# 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*

Copyright © 2011 Society for Music Theory

[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—^{7}, the Tristan chord with resolution, ^{7}–^{7} 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 _{1},_{2},_{4},_{5},*w*.

[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 _{w}*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—_{1},_{2},_{4},_{5}_{w}_{w}.

[3] The paper provides an organized view of the entire _{w}/MORRIS+_{w}_{w}/MORRIS+_{w},_{w}/MORRIS+_{w}^{(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 _{5}_{4}_{2}

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[4] For an introduction to the variety exhibited by the voice-leading transformations within the _{w}**Example 2a** provides a transformation from _{2},^{(3)} An example from _{1},*Faune* in
**Example 2b** includes a pair of 4-27 tetrachords (^{7})^{(4)} The excerpt from Berg’s *Wozzeck* in
**Example 2c** provides a case from _{2},**Example 2d** identifies a voice-leading transformation from _{4},**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

[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 _{n})_{n})^{(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 _{w}_{w}_{1},_{2},_{4},_{5},_{w}

**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 _{n}/W_{n}).

[9] A set is said to be in *prime orientation* if it is related to its prime form by _{n}_{n}I._{n}_{n}I.^{(10)}

[10] A *schritt* pcset transformation, _{n},_{n}_{−n}_{1}_{1},_{1}_{B}._{0},*wechsel* transformations, when _{n}_{n},_{−n}._{1}_{1};_{1}_{B}._{n});_{n}_{n}_{−n}^{(11)}

[11] In order to compare features of the different _{w}*w*, the warp index. For example, transformation _{w}_{1}_{1},_{2}_{2},_{4}_{4},_{5}_{5}._{w+3}_{1}_{4},_{2} W_{5},_{4}_{7},_{5}_{8}.

[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 _{n}_{n}*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 _{4}_{5}

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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 _{n}/W_{n}*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 _{4}_{4}_{5}^{(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 _{4}_{5}^{1}5^{1}8^{1}9^{1}.*unordered list of interval classes* (*uic*) converts ci to ic; for example _{4} W_{5}^{1}3^{1}4^{1}5^{1}.^{4}, x^{3}y^{1},^{2}y^{2},^{2}y^{1}z^{1},^{1}y^{1}z^{1}q^{1},^{1}5^{1}8^{1}9^{1}^{1}y^{1}z^{1}q^{1}.^{4},^{3}|y|^{1},^{2}|y|^{2},^{2}|y|^{1}|z|^{1},^{1}|y|^{1}|z|^{1}|q|^{1},^{1}3^{1}4^{1}5^{1} 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 ^{1}y^{1}z^{1}q^{1}^{1}5^{1}8^{1}9^{1}^{1}3^{1}4^{1}5^{1}._{4}_{5}^{1}4^{1}8^{1}A^{1}^{2}4^{2}.

**Example 3b**. VL and related concepts:

T-matrix for _{4}_{5}

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**Example 3c**. VL and related concepts:

All _{4}_{5}

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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 _{4} W_{5},**Example 3b**. The boldface and underlined ci within the matrix point out VL _{4}_{5}^{(15)} Notice that the notation _{4}_{5}*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. _{4}_{5}**Example 3c** lists the twenty-four VL that articulate transformation _{4}_{5},

[16] Two VL are *inverses* if and only if applying one then the other results in an overall _{0}_{4}_{5}_{4}_{5}_{4}_{1}_{B}_{4}_{6}_{4}_{4}**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, _{4}_{5}’s^{1}2^{1}4^{1}5^{1},**Example 3e**).^{(17)}

**Part II. VL within the MORRIS**

_{}

*w*

**Voice-Leading Spaces**

**Example 4**. Lattices for MORRIS_{w}

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[17] Part II derives and lists the _{w}**Example 4** features four lattices that facilitate deriving the _{w}_{w}*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 _{w}_{5}*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 ^{2}5^{2},^{2}7^{2},^{2}5^{1}7^{1}._{w}

