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A Study of Inter-Cardinality Voice Leading Using Voice-Leading Zones and the Extended 4-Cube Trio

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A Study of Inter-Cardinality Voice Leading Using Voice-Leading Zones and the Extended 4-Cube Trio Gamut: Online Journal of the Music Theory Society of the Mid-Atlantic Volume 8 Issue 1 Article 8 April 2018 A Study of Inter-Cardinality Voice Leading Using Voice-leading Zones and the Extended 4-Cube Trio Hon Ki Cheung University of Minnesota Follow this and additional works at: https://trace.tennessee.edu/gamut Recommended Citation Cheung, Hon Ki (2018) "A Study of Inter-Cardinality Voice Leading Using Voice-leading Zones and the Extended 4-Cube Trio," Gamut: Online Journal of the Music Theory Society of the Mid-Atlantic: Vol. 8 : Iss. 1 , Article 8. Available at: https://trace.tennessee.edu/gamut/vol8/iss1/8 This Article is brought to you for free and open access by Volunteer, Open Access, Library Journals (VOL Journals), published in partnership with The University of Tennessee (UT) University Libraries. This article has been accepted for inclusion in Gamut: Online Journal of the Music Theory Society of the Mid-Atlantic by an authorized editor. For more information, please visit https://trace.tennessee.edu/gamut. A STUDY OF INTER-CARDINALITY VOICE LEADING USING VOICE- LEADING ZONES AND THE EXTENDED 4-CUBE TRIO HON KI CHEUNG I. INTRODUCTION re triads and tetrachords living on such different planets, as Dmitri Tymoczko has A suggested, that we cannot represent the smooth voice leading between them in an effective manner? (2011, 97–98) Transformational, neo-Riemannian, and geometric techniques have created ways to relate smooth voice leading between chords of the same number of pitch classes. However, when the cardinalities of sonorities do not agree, such as in the relationship between a triad and a seventh chord, the mathematic and graphical formulation of such smooth voice-leading processes remains problematic. Problems arise with reconciling the cardinality difference and truthfully representing the role of pitch classes and voice-leading distances involved in the process. This work attempts to establish connections between certain types of triads and certain types of tertian seventh chords, building on the parsimonious graphs and voice-leading discussions from Jack Douthett and Peter Steinbach, Richard Cohn, and Joti Rockwell. The proposed model, which I call the Extended 4-Cube Trio, considers doublings within triads in a tetrachordal manner and introduces major seventh chords into Cohn’s original 4-Cube Trio (explained below). This model can be used to study progressions that involve chords of different cardinalities, such as the omnibus progression. Musical examples from Nikolas Medtner and popular songs from the Great American Songbook, which feature chromatic harmonic relationships involving both triads and lesser-even tetrachords, illustrate how the expanded tetrachordal universe facilitates the discussion of inter-cardinality voice leading. GAMUT 8/1 (2018) 189 © UNIVERSITY OF TENNESSEE PRESS, ALL RIGHTS RESERVED. ISSN: 1938-6690 CHEUNG: VOICE-LEADING ZONES AND THE EXTENDED 4-CUBE TRIO II. REVIEW OF PREVIOUS WORKS GRAPHICAL PRESENTATION OF SEMITONAL VOICE LEADING Richard Cohn proposes that minimal voice leading plays an essential role in the syntax of chromatic harmony. Nearly even sonorities can be derived from a perfectly even sonority with minimal perturbations (2012, 8–12, 34).1 Among triads, the augmented triad alone is perfectly even. The nearly even chords—major and minor triads—are connected to an augmented triad, and to two other triads of the opposite mode by the moving one of the voices by a semitone. These connections are effectively illustrated by Douthett and Steinbach’s Cube Dance, reproduced in Figure 1 (1998, 253–254). In the Cube Dance, major, minor, and augmented triads are arranged according to their idealized minimal voice-leading distance (i.e. the most direct semitonal movements between the two chords) and direction. Clockwise motion indicates an upward voice-leading movement, and the counterclockwise motion signifies a downward movement. Cohn expands the Cube dance by introducing “voice-leading zones” (2012, 102–104). Voice-leading zones are calculated by the mod-12 sum of the pitch classes in a triad. As seen in Figure 1, they are arranged as numbers on a clock-face and superposed onto the Cube Dance. The three triads in each voice-leading zone are derived from the closest augmented triad, and each triad in that zone is produced by moving one of the augmented triad’s voices by a semitone in the same direction. The algebraic difference between the voice-leading zones equals the net voice-leading distance between two triads, regardless of the actual voice-leading motion.2 1 Evenness is defined as the distribution of pitches within an octave. A set is maximally even when the pitches are separated by the same interval within an octave, such as a tritone for dyads and major thirds for trichords (Cohn 2013, 34; Tymoczko 2011: 62–63). 