Volume 17, Number 4, December 2011
Copyright © 2011 Society for Music Theory

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

Mark Sallmen


KEYWORDS: voice leading, transformations, harmony, pitch-class sets, networks, atonal music

ABSTRACT: Building on the work of Stephen Soderberg, Julian Hook, Robert Morris, and others, this article explores a wide variety of voice-leading transformations involving set types 4-27[0258], 4-18[0147], 4-13[0136], and 4-12[0236]. It considers tetrachordal connections between any two members of the same set class and all twenty-four ways to voice-lead each tetrachordal connection. The paper organizes these many possibilities and suggests compositional applications. It shows various ways to maintain control over the content of individual voices by constructing voice-leading spaces that involve a limited number of voice-leading transformations and rules for concatenating the transformations.

Received September 2009

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[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 “warp” of the diminished seventh chord. When the “warp index,” 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—ii4 3–V7, the Tristan chord with resolution, Asharp7Bflat7 at 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)

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[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 MORRISw hexachords—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 MORRISw is a subset of its corresponding MORRIS+w.

[3] The paper provides an organized view of the entire MORRISw/MORRIS+w system, but it also develops other voice-leading spaces within MORRISw/MORRIS+w, each of which involves only a few of the MORRISw/MORRIS+w voice-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 MORRIS5 voice-leading transformations can be concatenated in any order but remain within a single octatonic collection, in another space four MORRIS+4 transformations are strung together so that one voice descends chromatically while the other voices also move stepwise, and in a third example ten MORRIS+2 transformations are arranged so that each voice is saturated with a different interval class.

Example 2a. Introductory examples of MORRIS(+)w:
Scriabin, Vers la flamme, measures 14–15

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Example 2b. Introductory examples of MORRIS(+)w:
Debussy, Prélude à l’Après-midi d’une Faune, measure 16

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Example 2c. Introductory examples of MORRIS(+)w:
Berg, Wozzeck, Act III, measure 101

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Example 2d. Introductory examples of MORRIS(+)w:
One MORRIS+4 voice-leading transformation

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Example 2e. Introductory examples of MORRIS(+)w:
A realization of 2d; Berg, Altenberg Lieder, III, measures 2–8

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[4] For an introduction to the variety exhibited by the voice-leading transformations within the MORRIS+w universe, consider the musical excerpts in Example 2. Example 2a provides a transformation from MORRIS2, a pair of 4-18 tetrachords connected by ic-2 voice leading in two voices and by common tones in the other two, producing 6-30[013679].(3) An example from MORRIS+1, the excerpt from Debussy’s Faune in Example 2b includes a pair of 4-27 tetrachords (Gsharp6 5B7) with the same ic in each voice (ic 1) and an overall octatonic collection.(4) The excerpt from Berg’s Wozzeck in Example 2c provides a case from MORRIS+2, in which a pair of 4-18 tetrachords state ic 2 in two voices and ic 4 in the other two. Example 2d identifies a voice-leading transformation from MORRIS+4, a pair of 4-13 tetrachords with a different ic in each voice (ics 3, 0, 5 and 2). As shown in Example 2e, this four-voice model appears within the opening phrase of the vocal line in the third song of Berg’s Altenberg Lieder. Despite the linear realization, individual voices of the four-voice model are clear because each 4-13 articulates the same pitch contour 2–1–0–3. For example, ic 3 is clear in the “upper voice” because it is articulated by Bflat4, the last and highest pitch of the first 4-13, and Csharp5, the last and highest pitch of the second 4-13.

[5] The article builds on previous research in transformational theory and voice leading in several ways. First, following work of Julian Hook and others that deals with triads and seventh chords, the paper uses contextually-defined schritt (Sn) and wechsel (Wn) transformations to define mappings between tetrachords within the same set class.(5) Second, in his article on voice leading with set type 4-27, Adrian Childs (1998) develops transformation labels that identify contrary/parallel motion in the moving voices as well as the harmonic interval classes formed by the sustained and moving voice pairs; I employ an adapted version of this labeling system to address all four tetrachord types. Third, branching out from the research on parsimonious voice leading—which focuses on stepwise melodic motion and tends to emphasize situations where moving voices articulate the same melodic interval class—this paper embraces both stepwise and non-stepwise voice leading and includes cases where voices move by the same or by different interval classes.(6)

[6] Further, building on Robert Morris’s research in voice leading and compositional spaces, the paper utilizes T-matrices to investigate myriad voice-leading possibilities; develops several ways to measure voice leading, one of which is an unordered count of voice-leading interval classes that recalls Morris’s “voice-leading lists”; and suggests compositional applications of the theory.(7) All voice leading in the article is one-to-one and onto, that is, within each tetrachord each pc appears in exactly one voice and pcs are neither doubled nor omitted.(8) Finally, as is customary throughout much of the literature, networks graphically depict abstract and literal voice-leading spaces.(9)

[7] The paper is in five parts. Part I offers preliminary definitions of the transformation and voice-leading concepts used throughout the remainder of the paper. Part II introduces the MORRISw voice-leading spaces. Part III suggests how to concatenate the voice-leading transformations within each MORRISw while being sensitive to recurring patterns of moving and holding within individual voices, aggregate completion, and related issues. Part IV introduces MORRIS+1, MORRIS+2, MORRIS+4, and MORRIS+5, categorizes their voice-leading transformations according to various properties so that a coherent view of the entire system emerges, suggests several compositional applications, and provides an analysis of an excerpt from Bartok’s String Quartet No. 6. Part V shows how to put together lengthy strings of MORRIS+w voice-leading transformations while maintaining control over the intervallic content of individual voices.

Part I. Preliminary Definitions

[8] Part I defines several features of pitch-class sets (prime and inverted orientations, conventional ordering, set members) to prepare for a discussion of schritt and wechsel transformations (Sn/Wn). Three types of intervals are then defined; two are familiar (pc interval, interval class) and one is new (contextual interval). This sets the stage for the definition of voice-leading transformations (VL), as well as for ordered and unordered lists of interval classes and contextual intervals. Part I concludes with discussions of T-matrices and inverse- and Z-related VL.

[9] A set is said to be in prime orientation if it is related to its prime form by Tn and in inverted orientation if related to its prime form by TnI. For example, within set class 4-18[0147], {1, 4, 7, 0}, {2, 5, 8, 1}, and {3, 6, 9, 2} are in prime orientation because they are related by Tn to their set class prime form and {7, 4, 1, 8}, {6, 3, 0, 7}, and {5, 2, B, 6} are in inverted orientation because they relate to the prime form via TnI. We will adopt the conventional ordering of pcs within tetrachordal sets instanced by the sets just mentioned. That is, for sets in prime orientation the first three pcs articulate an ascending root-position diminished triad (e.g. {1, 4, 7, 0}), and for sets in inverted orientation the first three pcs articulate a descending root-position diminished triad (e.g. {7, 4, 1, 8}). Based on this conventional ordering, individual pcs within a set are designated as set members (sm) 1, 2, 3, and 4, respectively. That is, in {1, 4, 7, 0}, pc 1 is set member (sm) 1, pc 4 is sm 2, pc 7 is sm 3, and pc 0 is sm 4. Within {7, 4, 1, 8}, pc 7 is sm 1, pc 4 is sm 2, pc 1 is sm 3, and pc 8 is sm 4.(10)

[10] A schritt pcset transformation, Sn, is defined to articulate Tn when applied to a set in prime orientation, but T−n when applied to a set in inverted orientation. For example, S1 transforms {1, 4, 7, 0} into {2, 5, 8, 1}, which articulates T1, and S1 transforms {7, 4, 1, 8} into {6, 3, 0, 7}, which articulates TB. S0, the identity transformation, transforms any set onto itself. Concerning wechsel transformations, when Wn transforms a set in prime orientation into an inverted one the embedded diminished triads articulate Tn, but when it transforms an inverted set into a prime one the embedded diminished triads articulate T−n. For example, W1 transforms {0, 3, 6, 1} into {7, 4, 1, 6}, within which {0, 3, 6} and {1, 4, 7} articulate T1; W1 transforms {6, 3, 0, 5} into {B, 2, 5, 0}, in which {0, 3, 6} and {B, 2, 5} articulate TB. It would have been possible to define the W subscripts by chronicling the movement of any referential pc within the tetrachords. The use of the diminished triad strikes me as best because it allows a single rule for all four set types and has other advantages. For instance, it engages Straus’s 1997 notion of “near-transposition” (*Tn); that is, each Wn articulates *Tn when applied to a prime set and *T−n when applied to an inverted set because it moves all but one of the set’s pcs by the same pc interval.(11)

[11] In order to compare features of the different MORRIS+w voice-leading spaces, this paper often characterizes a set of transformations in terms of w, the warp index. For example, transformation Sw refers to MORRIS+1 S1, MORRIS+2 S2, MORRIS+4 S4, and MORRIS+5 S5. Similarly, Ww+3 refers to MORRIS+1 W4, MORRIS+2 W5, MORRIS+4 W7, and MORRIS+5 W8.

[12] This paper uses three types of voice-leading intervals. The first two types are familiar: given a voice in which pc x in tetrachordal set X is followed by pc y in tetrachordal set Y, the pc interval from pc x to pc y is y−x mod 12 and the interval class (ic) between pcs x and y is |y−x| mod 12. (Henceforth all arithmetic is considered to be mod 12 unless otherwise noted.) For example, the pc interval from 2 to 5 is 3, from 7 to 4 is 9, and from 1 to B is A, and the interval class between 2 and 5 is 3, between 7 and 4 is 3, and between 1 and B is 2. Like the subscripts of the Sn and Wn transformations, the third type of interval depends on the (prime or inverted) orientation of the initial set. If set X is in prime orientation the contextual interval (ci) from x to y is y−x (the same as the pc interval), but if X is in inverted orientation then the contextual interval from x to y is x−y (the inverse of the pc interval). For example, if pc 2 occurs within a set in prime orientation, such as {0, 3, 6, 2}, then the ci from pc 2 to pc 5 is 3; but if pc 2 appears within a set in inverted orientation, such as {2, B, 8, 0}, then the ci from pc 2 to pc 5 is 9. (The orientation of set Y does not affect the ci.)

Example 3a. VL and related concepts
Three realizations of MORRIS+4 W5 8259

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[13] Given any pair of tetrachordal sets X and Y, there are twenty-four ways to organize pcs into four voices so that each pc in each set appears precisely once. Therefore, there are twenty-four ways to voice-lead each Sn/Wn transformation. The paper identifies each of these possibilities using ordered lists of contextual intervals (oci). For example, oci 8259 indicates that sm 1 of X moves by ci 8, that sm 2 of X moves by ci 2, that sm 3 of X moves by ci 5, and that sm 4 of X moves by ci 9. A voice-leading transformation, hereafter VL, can be minimally defined by tetrachord type and oci.(12) For example, 4-13[0136] and oci 8259 define a VL that can be articulated by the set of voices {0–8, 3–5, 6–B, 1–A}. That is, sm 1 of X is pc 0, which moves by ci 8 to pc 8, sm 2 of X is 3, which moves by ci 2 to pc 5, sm 3 of X is 6, which moves by ci 5 to pc B, and sm 4 of X is 1, which moves by ci 9 to pc A. Throughout the paper, we will indicate tetrachord type with a voice-leading space (such as MORRIS+4 for 4-13) and include the pcset transformation that the VL articulates, creating VL labels such as MORRIS+4 W5 8259.(13) Example 3a provides three realizations of this VL. The first two use the set of voices {0–8, 3–5, 6–B, 1–A} and the third {5–9, 2–0, B–6, 4–7}, which begins with a chord in inverted orientation.(14)

[14] As various VL are compared and contrasted, it is useful to provide other ways to characterize them. The unordered list of contextual intervals (uci) uses superscripts to count the number of times each ci appears within a VL; for instance MORRIS+4 W5 8259 articulates uci 21518191. The unordered list of interval classes (uic) converts ci to ic; for example MORRIS+4 W5 8259 yields uic 21314151. On a larger level, these uci and uic organize into types based on their superscripts. There are five uci types: x4, x3y1, x2y2, x2y1z1, x1y1z1q1, where x, y, z, and q are different ci. For example, uci 21518191 (and all others with one instance of four different ci) are of the type x1y1z1q1. There are also five uic types: |x|4, |x|3|y|1, |x|2|y|2, |x|2|y|1|z|1, |x|1|y|1|z|1|q|1, where |x|, |y|, |z|, and |q| are different ic. For instance, uic 21314151 and all others with one instance each of four different ic are of the type |x|1|y|1|z|1|q|1. Concerning the relation of a VL’s uci and uic, when all ci in a uci articulate different ics (e.g. uci type x1y1z1q1 where x ≠ −y, x ≠ −z, x ≠ −q, y ≠ −z, y ≠ −q and z ≠ −q) the relation between uci type and uic type is straightforward, as in the case of uci 21518191 and uic 21314151. But when different ci articulate the same ic (e.g. x = −q), then the uci and uic superscripts will differ. For example, MORRIS+4 W5 824A articulates uci 214181A1 but uic 2242.

