Structural and Transformational Relations Between Z-Related Hexachords

Z-related pitch-class sets (pc-sets) have captured particular interest in set theory in music due to their similarity in intervallic structures alongside obscurity in transformational relations. This paper explores the structural and transformational relationships between Z-related hexachords in \(\mathbb {Z}_{12}\) . We demonstrate that common subsets, precisely those with prime forms [06] and [016], play a crucial role in defining the structure of complementary Z-related hexachords. We present a categorization of Z-related hexachords based on the discrete Fourier transform (DFT). The categorization aligns with the relative positions of the [06] subsets in the hexachord. Moreover, we introduce the method of dyad expansion for interpreting the transformational relationships between Z-related hexachords.