Recasting K-nets

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-T2").(2), (3) He asks, in effect, "what's T2-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 T7-like about the relationship. We might therefore ask whether it is possible to label the members of the hyper-TI group , such that, if Tx or Iy transforms the pitch classes in one K-net K into those of another K[variant prime], with arrows being updated accordingly, or transforms the arrow-labels in K into the arrow-labels in K[variant prime]? (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?Figure 1.Figure 2.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 defined in circular pitch-class space.[5] As shown in Figure 1, {C, E} + {G} relates to {G, B} + {D} by in ordinary K-net notation. I propose instead that we label this hyper-transposition . This is because 1 + 1 = 7 + 7 = 2 (mod 12). In other words, "2 divided by 2" can be either 1 or 7 in pitch-class space. Note that the label does not refer to a "dual transposition": it does not mean that one part of the K-net moves by 1 semitone while the other moves by 7 semitones. Instead, it says that the two parts of the K-net move by a total distance that is equal to both 1 + 1 and 7 + 7 (mod 12). Intuitively, the parts move by an average distance of 1 or 7 (mod 12): thus they might move by 1 and 1, 7 and 7, 0 and 2, 6 and 8, and so on.[6] Now what's T1 or T7-ish about the relation between C major and G major? The T7 relationship is obvious. What about T1? Well, {C, E} + {G} is strongly isographic () to {G, B} + {D} and the G major chord has to be transposed up one semitone to become G major. This is illustrated in Figure 3. The hyper-transpositional labels thus reflect actual transpositions: C major is related to G major by because C major is strongly isographic to chords that are T1- and T7-related to G major (G major and C major respectively). For the sake of clarity, I'm going to drop the clumsy notation and use whichever of the pair is most appropriate to the context.(4)[7] Q: In the new system, what hyper-transposition relates {C, E} + {G} and {C, E} + {A}? A: ("hyper-T one-half"). What could that mean? Here's an informal way to think about it: in 24-tone equal-temperament, {C, E} + {G} and {C , E} + {G} (C and E quarter-tone flat, G quarter-tone sharp) are strongly isographic, and hence related by . …