Proportionality transformations of melodic structure
The next class of absolute symmetry transformations of melodic structures consists of transformations of similarity (transformations of proportionality) which proportionally increase or decrease every interval and every melodic entity k times, maping it into interval k(i), where k is a positive rational number, i.e., a fraction [m/n] and where m and n are mutually prime positive integers. We must bear in mind that for n ¹1 all intervals belonging to the melodic entity subject to transformation of similarity must be divisible by n. In the case of k=1, the transformations of similarity are reduced to isometric transformations.
There are four types of transformations of similarity (proportionality):
The compositions of similarity transformations with isometric transformations, or of similarity transformations among themselves are completely analogous to compositions of isometric transformations. In this case the coefficient of proportionality of the resulting transformation is equal to the product of coefficients of proportionality of the initial transformations.
In the case of relative (approximate) symmetry transformations of the melodic structure, we may continue to search for invariants of transformations, i.e., for the occurrence of certain regularities within relative symmetry transformations. The solution to this problem is analogous to the solution of permutational equivalence of rhythmic structures.