**Example 5**. MORRIS_{w}VL

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[18] **Example 5** illustrates all of the resulting _{w}_{3}*w*00(−*w*),_{6}*w*0(−*w*),_{9}*w*(−*w*)^{2}*w*^{1}(−*w*)^{1};_{3}*w*)00(−*w*),_{6}*w*)0(−*w*),_{9}*w*)(−*w*)^{2}(−*w*)^{2};_{3+w}*ww*0,_{6+w}*w*0*w*0,_{9+w}*ww*00^{2}*w*^{2}._{5}_{3}_{6}_{9}^{2}5^{1}7^{1};_{3}_{6}_{9}^{2}7^{2};_{8}_{B}_{2}^{2}5^{2}._{w},^{2}*w*^{2}.^{(18)}

[19] The example also lists the hexachord types that are the focus of Soderberg’s article. Four set types appear in each _{w}._{3}*w*00(−*w*)_{9}*w*(−*w*)_{6}*w*0(−*w*)_{3}*w*)00(−*w*)_{9}*w*)( −*w*)_{6}*w*)0(−*w*)_{1}_{5}_{2}_{4}.

[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 _{4}_{3}*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 _{1}_{5},_{2}_{4}_{w},_{6}*w*0(−*w*)_{6}*w*)0(−*w*)_{5}’s_{6}__6__c__11___{6}__6__p__11__.

**Example 6**. Some obverse pairs of VL

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[21] Fourth, each Childs-derived label that includes ic 3 appears in two _{w}_{3},_{9},_{3},_{9},_{3+w},_{9+w}_{w}_{6−w}._{1}_{3}_{5}_{3}_{1}_{3}_{5}_{3}_{1}_{4}_{5}_{8}_{3+w}*ww*0_{3}_{3+w}_{6}_{6+w},_{9}_{9+w},*obverse VL pairs*.^{(19)} For example, _{2}_{9}_{B}_{w}*wwww* or _{−w}*w*)*w*)*w*)*w*)._{2}_{B}_{9}_{2}_{2}_{9}_{B}_{A}**Example 6**.

**Part III. Four Applications of MORRIS _{}**

*w*

**VL**

[22] There are innumerable ways to combine VL within a given _{w}._{1}_{5}_{2}_{4}_{w}_{1}_{4}_{1}_{4}_{1} S_{3}_{1}_{3}

**Example 7a**. A voice-leading space within _{1}

Network of members of 4–27 that include pc 2 and the _{1}

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**Example 7b**. A voice-leading space within _{1}

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 _{1}

Realizations of paths 3 and 10 from 7b

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**Example 8**. A series of octatonic _{5}

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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 _{1}_{1}^{7}^{7}_{4}^{7}^{}7_{3}_{9}_{3}^{7}^{7}_{9}^{7}^{7}).^{7}^{7}

[24] To reduce the possibilities further we can stipulate that the path begins and ends at ^{7}^{7},^{7},^{7},^{7},^{7},^{7}). 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 ^{7}–^{7}^{7}**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 _{5}_{3},_{6},_{9},_{8},_{B},_{2}.**Example 8** provides one such series, _{8}_{9}_{8}_{3}_{3}_{2}_{9}_{2}_{9}_{8}_{2},*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 _{2}^{(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 _{3},_{6},_{9},^{2}A^{2},_{5},_{8},_{B},^{2}2^{2}._{3}_{5},_{8},_{B};_{B},_{3}_{5},_{9}_{B},_{6}_{8}

**Example 9**. Seven MORRIS_{2} series that complete the aggregate as quickly as possible

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**Example 10a**. A twelve-tone compositional design saturated with _{4}

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**Example 10b**. A twelve-tone compositional design saturated with _{4}

**Example 11a**. _{w}

W_{w}_{0}_{2w}

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**Example 11b**. _{w}

W_{2w+3}_{2w+6}_{2w+9}

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[27] Musical realizations of these three complete series, _{3}_{B}_{3}_{5},_{3}_{B}_{9}_{B},_{3}_{B}_{6}_{8},**Example 9**. The second system presents the other four series that begin with _{3}._{9}_{6}._{2}^{(21)}

[28] _{4}**Example 10a**, whose rows and columns are saturated with _{4}_{3}_{3}_{4}_{8}, creating a twelve-tone cycle that wraps around to its starting point. Offset by one dyad, this row also thrice embeds _{7}_{3}_{4}_{8}_{9}_{1}_{0}/T_{4}/T_{8}^{(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 _{1},_{2},_{4},_{5}.

