“The End of Similarity?: Semitonal Offset as Similarity Measure”
Similarity relations, designed to measure the resemblances among set classes, have arrived in profusion in the past few decades, with more than twenty currently on the market. Despite the differences in their methodologies, it is now clear that all of these similarity relations say basically the same thing, that is, they arrange the set classes into predictable groups and orderings. From an entirely different perspective, a number of theorists have begun to create voice-leading spaces within which harmonies can progress by distances of varying and measurable size. Joseph Straus’s conception of such an atonal pitch space arranges set classes according to their degrees of optimal semitonal offset to display voice-leading connections between them. By exploring to what extent optimal offset and similarity really describe the same relationships, this paper asks whether offset can be a simple proxy for—perhaps even spell the end of—similarity. It answers, “yes,” but with an important reservation. Those set classes minimally offset from another are, in most cases, also maximally similar to it, while those deeply offset from it are also greatly dissimilar to it, but only with respect to “chromatic” set classes such as 3-1 or 4-1. It is as if the maximally uneven set classes are established in Straus’s atonal pitch space as vanishing points, points on the horizon toward which parallel lines recede. Offset is thereby in league with Ian Quinn’s Q(12,1) quality space as a qualitative measure, while retaining its usability as a voice-leading one.
“Subgroup Relations Among Pitch-Class Sets Within Tetrachordal K-Families”
The Klumpenhouwer network, or K-net, was formally introduced by David Lewin in 1990. Whereas his work emphasized the importance of recursive structures, Henry Klumpenhouwer employed K-nets primarily as a voice-leading model. Shaugn O’Donnell’s subsequent development of the K-class enabled the disclosure of large-scale harmonic consistencies and other important generalizing forces. Recently, Philip Lambert has examined the relationships among K-family, K-class and set-class membership for all trichordal K-families. Specifically, he suggests ways in which K-family networks defined by strong, positive or negative isography can be understood to form harmonic spaces comprised of pitch-class sets from disparate set–class types.
Our paper will introduce a new form of network correspondence based on the identification of equivalent subgroup operations. By associating subgroups to K-classes, we offer a novel set-class environment that is distinct from isographic space but intersects with it in meaningful and significant ways. We are able to establish correspondences between sonorities that cannot be reconciled in terms of conventional isographic relationships. Moreover, the algebraic relationships reveal additional perspectives concerning relevant voice-leading models. Some compelling aspects of associated subgroups will be demonstrated through analytical examples that include the four-part rotational arrays of Stravinsky and canons by Webern.
“Simplifying Complex Multiplication”
The relationships that exist in musical compositions that are generated by complex serial manipulations, such as Boulez’s multiplication operations, have often been equated to the random associations that result from chance operations. In this paper I will demonstrate that this conclusion has been reached because an imperfect understanding of the compositional process and resultant structures has obscured the specific correlations between them and the musical surface of these works.
Stephen Heinemann’s formula for complex multiplication, though arithmetically correct, provides an unnecessarily complex and abstract explanation of pitch-class generation in these pieces. This paper provides a clarification of Boulez’s multiplication technique in a way that is considerably less abstract and more intuitive (from a musical standpoint) than the current theoretical apparatus. It describes the structural properties of Boulez’s multiplication tables, demonstrating that the intrinsic transformational, serial and symmetrical relationships translate into meaningful qualities of the musical surface. It demonstrates, using Boulez’s sketches, that this theoretical approach reflects Boulez’s compositional process. Furthermore, by presenting concrete examples which apply this theoretical tool to analyses of several different works, this paper demonstrates its significance, practicality and potential for yielding new and fascinating insights into this music.