The symmetry of counterpoint

 

Symmetry of melodic line

 

The sequence of tones in a melody forms its melodic line. Within each melodic line we can follow a sequence of oriented intervals defined by neighboring tones. This sequence demonstrates the movement of the melodic line. Different forms of symmetry may occur in regard to the alternation of ascending (a) and descending (d) intervals which form a "melodic wave" (Mazel and Zuckerman, 1967, p. 79). The result of this are linear symmetry structures (friezes) which are made up of ascending and descending intervals.

By combining two melodies in regard to the symmetry of their ascending and descending intervals, globally symmetric two-voice structures representing an equivalent of plane ornaments may be obtained. Here we present an example of a symmetry diagram of ascending and descending intervals from the Toccata and Fugue in D Minor by J. S. Bach (Fig. 4.1) which corresponds to the ornament pm.

 

Figure 4.1 J. S. Bach, Toccata and Fugue in D Minor. The alternation of ascending and descending intervals gives the ornament pm (Donnini, 1986).

 

Two tones are denoted by the same name if their difference is the whole number of octaves (12k of half-tones, where k is a positive integer). Two intervals or two sequences of tones are denoted by the same name if they consist of same-name tones. For example, intervals C1-F1 and C2-F1, which consist of same-name tones C and F, triads C1-E1-G2 and C2-E3-G1 which consist of same-name tones C-E-G, and the sequences of tones D1-E2-C3-F3 and D4-E1-C2-F3 which consist of same-name tones D-E-C-F, are all treated as same-name tone sequences.

Each interval i has its value. This is the distance between its tones y1 and y2, in which case we disregard the orientation of the interval. The value of each interval i=y2-y1 is the absolute value of the number i: i=|y2-y1|.