Introduction

In the course of their history, musical theory and aesthetics have empirically and rationally developed a system of music laws. These are the laws of construction of a musical piece. Besides universal precepts, in all pieces of music one can perceive the stylistic marks of the period in which it was written, as well as the individual marks of its author. Because it represents a temporal form of art based on regular time division as to its meter, music also possesses a certain level of regularity and symmetry. A symmetry analysis of a musical piece will help us to describe the various aspects of symmetry which occur at different organizational levels, ranging from rhythmic repetitiveness, melody, counterpoint, and harmony symmetry, to symmetry of form.

In the existing literature on musical theory the problem of symmetry has been dealt with only superficially. In the rare cases where symmetry regularities are actually considered, their analysis is most often reduced only to the simplest forms of symmetry such as translational symmetry (literal repetition), transposition, and mirror reflection, one of the most commonly used symmetries in music.

As one of the basic laws of nature, symmetry is integrated into all material, intellectual and emotional creations of the human mind. The same is true of all art forms. Regularity and symmetry are the basis of "good form." They originate from the principle of economy, i.e., the tendency to use the minimum means (material, energy, etc.) necessary for the completion of a certain task. In such cases resulting symmetry is primarily the effect of natural causes. As an example, the construction of all scales is based on the principle of maximum simplicity, a division of the length of a string into two, three, four,... equal parts, the determination of the unit interval, and the attempt to measure all intervals with a common unit. At the same time, the way to a tempered system actually represents the symmetrization of the system, i.e., the embodiment of the principle of maximum simplicity which results in the division of an octave into twelve equal semitones. However, in many cases symmetry represents the rational context of the musical piece, or even a predetermined formal frame in which the author's creativity is expressed. A typical example of such predetermined symmetry conditions are the various ßtylistic exercises" found in fugues and canons (the "crab canon", the "endless canon", etc.).

The growing interest in music symmetry, influenced by the book Symmetry by Nobel-prize winning author H. Weyl (1952), has led to the appearance of several works which indicate the need for an exact analysis of symmetry in music. These are the monographic works of L.A. Mazel and V.A. Zuckerman, Analiz muzikalnyh prozvedenii (1967), and A.V. Shubnikov and V.A. Koptsik, Simmetriya v nauke i isskustve (1972), in which a part of the chapter on symmetry in art is dedicated to symmetry in music (pages 303-306). A further step in this direction was taken in the monographic work by G. Mazzolla, Geometrie der Töne (1990), and the works of M. Apagyi (1989), P. Escot (1989), S. Bruhn (1992), K. Fittler (1992), and others.

The goal of this thesis is the construction of necessary theoretical precepts, systems and methods which would represent the theoretical basis for the symmetry analysis of musical works. The effectiveness and appropriateness of such an approach may be verified by comparing the results thus obtained with other results which may be obtained by other standard and proven methods of analysis. In spite of their presence in all musical works, symmetry regularities have most often been ignored, as musical analysis rarely deals explicitly with symmetry. Their systematization and the determination of their appearance and frequency in music should provide an answer to the question whether the analyst and the composer who implicitly uses them have in many cases been ünconscious" of the symmetry in music.

We are witnesses to the changes brought about in the world of science and art by computers and electronic instruments. It should be stressed that the language of computers, as the language of numbers, makes possible the formalization of the language of music as well as the translation of musical works to the language of computers. The pioneer works in this area are the attempts to statistically analyze a large number of works by a certain composer and to register the basic patterns of his composition, in order to create new musical pieces in the style of this composer. Although it is based on a relatively simple empirical and statistical concept, this procedure points to the possibility of identifying musical patterns which occur in the works of a given composer.

Unlike the construction and analysis of musical works based on statistical parameters which, although stimulating, goes no further than merely registering certain common segments, symmetry analysis offers more complex insight into the structural characteristics of musical works, the symmetry organization of their basic musical components (rhythm, melody, harmony, etc.) and the correlation of corresponding symmetry structures. In order to obtain results, one must venture a step further from the level of statistics and design a complex analytical apparatus consisting of several systems appropriate for the symmetry analysis of the rhythmic, melodic and harmonic structure of a musical piece.

The preparation of the material for the symmetry analysis of a musical piece implies not only the mere translation of tonic intervals into numbers, a task which has already been performed in numerous instances, but primarily the possibility of numerical modelling of all relevant features of a given musical piece, i.e., the rhythmic, melodic and harmonic structure, none of which have been dealt with to date in this manner. The exact determination and study of various sub-entities and the symmetry transformations that connect them is practically impossible in any complex musical piece, unless each sub-entity (such as the measure in the rhythm level) is assigned a numeric value which precisely defines the duration of tones and their order. The assigning of numeric values from 0 to 11 to semitone pitch allows analogous numeric notation of the melodic structure to be performed, as well as the identification and classification of all aspects of its isometric transformations such as transposition, retrograde transposition, inversion, and retrograde inversion. This is what makes possible the symmetry analysis of musical basics (tonality and chord structure, the relationship between the tonalities, etc.). Illustrative examples of results obtained in this manner are the symmetry of the coefficients of similarity of tonalities, or the symmetry argumentation about the reasons for singling out the major and minor scales in the mode system.

In this thesis, complete systems have been constructed for the purpose of symmetry analysis, which allow the identification of all the forms of symmetry which may appear in the rhythmic, melodic, harmonic and other structures of a musical piece. Such formalization warrants that all possibilities of a given aspect of a musical piece have been put to use. For example, in the theory of modulation, all possible ways of direct modulation have been identified and classified based on the intervals between the first tones of the initial and final tonality. At the same time, this creates the basis for registering all types of indirect modulations. The construction of systems based on the complete classification of rules, regularities and symmetry that characterize each individual element of a musical piece (rhythm, melody, harmony, etc.) allows for the possibility of conducting symmetry analysis of all these elements in a musical piece.

A complete symmetry analysis implies not only the analysis of individual elements, but also their correlation which is the result of their superposition. It is possible to analyze this correlation by comparing any two individual elements, or by studying all elements simultaneously. In this manner it is possible to determine the extent to which the symmetry of the melodic or harmonic structure of a musical piece coincides with the symmetry of the musical form of the same piece.

Because it is our belief that the visualization of the various characteristics of a musical piece, especially its symmetry structure, may prove a valuable addition to theoretical analysis and play a crucial role in education, visual interpretations (diagrams) have frequently been used in this thesis, especially when presenting the results of symmetry analysis.

A great number of works in the field of psychology examines symmetry as a factor of auditory perception. A step in that direction, expected to prove or refute the significance of symmetry in music, is experimental research in this area.

Until the present, research of this kind has been limited exclusively to the visual perception of the symmetry of static objects (R. Arnheim, J. Deregowski). In non-temporal arts (e.g., painting), the work is presented to the viewer simultaneously as a whole and its rules of construction, regularity and symmetry are easily recognizable, but in the case of temporal arts it is necessary to register, memorize and recognize the patterns of symmetry within a time-frame. It is namely this specific characteristic of temporal arts (e.g., music) that offers a variety of interesting open questions such as, what forms of symmetry will be most noticeable and to what extent does this auditory perception depend on the level of musical education on the part of the individual. The method of symmetry analysis makes it possible to determine the exact level of regularity of each component of a musical piece and to conduct a quantitative analysis of the results of the experiment. In this sense, this thesis represents also a theoretical basis for future research in the field of auditory perception of symmetric patterns.

I would like to thank Dr. Nadezda Mosusova for her valuable help in writing this thesis. My sincere thanks also to Prof. Vlastimir Pericic and Dr. Milos Canak for their very useful comments and suggestions.