[31] **Example 11a** presents _{w},_{0},_{2w}_{w}._{w}*w* and the other pc by −*w*, so that all four pcs move by ic *w* (that is uic = *w*^{4})._{2}_{2}^{4})._{w}_{0}*w*;_{5}_{0}_{2w}*w*_{5}_{A}_{0}_{2w}_{5},**Example 11b** presents the remaining trio of wechsel transformations within each _{w}:_{3+2w},_{6+2w},_{9+2w}_{4} W_{B},_{2},_{5})._{w}_{1}’s_{5},_{8},_{B}_{4}_{5},_{5}_{4},_{2}_{1},

[32] The remaining nine transformations within each _{w}_{3},_{6},_{9},^{(23)} _{0}_{w}_{−w},_{|2w|}*w* = 1 or 5, *w*|_{1}’s_{1}_{B}_{5}’s_{5}_{7}_{2}’s*w* = 2 or 4, *w*|_{2}’s_{2}_{A}_{4}’s_{4}_{8}_{4}’s*w* = 1, 2, and 5—but not 4 because of the special case just mentioned—_{2w}_{–2w}_{2w}._{1}’s_{2},_{A},_{2}_{w}_{0}_{0},_{3},_{6},_{9};_{w}_{w},_{3+w},_{6+w},_{9+w};_{−w}_{−w},_{–3−w},_{–6−w},_{–9−w}._{0}_{0},_{3},_{6},_{9},_{w}_{w},_{3+w},_{6+w},_{9+w},_{2w}_{2w},_{3+2w},_{6+2w},_{9+2w}.^{(24)}

**Example 12a**. T-matrices: Generalized T-matrices for _{n}_{n}_{w}

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**Example 12b**. T-matrices: All _{w}

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**Example 13a**. Uic type |x|^{4} :

80 VL with uic type ^{4}

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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 ^{4}

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**Example 13d**. Uic type |x|^{4} :

Six-voice model constructed of _{4}^{4}

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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 ^{3}|y|^{1}

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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 ^{3}|y|^{1}

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**Example 15c**. Uic |x|^{2}|y|^{2} :

Sample VL from chart in 15a; uci type ^{2}y^{1}z^{1}

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**Example 15d**. Uic |x|^{2}|y|^{2} :

Sample VL from chart in 15a; uci type ^{1}y^{1}z^{1}q^{1}

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**Example 15e**. Uic |x|^{2}|y|^{2} :

Articulations of _{2}_{4}

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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 ^{2}|y|^{2}

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**Example 15h**. Uic |x|^{2}|y|^{2} :

VL pairs with uic ^{2}3^{2}

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

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[33] With all of the _{n}/W_{n}**Example 12a** provides generalized matrices in terms of n and
*w* (for *w* = 1, 2, 4, or 5), one for _{n}_{n},_{n} matrix features ten different ci, four that appear multiple times and six others that appear only once. Each _{n}_{n}_{n}^{2}y^{2})._{w}_{n}^{2}y^{2}*w*.^{(25)}_{n}*w*, which means that all four _{1}*w*-independent matrix positions, a set equal to their subscripts taken as a set. For example, the _{0}**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, ^{4},^{3}|y|^{1},^{2}|y|^{2},^{2}|y|^{1}|z|^{1},^{1}|y|^{1}|z|^{1}|q|^{1},

[35] The _{w}^{4},^{3}|y|^{1},^{2}|y|^{2},^{2}|y|^{1}|z|^{1},^{1}|y|^{1}|z|^{1}|q|^{1}.