Example 3b. VL and related concepts:
T-matrix for MORRIS+4 W5

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Example 3c. VL and related concepts:
All MORRIS+4 W5 VL and their uci and uic interpretations

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Example 3d. VL and related concepts:
Inverse VL

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Example 3e. VL and related concepts:
Z-quadruple of VL

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[15] A T-matrix provides an efficient way to consider all VL that articulate a given transformation. Set X appears to the left of the matrix in conventional ordering and set Y appears above the matrix in conventional ordering. Matrix rows are labeled 1–4 from top to bottom to correspond to the set members of X, and columns are labeled 1–4 from left to right to correspond to the sm of Y. Matrix position (j, k) contains the contextual interval from sm j of X to sm k of Y. A one-to-one and onto mapping of X onto Y involves four matrix positions that include each matrix row and column precisely once. As an example, consider the matrix for MORRIS+4 W5, where X = {0, 3, 6, 1} and Y = {B, 8, 5, A}, given as Example 3b. The boldface and underlined ci within the matrix point out VL MORRIS+4 W5 8259—ci 8, 2, 5, 9 at matrix positions (1, 2), (2, 3), (3, 1), and (4, 4), respectively.(15) Notice that the notation MORRIS+4 W5 8259 does not explicitly indicate the various sm of Y, but that viewing the VL in its matrix context does make them explicit—the sm of Y are the column numbers in the matrix positions. For example, ci 8 at matrix position (1, 2) means that sm 1 of X moves to sm 2 of Y via ci 8, ci 2 at matrix position (2, 3) means that sm 2 of X moves to sm 3 of Y, and so forth. Later in the paper, when the sm of Y is not vital to the discussion I use the compact notation (e.g. MORRIS+4 W5 8259), but when the goal sm is vital—as in Part V when it comes time to control the intervallic content of a single voice in a lengthy chain of VL—I list ci along with their matrix position (e.g. ci 8 @ (1, 2), etc.). Example 3c lists the twenty-four VL that articulate transformation MORRIS+4 W5, along with their uci and uic interpretations.

[16] Two VL are inverses if and only if applying one then the other results in an overall S0 0000. Inverse-related wechsel VL articulate the same pcset transformation and the same uci, whereas inverse-related schritt VL articulate inverse pcset transformations and inverse sets of ci; for example, the inverse of MORRIS+4 W5 8259 is MORRIS+4 W5 5829 and the inverse of MORRIS+4 S1 1186 is SB BB64. Some VL are their own inverses, as with MORRIS+4 S6 0606 and MORRIS+4 W4 8286 (Example 3d).(16) VL are said to be Z-related if and only if they articulate the same pcset transformation and uic. Inverse-related wechsel VL are Z-related by definition, but there are often non-trivial Z relations. For instance, MORRIS+4 W5’s 87BA, A8B7, B247, and B724 each articulate uic 11214151, two pairs of inverse VL that create a Z-quadruple (Example 3e).(17)

Part II. VL within the MORRISw Voice-Leading Spaces

Example 4. Lattices for MORRISw

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[17] Part II derives and lists the MORRISw VL, then introduces the Childs-derived labels, which help to compare and contrast various VL. Example 4 features four lattices that facilitate deriving the MORRISw VL—the nine ways to transform each MORRISw tetrachord into another member of the same set class by holding two pitch classes in common and by moving each of the other two pitch classes by interval class w. Each lattice has <w>-cycle rows and <3>-cycle columns. Within each lattice, each row represents a voice, so that voice leading by ci w is modeled by horizontal movement to the right to an adjacent pc within a row and voice leading by ci −w by movement to the left. Vertical pc moves are forbidden. Having one pc in each row creates a tetrachord, which is a member of the MORRISw tetrachordal set class if and only if three pcs are in one column and one pc is in an adjacent column. Taking MORRIS5 as an example, starting with {0, 3, 6, 2}, there are twenty-four possible ways to move two voices by ic 5—six possible pairs of moving voices and two possible directions for each moving voice—but only nine of these create another 4-12. Since pc 2 is in a column by itself it is helpful to organize these nine possibilities around what pc 2 does. First, if pc 2 leads to pc 7 a dead end is reached because pc 7 would be two columns away from the other pcs and with only one move remaining it is impossible to move all three remaining voices to an adjacent column. If pc 2 leads to the left, to pc 9, there are six other single moves that generate another 4-12. Three of these involve moves to the right (0–5, 3–8, and 6–B) and the other three involve moves to the left (0–7, 3–A, and 6–1). Third and finally, if pc 2 does not move, then two of the other voices need to be led to the right to generate another 4-12. There are three ways to do this (0–5 with 3–8, 0–5 with 6–B, and 3–8 with 6–B). Overall, there are three cases where both moving voices articulate ci 5 (move to the right) and so the uci is 0252, three cases where both moving voices articulate ci 7 (move to the left) and so the uci is 0272, and three cases where one moving voice articulates ci 5 and the other ci 7 resulting in uci 025171. In the same way one could generate the other three sets of MORRISw transformations.

Example 5. MORRISwVL

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[18] Example 5 illustrates all of the resulting MORRISw VL in staff notation, along with their uci and uic interpretations. Each set of nine VL divide into three groups based on uci. In each case, S3 w00(−w), S6 0w0(−w), and S9 00w(−w) articulate uci 02w1(−w)1; W3 (−w)00(−w), W6 0(−w)0(−w), and W9 00(−w)(−w) articulate uci 02(−w)2; and W3+w 0ww0, W6+w w0w0, and W9+w ww00 articulate uci 02w2. In MORRIS5 for example, S3 5007, S6 0507, and S9 0057 articulate uci 025171; W3 7007, W6 0707, and W9 0077 articulate uci 0272; and W8 0550, WB 5050, and W2 5500 articulate uci 0252. These three groups of three correspond to the three general lattice-movement possibilities mentioned above: one move in each direction, two moves to the left, and two moves to the right, respectively. In each MORRISw, all nine transformations articulate uic 02w2.(18)

[19] The example also lists the hexachord types that are the focus of Soderberg’s article. Four set types appear in each MORRISw. S3 w00(−w) and S9 00w(−w) state 6-27[013469], S6 0w0(−w) states 6-30[013679], and W3 (−w)00(−w) and W9 00(−w)( −w) state 6-Z42[012369] and 6-Z29[023679]. Two hexachords appear in only two constellations: W6 0(−w)0(−w) generates 6-Z28[013569] in MORRIS1 and MORRIS5 and 6-Z45[023469] in MORRIS2 and MORRIS4.

[20] The example also provides Childs-derived labels, in which each digit indicates an interval class, “h” stands for held, “c” for contrary motion and “p” for parallel. For example, the label for MORRIS4 S3 4008 is h3c15, which indicates that the held pcs (3 and 6) articulate ic 3, the moving voices (0–4 and 1–9) progress in contrary motion, and create what I will call the moving dyads ({0, 1} and {4, 9}), which articulate ic 1 then ic 5. Such labels highlight five features of the VL. First, ics 3 and 6 play a salient role: each VL features ic 3 or 6, either as its held dyad or as its parallel moving dyads. Second, in MORRIS1 and MORRIS5, if a held dyad articulates an odd ic the moving dyads articulate even ics, and if the held dyad is even the moving dyads are odd (e.g. h3c24, h1p66, h6c11, h4p33), but in MORRIS2 and MORRIS4 the held and moving dyads within the same VL are either all odd or all even (e.g. h3c15, h3p11, h6c22, h4p66). Third, within each MORRISw, even though S6 0w0(−w) and W6 0(−w)0(−w) are different VL—one involves contrary motion and the other parallel—they involve the same held ic and the same moving dyad ics, as with MORRIS5’s S6 0507 = h6c11 and W6 0707 = h6p11.

Example 6. Some obverse pairs of VL

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[21] Fourth, each Childs-derived label that includes ic 3 appears in two MORRISw systems; that is, VL that articulate S3, S9, W3, W9, W3+w, and W9+w in MORRISw have the same set of held and moving dyads as the corresponding transformation in MORRIS6−w. For example, MORRIS1 S3 100B and MORRIS5 S3 5007 both articulate h3c24; MORRIS1 W3 B00B and MORRIS5 W3 7007 both articulate h3p22; and MORRIS1 W4 0110 and MORRIS5 W8 0550—which are W3+w 0ww0 in both cases—articulate h2p33. Fifth and finally, W3 and W3+w VL swap their held and moving ics as do W6 and W6+w, and W9 and W9+w, creating what we will call obverse VL pairs.(19) For example, MORRIS2 W9 00AA = h3p55 and WB 2200 = h5p33 are obverses. Any VL followed by its obverse results in an overall Sw wwww or Sw (−w)(−w)(−w)(−w). MORRIS2 WB 2200 then W9 00AA articulates an overall S2 2222, while MORRIS2 W9 00AA then WB 2200 an overall SA AAAA. Three other obverse pairs are shown in Example 6.

Part III. Four Applications of MORRISw VL

[22] There are innumerable ways to combine VL within a given MORRISw. As an introduction to this topic, part III explores four situations: it creates series of MORRIS1 VL in which one voice is held throughout, concatenates MORRIS5 VL in an octatonic context, and generates twelve-tone structures from MORRIS2 and MORRIS4 VL. Since long strings of VL will be considered and since each pcset transformation within the MORRISw universe is associated with only one VL, I will employ a short hand notation where a pcset transformation label stands for its corresponding VL; for example MORRIS1 W4 stands for MORRIS1 W4 0110, MORRIS1 S3 stands for MORRIS1 S3 100B, and so forth.

Example 7a. A voice-leading space within MORRIS1:
Network of members of 4–27 that include pc 2 and the MORRIS1 VL that connect them

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Example 7b. A voice-leading space within MORRIS1:
List of paths through the network that begin and end at D7, that use each other node once, and that alternate holding and moving the voice that begins with pc 0

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Example 7c. A voice-leading space within MORRIS1:
Realizations of paths 3 and 10 from 7b

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Example 8. A series of octatonic MORRIS5 VL

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[23] One way to limit the possibilities for VL concatenation is to control the motion of one or more particular voices. For example, it is possible to arrange VL so that a particular voice is always held. As an example, the network in Example 7a suggests how to assemble MORRIS1 VL so that pc 2 is held throughout. Network nodes identify the eight members of set class 4-27 that include pc 2 and lines connecting the nodes show which chords may be connected by MORRIS1 VL. Each line is bidirectional, so that, for instance, D7 may lead to D7 and vice versa. In most cases, both directions articulate the same transformation, as with W4 which transforms D7 into D7 and vice versa; but in four cases, one direction articulates S3 and the other its inverse, S9 (e.g. S3 transforms G7 into E7 and S9 transforms E7 into G7). Since D7 and E7 each connect to three other chords, and the other six chords each connect to five others, there are still a huge number of possible paths through this network.

