Type and class vectors and matrices in ℤn. Application to ℤ6, ℤ7, and ℤ12

In post-tonal theory, set classes are normally elements of Z12 and are characterized by their interval-class vector. Those being non-inversionally-symmetrical can be split into two set types related by inversion, which can be characterized by their trichord-type vector. In this paper, I consider the general case of set classes and types in Zn and their m-class and m-type vectors, m ranging from 0 to n, which are properly grouped into matrices. As well, three relevant cases are considered: Z6 (hexachords), Z7 (heptatonic scales), and Z12 (chromatic scale), where all those type and class matrices are computed and provided in supplementary files; and, in the first two cases, also in the form of tables. This completes the corresponding information given in previous publications on this subject and can directly be used by researchers and composers. Moreover, two computer programs, written in MATLAB, are provided for obtaining the above-mentioned and other related matrices in the general case of Zn. Additionally, several theorems on type and class matrices are provided, including a complete version of the hexachord theorem. These theorems allow us to obtain the type and class matrices by different procedures, thus providing a broader perspective and better understanding of the theory.