Without a Safety (k)-Net

[1] When analyzing a passage of music with K-nets, the transformational operation from one K-net to the next is just as important as the construction of each individual K-net. To ensure that different K-nets are related, various authors have proposed that either an interval (Lewin 1990 and Klumpenhouwer 1991) or an inversional relationship (Stoecker 2002) from one K-net to the next must remain invariant. And with all the isographic tools that are available-strong, positive, negative, and axial isography-virtually every trichordal K-net can be isographically related to any other trichordal K-net. This high degree of isographic relatedness is a concern for Michael Buchler. In his critique of K-nets and K-net analyses, Buchler points out that"[s]ince the most promiscuous trichord classes include many of the most common and familiar melodic and harmonic structures found in a wide range of repertoire, trichordal isography generally comes easily to those who seek it. When the standard for pcset relatedness is this low, analysts ought to exercise particular diligence and discretion in making a strong case for the uniqueness and musicality of their readings."(1)[2] Though Buchler warns us about the promiscuous nature of trichordal isography, there are times when adjacent, trichordal collections do not share an interval or an inversional relationship. As a result, the K-nets that model these sonorities cannot be isographically related, creating a gap in our transformational pathway. These isographic holes are the focus of my article. Trichordal K-nets that cannot be conventionally and axially isographic are extremely rare in the literature. It is unusual to find two collections that do not share an interval class or an inversional relationship.(2) To better understand how these analytical gaps occur, I will focus on the voice leading between K-nets that cannot be isographically related.[3] In his first published study of K-nets, David Lewin analyzes a few brief passages from "Eine blasse Wascherin," the fourth song from Arnold Schoenberg's Pierrot Lunaire, op. 21.(3) The Pierrot passages that Lewin analyzes are good ones for illustrating the strengths of a K-net analysis. Lewin effectively demonstrates how to construct coherent, transformational pathways and introduces K-net recursion. But if we analyze the entire song with a K-net lens, we will find that the isographic tools that Lewin introduced cannot relate a few K-nets.[4] Example 1 includes the score for the opening instrumental phrase of "Eine blasse Wascherin." Most of the trichords in this passage contain a dyad of interval-class 3, and the K-nets that model these collections could be configured in such a way that T3 and T9 arrows are drawn. From an isographic perspective, virtually every trichordal K-net in this passage is strong, positively, or negatively isographic to its adjacent K-net. In m. 3, however, the K-net that interprets the [036] diminished trichord cannot be isographically related to the K-net that interprets the [048] augmented trichord. Since these two collections do not share an interval class, the K-nets cannot be related by strong, positive, or negative isography. And since these particular diminished and augmented trichords do not share an inversional relationship, they cannot be related by axial isography. As a result, the K-nets that interpret these two collections cannot, in any way, be isographically related. [5] An even more remarkable passage to explore non-isographic K-nets occurs in Schoenberg's song "Angst und Hoffen" from his Buch der hangenden Garten, op. 15, no. 7, given in Example 2.(4) The first three collections of the piano accompaniment are members of three different set classes: [048], [016], and [03] respectively. The augmented triad accompanies the word "Angst" while set-class [016] accompanies the word "Hoffen." The A4/C5 dyad on the downbeat of the second measure accompanies the word "wechselnd. …