[24] To reduce the possibilities further we can stipulate that the path begins and ends at D7 and includes each other chord precisely once, and that the voice beginning with pc 0 alternates holding and moving. Since each path begins with D7, and by holding pc 0, the only other chord that includes pc 0, D7, must be the second chord. Pc 0 must then move by ic 1. Since no chords on the network include Csharp, the next chord must be one of the four that includes pc B (B7, E7, G7, Gsharp7). Pc B then holds, and so the next chord must also be from this group. Since each chord in this group connects to two others in the group there are always two possibilities. Pc B must then move by ic 1. It may not move back to pc 0 (because there are no new chords that include pc 0) so it must move to pc A, which is harmonized by the only two chords that include it Bflat7E7 (or vice versa). In similar fashion the path returns to D7 via the two chords that have not yet been included. All in all, there are ten possible paths, given in Example 7b. Two paths contain only wechsel VL and eight paths contain a mixture of wechsel and schritt VL. Example 7c realizes two of these paths in staff notation. In each case the held pc 2 appears in the bass voice and the pc-0 voice in the soprano.

[25] Another way to restrict choice is to limit the possible VL to fewer than nine; say, for example, the six that produce hexachords that are subsets of the octatonic collection. In MORRIS5 for instance, these are S3, S6, S9, W8, WB, and W2. Any string involving these VL stays within the same octatonic collection. The polyrhythmic texture in Example 8 provides one such series, W8S9W8S3S3W2S9W2S9W8W2, which uses only four of the six octatonic VL but which includes all eight instances of 4-13 within one octatonic collection. In this and all such octatonic series each voice alternates pc intervals w and −w and therefore oscillates between two pitch classes; this occurs because consecutive w (or consecutive −w) would lead outside the octatonic collection.

[26] It is also possible to concatenate VL so that they create twelve-tone structures. For example, consider series of MORRIS2 VL that embrace all twelve pcs as quickly as possible, that is, four pcs in the first chord and two new pcs in each of the next four. Put another way, each series uses each pc exactly once—a pc may be held from one chord to the next but once left it is not reiterated. Fulfilling these conditions is only possible when all four voices move in the same direction and so schritt VL, which involve contrary motion, are forbidden.(20) Since a series of wechsel VL creates an alternation between sets in prime orientation and those in inverted orientation, there must also be an alternation of contextual intervals 2 and A in order to keep the voices moving by the same pc interval. This means that if the first transformation is W3, W6, or W9, which articulate uci 02A2, then the next transformation must be W5, W8, or WB, which articulate uci 0222. For example, transforming {0, 3, 6, B}, a set in prime orientation, by W3 creates motion by pc interval A (0–A and B–9), which leads to {A, 3, 6, 9}, a set in inverted orientation. To continue motion by pc interval A, we need to choose W5, W8, or WB; choosing WB, for instance, creates moving voices 6–4 and 9–7, which lead to {A, 3, 4, 7}. At this point there are three possible continuations that complete the aggregate: W3W5, W9WB, and W6W8.

Example 9. Seven MORRIS2 series that complete the aggregate as quickly as possible

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Example 10a. A twelve-tone compositional design saturated with MORRIS4 VL: Matrix

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Example 10b. A twelve-tone compositional design saturated with MORRIS4 VL: Musical realization of matrix

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[27] Musical realizations of these three complete series, W3WBW3W5, W3WBW9WB, and W3WBW6W8, appear in the first system of Example 9. The second system presents the other four series that begin with W3. There also exist seven series that begin with W9 and six that begin with W6. All twenty of these series feature the same set of four voices (0–A–8, 6–4–2, 3–1, and B–9–7–5) but with varying pc repetitions. For instance, in various series the 0–A–8 voice is articulated as 0–A–A–A–8, 0–A–A–8–8, 0–0–A–8–8, 0–0–A–A–8, and 0–0–0–A–8. Finally, all twenty of these series may also be played in retrograde, for a total of forty MORRIS2 series that complete the aggregate as quickly as possible.(21)

[28] MORRIS4 VL feature set type 4-13[0136] and voice leading by ic 4. Since 4-13 creates the pc aggregate when transposed by 4 and 8, these VL are ideally suited to produce twelve-tone designs. Consider the matrix in Example 10a, whose rows and columns are saturated with MORRIS4 VL, each expressed as a series of three dyads. For example, {0, 1}–{3, 6}–{8, 9} in the top row expresses W3 = h3p11. The held pcs, 3 and 6, appear in the middle dyad, surrounded by the moving dyads, so that both 4-13 are clear ({0, 3, 6, 1} and {3, 6, 9, 8}). The moving voices, 0–8 and 1–9, are articulated by pcs in the outer dyads. Completing the top row, {8, 9}–{B, 2}–{4, 5} and {4, 5}–{7, A}–{0, 1} replicate this W3 transformation at T4 and T8, creating a twelve-tone cycle that wraps around to its starting point. Offset by one dyad, this row also thrice embeds W7 = h1p33, the obverse of W3 = h3p11: {3, 6}–{8, 9}–{B, 2} and its T4 and T8 transformations. The second-highest row is a retrograde rotated circle-of-fifths transformation of the top row and therefore embeds W9 = h3p55 and W1 = h5p33. The remaining rows and columns are T0/T4/T8 transformations of these two.(22)

[29] Example 10b realizes the matrix for three pianists, one staff/hand/register for each matrix row and one quarter-note beat for each column. Instead of realizing the entire matrix at once, measure 1 articulates the upper-right portion of the matrix (including the main diagonal) and measure 2 the lower left (also including the main diagonal), so that each staff begins and ends with the dyad from the main diagonal, as with the top row’s {0, 1}, the second row’s {0, 5}, and so forth. As a result, the full columnar aggregate appears only twice, at the end of measure 1 and the beginning of measure 2.

Part IV. MORRIS+w: Other Voice-Leading Possibilities

[30] Part IV considers four larger sets of possibilities involving the same four tetrachord types, each of which embraces all twenty-four ways to transform a given set into another member of the same set class and all twenty-four VL for each transformation. These greatly expanded spaces are labeled MORRIS+1, MORRIS+2, MORRIS+4, and MORRIS+5. The discussion first introduces the additional schritt and wechsel transformations (with one VL for each) and then addresses the multiple VL possibilities.

Example 11a. MORRIS+w transformations:
Ww
, W0 and W2w

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Example 11b. MORRIS+w transformations:
W2w+3
, W2w+6, and W2w+9

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[31] Example 11a presents Ww, W0, and W2w within each MORRIS+w. Ww features a VL in which the pcs of the diminished triad move by ci w and the other pc by −w, so that all four pcs move by ic w (that is uic = w4). See, for example, MORRIS+2 W2 222A (uic 24). The total pc content of each Ww transformation is an octatonic collection. Each W0 features a VL that holds the diminished triad invariant and moves the other voice by ci –2w; for instance, MORRIS+5 W0 0002 sustains the pcs of the diminished triad while the other pc articulates ci 2. Conversely, each W2w features a VL that moves the diminished triad by ci 2w and sustains the other pc, as with MORRIS+5 WA AAA0, which moves the pcs of the diminished triad by ci A while the other pc is sustained. The total pc contents of W0 and of W2w are abstract complements of one another, as in MORRIS+5, where they articulate 5-8[02346] and 7-8[0234568], respectively. Example 11b presents the remaining trio of wechsel transformations within each MORRIS+w: W3+2w, W6+2w, and W9+2w (e.g. MORRIS+4 WB, W2, and W5). Note that each MORRIS+w connects to each of the three others via transformations that share the same total octachordal pc content. For example, MORRIS+1’s W5, W8, and WB share 8-23[0123578A], 8-24[0124568A], and 8-20[01245789] with MORRIS+4 W5, MORRIS+5 W4, and MORRIS+2 W1, respectively.

[32] The remaining nine transformations within each MORRIS+w are the schritt transformations other than S3, S6, and S9, which are easy to envision: transpositionally-related chords in which all voices move in parallel motion by ci n.(23) (S0 is the identity transformation, which transforms a set onto itself.) These schritt transformations articulate a few interesting features concerning total pc content. First, Sw and Sw, the only schritt transformations with octachordal total pc content, generate the complement of the MORRIS+|2w| tetrachord. When w = 1 or 5, |2w| = 2, which means that MORRIS1’s S1 and SB and MORRIS+5’s S5 and S7 each create 8-18[01235689], the complement of MORRIS+2’s 4-18. Similarly, when w = 2 or 4, |2w| = 4, so that MORRIS+2’s S2 and SA and MORRIS+4’s S4 and S8 generate 8-13[01234679], the complement of the MORRIS+4’s 4-13. The latter case is special because pairs of 4-13 tetrachords are generating their own complement. Second, for w = 1, 2, and 5—but not 4 because of the special case just mentioned—S2w and S–2w generate the same total septachordal pc content as W2w. For example, MORRIS1’s S2, SA, and W2 each generate 7-34[013468A]. All in all, it will be helpful to divide the twenty-four transformations within each MORRIS+w into six groups as follows: The S0-group includes S0, S3, S6, and S9; the Sw-group includes Sw, S3+w, S6+w, S9+w; and the Sw-group includes Sw, S–3−w, S–6−w, and S–9−w. The W0-group includes W0, W3, W6, and W9, the Ww-group includes Ww, W3+w, W6+w, and W9+w, and the W2w-group W2w, W3+2w, W6+2w, and W9+2w.(24)

Example 12a. T-matrices: Generalized T-matrices for Sn and Wn within MORRIS+w

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Example 12b. T-matrices: All MORRIS+w T-matrices

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[33] With all of the Sn/Wn transformations in place we now consider multiple VL possibilities for each, which, as outlined in part I, can be derived from T-matrices. Example 12a provides generalized matrices in terms of n and w (for w = 1, 2, 4, or 5), one for Sn and one for Wn, which allow general properties of the matrices and resulting VL to emerge. For example, each Sn matrix features ten different ci, four that appear multiple times and six others that appear only once. Each Wn matrix features eight different ci, one that appears once, one that appears three times, and six others that appear twice each. Each Wn matrix is symmetrical around the main diagonal. Many consistent patterns may be drawn from the matrices but I will mention only three. First, in every Wn matrix there are four sets of matrix positions that yield an oci with two instances each of two different ci (uci type x2y2). For instance, with the matrix for MORRIS+w Wn the set of positions {(1, 1), (2, 4), (3, 3), (4, 2)} yields oci xyxy, with uci x2y2 where x = 6+n and y = 6+n−w.(25) Second, the upper-left 3 × 3 square of a Wn matrix depends on n but not w, which means that all four W1 matrices have an identical upper-left 3 × 3 square. Third, transformations in the same group feature the same set of ci in these w-independent matrix positions, a set equal to their subscripts taken as a set. For example, the W0-groups have ci 0, 3, 6, and 9 in their upper-left 3 × 3 squares. These features can be verified by consulting Example 12b, which lists all ninety-six matrices.

[34] I will now organize the voice-leading possibilities within this expanded universe by uic type. VL articulate one of five uic types, |x|4, |x|3|y|1, |x|2|y|2, |x|2|y|1|z|1, and |x|1|y|1|z|1|q|1, all but the fourth of which are addressed below. The discussion of each uic type is in two parts. The first addresses the uci types that generate the uic type and the transformations and VL that create each uci type. The second identifies the possible uic within the uic type, lists the VL within each uic and groups the various uic into categories based on the presence/absence of ics 0, 3, and/or 6. Different uic within the same category contain the same (or similar) numbers of VL, which articulate the same (or similar) number and type of schritt/wechsel transformations. The discussion outlines the organization of the possible VL and suggests several compositional applications.

[35] The MORRIS+w system includes 2,304 VL. Of these, eighty articulate uic type |x|4, 136 articulate uic type |x|3|y|1, 432 uic type |x|2|y|2, 904 uic type |x|2|y|1|z|1, and 752 uic type |x|1|y|1|z|1|q|1.

