The symmetry of rhythmic structure Most natural laws and occurrences, such as the coming and going of waves, the change of day and night, the changing of seasons, tides, breathing, heartbeat, pendulum movements, etc., are all different manifestations of periodicity in time. Rhythm is the repetition of occurrences or states in identical time intervals (Ghyka 1977, p. 6; Alyakrinski and Stepanova, 1985). By looking at the fiveline staff as a specific form of a twodimensional orthogonal coordinate system, the rhythmic structure represents a projection of note durations on the time axis (horizontal xaxis). In this instance, the total sum of the durations within a unit of distance (measure) is constant. Studied from a geometric point of view, rhythmic structure is onedimensional (linear). If we take the motif as a basic rhythmic and melodic entity and explore the changes of its rhythmic structure within the study of a motif (Lemacher and Schroeder, 1967, p. 18), we can follow different symmetry transformations. Besides the motif, a similar analysis may be applied to any metricrhythmic entity (e.g., measure, halfphrase, etc.).
Absolute and relative symmetry transformations of rhythmic structure All symmetry transformations of the rhythmic structure can be classified into two basic types:
In the first case we can examine two kinds of transformations: isometric transformations (transformations of congruence) and transformations of similarity (proportionality transformations). Isometric transformations of rhythmic structure are:
Translational repetition (translation) is precisely defined by the minimal distance l between the beginning element (original) and the transformed element (image). The vector l^{®} is called the translation period. Retrograde repetition (mirror reflection) is defined by the position of the axis of reflection m which is perpendicular in the center of the line segment which connects the end of the original with the beginning of the image (Fig. 2.1).
(a)
(b) Figure 2.1 a) Translational repetition (translation t); b) retrograde repetition (mirror reflection m). Transformations of similarity (proportionality) are:
In both cases all the tone durations are proportionally augmented or diminished, i.e., multiplied by the coefficient k and repeated in this manner in an identical or reverse order, respectively (Fig. 2.2). Specifically, in the case of proportional repetition with coefficient k=2, the result is augmentation, and for k=^{1}/_{2} the result is diminution. Since similarity symmetry transformations involve the preservation of metrics given by a measure, coefficient k is in most cases a degree of the number 2, or a number of the form 2^{n}, where n is an integer. (a)
(b) Figure 2.2 a) Proportional repetition (dilatation d_{k}); b) retrograde proportional repetition (dilatational reflection m_{k}). All of the above mentioned symmetries are studied within the bounds of geometric theory of symmetry (see Chapter 1). Owing to the fact that the composition (product) of two symmetries, i.e., their consecutive action, is a third symmetry transformation, the next step in the study of rhythmic symmetry structure (with the aim of defining all its symmetries), is the combination of the mentioned symmetry transformations. In order to achieve this, it is necessary to determine all the possible products of the mentioned symmetries, i.e., their consecutive actions. For the sake of clarity this process may be accompanied by corresponding illustrative diagrams and examples from musical literature. In the case of congruence transformations:
In the case of transformations of proportionality:
In all of these cases, the coefficient of proportionality k of the resulting transformation is equal to the product of the coefficients of the original transformations. Finally, by combining the transformations of congruence with transformations of proportionality:
In all these cases, the coefficient k of the resulting transformation is equal to the coefficient of the original transformation of proportionality. Already on this level we note the appearance of huge differences between the geometric theory of symmetry and its potential applications in music. Within the geometric theory of symmetry, the result of the combination of the said isometries are groups of symmetries, i.e., infinite regular linear symmetric structures. For a structure to represent a group it is necessary for the product of every two elements of the group to also belong to the group (closure). This means that, for example, the presence of translational repetition (translation) causes translational repetition ad infinitum. Thus, when studying the elements of symmetry in music, only local symmetries and their finite products are obvious (i.e., every symmetry transformation may be used once or many times, but the number of times it is used is always finite). This is not the case with geometric theory of symmetry in which the appearance of a local element of symmetry implies global symmetry. A special effect may be achieved by a multiple consecutive application of the same transformation to an original rhythmic entity, e.g., a diminuendo. Absolute transformations of a rhythmic structure have their own geometric equivalents and allow complete and exact classification. Relative transformations, on the other hand, are significantly more complicated than that. Of course, even in this case the method of study is based on the same concept  the determination of invariants, i.e., elements, structures (entities) or substructures, which remain unchanged after a transformation. This is one of many possible applications of the universal principle of conservation.