[36] The eighty VL that articulate uic type ^{4}^{4},^{3}y^{1}^{2}y^{2}^{4};^{3}y^{1}_{w}_{6+w}_{w}_{1}’s_{1}_{7}^{2}y^{2}_{w}_{3}_{9}^{2}9^{2}^{4}._{w}_{2w}_{2}_{4}._{2},_{4}^{4},^{4},^{4},_{A}_{7}_{1}_{4}_{B}_{5}_{8}_{2}**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 ^{4},^{4},^{4},^{4},^{4},^{4},^{4})^{(26)} The first category includes uic ^{4}^{4},^{4},_{3}_{9}_{w}.^{4},^{4},^{4},^{4}^{4}^{4}_{w}^{4}^{4},

[38] **Example 13d** provides a seven-chord, six-voice model created from the _{4}^{4},^{(27)} This model demonstrates variety in several respects. It incorporates all six _{4}^{4}_{2}_{A}_{5}_{8}_{A}_{B}_{2}_{A}**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 ^{3}|y|^{1};^{3}y^{1}^{2}y^{1}z^{1}_{n}*w*,^{3}y^{1}*w*).^{3}|y|^{1}._{n},_{1}_{1} B113,_{4}_{1}_{5}_{1}^{3}3^{1};_{2}_{1}^{3}3^{1};_{4} S_{1}_{1}^{3}0^{1}^{3}6^{1},_{3},_{9},_{w+3},_{w+9}_{w}._{5}_{3}_{2w}_{1}_{5}^{3}|y|^{1}._{5}_{A}^{3}4^{1}*Z-triple*—and the last one uic ^{3}2^{1}.**Example 14a** provides a complete list of VL with uic type ^{3}|y|^{1}**14b** realizes the _{1}_{5}_{A}

[40] **Example 14c **organizes all 136 VL with uic type ^{3}|y|^{1}^{3}3^{1},^{3}6^{1},^{3}0^{1},^{3}3^{1}._{n}_{n}

[41] The 432 VL with uic type ^{2}|y|^{2}^{2}y^{2}^{2}y^{1}z^{1}^{1}y^{1}z^{1}q^{1}

[42] Of the 216 VL that articulate uci type ^{2}y^{2}_{n}_{0}_{1}_{2}_{4}_{n}_{w}_{n}^{(29)} For example, _{5}_{4}^{2}8^{2},^{2}7^{2},^{2}A^{2},^{2}5^{2},^{2}4^{2},^{2}5^{2},^{2}5^{2},^{2}5^{2},**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)}

(click to enlarge and see the rest) |
(click to enlarge) |

[43] The 136 VL with uci type ^{2}y^{1}z^{1}_{w}’s_{0}_{6}_{3}_{9}_{w}_{6+w}_{3},_{9},_{3+w}_{9+w}_{0}_{2}_{0}_{6}_{2}_{6}**Example 15c** realizes these and other sample VL in musical notation.

[44] The 80 VL with uci type ^{1}y^{1}z^{1}q^{1}_{0}_{w}_{2w}_{2}_{4}._{0},_{3},_{6},_{9}_{5}_{0}_{3}_{6}_{9}_{2w}_{2}_{4}^{2}5^{2}),^{2}4^{2}),^{2}5^{2}).**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: _{2}_{4}*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 ^{2}|y|^{2},^{2}3^{2},^{2}6^{2},^{2}6^{2}._{0}_{w}.^{2}n^{2}^{2}1^{2},^{2}2^{2},^{2}4^{2},^{2}5^{2})_{n}_{0}_{6−n}.^{2}n^{2}^{2}1^{2},^{2}2^{2},^{2}4^{2},^{2}5^{2})_{6−n},_{6}_{n}.

[47] The fourth category involves ic 3 but not ics 0 and 6. Each uic ^{2}n^{2}^{2}1^{2},^{2}2^{2},^{2}4^{2},^{2}5^{2})_{n},_{6−n},_{2}_{2}_{4}_{4}**Example 15h** realizes this pair and the three other pairs within uic 1^{2}3^{2}.

[48] Categories 5 and 6 do not involve ics 0, 3, and 6. In category 5, one ic is odd and the other even ^{2}2^{2},^{2}4^{2},^{2}5^{2},^{2}5^{2})^{2}5^{2}^{2}4^{2}).*pair of Z-quadruples* (e.g. _{2}’s_{1}_{7}^{2}4^{2}).

