Dyadic TC Lattices: Revisiting the TC/IS Relationship

[1] In an article from 1988, Richard Cohn explores the relationship between inversional symmetry and transpositional combination, especially as it relates to Bartok analysis (Cohn 1988). He argues that while it is well known that many of the pitch collections most important to Bartok's music are inversionally symmetric, it is also significant that they can all be explained using transpositional combination (TC). He makes this clear in the illustration shown in Example 1. Here we see both the inversional axis (represented with a dotted line) and the TC-property (shown beneath each figure) of several noteworthy collections. Indeed, collections such as Antokoletz's "x" and "z" cells, along with the diatonic, whole-tone, and octatonic collections, all parse into two transpositionally related pitch-class sets. As Cohn argues, while these collections are all inversionally symmetric, the presence of the TC-property is not a mere coincidence that can be ignored.Example 1. Cohn's illustrations of IS and TC collections used in Bartok[2] While the crux of Cohn's article is analytic in nature, he also mentions some of the more abstract elements of TC. Some of this comes in the form of simple statistics. As he writes:Of the 223 set-classes, 137 bear the TC-property. Of particular interest for the present purposes is the relationship of the contents of this group to the contents of the group of ninety-four bearing the IS-property. Seventy-two set-classes bear both properties; that is, about three-quarters of all IS-set-classes are also TC-set-classes (Cohn 1988, 24-25).He goes on to explain that the list of set-classes containing both IS and TC includes most of the important set-classes for Bartok analysis:These include the thirteen tetrachord-types which can be realized as inversionally-symmetric structures of exactly four pitches; all of the partial and complete interval-cycles, including chromatic, diatonic, and whole-tone sets; and all of the Tn-invariant sets such as the octatonic collection and the "1:3 model" (Cohn 1988, 25).Since his goals for the article are primarily analytic, Cohn stops short of traveling further down this abstract path. But for those interested in this topic, there are still worthwhile questions to ask. Just what are the set-classes that have IS but not TC? Also, are there any distinctions among those sets that do have both IS and TC?[3] Cohn answers some of these questions in his dissertation (Cohn 1987). In this exhaustive study, he provides the theoretic groundwork for TC and illustrates its analytic importance in contemporary music. In addition to defining the operation and showing its role in music, this document contains appendixes listing various TC properties of all 223 scs. Of primary interest here are the categories into which Cohn places set-classes based on their TC properties.(1) He uses four categories: 1) prime; 2) semi-prime; 3) fully factorable; and 4) semi-factorable. Prime scs-such as 4-12 (0236)-do not possess the TC-property at all, while fully factorable scs-such as 4-17 (0347)-have a TC path stemming from a "simple operation" (TC3 of (04) in the case of 4-17). Semi-factorable scs have TC, but their generative path traces back to a "mixed operation," meaning one of the initial operands is a trichord or larger. The example Cohn gives is 6-19 (013478), which comes from TC1 of 3-11 (037). Finally, semi-prime sets are those whose TC operands are both trichords or larger, resulting in a "complex operation." This is the case for 6-31 (014579), which is generated when 3-11 (037) is transpositionally combined with itself.[4] Armed with further information about the role TC plays in the set universe, we may ask even more questions regarding the relationship between TC and IS. As stated earlier, it will certainly be interesting to know exactly which sets have IS but not TC. Further, now that we know that scs can be categorized based on their TC properties, it will be useful to see if there are differences among those sets that do have both IS and TC. …