In music theory, what are the two common transformations applied to rows in 12-tone compositions?

In the context of 12-tone composition, the two common transformations applied to rows are inversion and retrograde.

Inversion refers to the process of flipping the intervals of a melody or row, where each interval is mirrored. For example, if the original row ascends by a minor third, the inverted row would descend by the same interval. This transformation allows composers to explore different textures and harmonies while maintaining the original row's identity.

Retrograde involves reversing the order of the pitches in the row. A retrograded row plays the notes in the opposite sequence, which can create a distinct musical effect and can deepen the thematic material by presenting it in a new light.

These two transformations are essential in 12-tone music, guiding composers in developing their motifs while adhering to the principles established by Arnold Schoenberg, who pioneered this technique. Thus, inversion and retrograde are fundamental tools for achieving variety and cohesion within the structure of a 12-tone piece.