Contextual-Inversion Spaces

Abstract

This article creates transformational spaces for interpreting progressions of sets belonging to the same T/I equivalence class. Within these spaces, sets are placed in proximity based upon contextual inversion, specifically those contextual inversions that (like the familiar neo-Riemannian L, P, and R) preserve common tones. Sequential enchaining of contextual inversions will generally define simple, straight-line motions within the spaces described in this article. The musical motions of works by Webern, Schoenberg, and Stravinsky are measured against the systematic standard enshrined in the spaces, with both conformity and deviation suggesting interpretive possibilities. All appendixes for this article are available as supplementary material at http://dx.

Neo-Riemannian Graphs Beyond Triads and Seventh Chords

“…All the three sets are also in sum classes 1 and 2 because the sum of either the directed pitch intervals or the direct pitch-class intervals is equal to 1, but in these example only [3,4,6] and [3,5,6] are connect by the parsimony voice-leading. Figure 5c) shows that the set [t,0,4] connects by J with [9,1,3], and by F with [1,5,7] and, even neither of these connections are by parsimonious voice-leading, all the three sets are in sum classes 1 or 2 again, since the sum of either the directed pitch intervals or the direct pitch-class intervals is equal to 11. With these kind of connections it is possible to make cycles with members of trichords (013), (014), (016), (025), (026), and (027) which, as in hexatonic cycles, are divided into two adjacent sum classes.…”

Decontextualizing Contextual Inversion

“…It has been an important analytical resource to many theorists in a variety of analytical contexts. Exemplary analysis using contextual inversions, as well as references to many other applications, can be found in articles by Lambert [17] and Straus [24] and Kochavi's dissertation [15].…”

“…Straus [24] takes on the task of satisfying (1) perfectly while preserving some of the common-tone meaning of the neo-Riemannian transformations, by generalizing the two-common-tone property of P, L, and R to all trichords. An important flaw of this strategy was identified, however, by Fiore and Noll [8], which is that there is not a consistent group structure on P, L, and R so defined.…”

“…Inversionally symmetrical trichords have a single maximal common-tone preserving operation, but other trichords have at least three inversions that preserve two pitch-classes (four if one of its intervals is a tritone). For this reason, Straus [24] defines three contextual inversions for all trichords. But defining multiple contextual inversions poses an additional danger: for two operations to be understood as the same when applied to different sets, they must generate the same groups.…”

ATonnetzmodel for pentachords

“…For instance if τ " R is the retrograde then we will have the RI-chains. Geometrical representations with RI-chains involved are in the paper by Joseph Straus[31].…”