2 The hexatonic poles, such as the A minor triad in zone 1 and the C major triad in zone 2, require three semitonal movements to reach each other. However, two of those movements are in opposite directions and are therefore GAMUT 8/1 (2018) 190 CHEUNG: VOICE-LEADING ZONES AND THE EXTENDED 4-CUBE TRIO FIGURE 1: Douthett and Steinbach’s Cube Dance, reprinted from Cohn (2012, 158) FIGURE 2: Cohn’s 4-Cube Trio, corrected by Scott Murphy (2014, 91) The same concept applies to tetrachords. Semitonal displacements from the perfectly even fully-diminished seventh chord produce four half-diminished chords and four dominant seventh chords through upward or downward shifts, respectively. However, to represent the cancelled out in the calculation of net voice-leading distance. The net result is a movement of one semitonal displacement. GAMUT 8/1 (2018) 191 CHEUNG: VOICE-LEADING ZONES AND THE EXTENDED 4-CUBE TRIO voice-leading procedure of seventh chords using semitonal movements, French sixths and minor seventh chords are introduced to the graph to connect the two nearly even chords. Figure 2 reproduces Cohn’s 4-Cube Trio, with corrections by Scott Murphy, which illustrates the relationships between the five types of tetrachords (2014, 91). These figures focus on how triads and tetrachords are each connected in their own universes, but they do not show any interactions between the two spaces. The voice-leading process between two differently-sized chords can be as smooth as the motion between triads or tetrachords, but it is problematic to explain how each voice moves using existing models. WORKS ON THE STUDY OF INTER-CARDINALITY VOICE LEADING Douthett and Steinbach 1998, as well as Rockwell 2009 use the τ function to describe movement from one pitch class to its nearest destination in a given progression, with a maximum distance of a whole tone. The three pairs of chords in Figure 3 all relate by smooth voice-leading processes. For chords with the same number of distinct pitch classes (intra-cardinality voice leading), the model demonstrates a one-to-one relationship between pitch classes that can be described by a function, as each pitch class in the first chord is mapped onto one unique value in the second chord.3 3 While a one-to-one correspondence between chords of equal cardinality is possible, it is not necessarily required, as in contexts where a single voice can rest or sound more than one note. See Morris 1998, 175–208, which allows for such contexts. Douthett and Steinbach’s definition of τ-function only deals with voice leading between two pitch class sets of the same size, and Rockwell expanded the definition to include inter-cardinality voice-leading motions. GAMUT 8/1 (2018) 192 CHEUNG: VOICE-LEADING ZONES AND THE EXTENDED 4-CUBE TRIO FIGURE 3: Intra- and inter-cardinality voice leading The formulation of the function is less definitive in the case of inter-cardinality voice leading, which involves a difference in the number of distinct pitch classes between two chords. In the second example of Figure 3, it is possible to interpret both pitch classes B and D mapping onto C, as this commonly happens in the resolution of a dominant seventh chord to a tonic triad. Alternatively, resolving D to E is an equally valid mapping. Either way, two of the pitch classes in the G dominant seventh chord are mapped onto a duplicated pitch in the C major triad. In this case, the second example displays the surjective property of functions, which means the function values from different inputs can be the same. There are multiple problems with the definition of the τ function. It is defined so loosely that the same set of input values may suggest different outcomes depending on voice-leading context. Ambiguity in the definition results in an inability to express a unique solution for any given voice-leading process. Since it takes a whole step for D to reach either C or E in the second example of Figure 3, the τ function does not offer a definite voice-leading solution, and therefore it requires some subjective interpretation to determine mappings. Additionally, while the τ function allows many-to-one mapping works, its one-to-many inversion does not hold. In the third example in Figure 3, the τ function with input C in the C major triad is an ambiguously defined function. Mathematically speaking, a single input cannot produce multiple outputs; a parallel would be as if both x + 2 = 3 and x + 2 = 4 were GAMUT 8/1 (2018) 193 CHEUNG: VOICE-LEADING ZONES AND THE EXTENDED 4-CUBE TRIO simultaneously true. Two separate inputs are required—even if they are the same pitch class that has been split—to form a valid algebraic expression. To truthfully represent the behavior of the pitch class, the D and F need to be mapped with two distinct input values representing the pitch class to formulate an algebraic expression.
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