Exploring Tetrachordal Voice-Leading Spaces Within and Around the MORRIS Constellation

Abstract

[1] Building on Robert Morris's (1990) research on hexachordal ZC-relations, Stephen Soderberg (1998) identifies a constellation of ten hexachords that embed either one diminished seventh chord or two diminished triads. Soderberg divides the constellation, called MORRIS (or T-HEX), into four overlapping eight-hexachord sub-constellations based on tetrachordal subset content. The first of these sub-constellations, TRISTAN, includes the hexachords that embed two instances of set class 4-27[0258], the set class of the major-minor and half-diminished seventh chords. Similarly, constellations ZAUBER, AGITATION, and BROODING include the hexachords that embed two instances of set classes 4-18[0147], 4-13[0136], and 4-12[0236], respectively. Soderberg characterizes each of these tetrachordal set types as a of the diminished seventh chord. When the w, is 1, the result is set class 4-27-that is, moving any pc of a diminished seventh chord by interval class (ic) 1 creates a member of set class 4-27. Similarly, setting w = 2, 4, and 5 creates set classes 4-18, 4-13, and 4-12, respectively. The article goes on to point out a general property of voice leading: in each hexachord the pair of tetrachords can be connected by holding two pitch classes in common and by moving two others by ±w. The cases involving 4-27 and ic 1 voice leading (TRISTAN) are familiar-ii-V7,the Tristan chord with resolution, A7-B7at the beginning of Debussy's Faune, and others-and have been addressed in the theoretic literature by several authors.(1) Example 1 presents the MORRIS constellation and its four overlapping sub-constellations, henceforth called MORRIS1,MORRIS2,MORRIS4,and MORRIS5,with each subscript indicating the warp index, w.Example 1. The MORRIS Constellation (after Soderberg 1998)[2] Taking Soderberg's MORRIS constellation as a starting point, this study explores a wide variety of voice-leading transformations involving set types 4-27, 4-18, 4-13, and 4-12. It starts with the tetrachordal voice-leading transformations that produce the MORRISwhexachords-the nine ways that a tetrachord may be connected to another in the same set class by holding two pitch classes in common and by moving two others by ±w. (For each value of w there are nine transformations but only eight hexachord types because two of the transformations yield the same hexachord type.) The article then greatly expands the scope of inquiry, not only by allowing a given tetrachord to connect to any member of the same set class, but also by considering all twenty-four ways to voice-lead each of these twenty-four tetrachordal connections. As a result, each of these much larger voice-leading spaces-MORRIS+1,MORRIS+2,MORRIS+4,and MORRIS+5-contains 576 (= 24 × 24) voice-leading transformations. Each MORRISwis a subset of its corresponding MORRIS+w.[3] The paper provides an organized view of the entire MORRISw/MORRIS+wsystem, but it also develops other voice-leading spaces within MORRISw/MORRIS+w,each of which involves only a few of the MORRISw/MORRIS+wvoice-leading transformations. The additional voice-leading spaces usually also include rules for requiring, preferring, allowing, or forbidding one transformation to follow another. Although the number of potential voice-leading spaces is infinite,(2) my reasons for creating the ones I do are simple-to exercise melodic control over intervallic characteristics of the voices and to exercise harmonic control, not only of the individual tetrachords, but over the total pc content of sets of adjacent tetrachords. For example, in one space defined below a set of six MORRIS5voice-leading transformations can be concatenated in any order but remain within a single octatonic collection, in another space four MORRIS+4transformations are strung together so that one voice descends chromatically while the other voices also move stepwise, and in a third example ten MORRIS+2transformations are arranged so that each voice is saturated with a different interval class. …