Example 13a. Uic type |x|4 :
80 VL with uic type |x|4 organized by uci type and pc transformation

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Example 13b. Uic type |x|4 :
Realizations of some VL from chart in 13a

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Example 13c. Uic type |x|4 :
80 VL with uic type |x|4 organized by uic in categories

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Example 13d. Uic type |x|4 :
Six-voice model constructed of MORRIS+4 VL with uic 24

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Example 13e. Uic type |x|4 :
Musical excerpt based on model in 13d

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Example 14a. Uic type |x|3|y|1:
136 VL with uic type |x|3|y|1 organized by uci type and pc transformation

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Example 14b. Uic type |x|3|y|1 :
Sample VL from chart in 14a

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Example 14c. Uic type |x|3|y|1 :
136 VL with uic type |x|3|y|1 organized by uic in categories

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Example 15a. Uic |x|2|y|2 :
432 VL with uic type |x|2|y|2 organized by uci type and pc transformation

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Example 15b. Uic |x|2|y|2 :
Sample VL from chart in 15a; uci type x2y2 where x ≠ −y

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Example 15c. Uic |x|2|y|2 :
Sample VL from chart in 15a; uci type x2y1z1 where z = −y

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Example 15d. Uic |x|2|y|2 :
Sample VL from chart in 15a; uci type x1y1z1q1 where z = −x and q = −y

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Example 15e. Uic |x|2|y|2 :
Articulations of MORRIS+2 W4 715B/B715 from 15d

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Example 15f. Uic |x|2|y|2 :
Musical passage based on chords in 15e

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Example 15g. Uic |x|2|y|2 :
432 VL with uic type |x|2|y|2 organized by uic in categories

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Example 15h. Uic |x|2|y|2 :
VL pairs with uic 1232 that organize into VL pairs based on shared harmonic ics

Example 15h thumbnail

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[36] The eighty VL that articulate uic type |x|4 arise from three uci types: x4, x3y1 where x = −y, and uci x2y2 where x = −y. Forty-eight VL articulate uci type x4; these are the straightforward cases involving parallel voice leading. Eight VL arise from uci type x3y1 where x = −y: one VL from Ww and one from W6+w within each MORRIS+w (e.g. MORRIS+1’s W1 111B and W7 7775). The other twenty-four VL have uci type x2y2 where y = −x. Each MORRIS+w includes S3 9393 and S9 3939, a total of eight VL with uci 3292 and uic 34. (The oci is identical for each MORRIS+w because it involves array positions (3,1), (2,2), (1,3) and (4,4), whose values do not depend on the pc that differentiates the tetrachords from one another.) The remaining sixteen VL occur within the W2w-groups of MORRIS+2 and MORRIS+4. In MORRIS+2, W4 contains B11B, 4848, and 7755, which produce uic 14, 44, and 54, respectively. WA also articulates three such VL, and W7 and W1 one each. In MORRIS+4 it is WB and W5 that include three each and W8 and W2 that have one. Example 13a lists all of these VL and Example 13b provides realizations of some of them in staff notation.

[37] Example 13c divides the seven uic (04, 14, 24, 34, 44, 54, 64) into three categories.(26) The first category includes uic 04 and 64, each of which includes only four VL. The second category includes only uic 34, which includes sixteen VL, the Z-pairs S3 3333/9393 and S9 9999/3939 in each MORRIS+w. Category 3 uic 14, 24, 44, and 54 each contain fourteen VL. In uic 14 and 54 these are distributed 3+5+3+3 across the four MORRIS+w spaces and in uic 24 and 44, 2+4+6+2.

[38] Example 13d provides a seven-chord, six-voice model created from the MORRIS+4 VL within uic 24, with the top four voices articulating one string of VL and the bottom four voices another. Such models are easy to generate because any two voices saturated with ic 2 that do not form harmonic ic 4 (the ic that 4-13 does not contain) can participate in multiple statements of these VL.(27) This model demonstrates variety in several respects. It incorporates all six MORRIS+4 VL with uic 24 (S2 2222, SA AAAA, W5 A22A, W8 2A2A, WA AAA2, and WB 22AA) and mixes prime and inverted sets in various ways. At times it unfolds the same VL in both layers simultaneously while at other times the VL differ. The voice leading from one chord to the next usually mixes pc intervals 2 and A, but the third and fourth chords articulate inverse-related VL, S2 2222 and SA AAAA, which, since they are applied to sets of contrasting orientation, create pc interval 2 in all voices. Moreover, the seven chords articulate six different set types; each of the first six chords articulates a MORRIS-constellation hexachord because it is comprised of two 4-13s that share two pcs, and the final chord has only four pcs—the same 4-13 in both layers. Example 13e provides a musical realization based on the model: the middle two voices of the model appear in the cello part, a mixture of double stops and compound melody, and the other model voices appear in the piano part, all in the treble register. The middle of the excerpt articulates the complete model, while the beginning and end offer fragments.(28)

[39] There are 136 VL with uic type |x|3|y|1; forty derive from uci x3y1 where y ≠ −x and ninety-six from uci = x2y1z1 where z = −x. Each Wn has precisely one VL in which the diminished triad moves by ci n and the other pc moves by ci n–2w, which creates forty-eight uci of the form x3y1 (where y = n–2w). Omitting the eight instances where y = −x leaves forty VL with uic = |x|3|y|1. Half of the remaining ninety-six VL articulate Sn, six for each of eight values of n (1, 2, 4, 5, 7, 8, A and B). Considering n = 1 as an example, MORRIS+1 S1 B113, MORRIS+4 S1 1B13, and MORRIS+5 S1 311B generate uic 1331; MORRIS+2 S1 7975 generates uic 5331; and MORRIS+4 S1 4480 and S1 2AA6 generate uic 4301 and 2361, respectively. Another sixteen arise because W3, W9, Ww+3, and Ww+9 articulate one VL of this type within each MORRIS+w. These VL involve voice exchange, indicated by *. For example, MORRIS+5 W3 9395* can be realized by the set of voices {0–9, 3–6, 6–3, 2–7}, in which 3–6 and 6–3 articulate a voice exchange. Accounting for the remaining thirty-two VL, each transformation within the W2w group of MORRIS+1 and MORRIS+5 articulates four VL with uic |x|3|y|1. For instance, MORRIS+5 WA includes AA82, 2A42, 2AA8, and 4A88; the first three articulate uic 2341—forming a Z-triple—and the last one uic 4321. Example 14a provides a complete list of VL with uic type |x|3|y|1 and 14b realizes the S1 and MORRIS+5 WA VL in musical notation.

[40] Example 14c organizes all 136 VL with uic type |x|3|y|1 by uic. There are four categories. Category 1 includes the four uic for which |y| = 3 × |x|: these are 1331, 2361, 4301, and 5331. Each of these four uic includes twelve Sn-based VL along with two or four Wn-based VL. The remaining three categories include only wechsel-derived VL. Each uic in category 2 includes eight VL, each one in category 3 twelve VL, and each one in category 4 only two.

[41] The 432 VL with uic type |x|2|y|2 articulate one of three uci types: uci type x2y2 where x ≠ −y (216 VL), uci type x2y1z1 where z = −y (136 VL), and uci type x1y1z1q1 where x = −z and y = −q (80 VL).

[42] Of the 216 VL that articulate uci type x2y2 where x ≠ −y, forty are Sn (6+n)n(6+n)n where n ≠ 3 or 9—S0 6060*, S1 7171, S2 8282, S4 A4A4, and so forth. The remaining ones arise from Wn transformations. Most MORRIS+w Wn articulate four such VL, one for each of the two dyad partitions that involves [03] and two for the partition that involves [06].(29) For example, MORRIS+5 W4 features 8118, 7722, A5A5, and 4545, which articulate uci 1282, 2272, 52A2, and 4252, and uic 1242, 2252, 2252, and 4252, respectively. In 8118, the embedded [03] dyad formed by sm 2 and sm 3 moves by ci 1 and the [02] dyad formed by sm 1 and sm 4 moves by ci 8; and in 7722, the [03] dyad formed by sm 1 and sm 2 moves by ci 7 and the [04] dyad (sm 3 / sm 4) moves by ci 2; in A5A5 and 4545 the [01] dyad (sm 2 / sm 4) moves by ci 5 and the [06] dyad (sm 1 / sm3) moves by ci A in one case and by ci 4 in the other. The upper portion of Example 15a lists all of these VL. In Example 15b the VL mentioned in the discussion are realized so that the top two voices move in parallel motion by one ci and the bottom two voices move in parallel motion by another.(30)

[43] The 136 VL with uci type x2y1z1 where z = −y arise as follows: each MORRIS+w’s S0 and S6 include six such VL, each S3 and S9 includes four, each Ww and W6+w three, and each W3, W9, W3+w and W9+w two. Each S0 includes five VL that hold two pcs (ci 0) while the other two voices articulate a voice exchange (*) and one VL with a double voice exchange (**) (e.g. MORRIS+2 S0 0057*, 3900*, 0804*, B001*, 0390*, and 6864**), each S6 includes one VL with a pair of ic 0 and five with a pair of ic 6 (e.g. MORRIS+2 S6 020A, 5667, 66B1, 9366, 6936, 626A*), and so forth. The middle part of the chart in Example 15a provides the complete list and Example 15c realizes these and other sample VL in musical notation.

[44] The 80 VL with uci type x1y1z1q1 where z = −x and q = −y arise only from the S0 group of each MORRIS+w and from the W2w group of MORRIS+2 and MORRIS+4. Each S0, S3, S6, and S9 includes two VL, both of which involve ic 3 (e.g. MORRIS+5 S0 3984** and 239A**, S3 5397* and 93B1, S6 932A and 8934, S9 3957*and B931). Each W2w-group transformation includes six VL organized into three Z-pairs; for instance, MORRIS+2 W4 includes 715B/B715 (uic = 1252), B148/481B (uic = 1242), and 4758/7845 (uic = 4252). The lower part of Example 15a includes the complete list and 15d lists most of the cases just mentioned in staff notation.

[45] Examples 15e and 15f illustrate a compositional application involving one of these Z-pairs: MORRIS+2 W4 715B/B715. Example 15e includes four two-chord, four-voice models, labeled X 715B, Y 715B, X B715, and Y B715. X and Y denote tetrachord pairs (X = {0, 3, 6, B}–(A, 7, 4, B} and Y = {1, A, 7, 2}–(3, 6, 9, 2}) and 715B and B715 signify the motion of the voices. Each model is arranged so that pc interval B appears in the soprano register, pc interval 1 in the alto, pc interval 5 in the tenor, and pc interval 7 in the bass. Example 15f provides a short musical excerpt for piano that includes five different combinations of these models, in which voices sometimes appear an octave higher or lower than in the model in order to achieve more sonorous spacing and/or to avoid registral overlap. The excerpt first explores the four ways to combine either X 715B or X B715 with either Y 715B or Y B715. Although each combination results in the same pair of octachordal pcsets and the same melodic motion within each register, there is considerable variety because each combination creates a different pair of parallel dyads within each register, as with the soprano’s {B5, G6}–{Asharp5, Fsharp6}, {B5, Bflat6}–{Asharp5, A6}, {C7, G7}–{B6, Fsharp7}, and {C6, Bflat6}–{B5, A6}, which articulate harmonic ics 4, 1, 5, and 2, respectively. The excerpt ends with the simultaneous unfolding of X 715B, X B715, Y 715B, and Y B715, sixteen-voice harmony in which each pc of X and Y appears in two registers. Each register contains parallel tetrachords, 4-4[0125] in soprano and alto and 4-14[0237] in bass and tenor.

[46] Example 15g lists all of the VL of uic type |x|2|y|2, organized by uic. The twenty-one possible uic organize into six categories. Category 1 involves only ics 0, 3, and/or 6, that is, uic 0232, 0262, and 3262. All VL within these uic arise from S0-group transformations and have identical oic in each MORRIS+w. Category 2 involves ic 0 but not ics 3 and 6. Each uic 02n2 (0212, 0222, 0242, 0252) includes fourteen VL: nine VL that constitute the MORRISn VL discussed at length in parts II and III of the paper, one S0 VL in each of the other three spaces, and a further pair in MORRIS+6−n. Category 3 involves ic 6 but not ics 0 and 3. Mirroring category 2, each uic 62n2 (6212, 6222, 6242, and 6252) embeds fourteen VL: nine within MORRIS+6−n, one S6 VL in each of the other three spaces, and a further pair in MORRIS+n.

[47] The fourth category involves ic 3 but not ics 0 and 6. Each uic 32n2 (3212, 3222, 3242, and 3252) embeds thirty-two VL, ten within MORRISn, ten within MORRIS6−n, and six in each of the other two spaces. A fascinating subset of each uic in this category is the presence of four VL pairs that share not only melodic ics but harmonic ones. Consider as an example MORRIS+2 W2 9BB9 and MORRIS+4 W4 9119. In each case, sm 2 and sm 3 create ic 3 harmonically and move by ic 1 melodically, and sm 1 and sm 4 create ic 1 harmonically and move by ic 3 melodically. Example 15h realizes this pair and the three other pairs within uic 1232.

