Static symmetry - mirror reflection is probably the most common form of symmetry in the thematic plane, beginning with binary form a-a (Fig 7.1), ternary form a-b-a, or its variation a-b-b-a, the form of Italian and French overtures of the old masters a-b-a and the da-capo arias of the old masters, the transitional form between a simple and a complex ternary form a-b-c-b-a (e.g., Waltz in A-flat major, op. 40, no. 1 by P.I. Tschaikovski, or the Nocturno in B-flat minor, op. 9, no. 1 by F. Chopin), complex ternary forms A-B-A (aba-cdc-aba) (e.g., Sonata, op. 2, no. 3 by L. van Beethoven), the more liberate ternary forms A-b-A, or a-B-a, the minuet A-B-A, the arrangement of dances within a suite (minuet 1 - minuet 2 - minuet 1, e.g., The English Suite in F major by J.S. Bach, or gavotte 1 - gavotte 2 - gavotte 1, e.g., The English Suite in G minor and The English Suite in D minor by J.S. Bach), the rondo A-B-A-B-A, the sonata form A-B-A, the sonata form with an inverse reprise A-B-C-B-A, the reversible ("palindromic", Lemacher and Schroeder, 1967, p. 46) forms of the rondo A-B-C-B-A (e.g., the adagio movement of A. Bruckner's Third Symphony), or the form of Chopin's Mazurka in E minor, op. 41, no. 2 which Mazel (1979, p. 499) denotes as concentric (Fig. 7.2), a sonata rondo form A-B-C-A-B-A (rondo from the Violin Concerto in D major of L. van Beethoven), to the complex sonata form of A-B-C-D-C-B-A. The overture Tanheuser by R. Wagner has the same scheme:
A (introduction) - B (main theme) - C (secondary theme) - D (episode) - C (secondary theme) - B (main theme) - A (coda).
Figure 7.1 Binary form a-a (Lemacher and Schroeder, 1967).
Figure 7.2 Concentric form A-B-C-B-A.
Mirror symmetry is sometimes consistently put into effect in the structural plane as well. In this case the equivalent thematic and tonal entities have the same duration. For example the thematic plane of Ländler by F. Schubert corresponds to the scheme ||:A:|| ||:BA:|| (Fig. 7.3) which by the absence of repetition is reduced to the mirror reflection scheme of A-B-A, and in the tonal plane to the mirror symmetric scheme of E-flat -B-flat - E-flat , in which case the thematic and the tonal entities have the same duration (8 measures each). Instead of establishing the musical piece completely on the basis of static symmetry, we much more frequently come across combinations of static and dynamic symmetry, i.e., the presence of mirror reflection in one of the planes and its absence in the other planes.
Figure 7.3 Ländler by F. Schubert (Apagyi, 1989).
For example, the Gregorian chant a-b-a (Kyrie-Christe-Kyrie) (Fig. 7.4) is symmetric in the thematic and textual plane, but asymmetric in the tonal and melodic. The local symmetries that occur on its tonal and melodic plane are pronounced as we break it down further: A(aba)-B(cdc)-C(efg).
Figure 7.4 (Lemacher and Schroeder, 1967).
In the sonata allegro form, the asymmetric tonal plane A(TD)-B-A(TT) corresponds to the mirror symmetric thematic plane A-B-A; the asymmetric tonal plane
corresponds to the mirror symmetric thematic plane of the sonata rondo form A-B-A-C-A-B-A (rondo from the Violin Concerto in D major by L. van Beethoven); an asymmetric tonal plane corresponds to the mirror symmetric thematic plane of the Third Symphony finale by L. van Beethoven, as in this case we are dealing with a non-standard combination of sonata form and variations (Fig. 7.5); an antisymmetric tonal plane T-D-T-S-T corresponds to the mirror symmetric thematic plane of the rondo form A-B-A-B-A (Mazel, 1979, p. 277). Contrary to that, the antisymmetric thematic plane of Scarlatti's sonata A-B has a corresponding antisymmetric tonal plane A(TDTD)-B(DTDT), and the antisymmetric tonal plane of the dance in a baroque suite a-b has a corresponding antisymmetric tonal plane a(TDTD)-b(DTDT).
In the development part a mirror symmetric distribution of tonalities occurs frequently, for example in the first movement of the Sonata no. 1 (A-flat -B-flat minor-C minor-B-flat minor-A-flat ), in the first movement of the Sonata no. 5 (C minor-F minor-B-flat minor-D-flat major-B-flat minor-F minor-C minor), or in the finale of the Moonlight Sonata (C-sharp minor-F-sharp minor-G major- F-sharp minor-C-sharp minor) by L. van Beethoven. These are particular examples of a more general rule: in many cases the development of the piece in the beginning contains a gradual approach towards a certain tonality (which is often not in the first degree of relation) and then the gradual return to the fundamental tonality (Mazel, 1979, p. 398).
Figure 7.5 (Mazel, 1979, p. 320).