Optimization of melodic line
Each melodic line satisfies, in a greater or lesser degree, the principle of economy: the values of intervals between neighboring tones are within certain limits. If a sequence of intervals between neighboring tones i_{1}, i_{2},...,i_{n}, corresponds to a melodic line, we call the sum of their values D=i_{1}+i_{2}+...+i_{n} the deviation of the melodic line. The deviation of the melodic line shows its deviation from a straight line parallel to the xaxis. In the case of the straight line that corresponds to the constant tonal pitch, the deviation equals zero. Deviation can be decreased if each tone of a melodic line is substituted with an adequately chosen samename tone. Among all samename melodic lines there will occur a melodic line with a minimum (least possible) deviation D_{m}. We will call this melodic line optimal (with the highest possible degree of economy) (Fig. 4.2). In order for a melodic line to be optimal, it is necessary for the values of all intervals between neighboring tones to be minimal. Finding a samename minimum interval for the given interval i comes down to the following procedure:
In the previous chapter (p. 52) we classified intervals according to their sonority into consonant and dissonant. Note that in transition from any one interval to a samename interval, the sonority of the interval does not change: consonant intervals become consonant, and dissonant intervals become dissonant. This means that the sonority of intervals is an invariant of this transition.
Figure 4.2 Different samename melodic lines (a,b) and the optimal samename melodic line (c).