[48] Categories 5 and 6 do not involve ics 0, 3, and 6. In category 5, one ic is odd and the other even (1222, 1242, 2252, and 4252) and in category 6 both are odd or both are even (1252 and 2242). Within category 5, each uic includes twenty VL, all of which articulate wechsel transformations. Within category 6, each uic consists of thirty-six VL, including a pair of Z-quadruples (e.g. MORRIS+2’s W1 4A28/84A2/4422/8AA8 and W7 A482/2A48/2442/AA88 within uic 2242).

Example 16. VL of uic type |x|2|y|2 in Bartok, String Quartet No. 6, IV, measures 1–13

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[49] The discussion of uic type |x|2|y|2 concludes by pointing out a few such VL within an excerpt from Bartok’s String Quartet No. 6. The excerpt in Example 16 is the opening of the piece, in which the outer voices present overlapping statements of the opening 3½-measure melody (bracketed on the score) and then diverging sequences based on fragments of that melody. The inner voices are also primarily stepwise, moving first in free counterpoint and then in parallel tritones. Harmonically, the excerpt features five triadic anchor points, nineteen instances of the MORRIS tetrachords, and a variety of other sonorities. The MORRIS tetrachords create six VL of uic type |x|2|y|2: two from MORRIS1 (W9 00BB and W7 1010), one from MORRIS2 (S3 200A), a side-by-side pair from MORRIS+4 (SB 5B5B and WA A0A0), and one from MORRIS+5 (S0 6B61**). Two of these VL articulate uic 0212, two uic 0222 and the others 1252 and 1262. Four involve ic 0, which is not surprising given the frequency with which two voices hold while two others move, and all six involve either ic 1 or ic 2, which makes sense given the stepwise thematic material. In all but two cases “passing sonorities” complicate the musical realizations. The rhythmic offset of the moving voices of MORRIS2 S3 200A creates an intervening 4-17[0347], parallel tritones moving by half step decorate W7 1010 with a pair of 4-16s [0157], and further stepwise motion during MORRIS+4 SB 5B5B and MORRIS+5 S0 6B61** creates embellishing 4–5 [0126] and 4–6 [0127] sonorities. Moreover, each VL features precisely two voices moving stepwise (either by ic 1 or ic 2), which creates contrast with the approaches to the Fsharpm, GM, EflatM and Cm triadic anchors, each of which involves stepwise motion in all four voices. (The approaches to Fsharpm and GM mix half- and whole-step voice leading and the approaches to EflatM and Cm feature all four voices moving by half step.) Finally, taken as a group, the roots of all five triadic anchors (A, Fsharp, G, Eflat, C) articulate 5-31 [01369], the sole pentachordal set type that embeds all four of the MORRIS tetrachords.

[50] There are 752 VL that articulate uic type |x|1|y|1|z|1|q|1, all of which derive from uci type |x|1|y|1|z|1|q|1 where x ≠ −y, x ≠ −z, x ≠ −q, y ≠ −z, y ≠ −q, and z ≠ −q. Each Sn and Wn generates between 20 and 48 VL of this uic type (consult Example 17a) and the VL organize into nineteen uic (which divide into six categories) as shown in Example 17b. The reader may wish to step through these charts on their own as I have done with similar charts above, or simply proceed with the discussion here, which provides a compositional application involving one sample uic and then highlights another special uic.

Example 17a. Uic |x|1|y|1|z|1|q|1 :
752 VL with uic type |x|1|y|1|z|1|q|1 organized by uci type and pc transformation

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Example 17b. Uic |x|1|y|1|z|1|q|1 :
752 VL with uic type |x|1|y|1|z|1|q|1, organized by uic in categories

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Example 17c. Uic |x|1|y|1|z|1|q|1 :
Twelve MORRIS+4 VL with uic 01213151 (see chart in 17b)

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Example 17d. Uic |x|1|y|1|z|1|q|1 :
Musical excerpt featuring three MORRIS+4 VL with uic 01213151

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Example 17e. Uic |x|1|y|1|z|1|q|1 :
MORRIS+2
VL with uic 11214151 (see chart in 17b)

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[51] VL with uic type |x|1|y|1|z|1|q|1 can be used to differentiate voices from one another. As an example, consider the twelve MORRIS+4 VL with uic 01213151: S0 0372/0A95, S3 90A5, S9 0732, W0 0295/5902, W3 9025/9502, WA 37A0/A073, and W1 73A0/7A03. Example 17c realizes these in staff notation so that ic 0 always appears in the soprano voice, ic 2 in the alto, ic 3 in the tenor, and ic 5 in the bass. Example 17d constructs a short excerpt for string quartet that incorporates three 01213151 VL. The passage’s six 4-13 chords articulate S9 0732–(S9 3669)–W0 0295–(W7 777B)–S0 0372. For each 01213151 VL, ic 0 appears in violin I, ic 2 in violin II, ic 3 in viola, and ic 5 in the cello. Since every second VL is part of 01213151, each pc in the passage participates in a statement of its instrument’s primary ic; for instance, violin II’s DflatEflat, A–B, and C–D state ic 2. The parenthesized VL, which do not articulate uic 01213151, arise as a result of the consistent ic/instrument pairing within the 01213151 VL. Three other features of the passage bear mentioning. First, the 01213151 VL involve more shared pcs as the passage progresses; that is, at S9 0732 the 4-13s share two pcs, at W0 0295 three, and at S0 0372 all four. This is reflected by gradual decreases in dynamic level, rhythmic activity, and register. Second, all 4-13s but the fourth are in prime orientation; these articulate S9S9S5S0, which is a large-scale projection of 4-13. Finally, the staggered pc entrances within each 4-13 are calculated to reinforce pitch/rhythmic relationships; that is, each ic 2 in violin II articulates an attack-point duration of 2 (taking the eighth note as the unit), each ic 3 in the viola duration 3, and each ic 5 in the cello duration 4. The simultaneous arrival of the pcs in the final chord strengthens the sense of cadence.

[52] We conclude the discussion of uic type |x|1|y|1|z|1|q|1 by pointing out uic 11214151, whose sixty-four VL are the most of any uic we have seen. These sixty-four organize into sixteen Z-pairs, with a further quartet of Z-quadruples in both MORRIS+2 and MORRIS+4. Example 17e realizes the twenty MORRIS+2 VL within this extraordinary uic.

Part V. Concatenating MORRIS+w VL and Controlling Individual Voices

[53] Part V focuses on several ways to concatenate MORRIS+w VL while controlling the content of one or more individual voices. First, it concatenates repeated instances of a single VL and alternates Z-related VL. Second, a set of VL within a single uic are put together so that they saturate each voice with a different ic and then so that each voice alternates between two ics. Two further examples saturate a given voice with a particular pc interval (not just interval class) and another shows how to maintain a consistent relationship between two voices.

[54] In order to study the content of individual voices it is crucial to consider the following general feature of VL concatenation: given a series of three chords that articulate VLa then VLb, ci xa at matrix position (ra,ca) within the matrix of VLa, and ci xb at matrix position (rb,cb) within the matrix of VLb, ci xb will follow ci xa in the same voice if and only if ca = rb.

Example 18. Strings of identical and Z-related VL

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[55] For example, if VLa and VLb are both equal to MORRIS+2 S2 8273, ci 8 at matrix position (1,3) within VLa will be followed in the same voice by ci 7 at matrix position (3,4) because the column of (1,3) and the row of (3,4) have the same value, 3. Similarly, ci 7 at matrix position (3,4) within VLa will be followed in the same voice by ci 3 at matrix position (4,1) within VLb, ci 3 at matrix position (4,1) within VLa will be followed in the same voice by ci 8 at matrix position (1,3) within VLb, and ci 2 at matrix position (2,2) within VLa will be followed in the same voice by another statement of ci 2 within VLb. Not only does this information give a complete description of the voices for the three chords that articulate VLa–VLb, but it also dictates that additional consecutive applications of MORRIS+2 S2 8273 will result in three voices saturated with the recurring ci pattern 8–7–3–... and the other voice with repetitions of ci 2. Three consecutive statements of S2 8273 yields an overall S6 6666 (because 8 + 7 + 3 = 2 + 2 + 2 = 6) and six consecutive statements creates an overall S0 0000, a return to the original chord with the same pcs in the same voices. The four-voice realization in Example 18a illustrates this. The bass, soprano, and tenor voices are saturated with ci-succession 8–7–3. The bass voice is composed of two statements of it (Csharp–A–E–G and G –EflatBflatCsharp) and the soprano and tenor voices are transposed rotations of the bass voice. Differentiating itself from the other voices, the ci-2 saturated alto voice articulates a complete whole-tone scale.

[56] Example 18b provides a slightly different situation: twelve consecutive statements of MORRIS+2 S2 5A89 saturate all four voices with a repeating ci pattern, 5–A–9–8, which creates S8 8888 every fourth chord and an overall S0 0000. Example 18c alternates the Z-related VL that appear separately in Examples 18a and 18b. The result is a pair of voices saturated with ci-succession 3–5–2–A (soprano imitated two beats later at T4 by alto), another voice that alternates ci 7 and 9 (bass), and another that repeats ci 8 (tenor). Overall, these sequential patterns create voices that articulate symmetrical pc collections: the augmented triad (3-12[048]), the whole-tone, hexatonic, and enneatonic collections (6-35 [02468A], 6-20[014589], 9-12[01245689A]), and 6-30[013679], which is invariant under T6.

[57] There are a few notable differences when wechsel transformations are involved. First, by definition, the juxtaposition of a VL with its inverse creates an overall S0 0000, a simple neighbor-like return to the original chord as illustrated by MORRIS+1 WB 28B9 and B289 in Examples 18d and 18e. Second, consecutive statements of a single Wn create an oscillation between two chords, as with Example 18f’s series of WB 28B9, which alternates D7 and A7. Highlighting chordal identity, the musical realization places all four voices in the same register so that each D7 features the same set of pitches as does each A7. Third, any series of wechsel transformations causes an alternation between prime and inverted orientations of the chords, which means that if a voice is saturated with a single ci there is an alternation of inversionally-related pc intervals and therefore an oscillation between two pcs. For example, the series of WB 28B9 in Example 18f creates a soprano line saturated with ci 9, which causes an alternation between pc intervals 9 and 3 articulated by C–A–C–A– . . . Indeed, any repeating pattern involving an odd number of ci creates abutting inversionally-related segments. For instance, Example 18f’s bass voice features ci succession 2–8–1 twice, once articulated by D–E–Aflat–G (pc intervals 2–4–B) and once by G–F–Csharp–D (pc intervals A–8–1). Fourth, rotationally-related voices in these wechsel series are related to one another by rotation and T0, as with the tenor and alto voices which imitate the bass at T0, two and four chords later, respectively. Finally, Z-triples and quadruples can be used to generate interesting examples such as 18f (and avoid brief neighbor-like figures such as 18d and 18e) as long as inverse VL are not placed side-by-side. For instance, one could arrange the VL in the Z-triple MORRIS+5 W1 11B5/511B/71BB into a recurring pattern in which statements of 71BB separate the inverses 11B5 and 511B from one another: 11B5–71BB–511B–71BB.

Example 19a. MORRIS+2 VL with uic 11214151; each voice saturated with one ic: VL graph

Example 19a thumbnail

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Example 19b. MORRIS+2 VL with uic 11214151; each voice saturated with one ic: Path through left half of graph in 19a; ci and matrix position listings

Example 19b thumbnail

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Example 19c. MORRIS+2 VL with uic 11214151; each voice saturated with one ic: Realization of path in 19b

Example 19c thumbnail

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[58] The general feature of VL connection can also be used to create voices consisting of a single ic. For instance, given the twenty MORRIS+2 VL within uic 11214151, it is possible to concatenate VL so that each voice is limited to a single ic. Example 19a graphs all such connections. The graph is in two disjoint but congruent halves, dividing the twenty VL into two groups of ten. Single-direction arrows identify situations where one VL may follow another (e.g. W1 15A8 may follow W4 78AB) and double-direction arrows identify (inverse-related) VL that may precede or follow one another (e.g. W1 15A8 and W1 8A15). Twelve VL have precisely two VL that could precede it and two that could follow it, four others have two VL that could precede it but only one that could follow it, and the remaining four have only one VL that could precede it but two that could follow. Example 19b provides one path through the left half of the graph, along with a chart that shows that, for each ic, the matrix column value for one VL is the same as the row value for the next.(31) Example 19c provides a four-voice model that articulates the path given in Example 19b. Example 19d provides a short musical excerpt based on the model. The ic-1 voice appears in the flute, the ic-5 voice in the clarinet, and the ic-2 and ic-4 voices in the vibraphone dyads.

