A 5-Dimensional Tonnetz for Nearly Symmetric Hexachords
AA 5-Dimensional
Tonnetz for Nearly Symmetric Hexachords ∗ Vaibhav Mohanty † Quincy House, Harvard University, Cambridge, MA 02138
The standard 2-dimensional
Tonnetz describes parsimonious voice-leading connections betweenmajor and minor triads as the 3-dimensional
Tonnetz does for dominant seventh and half-diminishedseventh chords. In this paper, I present a geometric model for a 5-dimensional
Tonnetz for parsimo-nious voice-leading between nearly symmetric hexachords of the mystic-Wozzeck genus. Cartesiancoordinates for points on this discretized grid, generalized coordinate collections for 5-simplices cor-responding to mystic and Wozzeck chords, and the geometric nearest-neighbors of a selected chordare derived.
I. INTRODUCTION
In this paper, I constrct a 5-dimensional
Tonnetze for nearly symmetric hexachords, known as the mystic andWozzeck chords. Cohn (1996) describes that it should be possible for chords of the T n /T n I set class of 6-34 toexhibit voice-leading parsimony in ways similar to the major and minor triads of 3-11. He showed that the majorand minor triads, which can be viewed as perturbations of the (symmetric) augmented triad, exhibit smooth chordtransitions which are achieved via two separate voice-leading regions: the hexatonic region and the Weitzmannwaterbug region (Cohn 2012). Separate Neo-Riemannian transformations exist within each of these regions, and allof the transformations contained in the union of these two sets, with the exception of the hexatonic pole relation H ,are visualizable on the well-known 2D Tonnetz .The Boretz spider region and Childs’ (1998) octatonic region similarly contain Neo-Riemannian transformationsthat appropriately describe parsimonious voice-leading between dominant seventh and half-diminished seventh chords,which are perturbations of the symmetric fully diminished chord. These chords are visually reprented as regulartetrahedra (3-simplices) in Gollin’s (1998) 3D
Tonnetz , and the Neo-Riemannian transformations appear naturally asnearest-neighbor relations.The author has previously developed the dodecatonic and centipede regions for voice-leading between chords ob-tained from the perturbation of the symmetric whole-tone scale: the mystic and Wozzeck chords (Mohanty 2018). Inthis paper, I mathematically construct the 5D
Tonnetz for voice-leading between mystic and Wozzeck chords. First,the positions of the pitch classes in R must be established. From there, the Wozzeck and mystic chords can be definedby the positions of their vertices. Lastly, nearest-neighbor chords can be found, and Neo-Riemannian transformationsfrom the previous work (Mohanty 2018) can be assigned where applicable. ∗ I thank Professor Suzannah Clark for discussions during the preparation of this paper. † E-mail: [email protected] a r X i v : . [ m a t h . HO ] J un II. THE COORDINATE SPACE
In this section, I construct the geometric positions and identities of individual pitch classes in 5D space. Like the2D and 3D
Tonnetze , equally spaced points in the space represent pitch classes, and simplices bounded by n − R n correspond to nearly symmetric chords. A. The Tonnetz basis of R In both the 2D and 3D note spaces, the axes along which individual pitch classes lie are not mutually orthogonal.Writing the unit vectors pointing along these axes as { ˆq , . . . , ˆq N − } , where N is the cardinality of the chord, it iseasy to see that the i -th and j -th unit vectors will satisfy ˆq i · ˆq j = 12 . (1)I generalize this relation to the N = 6 case so that all 5 axes in the 5-dimensional note space will be oriented at 60degrees with respect to one another. Imposing the conditions in eq. (1) separately on the 5 axes, we find that theunit vectors { ˆq i } can be written in the Cartesian basis { ˆe i } as ˆq = , ˆq = / √ / , ˆq = / / (2 √ (cid:112) / , ˆq = / / (2 √ / (2 √ (cid:112) / , ˆq = / / (2 √ / (2 √ / (2 √ (cid:112) / . (2)Any vector [ v ] q ∈ R in the Tonnetz basis { ˆq i } can be represented in the Cartesian basis as v = U [ v ] q by the unitarytransformation U = (cid:0) ˆq ˆq ˆq ˆq ˆq (cid:1) = / / / / √ / / (2 / √
3) 1 / (2 / √
3) 1 / (2 / √ (cid:112) / / (2 √
6) 1 / (2 √ (cid:112) / / (2 √ (cid:112) / . (3) B. Pitches in the coordinate space
