Antisymmetry serves as the basic principle of the construction of counterpoint mono-thematic compositions: canons (proposta - risposta) and fugues (dux - comes). It can also occur in dynamic symmetry. The thematic plane of a binary form a-b, the five-part form a-b-a-c-a (for example, the march from S. Prokofiev's opera In Love with Three Oranges), the rondo form A-B-A-C-A (i.e., the relationships b-c or B-C), or the non-standard sonata rondo form A-B-A-C-A-D-A-B-A (i.e, the relationship C-D). The previously mentioned Scarlatti sonata (A-B), or the dance in a baroque suite (a-b) may be treated also as the occurrence of antisymmetry. A similar case is presented by periodic structures of the type abbc which Mazel (1979, p. 147) calls semi-symmetric, the structure of the eighth measure ab(2)+b(1)+b(1)+b(1)+b(1)+bc(2) (Fig. 7.10), or by the letter scene from the second act of the opera Eugene Onegin whose corresponding thematic scheme is AbA-BcB-CbC, in which antisymmetric relationships a-c and A-C occur.
In the tonal plane antisymmetry is manifested through the appearance of tonal opposites: major and minor tonalities, and unrelated or polar tonalities.
Figure 7.10. Chopin: Mazurka, op. 59, no. 3 (Mazel, 1979).
Antisymmetry occurs in all structures based on the principle of doubling. In the melodic plane doubling (the equal alternation of ascending and descending parts) makes up a melodic wave (Fig. 7.11). In most cases the melodic wave begins in an ascending direction, but there are contrary examples as well (Fig. 7.12). Very often the result is a symmetric melodic image (Fig. 7.13).
Figure 7.11. F. Chopin, Etude, op. 10, no. 2" (Mazel and Zuckerman, 1967).
Figure 7.12 E. Waldteufel, Birdsong waltz (Mazel and Zuckerman, 1967).
Figure 7.13 R. Schumann Abegg Variations, op. 1" (Mazel and Zuckerman 1967).
In some instances, as is the case with N. Rimsky-Korsakov who more frequently than other composers uses symmetry in the tonal-melodic plane (Fig. 7.14) one can look for its roots in the elements of folk melodies or, in a broader sense, in ornamental folk art (Mazel and Zuckerman, 1967, p. 100). In many cases the identification of symmetry in the tonal and melodic plane is possible only after reducing the melodic structure. As Mazel (1979) points out, by singling out the tones that overlap with the stressed syllables of the text of Pesnya o vstrechnom by D. Shostakovich (Fig. 7.15a) we get a simplified skeleton of the melody that contains only three tones and has a high degree of symmetry (Fig. 7.15b). Tutti-concertino doubling also characterizes the construction of the first movement of J. S. Bach's Brandenburg Concerto no. 2 (Holopov, 1974, p. 141) (Fig. 7.16).
Figure 7.14 N. Rimsky-Korsakov, The Tale of the Invisible City of Kitezh (Mazel and Zuckerman, 1967).
Figure 7.15 (Mazel, 1979).
Figure 7.16 (Holopov, 1974, p. 141).