A New Approach to Transformational Analysis: Reductive Isography for Associating K-nets with Different Numbers of Nodes

Abstract

After Buchler`s criticism of the use of K-nets and the vigorous debate that followed, K-nets may have lost some of their appeal in the analysis of post-tonal music. In particular, two questions raised by Buchler, the first concerning the reliance of K-nets on inversion which many consider a less compelling transformation than transposition in both formulating K-nets and determining network isographies and the second concerning the limited isographic relations drawn by K-nets of the same size, lead many theorists to believe that deploying this analytical tool is inaccessible in any practical way or they may put it aside as being very restrictive. Thus, in this study I introduce a new method, reductive isographies, for associating K-nets with different numbers of nodes to embrace the varied transpositional and inversional gestures exposed in the musical surface. A new approach to Klumpenhouwer Networks through the analyses of two short passages allow us to eliminate some of the problems which emerged in earlier debates about K-nets. In the analyses of two short passages, the transformations capable of associating different numbers of nodes help us to better understand the nature of transpositions and inversions, especially in musical surfaces, thusavoiding analytic promiscuity. In this type of analysis, while positive and positively reductive isographies relate K-nets derived from different zones, strong and strongly reductive isographies reinforce or extend a single zone. Axially and axially reductive isographies, however, may prove to be even more useful since they serve either to extend a zone or to introduce a new zone, or may in fact perform both roles at the same time. Of even greater importance, a zone may be established by an inversional tetrachord, though this depends on the musical context. Also, in a higher-level structure, it may be possible to select only some of the chords as structural ones. I do not mean that this approach solves all the serious problems or provides a method without any flaws, but since I often recognize both the cooperation and the competition between transpositional and inversional relations when hearing works of music, it seems foolhardy to simply deny or discard K-nets in pursuing a comprehensive approach in the analysis of those compositions which employ both transpositions and inversions. K-nets remain both valid and effective as analytical devices.