Let S denote the set of tones in the Tonnetz coordinate space; in particular, S includes all linear combinations ofthe Tonnetz basis vectors { q i } with integer coefficients. That is, S = { i ˆq + j ˆq + k ˆq + (cid:96) ˆq + m ˆq | i, j, k, (cid:96), m ∈ Z } . (4)We define a map ϕ : S → Z such that ϕ ( s ) for s ∈ S returns an integer ϕ ( s ) ∈ { , . . . , } that corresponds toa particular pitch class { C, . . . , B } , and assignment is inherently arbitrary. However, throughout this paper, I usethe standard convention of 0 = C , 1 = C(cid:93) , etc. I will also use an ordered pair of integers ( i, j, k, (cid:96), m ) to representelements of S instead of the standard column vector; this notation should not be confused with my notation for rowvectors, which are not written with commas.Now, I explicitly construct ϕ by following the conventions of the 2D and 3D Tonnetze . We can succinctly state that ϕ ( i ˆq + j ˆq + k ˆq + (cid:96) ˆq + m ˆq ) = mod (4 i + 8 j + 10 k + (cid:96) + 6 m ) (5)where mod : Z → Z returns the remainder of the argument divided by 12. From this definition, one may see that,starting at the origin, the notes along the ˆq in the positiive direction are C , E , G(cid:93) , etc. The notes along the ˆq inthe positive direction are C , A(cid:91) , E , etc. Similar logic can be applied to the other three axes. Notes that are not alongany axis are determined simply by linearity. III. NEARLY SYMMETRIC HEXACHORDS
Now that the coordinate space has been constructed precisely, I now introduce the geometric definitions of themystic and Wozzeck chords. As described in section 1.3, the mystic and Wozzeck chords are inversionally relatednearly symmetric hexachords, and I will show that the particular definition of ϕ in the previous section has beenprovided so that each mystic chord and each Wozzeck chord forms a 5-simplex in R . A. Wozzeck chords
A Wozzeck chord is obtained from the downward perturbation of any tone in a whole-tone scale and will be denotedwith a (+) symbol such that “ C Wozzeck” can be written as C +. By the convention presented in an earlier work(Mohanty 2018), a Wozzeck chord will be labeled by the lower of the two tones comprising a minor 2nd. This is tosay that C + is the collection of pitch classes { C, D(cid:91), E, F (cid:93), G(cid:93), B(cid:91) } .In the coordinate space defined in the previous section, a Wozzeck chord which has its root at the point( i, j, k, (cid:96), m ) is given by the collection of tones { ( i, j, k, (cid:96), m ) , ( i +1 , j, k, (cid:96), m ) , ( i, j +1 , k, (cid:96), m ) , ( i, j, k +1 , (cid:96), m ) , ( i, j, k, (cid:96) +1 , m ) , ( i, j, k, (cid:96), m + 1) } . This corresponds to the collection of vertices of a 5-simplex in R with orientation we willrefer to as (+). B. Mystic chords
A mystic chord is given by a upward perturbation of a tone within the whole-tone scale; these chords are denoted withthe ( − ) symbol. Thus, C − refers to the “ C mystic” chord and is comprised of the pitch classes { C, D(cid:91), E(cid:91), F, G, A } .A mystic chord is given by the collection of 6 tones { ( i, j, k, (cid:96), m ) , ( i + 1 , j, k, (cid:96), m ) , ( i + 1 , j − , k, (cid:96), m ) , ( i + 1 , j, k − , (cid:96), m ) , ( i + 1 , j, k, (cid:96) − , m ) , ( i + 1 , j, k, (cid:96), m − } . These points directly correspond to the set of vertices of a 5-simplexwith orientation opposite to that of the Wozzeck chords—the ( − ) orientation. C. Duality in the mystic-Wozzeck genus
As major and minor triads—represented by triangles (or 2-simplices) in the 2D
Tonnetz —have opposite graphicalorientations, the dominant seventh and half-diminished seventh chords in Gollin’s (1998) 3D
Tonnetz also are “upsidedown” images of each other. This notion of orientation is well-described mathematically and can easily be obtained bycomparing the set of R coordinates with the general forms of the Wozzeck collection from section 3.1 and the mysticcollection from section 3.2. A nearly symmetric hexachord can only be represented in one of the two inversionallyrelated forms, so the notions of (+) or ( − ) orientation holds for the mystic-Wozzeck genus as it does for the majorand minor triads as well as the Tristan genus. The mathematical notion of orientation, which is a signed quantity,preserves this analogy as well. IV. NEIGHBORS IN THE 5D
TONNETZ
As described by Cohn (2012), the nearly symmetric hexachords exhibit parsimonious voice leading, and small-displacement chord transitions are fully described by a set of Neo-Riemannian transformations, which are definedin the author’s previous work (Mohanty 2018). The table of these Neo-Riemannian transformations and the resultof applyinng these transformations to C + are displayed in Table 1 . Starting with some arbitary chord on the 2D
Tonnetz , it is easy to see that applying the standard triadic Neo-Riemannian transformations R , P , L , S , and N tothe starting chord result in a chord that shares either an edge or a corner with the starting chord. In the 2D and3D Tonnetze , not all of the neighboring corner or edge chords are represented by the above transformations, but forthe 5D