[49] The discussion of uic type ^{2}|y|^{2}**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 ^{2}|y|^{2}:_{1}_{9}_{7}_{2}_{3} 200A),_{4}_{B}_{A}_{5}_{0}^{2}1^{2},^{2}2^{2}^{2}5^{2}^{2}6^{2}._{2}_{3}_{7}_{4}_{B}_{5}_{0}*two voices* moving stepwise (either by ic 1 or ic 2), which creates contrast with the approaches to the *all four voices*. (The approaches to

[50] There are 752 VL that articulate uic type ^{1}|y|^{1}|z|^{1}|q|^{1},^{1}|y|^{1}|z|^{1}|q|^{1}_{n}_{n}**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.

(click to enlarge and see the rest) |
(click to enlarge and see the rest) |

**Example 17c**. Uic ^{1}|y|^{1}|z|^{1}|q|^{1}

Twelve _{4}^{1}2^{1}3^{1}5^{1}

(click to enlarge)

**Example 17d**. Uic ^{1}|y|^{1}|z|^{1}|q|^{1}

Musical excerpt featuring three _{4}^{1}2^{1}3^{1}5^{1}

(click to enlarge)

**Example 17e**. Uic ^{1}|y|^{1}|z|^{1}|q|^{1}

MORRIS+_{2}^{1}2^{1}4^{1}5^{1}

(click to enlarge)

**Example 18**. Strings of identical and Z-related VL

(click to enlarge and see the rest)

[51] VL with uic type ^{1}|y|^{1}|z|^{1}|q|^{1}_{4}^{1}2^{1}3^{1}5^{1}:_{0}_{3}_{9}_{0}_{3}_{A}_{1}**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 ^{1}2^{1}3^{1}5^{1}_{9}_{9}_{0}_{7}_{0}^{1}2^{1}3^{1}5^{1}^{1}2^{1}3^{1}5^{1},^{1}2^{1}3^{1}5^{1},^{1}2^{1}3^{1}5^{1}^{1}2^{1}3^{1}5^{1}_{9}_{0}_{0}_{9}_{9}_{5}_{0},

[52] We conclude the discussion of uic type ^{1}|y|^{1}|z|^{1}|q|^{1}^{1}2^{1}4^{1}5^{1},*quartet of Z-quadruples* in both _{2}_{4}.**Example 17e** realizes the twenty _{2}

**Part V. Concatenating MORRIS+**

_{}

*w*

**VL and Controlling Individual Voices**

[53] Part V focuses on several ways to concatenate _{w}

[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 _{a}_{b},_{a}_{a},c_{a})_{a},_{b}_{b},c_{b})_{b},*ci x*

_{b}will follow ci x

_{a}in the same voice if and only if c

_{a}= r

_{b}

[55] For example, if _{a}_{b}_{2}_{2}_{a}_{a}_{b},_{a}_{b},_{a}_{b}._{a}–VL_{b},_{2}_{2}_{2}_{6}_{0}**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 (

[56] **Example 18b** provides a slightly different situation: twelve consecutive statements of _{2}_{2}*all four voices* with a repeating ci pattern, 5–A–9–8, which creates _{8}_{0}**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 _{4}_{6}.

[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 _{0}_{1}_{B}**Examples 18d** and **18e**. Second, consecutive statements of a single _{n}**Example 18f**’s series of _{B}^{7}^{7}.^{7}^{7}._{B} 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–__T _{0}__,

_{0},

_{5}

_{1}

**Example 19a**. _{2}^{1}2^{1}4^{1}5^{1};

(click to enlarge and see the rest)

**Example 19b**. _{2}^{1}2^{1}4^{1}5^{1};

(click to enlarge)

**Example 19c**. _{2}^{1}2^{1}4^{1}5^{1};

(click to enlarge)

[58] The general feature of VL connection can also be used to create voices consisting of a single ic. For instance, given the twenty _{2}^{1}2^{1}4^{1}5^{1},**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. _{1}_{4}_{1}_{1}**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.

