Symmetry transformations of chords as the basis of chromatic modulations
Unlike the diatonic and enharmonic modulations which are based on the existence of a chord or its inversion common to the initial and final tonality, in the case of chromatic modulation, such a chord never occurs. Here we can single out the initial chord - the last chord of the initial tonality and the one that follows after it, the transformed chord generated by the chromatic transformation of the initial chord, representing the first chord in the final tonality. In this sense there is a resemblance in the tone shift. However, the similarity between the initial and transformed chord is what makes this change of tonality a modulation, which is not the case with the tonal shift.
We can classify chromatic modulations based on the properties of the chromatic transformation that links the initial with the transformed chord. Here we identify three possibilities:
From the aspect of the theory of symmetry the type change of triads and the apparent relationship between keys a third apart are all retrograde inversions (central reflections), and the chromatic relationship between keys a third apart is transposition (translation).
From the point of view of symmetry it is possible to analyze the tonality sequence in any musical piece. Since we are dealing with a linear structure, potential symmetries are translations and reflections perpendicular on the axis of translation. As opposed to most previously studied forms of symmetry which are of a more local character, the symmetry of tonality refers to larger entities, thus approaching the symmetry of form from the aspect of global action.