Example 19d. MORRIS+2 VL with uic 11214151; each voice saturated with one ic: Musical passage based on four-voice model in 19c

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Example 20a. MORRIS+2 VL with uic 11214151; each voice alternates between two ics: VL graph

Example 20a thumbnail

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Example 20b. MORRIS+2 VL with uic 11214151; each voice alternates between two ics: Realization of one path within the graph of 20a

Example 20b thumbnail

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Example 20c. MORRIS+2 VL with uic 11214151; each voice alternates between two ics: Ci and matrix positions for each voice in 20b

Example 20c thumbnail

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Example 21. MORRIS+4 Sn with uic 01112151; one voice saturated with pc intervals 5 (and 0)

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Example 22. MORRIS+4 WB B1B1, W7 AA1B, W8 1B2A; one voice descends chromatically

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[59] Using the same set of VL, it is possible to create alternation between two ics within each voice. The graph in Example 20a illustrates how to concatenate VL so that two voices alternate ics 1 and 2 and the other two alternate ics 4 and 5. The graph is radically different in shape from the previous one: four VL are dead ends and four others are unapproachable beginnings (these eight appear in the center of the graph), and the remaining twelve each have two possible antecedents and two possible consequents that create a complex web with no double-direction arrows. Example 20b provides a four-voice realization for one nine-VL path through the graph and 20c lists ci and matrix position connections for each voice.

[60] We now consider how to control pc interval (not just interval class) within an individual voice. Since having twenty VL within the same uic and MORRIS+w is a rare luxury, we study a situation with fewer possible VL, the set of eight MORRIS+4 schritt transformations that articulate uic 01112151 (S1 7BA0, SB 0251, S2 20B7, S2 5021, SA A150, SA B7A0, S5 B207, and S7 A510). With these eight VL, even the seemingly modest goal of saturating a single voice with pc interval 5 reaches a dead end after, at most, three VL. One reasonable solution is to permit the featured voice to articulate pc interval 0, but only whenever continuation by pc interval 5 is impossible. The graph in Example 21a illustrates the possibilities for VL concatenation assuming the use of chords in prime orientation. Since the featured voice may contain side-by-side ci 5 but not side-by-side ci 0, it is necessary to track whether the featured voice is articulating ci 5 (e.g. SA A150) or ci 0 (e.g. SA A150).(32) Examples 21b–d provide musical realizations for three paths within the graph, focusing on SA A150 and S7 A510 and avoiding dead-end nodes such as S2 5021. These series articulate overall SA AAAA, S8 8888, and S6 6666, respectively, which means that sequential repetition of each string of VL would eventually lead to an overall S0 0000. Example 21e shows the matrix position connections for the featured voice of the path in 21b.

[61] Of course if VL and featured pc interval are carefully chosen, it is possible to saturate a voice with one pc interval with only a few VL and without resorting to interspersed pc interval 0. For example, MORRIS+4’s WB B1B1, W7 AA1B, and W8 1B2A can be combined to create an infinitely long series of pc interval B in one voice—no matter the initial chord’s orientation or featured-voice sm. If the initial chord is in prime orientation ci B will create pc interval B; if sm 1 is in the featured voice then WB B1B1 is needed, if sm 2 then W8 1B2A, if sm 3 WB B1B1, and if sm 4 W7 AA1B. If the initial chord is in inverted orientation ci 1 will create pc interval B; if sm 1 is in the featured voice then W8 1B2A is needed, if sm 2 then WB B1B1, if sm 3 W7 AA1B, and if sm 4 WB B1B1. Example 22a presents a graph showing how the string of pc B would be continued. The graph is somewhat strange because any starting point leads, sooner or later, to the same (potentially endless) circuit of four nodes. The ability to generate an infinite series of pc interval B with so few VL is a result of the multiple occurrences of ci 1/B within the VL and the freedom to choose VL with different uic. Added benefits include that W7 AA1B and W8 1B2A articulate the same uic (1222), and that the three VL articulate only two different ics (1 and 2), which saturates the texture with stepwise motion. Example 22b provides a realization of one path through the graph, 22c shows the matrix position connections for the voice that articulates pc interval B, and 22d offers a straightforward musical realization of the model for string quartet.

Example 23. MORRIS+5 VL in which two voices move in parallel ic 3, by ic 0, 1, or 2

Example 23 thumbnail

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[62] It is also possible to maintain a consistent relationship between two voices, such as a series of VL that keeps two voices ic 3 apart, creating parallel motion. Example 23a lists the thirty-seven MORRIS+5 VL for which two voices that are ic 3 apart proceed by ci 0, 1, 2, A, or B. In all such VL the first two or middle two ci of the oci are the same (e.g. W9 1195, W0 400A, W5 B227). These thirty-eight VL are divided into four boxes based on the matrix positions of the repeated ci: in the upper left box positions (1,2)/(2,1), in the upper right (2,3)/(3,2), in the lower left either (1,3)/(2,2) or (1,2)/(2,3), and lower right (2,2)/(3,1) or (2,1)/(3,2). The method of connection is the same as in previous examples, that is, to keep the parallel ic 3 within the same pair of voices the column values for one VL’s repeated ci must match the row values for the next. For example, the repeated ci of VL in the upper left box appear in columns 1 and 2 and so they must be followed by VL whose repeated ci appear in rows 1 and 2, that is, by a VL from that same box or by one from the lower left box. Examples 23b–23e demonstrate a few possible ways to concatenate these VL. In 23b the upper voices ascend in parallel minor thirds by whole step and in 23c they descend by half step using only schritt transformations. In Example 23d, the lower voices oscillate between two major sixths, {Eflat3, C4} and {F3, D4}, in an unpredictable rhythm caused by VL featuring pc interval 0, while the upper voices explore the A major diatonic collection. In Example 23e, parallel minor thirds articulate a repeating pattern of pc intervals, 0–B–A–0–B–A–0.

[63] Examples 23f and 23g provide musical passages based on the models in 23c and 23e, respectively. Example 23f features a single melodic line for flute that incorporates all four voices of the model (transposed up a major third). Example 23g, for piano six hands, creates a series of twelve-tone chords that includes the four-voice model of Example 23e unfolding simultaneously with its T7 duplicate; this creates a series of 8-18[01235689] octachords over which is superimposed the series of 4-18[0147] that completes each vertical aggregate. Player II plays a series of parallel perfect fifths that articulate the soprano voice of the model and its T7 relative (right hand) and a series of parallel perfect fourths that articulate the alto voice of the model and its T7 relative (left hand). Player III treats the tenor and bass voices of the model in a similar fashion. Complementing this MORRIS+5based music and completing the twelve-tone structure, player I plays a MORRIS+2 series (with a few passing and neighbor tones) that emphasizes voice-leading ics 0, 1, and 2.

[64] This paper defines contextual intervals (ci), ordered/unordered lists of contextual intervals and interval classes (oci, uci, uic), voice-leading transformations (VL), and related concepts, and uses them to identify and organize the one-to-one and onto voice-leading possibilities for each of four tetrachordal set types. In this context, it presents the MORRISw VL possibilities suggested by Soderberg 1998, demonstrates a few applications, expands the universe to create MORRIS+w, and shows various ways to concatenate VL while controlling the motion of one or more individual voices. It also categorizes VL in several ways and suggests a variety of compositional applications, in which four-, six-, and twelve-voice pitch-class models yield various monophonic, homophonic, and contrapuntal musical textures, within octatonic, twelve-tone, and other pitch contexts.

[65] A variety of future directions are suggested. There are other ways to define voice-leading spaces within the MORRIS+w universe, including the following: First, gather together VL with similar (or identical) uic; for example, VL that come from uic 1341, 4311, 1242, 021141, 120141, and 420111 include at least one ic 1 and at least one ic 4, may also include one or two ic 0, and do not include any other ics. Second, limit VL by transformation; for instance, explore concatenations of the forty-eight MORRIS+5 VL that articulate either W1 or W4. Third, explore situations where one or more set members move by a particular ic or ci; for instance there are eight MORRIS+2 VL in which sm 1 moves by ci 1 and sm 4 by ci B.(33)

[66] There are also many ways to branch out from the MORRIS+w universe. For example, I plan to study the transformations and VL that arise from moving directly from one MORRIS tetrachord to another; for example, a given member of 4-18 can be followed by any one of twenty-four members of 4-27 and there are twenty-four ways to voice lead each of these, for a total of 576 VL. Such study suggests relating the MORRISw tetrachords and resulting chord progressions to (extended) tonal practice.(34) Also, the VL machinery may be applied to other individual set types. Non-symmetrical tetrachords have the same number of VL (576) but symmetrical ones have fewer. Finally, it is possible to connect the MORRIS tetrachords to others, address sets of other cardinalities, and consider VL that are not one-to-one and onto. All of this, along with the potential analytic and compositional applications of these theoretic possibilities, makes it clear that (tetrachordal) voice leading includes an infinitely interesting set of options that we have only begun to explore.

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Mark Sallmen
University of Toronto

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—————. 2003. “A Tetrahedral Graph of Tetrachordal Voice-Leading Space.” Music Theory Online 9.4.
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—————. 1996. “Cohn Functions.” Journal of Music Theory 40, no. 2: 181–216.

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—————. 1998. “Some Ideas about Voice-Leading Between Pcsets.” Journal of Music Theory 42, no. 1: 15–72.

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—————. 2001. “Special Cases of the Interval Function Between Pitch-Class Sets X and Y.” Journal of Music Theory 45, no. 1: 1–29.

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—————. 1995. “Compositional Spaces and Other Territories.” Perspectives of New Music 33: 328–359.

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—————. 1998. “Voice-Leading Spaces.” Music Theory Spectrum 20, no. 2: 175–208.

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—————. 1998. “Klumpenhouwer Networks, Isography, and the Molecular Metaphor.” Intégral 12: 53–80.

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Straus, Joseph N. 1997. “Voice Leading in Atonal Music.” In Music Theory in Concept and Practice, ed. James M. Baker, David W. Beach, and Jonathan W. Bernard, 237–274. Rochester: Rochester University Press.

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—————. 2003. “Uniformity, Balance, and Smoothness in Atonal Voice Leading.” Music Theory Spectrum 25, no. 2: 305–365.

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—————. 2010. “Geometrical Methods in Recent Music Theory.” Music Theory Online 16.1.
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Waters, Keith J. and J. Kent Williams. 2010. “Modeling Diatonic, Acoustic, Hexatonic, and Octatonic Harmonies and Progressions in Two- and Three-Dimensional Pitch Spaces; or Jazz Harmony after 1960.” Music Theory Online 16.3.
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Footnotes

1. See the discussion of DOUTH2 in Lewin 1996, as well as Childs 1998, Cohn 1998, Gollin 1998, Douthett and Steinbach 1998, and Hook 2002 & 2007. All of this research concerning 4-27 grows from similar work with major and minor triads, which appears in all of the studies cited above (except Soderberg 1998) and many others, including Hyer 1995, Cohn 1996, 1997, & 2000, Capuzzo 2004, Engebretsen 2008, and Kochavi 2008.