(click to enlarge and see the rest) |
(click to enlarge) |

**Example 20b**. _{2}^{1}2^{1}4^{1}5^{1};

(click to enlarge)

**Example 20c**. _{2}^{1}2^{1}4^{1}5^{1};

(click to enlarge)

**Example 21**. _{4}_{n} with uic ^{1}1^{1}2^{1}5^{1};

(click to enlarge and see the rest)

**Example 22**. _{4}_{B}_{7}_{8}

(click to enlarge and see the rest)

**Example 23**. _{5}

(click to enlarge and see the rest)

[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 _{w}_{4}^{1}1^{1}2^{1}5^{1}_{1}_{B}_{2}_{2}_{A}_{A}_{5}_{7}**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. _{A}** 5**0)

_{A}

**).**

__0__^{(32)}

**Examples 21b–d**provide musical realizations for three paths within the graph, focusing on S

_{A}A150 and S

_{7}A510 and avoiding dead-end nodes such as

_{2}

**21.**

__0___{A}

_{8}

_{6}

_{0}

**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, _{4}’s_{B}_{7}_{8}_{B}** B**1B1

_{8}

**2A,**

__B___{B}

**1,**

__B___{7}

**.**

__B___{8}

**B2A**

__1___{B}

**B1,**

__1___{7}

**B,**

__1___{B}

**.**

__1__**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

_{7}

_{8}1B2A articulate the same uic

^{2}2

^{2}),

**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.

[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 _{5}_{9}_{0}_{5}*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, {

[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 _{7}_{7}_{7}_{5}–_{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 _{w}_{w},

[65] A variety of future directions are suggested. There are other ways to define voice-leading spaces within the _{w}^{3}4^{1},^{3}1^{1},^{2}4^{2},^{2}1^{1}4^{1},^{2}0^{1}4^{1},^{2}0^{1}1^{1}_{5}_{1}_{4}._{2}^{(33)}

[66] There are also many ways to branch out from the _{w}_{w}^{(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.

Mark Sallmen

University of Toronto

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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

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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 _{w}

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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 4–18[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 _{n}_{n}_{1}_{(4−n)}^{7}–C^{7}^{7}–^{7}, and C^{7}–E_{0}, W_{1}, and _{4}_{1}_{4}_{3}_{0}

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12. Oci alone is insufficient to define individual voice leading situations. First, some oci appear in multiple _{w}_{1}_{3}_{2}_{3}_{4}_{3}_{5}_{3}_{w}

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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 →

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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 _{4}_{5}

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 _{5}_{5}

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17. For more practice with the notions of _{n}/W_{n}_{2}_{6}^{2}2^{2}_{1}_{1}^{4}),_{2}_{1}^{2}4^{2}),_{5}_{1}^{1}2^{1}3^{1}5^{1}),_{2}_{1}

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18. Within each _{w},_{3}*w*00(−*w*) and _{9}*w*(−*w*)_{0}_{0}

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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: _{3}–W_{B}–W_{3}–W_{5}_{3}–W_{B}–W_{9}–W_{B},_{3}–W_{B}–W_{6}–W_{8},_{3}–W_{5}–W_{3}–W_{B},_{3}–W_{5}–W_{9}–W_{5},_{3}–W_{8}–W_{6}–W_{B},_{3}–W_{8}–W_{3}–W_{8};_{9}–W_{5}–W_{9}–W_{B},_{9}–W_{5}–W_{6}–W_{8},_{9}–W_{5}–W_{3}–W_{5},_{9}–W_{B}–W_{9}–W_{5},_{9}–W_{B}–W_{3}–W_{B},_{9}–W_{8}–W_{9}–W_{8},_{9}–W_{8}–W_{6}–W_{5};_{6}–W_{8}–W_{3}–W_{B},_{6}–W_{8}–W_{9}–W_{5},_{6}–W_{5}–W_{9}–W_{8},_{6}–W_{5}–W_{6}–W_{5},_{6}–W_{B}–W_{3}–W_{8},_{6}–W_{B}–W_{6}–W_{B},_{w},_{1}’s^{7}–D^{7}–C^{7}–_{4}–W_{6}–W_{7}–W_{6}.

