Visual modeling of musical works and their symmetry structure

When conducting a symmetry analysis of different structures it is almost always necessary to recur to their visual modeling. In the case of music, its visual model - the notes, is inseparable from the musical piece itself and represents its concept. The line connecting a musical piece and its visual diagram interpretation, which allows the study of its symmetry structure, is sometimes almost direct (Fig. 1.9) (Donnini, 1986).

 

 

(a)

(b)

(c)

Figure 1.9 a) The first measures of Symphony no. 5 by L. van Beethoven according to the visual transcription of V. Kandinsky - an example of relative transposition; b) D. Lombardi, Tumbling Tumbleweed, the symmetry of similarity (spiral triangular structure); c) K.H. Stockhausen, Stimmung, symmetry of similarity (spiral structure); d) L. Donnini, Stukelein.

 

Different musical elements or the structure of a piece is frequently successfully modeled by elementary geometric symmetric figures such as a circle (Figs. 5.1-5.8), torus (e.g., the models of an interval-harmonic structures of tonality) (Fig. 1.10), or the spiral (Mazzola, Wieser, Bruner and Mazzulini, 1989). A good example of this type of structure with a spiral tonal organization (Fig. 1.11) is J.S. Bach's Wohltemperiertes Klavier, F. Chopin's 24 preludes, or P. Hindemith's Ludus Tonalis (Bruhn, 1992).

 

Figure 1.10. Modeling of 12 semitones in a torus. The smaller circles correspond to cycles of major thirds and the larger circles to the cycles of minor thirds in a chromatic scale. The 180 rotation axis C1 corresponds to the complementarity of the intervals, the plane reflection C2 corresponds to the cycle of fourths, the 90 rotation axis corresponds to the addition of a minor third interval, and the 120 rotation C4 on the torus corresponds to the addition of a major third interval (according to G. Mazzola).

 

Figure 1.11 P. Hindermith's Ludus Tonalis, a spiral organization of the tone scheme.

In the following chapters we have translated the terminology of the theory of symmetry to the language of music and have established the relationship between analogous rules. Because of the specific nature of the laws of symmetry in music, the theory of symmetry laws do not permit their direct (mechanical) transposition and application. This is why it is necessary to first define all specific properties, in the first place those dealing with local and global symmetry, and to create a special system of symmetry laws for the purpose of the symmetry analysis of musical works. An individual analysis of various components (rhythm, melody, harmony) of a musical piece has been conducted, as well as their superposition. The result of this is a complete symmetry analysis of a musical piece.