Two letters by Jack Douthett on uniform triadic transformations

Two letters that Jack wrote soon after I introduced uniform triadic transformations in 1999 exemplify the energy with which he threw himself into work on subjects that stirred his interest. As I note in a tribute elsewhere in this issue (Krantz 2022), I first met Jack at a meeting of the Society for Music Theory in the fall of 1999 – a conference at which I also presented my work on UTTs (later published in Hook 2002). By April of 2000, Jack had distributed two substantial letters with his evolving thoughts on the subject. The wreath-product structure of the UTT group, the simply transitive subgroups of that group, and “skew groups” (which combine mode-preserving transformations of one kind with mode-reversing transformations of another kind) were subjects of special fascination to him from the start, as they remained a few years later when we produced our joint paper on applications of UTTs to serialism (Hook and Douthett 2008). The second letter is particularly notable for a few ideas that have not been pursued elsewhere. Here Jack relates UTTs to the seventh-chord transformations that he had studied with Peter Steinbach (Douthett and Steinbach 1998), and also to Jonathan Kochavi’s work (Kochavi 1998) on contextual inversions (presenting what Jack calls “Kochavi diagrams” for some groups of UTTs). He also refines my embryonic suggestion for generalizing UTTs to situations where there are more than two classes of objects (such as inversionally related chord qualities).