This study explores how mathematical group theory can reveal the structure of musical compositions, particularly focusing on transformations such as transposition, inversion, retrograde, and translation in musical works. By reducing melodies to their pitch classes and applying algebraic operations, we uncover new insights into the symmetrical properties of music. This method simplifies the connection between advanced mathematical theories and practical music analysis, enabling the detection of complex patterns and symmetries that traditional techniques may overlook. The results explicitly reveal the duality between tonal and positional symmetry aspects of melodic transformations, acting on the space of abstract melodies including repeated notes and rests. We give an explicit form for a melody of the given length L resulting from application of a symmetry separating the positional and tonal aspects, and explicit recipes for construction and enumeration of symmetric melodies using this duality of structure.