Tonnetz every neighbor has an associated transformation. The polar relation H does not—and should—notshare any common tones with the starting chord, so it is not a neighbor. Examining Gollin’s (1998) diagrams, it isclear that the same rule holds for the 3D Tonnetz ; all of the well-defined Neo-Riemannian transformations except theoctatonic pole O transformation correspond either to an edge-preserving or corner-preserving neighbor chord. Onemay expect the rule to hold for the 5D Tonnetz , and indeed it does, as I will show.Since the full 5-dimensional space cannot be directly visualized in spatial coordinates, I have produced severalreduced images in
Figures 1 through . In each figure 1-7, the central chord, which appears as a hexagon with TABLE 1. Summary of Neo-Riemannian transformations for nearly symmetric hexachords and results of operations on C +.Starting Chord Transformation Resulting ChordFor P , -related chords C + R ∗∗ D(cid:93) − For P , -related chords C + S A ( ) B − C + S A ( ) G − C + S F A − C + S W ( ) C(cid:93) − C + S W ( ) F − For P n − , -related chords C + S C − C + S ( A ) A(cid:93) − C + S ( W ) E − C + S ( A ) F (cid:93) − C + S ( F ) G(cid:93) − Polar Relation C + Z D − several diagonal lines, is the arbitrarily chosen C +. This hexagon represents an orthographic projection of the C +5-simplex onto 2 dimensions, and every solid line represents the edge of a 5-simplex in the R Tonnetz space. Despitevarying lengths of the solid lines in this projection, each line represents the same R distance, which is precisely unitdistance using the standard Euclidean metric.The permutation of the vertex labels of a given chord in different Figures 1 through allow for easy visualization ofneighbors. A 5-simplex has 15 edges and 6 corners, so “true” picture of the 5D Tonnetz is a simultaneous superpositionof all 7 panels shown in
Figures 1 through . In the figure, the Neo-Riemannian transformation relating C + andthe neighboring chord—if such a transformation is well-defined—is given in bold next to the neighboring chord.The rules for the chord neighbors shown in Figures 1 through generally hold for any Wozzeck chord, and theneighbors for any mystic chord can be quickly deduced by symmetry properties. A limitation of the 5D Tonnetz isthe inevitable fact that the entire
Tonnetz cannot be visualized with accurate representation of all spatial dimensionssimultaneously. The orthographic projections used in this paper, moreover, cause the Wozzeck and mystic chordsto appear geometrically as identical objects, whereas the 2D and 3D
Tonnetz clearly distinguish between chords ofopposite quality by clearly displaying orientation of simplices. For the 5D
Tonnetz , the reader must actively examinethe identities of the vertex pitch classes of a particular chord to parse whether the examined chord is a mystic orWozzeck chord.
V. CONCLUSION
In this article, I have presented an explicit construction of the 5D
Tonnetz for voice-leading between nearly sym-metric hexachords. As discussed in previous work (Mohanty 2018; Cohn 2012), Mystic and Wozzeck chords obeyvoice-leading rules similar to those for the major-minor triadic complex as well as the Tristan genus. The 5D
Ton-netz presented here is intended as an analogy and extension of the 2D and 3D
Tonnetze to the remaining class ofperturbatively constructed chords of cardinality n = 6. As Cohn’s (1996) hexatonic and Weitzmann waterbug regionsdefine Neo-Riemannian transformations for major and minor chords that can be represented on the 2D Tonnetz ,Childs’s (1998) octatonic and Boretz spider regions present Neo-Riemannian transformations that are used to voice-lead between dominant seventh and half-diminished seventh chords that can be represented on Gollin’s (1998) 3D
Tonnetz . For mystic and Wozzeck chords, the dodecatonic and centipede regions (Mohanty 2018) are comprised ofNeo-Riemannian transformations that can be depicted within the 5D
Tonnetz presented here.
REFERENCES
Childs, Adrian P. 1998. “Moving beyond Neo-Riemannian Triads: Exploring a Transformational Model for Seventh Chords.”
Journal of Music Theory
42, no. 2: 181-193.Cohn, Richard. 1996. “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progres-sions.”
Music Analysis
15, no. 1: 9-40.
Cohn, Richard. 2012.
Audacious Euphony: Chromatic Harmony and the Triad’s Second Nature . 2nd Edition. New York:Oxford University Press.Gollin, Edward. 1998. “Some Aspects of Three-Dimensional ‘
Tonnetze ’.”
Journal of Music Theory
42, no. 2: 195-206.Mohanty, Vaibhav. 2018. “Dodecatonic Cycles and Parsimonious Voice-Leading in the Mystic-Wozzeck Genus.” Submittedfor publication. Preprint: arXiv:1805.11087.
APPENDIX: FIGURES
FIG. 1. Edge-sharing chords in the 5D
Tonnetz . The central chord is C +. FIG. 2. Edge-sharing chords in the 5D
Tonnetz . The central chord is C +. FIG. 3. Edge-sharing chords in the 5D
Tonnetz . The central chord is C +. FIG. 4. Edge-sharing chords in the 5D
Tonnetz . The central chord is C +. FIG. 5. Edge-sharing chords in the 5D
Tonnetz . The central chord is C +. FIG. 6. Corner-sharing chords in the 5D
Tonnetz . The central chord is C +. FIG. 7. Corner-sharing chords in the 5D