Aligned-Cycle Spaces

Abstract

In this article I investigate aligned cycles. First identified by Alban Berg in a letter to Arnold Schoenberg and later discussed by George Perle and David Headlam, an aligned cycle is when two (or more) interval cycles unfold in the same direction in a note-against-note alignment, for instance, a rising whole-tone scale (an interval 2-cycle) simultaneously with a rising chromatic scale (an interval 1-cycle). This article extends beyond prior discussions of aligned cycles by exploring the different combinations and rotations of these cyclic structures and investigates how these permutations create different families of harmonies. In particular, this article focuses on three-voice aligned cycles. I show how the different combinations, alignments, and rotations of the individual voices of an aligned cycle create a unique collection of aligned-cycle classes and, at a more large-scale level, aligned-cycle families. To conclude, I reexamine a three-voice aligned cycle from the second movement of Berg's String Quartet, op. 3, and investigate cyclic progressions in Thomas Adès's Piano Quintet, Lieux retrouvés, and The Tempest.