piece is one of the manifestations of general intelligence. On the one hand, the structural organization of a musical piece is an indicator of the composer's intelligence. In the words of C. Seashore, "A person's music is in direct proportion to his intelligence. The type and the degree of a person's intelligence can set the limits of a musical accomplishment. A great composer, conductor or interpreter are ever in immense intellectual movements and have the ability to elaborate and to control their creativity on a very high intellectual level" (Meyer, 1986). For the listener, on the other hand, the musical piece is also a specific ïntelligence test" of his ability to predict, anticipate or reconstruct the composer's ideas and to perceive the organization principles upon which the musical piece is based. Of course, the basic structures of this type are symmetry structures which are abundantly used in all kinds of IQ tests. The reproducibility of a musical piece can be understood in two ways: as objective reproducibility, i.e., the possibility of its repeated performance, its "reconstruction" and preservation in time, and also as subjective reproducibility, the possibility that the same musical piece will produce the same or a similar emotional effect in different listeners. In this case, too, the study of symmetry, i.e., of musical patterns and their effect on listeners may serve as the basis for a more exact study of the universality of musical language and the degree of its meaning span. Directly connected to this problem is also the question of the informational value of a musical piece and its possibilities of being informationally self-contained. In comparing the informational value of one of Bach's works and J. Cage's piece of aleatory music Imaginary Landscape no. 4 in which the sounds from 24 radio sets from randomly chosen stations are constantly merging in a variety of volumes, D. Hofstadter asserts that the information that Cage's work conveys is not self-contained. One can only understand it in the whole context of the development of music, as a negation of all previous cultural heritage. In the words of Cage himself, "Let us allow sounds to be what they are, as opposed to tools for expressing man-made theories or human emotions." The only piece of information that the listener can derive from this work, without reference to the cultural or historical context in which it was written, is the elementary fact: "Coincidence exists." Consequently, his work is devoid of almost any inner meaning. Unlike Cage's work, Bach's compositions are self-contained and offer, already on the structural organization level, a significantly more abundant informational content. As a final argument for the conclusion that "intelligence is fond of patterns" we will observe the development of automatic composing by means of computers in the period between 1956 until today (Ames, 1987). Unlike the early generations of computers which served merely as the generators of stochastic elements in the composition process (as, for example, in the works of I. Xenakis), the new generations of computers with the beginnings of artificial intelligence (computers at every moment capable of evaluating a variety of possibilities and priorities - the ranking of elements and building a ßystem of values") have the prerequisites for the construction and analysis of pieces with a complex structural organization. As an antithesis to the earlier domination of randomness is the Expert System for the Construction of Chorals in the Style of J. S. Bach (K. Ebicoglu), or the TEMPER system of G. Haus and P. Morini (1992) which allows the transformation of plane tessellations into music and represents a part of the authors' attempts, in the quest for common parameters in different forms of art, to translate the symmetry structures of one kind of media to another (poetry or visual arts into music) and to register the common esthetic parameters of structures obtained in this manner. In accepting the very wide and expansive treatment of symmetry as the "law of construction of structural objects" (A. V. Shubnikov), we hope that the method of symmetry analysis can serve as an exact complementary tool to existing esthetic and formal methods of analysis of musical works.
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