For voice-leading studies that address these and other set types, see Roeder 1989, Straus 1997 & 2003, Callender 1998, Morris 1998, Lewin 1998 & 2001, Alegant 2001, Cope 2002, Cohn 2003, Tymoczko 2005 & 2010, Childs 2006, Callender, Quinn, and Tymoczko 2008, Rockwell 2009, and Waters and Williams 2010.
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2. Since each MORRIS+w space includes 576 voice-leading transformations, the number of ways to choose one or more of these possibilities is vast: there are 576 ( = 576!/(575!1!)) ways to choose one, 165,600 ( = 576!/(574!2!)) ways to choose two, 31,684,800 ( = 576!/(573!3!)) ways to choose three, and so on. Coupled with this huge number, the potential rules for concatenation involve an infinite number of ways of requiring, preferring, allowing, or forbidding.
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3. The pair of 4-18 tetrachords provides double-neighbor-like ornamentation of a French sixth chord, which is not part of the MORRIS(+)w universe.
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4. Pairs of 4-27 tetrachords connected by ic 1 in each voice are addressed by Cohn 1998, 295.
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5. See Hook 2002, Lewin 1993, 25–30, Klumpenhouwer 1994, Clough 1998, Kochavi 1998, Roeder and Cook 2006, and Cook 2009. Sallmen 2009 uses schritt and wechsel transformations to study series of 418[0147] tetrachords in Elliott Carter’s “Dolphin.”
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6. Parsimonious voice leading is addressed by the 4-27– and triad–centered literature cited above, as well as by Roeder 1989, Callender 1998, Cohn 2003, Tymoczko 2005 & 2010, Roeder and Cook 2006, and Rockwell 2009.
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7. For T-matrices and voice-leading lists see Morris 1998. For other ways to consider the voice-leading possibilities between two sets consult “IFUNC(X, Y)i” in Lewin 1987 & 2001, “the interval vector between X and Y” in Morris 1987, and the “progression vector” in Nauert 2003. For sample compositional applications of compositional designs, compositional spaces, and voice-leading spaces, consult Morris 1987, 1995, & 1998, respectively.
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8. Morris 1998 calls this definitive R3-type voice leading. For careful consideration of voice-leading types that include pc omission and/or doubling see Morris 1998, 203–206, and Lewin 1998.
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9. See, for instance, Morris 1995 & 1998 and Cohn 2000 & 2003.
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10. The concept of sm is crucial to a detailed discussion of voice leading and a conventional ordering is a clear way to identify sm. This particular choice of sm and conventional ordering highlights consistencies from one tetrachordal space to another because the diminished triad always involves sm 1–3 and the “warped pc” sm 4. It would have been possible to take sm from the familiar set-class prime forms—and I tried this in my first drafts of this work—but such a system conceals inter-space consistency because of the varying placement of the diminished triad. That is, the diminished triad appears as the second, third and fourth digits in [0258] and [0147] but as the first, third and fourth in [0136] and [0236], and it does not help that the diminished triad variously appears as “036,” “147” and “258.”
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11. The W subscripts in Hook’s 2002 study of set class 4-27[0258] reflect the motion of the roots of the major-minor and half-diminished seventh chords, which has the advantage of connecting the Wn to conventional tonal analysis. Hook’s Wn corresponds to my MORRIS+1 W(4−n). As examples, C7–C7, C7Csharp7, and C7–E articulate Hook’s W0, W1, and W4, respectively, and my MORRIS+1 W4, W3, and W0, respectively.
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12. Oci alone is insufficient to define individual voice leading situations. First, some oci appear in multiple MORRISw spaces and we need a way to distinguish these situations from one another. For example oci 9003 is part of four different VL: MORRIS1 S3 9003, MORRIS2 S3 9003, MORRIS4 S3 9003 and MORRIS5 S3 9003. Second, some oci satisfy the harmonic requirements of only one of the four tetrachord types and there is usually no way quick way to determine this (other than trial and error). For example, when applied to a member of 4-13[0136], oci 8259 produces another member of the same set class, but applied to any other MORRIS+w set type oci 8259 produces a second chord that is of a different set type and therefore not within the scope of this paper.
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13. Although the notational conventions of my VL and Lewin’s 1998 “a voice leading from X into Y” (e.g. {A → Gsharp, D → E, F → E}) are somewhat different, the concepts share much in common. Both identify—either explicitly or implicitly—the chord/set members of X and Y involved in each voice and the melodic intervals they form. The primary difference is that all of my VL are one-to-one and onto whereas some of Lewin’s voice leadings are not. For an approach that defines voice leading by explicitly listing, within each voice, the set member of X, the set member of Y, and the melodic interval, consult Cope 2002, 125.
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14. For X = {5, 2, B, 4}, a set in inverted orientation, sm 1 is pc 5, which moves by ci 8 to pc 9, sm 2 is pc 2, which moves by ci 2 to pc 0, sm 3 is pc B, which moves by ci 5 to pc 6, and sm 4 is pc 4, which moves by ci 9 to pc 7.
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15. Since the T-matrix features contextual intervals (not pc intervals) any pair of sets that articulate the pcset transformation—no matter the orientation—generates the same matrix. For example, X = {6, 3, 0, 5} and Y = {7, A, 1, 8}, which also articulate MORRIS+4 W5, would also generate the matrix in Example 3b, keeping in mind that in this case the contextual intervals are the inverses of the pc intervals.

By definition, all VL in this paper involve each pc in each set exactly once, but the matrix gives all voice leading possibilities, including those that would double and/or omit pcs from one or both sets. For instance, with looser restrictions one could consider the set of voices {0–5, 3–8, 6–B, 3–A}, which includes each sm of Y precisely once but which doubles sm 2 of X and omits sm 4 of X and which articulates ci 5 @ (1, 3), (2, 2), and (3, 1), and ci 7 @ (2, 4).
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16. A more detailed explanation of inverses: For each ci at matrix position (j, k) within a given wechsel VL, the inverse VL includes the same ci at matrix position (k, j), which ensures that, if the VL are juxtaposed, both instances of the ci appear in the same voice. Since wechsel transformations alternate prime and inverted orientations, consecutive instances of the same ci in a given voice result in an overall ci 0. For example, ci 8 appears at (1, 2) in W5 8259 but at (2, 1) in W5 5829, which means that, in the example, the soprano voice moves by ci 8 from sm 1 (pc 0) of the first chord to sm 2 (pc 8) of the next chord and then by ci 8 back to sm 1 (pc 0) of the third chord. For inverse-related schritt VL, the row/column swapping works the same way but, since schritt transformations do not alternate orientations, inverse transformations and ci are involved.
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17. For more practice with the notions of Sn/Wn, VL, uic and inverse, the reader may return to the musical excerpts in Example 2, which articulate MORRIS2 S6 020A (uic 0222), MORRIS+1 W1 111B (uic 14), MORRIS+2 W1 8AA8 (uic 2242), and MORRIS+5 W1 73A0 (uic 01213151), respectively. Example 2c’s MORRIS+2 W1 8AA8 is its own inverse, which means that the chordal alternation in the excerpt articulates side-by-side statements of this VL.
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18. Within each MORRISw, S3 w00(−w) and S9 00w(−w) are inverses because applying one then the other results in an overall S0 0000. The remaining VL are involutions because they are their own inverses and so applying the VL twice yields an overall S0 0000.
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19. The term “obverse” is borrowed from Morris 1998, 185, where it is used to organize six triadic transformations into three pairs (P/P’, L/L’, and R/R’).
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20. To see why contrary motion is forbidden, consider starting with {0, 3, 6, B} and have two voices move in contrary motion. For example, if pc 0 moves to pc 2 and pc 6 moves to pc 4 a dead end is reached because there will be no way to reach pcs 8 and A via ic 2 motion without first returning to a previously left pc (0 or 6), which is forbidden. Another example, again starting with {0, 3, 6, B}: if pc 0 moves to pc 2 and pc B moves to 9, creating {2, 3, 6, 9}, one could then move pc 3 to 5 and 6 to 8, but with the B–9 and 3–5 voices moving in opposite directions, pc 1 becomes unreachable.
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21. The complete list of series: W3–WB–W3–W5, W3–WB–W9–WB, W3–WB–W6–W8, W3–W5–W3–WB, W3–W5–W9–W5, W3–W8–W6–WB, W3–W8–W3–W8; W9–W5–W9–WB, W9–W5–W6–W8, W9–W5–W3–W5, W9–WB–W9–W5, W9–WB–W3–WB, W9–W8–W9–W8, W9–W8–W6–W5; W6–W8–W3–WB, W6–W8–W9–W5, W6–W5–W9–W8, W6–W5–W6–W5, W6–WB–W3–W8, W6–WB–W6–WB, and all of their retrogrades. Such series can also be constructed in other MORRISw, such as MORRIS1’s D7–D7–CEflat7Csharp, which articulates W4–W6–W7–W6.
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22. The circle-of-fifths transformation, which amounts to pc multiplication by 5, “maps the chromatic scale onto the circle of fifths and vice versa” (Mead 1994, 36).
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23. Such VL exhibit “entirely uniform” voice leading, as defined by Straus 2003, 315.
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24. Two notes on this terminology. First, these “groups” are not, strictly speaking, mathematical groups. Second, notice that the third schritt group employs –w in its subscripts whereas the third wechsel group employs 2w. This apparent inconsistency is vital because it facilitates the discussion of similarities between transformations in different MORRIS+w. For instance, W2w refers simultaneously to four transformations—MORRIS+1 W2, MORRIS+2 W4, and MORRIS+4 W8, MORRIS+5 WA—which share important voice leading features with one another. For the same reason, subscripts including –w are necessary in the third schritt group.
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25. The other matrix position sets are {(1, 4), (2, 3), (3, 2), (4, 1)}, {(1, 3), (2, 4), (3, 1), (4, 2)}, and {(1, 2), (2, 1), (3, 4), (4, 3)}.
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26. Category numbering is arbitrary; that is, category 1 in Example 13c does not correspond to category 1 in other examples.
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27. This is true because any pair of dyads has at least two sm interpretations and each sm interpretation has at least one VL that articulates it. For example, given the voice pair {Dflat–B, C–D}, C and Dflat can be interpreted as sm 1 and sm 4, respectively, of a 4-13 in prime orientation, or as sm 4 and sm 1, respectively, of an inverted 4-13. Given the former interpretation, sm 1 (C) must move by ci 2 (to pc D) and sm 4 (Dflat) must move by ci A (to pc B) and so a VL with oci 2 _ _ A is needed, either W8 2A2A or WB 22AA. Given the latter sm interpretation, sm 4 (C) must move by ci A (to pc D) and sm 1 (Dflat) must move by ci 2 (to pc B) and so, once again a VL with oci 2 _ _ A is needed, either W8 2A2A or WB 22AA.
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28. The model’s voice leading is strictly maintained throughout, except between the second and third piano chords of measure 3.
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29. The W2w transformations of MORRIS+2 and MORRIS+4 have fewer than four in this chart because they are the special cases that result in uic |x|4 mentioned above.
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30. Chord progressions in which two voices move by one pc interval and the other two voices move by another are addressed by “dual transformations” in O’Donnell 1997 & 1998.
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31. The reader may find it helpful to refer back to the T-matrices, which are provided in Example 12b.
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32. With all of the chords in prime orientation and the featured voice stating pc interval 5, S1 7BA0, S2 20B7, SA B7A0, and S5 B207 are of no use. Moreover, S5 B207 is useless because no VL have 5 in matrix column 3, which leaves them unable to connect to S5 B207, whose 0 is in matrix row 3.
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33. These are S7 140B, S7 131B, SA 16AB, SA 113B, W7 1B1B, W7 148B, WA 11BB, and WA 12AB.
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34. For example, any 4-12 in prime orientation can be realized as a dominant seventh chord with missing fifth and added minor ninth (C–E–BflatDflat ordered low to high) or as a minor triad with missing fifth and added major ninth and augmented eleventh (BflatDflat–C–E). Another example, the set of voices {C–C, F–E, Aflat–G, B–Bflat} articulates 4-18 then 4-27, or, in tonal terms, V7 of F minor with appoggiaturas that articulate an incomplete applied vii°7, which is especially clearly related to conventional tonal practice if C–C appears in the bass voice.
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See the discussion of DOUTH2 in Lewin 1996, as well as Childs 1998, Cohn 1998, Gollin 1998, Douthett and Steinbach 1998, and Hook 2002 & 2007. All of this research concerning 4-27 grows from similar work with major and minor triads, which appears in all of the studies cited above (except Soderberg 1998) and many others, including Hyer 1995, Cohn 1996, 1997, & 2000, Capuzzo 2004, Engebretsen 2008, and Kochavi 2008.

