Recasting K-nets
Dmitri Tymoczko · Music Theory Online · 2007
I. The Problem[1] In a recent article, Michael Buchler observes that K-nets such as those in Figure 1, which I will notate as {C, E} + {G} and {G, B} + {D}, are related by ("hyper-T "). ( 1) , (2) He asks, in effect, "what's T -like about the relation between C major and G major triads?"[2] This question is worth taking seriously.Although hyper-transposition is different from ordinary transposition, being a function over functions rather than a function over pitch classes, comparisons between these two types of "transposition" are intrinsic to the practice of K-net analysis.And although the primary analytical use of K-nets is to relate sets belonging to different set classes, the technology applies equally well to chords such as C major and G major.Buchler has uncovered an example that seems to demonstrate that there is only a tenuous analogy between the two sorts of transposition.It will not do simply to reiterate that they are different.For Buchler's challenge is, given that they disagree so dramatically about such a simple case, what's the musical value of comparing them?[3] It is possible, however, that a simple change in notation might help meet his objection.For suppose we used the label to refer to the hyper-transposition linking {C, E} + {G} to {G, B} + {D}.In that case, the force of Buchler's worry would be significantly ameliorated, since there is obviously something T -like about the relationship.We might therefore ask whether it is possible to label the members of the hyper-TI group , such that, if T or I transforms the pitch classes in one K-net K into those of another K′, with arrows being updated accordingly, or transforms the arrow-labels in K into the arrow-labels in K′? (See Figure 2.) In other words, can we label the elements of the hyper-TI group in a way that is consistent with the TI group? II. The Solution[4] Yes. Simply divide the current hyper-T and hyper-I labels by 2. The only complication is that division is not uniquely