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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 2*w*. This apparent inconsistency is vital because it facilitates the discussion of similarities between transformations in different MORRIS+_{w}. For instance, W_{2w} refers simultaneously to four transformations—_{1} W_{2},_{2} W_{4},_{4} W_{8},_{5} W_{A}*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 {_{8}_{B}_{8} 2A2A or W_{B} 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 _{2w}_{2}_{4}^{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, _{1}** 7**BA0,

_{2}

**,**

__7___{A}

**A0,**

__7___{5}

__7___{5}

**7**

__0__

__5___{5}

**7,**

__0__

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33. These are _{7}_{7}_{A}_{A}_{7}_{7}_{A}_{A}

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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–^{7}^{7},

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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.

_{w}space includes 576 voice-leading transformations, the number of ways to choose one or more of these possibilities is vast: there are 576

_{w}

_{n}

_{n}

_{1}

_{(4−n)}

^{7}–C

^{7}

^{7}–

^{7}, and C

^{7}–E

_{0}, W

_{1}, and

_{4}

_{1}

_{4}

_{3}

_{0}

_{w}

_{1}

_{3}

_{2}

_{3}

_{4}

_{3}

_{5}

_{3}

_{w}

*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

_{4}

_{5}

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).

_{5}

_{5}

_{n}/W

_{n}

_{2}

_{6}

^{2}2

^{2}

_{1}

_{1}

^{4}),

_{2}

_{1}

^{2}4

^{2}),

_{5}

_{1}

^{1}2

^{1}3

^{1}5

^{1}),

_{2}

_{1}

_{w},

_{3}

*w*00(−

*w*) and

_{9}

*w*(−

*w*)

_{0}

_{0}

_{3}–W

_{B}–W

_{3}–W

_{5}

_{3}–W

_{B}–W

_{9}–W

_{B},

_{3}–W

_{B}–W

_{6}–W

_{8},

_{3}–W

_{5}–W

_{3}–W

_{B},

_{3}–W

_{5}–W

_{9}–W

_{5},

_{3}–W

_{8}–W

_{6}–W

_{B},

_{3}–W

_{8}–W

_{3}–W

_{8};

_{9}–W

_{5}–W

_{9}–W

_{B},

_{9}–W

_{5}–W

_{6}–W

_{8},

_{9}–W

_{5}–W

_{3}–W

_{5},

_{9}–W

_{B}–W

_{9}–W

_{5},

_{9}–W

_{B}–W

_{3}–W

_{B},

_{9}–W

_{8}–W

_{9}–W

_{8},

_{9}–W

_{8}–W

_{6}–W

_{5};

_{6}–W

_{8}–W

_{3}–W

_{B},

_{6}–W

_{8}–W

_{9}–W

_{5},

_{6}–W

_{5}–W

_{9}–W

_{8},

_{6}–W

_{5}–W

_{6}–W

_{5},

_{6}–W

_{B}–W

_{3}–W

_{8},

_{6}–W

_{B}–W

_{6}–W

_{B},

_{w},

_{1}’s

^{7}–D

^{7}–C

^{7}–

_{4}–W

_{6}–W

_{7}–W

_{6}.

*w*in its subscripts whereas the third wechsel group employs 2

*w*. This apparent inconsistency is vital because it facilitates the discussion of similarities between transformations in different MORRIS+

_{w}. For instance, W

_{2w}refers simultaneously to four transformations—

_{1}W

_{2},

_{2}W

_{4},

_{4}W

_{8},

_{5}W

_{A}

*w*are necessary in the third schritt group.

_{8}

_{B}

_{8}2A2A or W

_{B}22AA.

_{2w}

_{2}

_{4}

^{4}mentioned above.

_{1}

**BA0,**

__7___{2}

**,**

__7___{A}

**A0,**

__7___{5}

__7___{5}

**7**

__0__

__5___{5}

**7,**

__0__

__0___{7}

_{7}

_{A}

_{A}

_{7}

_{7}

_{A}

_{A}

^{7}

^{7},

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