For voice-leading studies that address these and other set types, see Roeder 1989, Straus 1997 & 2003, Callender 1998, Morris 1998, Lewin 1998 & 2001, Alegant 2001, Cope 2002, Cohn 2003, Tymoczko 2005 & 2010, Childs 2006, Callender, Quinn, and Tymoczko 2008, Rockwell 2009, and Waters and Williams 2010.
Since each MORRIS+w space includes 576 voice-leading transformations, the number of ways to choose one or more of these possibilities is vast: there are 576 ( = 576!/(575!1!)) ways to choose one, 165,600 ( = 576!/(574!2!)) ways to choose two, 31,684,800 ( = 576!/(573!3!)) ways to choose three, and so on. Coupled with this huge number, the potential rules for concatenation involve an infinite number of ways of requiring, preferring, allowing, or forbidding.
The pair of 4-18 tetrachords provides double-neighbor-like ornamentation of a French sixth chord, which is not part of the MORRIS(+)w universe.
Pairs of 4-27 tetrachords connected by ic 1 in each voice are addressed by Cohn 1998, 295.
See Hook 2002, Lewin 1993, 25–30, Klumpenhouwer 1994, Clough 1998, Kochavi 1998, Roeder and Cook 2006, and Cook 2009. Sallmen 2009 uses schritt and wechsel transformations to study series of 418[0147] tetrachords in Elliott Carter’s “Dolphin.”
Parsimonious voice leading is addressed by the 4-27– and triad–centered literature cited above, as well as by Roeder 1989, Callender 1998, Cohn 2003, Tymoczko 2005 & 2010, Roeder and Cook 2006, and Rockwell 2009.
For T-matrices and voice-leading lists see Morris 1998. For other ways to consider the voice-leading possibilities between two sets consult “IFUNC(X, Y)i” in Lewin 1987 & 2001, “the interval vector between X and Y” in Morris 1987, and the “progression vector” in Nauert 2003. For sample compositional applications of compositional designs, compositional spaces, and voice-leading spaces, consult Morris 1987, 1995, & 1998, respectively.
Morris 1998 calls this definitive R3-type voice leading. For careful consideration of voice-leading types that include pc omission and/or doubling see Morris 1998, 203–206, and Lewin 1998.
See, for instance, Morris 1995 & 1998 and Cohn 2000 & 2003.
The concept of sm is crucial to a detailed discussion of voice leading and a conventional ordering is a clear way to identify sm. This particular choice of sm and conventional ordering highlights consistencies from one tetrachordal space to another because the diminished triad always involves sm 1–3 and the “warped pc” sm 4. It would have been possible to take sm from the familiar set-class prime forms—and I tried this in my first drafts of this work—but such a system conceals inter-space consistency because of the varying placement of the diminished triad. That is, the diminished triad appears as the second, third and fourth digits in [0258] and [0147] but as the first, third and fourth in [0136] and [0236], and it does not help that the diminished triad variously appears as “036,” “147” and “258.”
The W subscripts in Hook’s 2002 study of set class 4-27[0258] reflect the motion of the roots of the major-minor and half-diminished seventh chords, which has the advantage of connecting the Wn to conventional tonal analysis. Hook’s Wn corresponds to my MORRIS+1 W(4−n). As examples, C7–C7, C7Csharp7, and C7–E articulate Hook’s W0, W1, and W4, respectively, and my MORRIS+1 W4, W3, and W0, respectively.
Oci alone is insufficient to define individual voice leading situations. First, some oci appear in multiple MORRISw spaces and we need a way to distinguish these situations from one another. For example oci 9003 is part of four different VL: MORRIS1 S3 9003, MORRIS2 S3 9003, MORRIS4 S3 9003 and MORRIS5 S3 9003. Second, some oci satisfy the harmonic requirements of only one of the four tetrachord types and there is usually no way quick way to determine this (other than trial and error). For example, when applied to a member of 4-13[0136], oci 8259 produces another member of the same set class, but applied to any other MORRIS+w set type oci 8259 produces a second chord that is of a different set type and therefore not within the scope of this paper.
Although the notational conventions of my VL and Lewin’s 1998 “a voice leading from X into Y” (e.g. {A → Gsharp, D → E, F → E}) are somewhat different, the concepts share much in common. Both identify—either explicitly or implicitly—the chord/set members of X and Y involved in each voice and the melodic intervals they form. The primary difference is that all of my VL are one-to-one and onto whereas some of Lewin’s voice leadings are not. For an approach that defines voice leading by explicitly listing, within each voice, the set member of X, the set member of Y, and the melodic interval, consult Cope 2002, 125.
For X = {5, 2, B, 4}, a set in inverted orientation, sm 1 is pc 5, which moves by ci 8 to pc 9, sm 2 is pc 2, which moves by ci 2 to pc 0, sm 3 is pc B, which moves by ci 5 to pc 6, and sm 4 is pc 4, which moves by ci 9 to pc 7.
Since the T-matrix features contextual intervals (not pc intervals) any pair of sets that articulate the pcset transformation—no matter the orientation—generates the same matrix. For example, X = {6, 3, 0, 5} and Y = {7, A, 1, 8}, which also articulate MORRIS+4 W5, would also generate the matrix in Example 3b, keeping in mind that in this case the contextual intervals are the inverses of the pc intervals.

By definition, all VL in this paper involve each pc in each set exactly once, but the matrix gives all voice leading possibilities, including those that would double and/or omit pcs from one or both sets. For instance, with looser restrictions one could consider the set of voices {0–5, 3–8, 6–B, 3–A}, which includes each sm of Y precisely once but which doubles sm 2 of X and omits sm 4 of X and which articulates ci 5 @ (1, 3), (2, 2), and (3, 1), and ci 7 @ (2, 4).
A more detailed explanation of inverses: For each ci at matrix position (j, k) within a given wechsel VL, the inverse VL includes the same ci at matrix position (k, j), which ensures that, if the VL are juxtaposed, both instances of the ci appear in the same voice. Since wechsel transformations alternate prime and inverted orientations, consecutive instances of the same ci in a given voice result in an overall ci 0. For example, ci 8 appears at (1, 2) in W5 8259 but at (2, 1) in W5 5829, which means that, in the example, the soprano voice moves by ci 8 from sm 1 (pc 0) of the first chord to sm 2 (pc 8) of the next chord and then by ci 8 back to sm 1 (pc 0) of the third chord. For inverse-related schritt VL, the row/column swapping works the same way but, since schritt transformations do not alternate orientations, inverse transformations and ci are involved.
For more practice with the notions of Sn/Wn, VL, uic and inverse, the reader may return to the musical excerpts in Example 2, which articulate MORRIS2 S6 020A (uic 0222), MORRIS+1 W1 111B (uic 14), MORRIS+2 W1 8AA8 (uic 2242), and MORRIS+5 W1 73A0 (uic 01213151), respectively. Example 2c’s MORRIS+2 W1 8AA8 is its own inverse, which means that the chordal alternation in the excerpt articulates side-by-side statements of this VL.
Within each MORRISw, S3 w00(−w) and S9 00w(−w) are inverses because applying one then the other results in an overall S0 0000. The remaining VL are involutions because they are their own inverses and so applying the VL twice yields an overall S0 0000.
The term “obverse” is borrowed from Morris 1998, 185, where it is used to organize six triadic transformations into three pairs (P/P’, L/L’, and R/R’).
To see why contrary motion is forbidden, consider starting with {0, 3, 6, B} and have two voices move in contrary motion. For example, if pc 0 moves to pc 2 and pc 6 moves to pc 4 a dead end is reached because there will be no way to reach pcs 8 and A via ic 2 motion without first returning to a previously left pc (0 or 6), which is forbidden. Another example, again starting with {0, 3, 6, B}: if pc 0 moves to pc 2 and pc B moves to 9, creating {2, 3, 6, 9}, one could then move pc 3 to 5 and 6 to 8, but with the B–9 and 3–5 voices moving in opposite directions, pc 1 becomes unreachable.
The complete list of series: W3–WB–W3–W5, W3–WB–W9–WB, W3–WB–W6–W8, W3–W5–W3–WB, W3–W5–W9–W5, W3–W8–W6–WB, W3–W8–W3–W8; W9–W5–W9–WB, W9–W5–W6–W8, W9–W5–W3–W5, W9–WB–W9–W5, W9–WB–W3–WB, W9–W8–W9–W8, W9–W8–W6–W5; W6–W8–W3–WB, W6–W8–W9–W5, W6–W5–W9–W8, W6–W5–W6–W5, W6–WB–W3–W8, W6–WB–W6–WB, and all of their retrogrades. Such series can also be constructed in other MORRISw, such as MORRIS1’s D7–D7–CEflat7Csharp, which articulates W4–W6–W7–W6.
The circle-of-fifths transformation, which amounts to pc multiplication by 5, “maps the chromatic scale onto the circle of fifths and vice versa” (Mead 1994, 36).
Such VL exhibit “entirely uniform” voice leading, as defined by Straus 2003, 315.
Two notes on this terminology. First, these “groups” are not, strictly speaking, mathematical groups. Second, notice that the third schritt group employs –w in its subscripts whereas the third wechsel group employs 2w. This apparent inconsistency is vital because it facilitates the discussion of similarities between transformations in different MORRIS+w. For instance, W2w refers simultaneously to four transformations—MORRIS+1 W2, MORRIS+2 W4, and MORRIS+4 W8, MORRIS+5 WA—which share important voice leading features with one another. For the same reason, subscripts including –w are necessary in the third schritt group.
The other matrix position sets are {(1, 4), (2, 3), (3, 2), (4, 1)}, {(1, 3), (2, 4), (3, 1), (4, 2)}, and {(1, 2), (2, 1), (3, 4), (4, 3)}.
Category numbering is arbitrary; that is, category 1 in Example 13c does not correspond to category 1 in other examples.
This is true because any pair of dyads has at least two sm interpretations and each sm interpretation has at least one VL that articulates it. For example, given the voice pair {Dflat–B, C–D}, C and Dflat can be interpreted as sm 1 and sm 4, respectively, of a 4-13 in prime orientation, or as sm 4 and sm 1, respectively, of an inverted 4-13. Given the former interpretation, sm 1 (C) must move by ci 2 (to pc D) and sm 4 (Dflat) must move by ci A (to pc B) and so a VL with oci 2 _ _ A is needed, either W8 2A2A or WB 22AA. Given the latter sm interpretation, sm 4 (C) must move by ci A (to pc D) and sm 1 (Dflat) must move by ci 2 (to pc B) and so, once again a VL with oci 2 _ _ A is needed, either W8 2A2A or WB 22AA.
The model’s voice leading is strictly maintained throughout, except between the second and third piano chords of measure 3.
The W2w transformations of MORRIS+2 and MORRIS+4 have fewer than four in this chart because they are the special cases that result in uic |x|4 mentioned above.
Chord progressions in which two voices move by one pc interval and the other two voices move by another are addressed by “dual transformations” in O’Donnell 1997 & 1998.
The reader may find it helpful to refer back to the T-matrices, which are provided in Example 12b.
With all of the chords in prime orientation and the featured voice stating pc interval 5, S1 7BA0, S2 20B7, SA B7A0, and S5 B207 are of no use. Moreover, S5 B207 is useless because no VL have 5 in matrix column 3, which leaves them unable to connect to S5 B207, whose 0 is in matrix row 3.
These are S7 140B, S7 131B, SA 16AB, SA 113B, W7 1B1B, W7 148B, WA 11BB, and WA 12AB.
For example, any 4-12 in prime orientation can be realized as a dominant seventh chord with missing fifth and added minor ninth (C–E–BflatDflat ordered low to high) or as a minor triad with missing fifth and added major ninth and augmented eleventh (BflatDflat–C–E). Another example, the set of voices {C–C, F–E, Aflat–G, B–Bflat} articulates 4-18 then 4-27, or, in tonal terms, V7 of F minor with appoggiaturas that articulate an incomplete applied vii°7, which is especially clearly related to conventional tonal practice if C–C appears in the bass voice.
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