A 5-Dimensional Tonnetz for Nearly Symmetric Hexachords

A 5-Dimensional T onnetz for Nearly Symmetric Hexac hords ∗ V aibhav Mohan ty † Quincy House, Harvar d University, Cambridge, MA 02138 The standard 2-dimensional T onnetz describ es parsimonious v oice-leading connections b etw een ma jor and minor triads as the 3-dimensional T onnetz does for dominant sev enth and half-diminished sev enth c hords. In this pap er, I present a geometric mo del for a 5-dimensional T onnetz for parsimo- nious v oice-leading b etw een nearly symmetric hexac hords of the mystic-W ozzeck gen us. Cartesian co ordinates for points on this discretized grid, generalized co ordinate collections for 5-simplices cor- resp onding to mystic and W ozzeck chords, and the geometric nearest-neighbors of a selected chord are derived. I. INTR ODUCTION In this paper, I constrct a 5-dimensional T onnetze for nearly symmetric hexac hords, kno wn as the m ystic and W ozzeck chords. Cohn (1996) describ es that it should be p ossible for chords of the T n /T n I set class of 6-34 to exhibit v oice-leading parsimon y in w ays similar to the ma jor and minor triads of 3-11. He sho wed that the ma jor and minor triads, whic h can b e viewed as p erturbations of the (symmetric) augmen ted triad, exhibit smo oth chord transitions which are achiev ed via tw o separate voice-leading regions: the hexatonic region and the W eitzmann w aterbug region (Cohn 2012). Separate Neo-Riemannian transformations exist within eac h of these regions, and all of the transformations contained in the union of these t wo sets, with the exception of the hexatonic p ole relation H , are visualizable on the w ell-known 2D T onnetz . The Boretz spider region and Childs’ (1998) o ctatonic region similarly contain Neo-Riemannian transformations that appropriately describe parsimonious v oice-leading betw een dominant seven th and half-diminished sev en th c hords, whic h are perturbations of the symmetric fully diminished chord. These c hords are visually reprented as regular tetrahedra (3-simplices) in Gollin’s (1998) 3D T onnetz , and the Neo-Riemannian transformations appear naturally as nearest-neigh b or relations. The author has previously developed the dodecatonic and centipede regions for voice-leading betw een chords ob- tained from the p erturbation of the symmetric whole-tone scale: the m ystic and W ozzeck chords (Mohant y 2018). In this pap er, I mathematically construct the 5D T onnetz for voice-leading betw een mystic and W ozzeck c hords. First, the positions of the pitc h classes in R 5 m ust be established. F rom there, the W ozzeck and m ystic c hords can be defined b y the p ositions of their v ertices. Lastly , nearest-neighbor c hords can b e found, and Neo-Riemannian transformations from the previous w ork (Mohant y 2018) can b e assigned where applicable. ∗ I thank Professor Suzannah Clark for discussions during the preparation of this pap er. † E-mail: mohant y@college.harv ard.edu 2 I I. THE COORDINA TE SP ACE In this section, I construct the geometric positions and identities of individual pitch classes in 5D space. Like the 2D and 3D T onnetze , equally spaced p oints in the space represent pitc h classes, and simplices b ounded by n − 1 v ertices in R n corresp ond to nearly symmetric c hords. A. The T onnetz basis of R 5 In b oth the 2D and 3D note spaces, the axes along which individual pitc h classes lie are not mutually orthogonal. W riting the unit v ectors p ointing along these axes as { ˆ q 1 , . . . , ˆ q N − 1 } , where N is the cardinality of the c hord, it is easy to see that the i -th and j -th unit v ectors will satisfy ˆ q i · ˆ q j = 1 2 . (1) I generalize this relation to the N = 6 case so that all 5 axes in the 5-dimensional note space will b e orien ted at 60 degrees with resp ect to one another. Imposing the conditions in eq. (1) separately on the 5 axes, we find that the unit v ectors { ˆ q i } can b e written in the Cartesian basis { ˆ e i } as ˆ q 1 =      1 0 0 0 0      , ˆ q 2 =      1 / 2 √ 3 / 2 0 0 0      , ˆ q 3 =       1 / 2 1 / (2 √ 3) p 2 / 3 0 0       , ˆ q 4 =       1 / 2 1 / (2 √ 3) 1 / (2 √ 6) p 5 / 8 0       , ˆ q 5 =       1 / 2 1 / (2 √ 3) 1 / (2 √ 6) 1 / (2 √ 10) p 3 / 5       . (2) An y vector [ v ] q ∈ R 5 in the T onnetz basis { ˆ q i } can b e represented in the Cartesian basis as v = U [ v ] q b y the unitary transformation U =  ˆ q 1 ˆ q 2 ˆ q 3 ˆ q 4 ˆ q 5  =       1 1 / 2 1 / 2 1 / 2 1 / 2 0 √ 3 / 2 1 / (2 / √ 3) 1 / (2 / √ 3) 1 / (2 / √ 3) 0 0 p 2 / 3 1 / (2 √ 6) 1 / (2 √ 6) 0 0 0 p 5 / 8 1 / (2 √ 10) 0 0 0 0 p 3 / 5       . (3) B. Pitc hes in the coordinate space Let S denote the set of tones in the T onnetz co ordinate space; in particular, S includes all linear combinations of the T onnetz basis v ectors { q i } with in teger co efficients. That is, S = { i ˆ q 1 + j ˆ q 2 + k ˆ q 3 + ` ˆ q 4 + m ˆ q 5 | i, j, k , `, m ∈ Z } . (4) W e define a map ϕ : S → Z 12 suc h that ϕ ( s ) for s ∈ S returns an integer ϕ ( s ) ∈ { 0 , . . . , 11 } that corresp onds to a particular pitch class { C , . . . , B } , and assignment is inheren tly arbitrary . How ev er, throughout this paper, I use the standard conv en tion of 0 = C , 1 = C ] , etc. I will also use an ordered pair of in tegers ( i, j, k , `, m ) to represen t elemen ts of S instead of the standard column v ector; this notation should not be confused with m y notation for row v ectors, which are not written with commas. No w, I explicitly construct ϕ by follo wing the conv en tions of the 2D and 3D T onnetze . W e can succinctly state that ϕ ( i ˆ q 1 + j ˆ q 2 + k ˆ q 3 + ` ˆ q 4 + m ˆ q 5 ) = mo d 12 (4 i + 8 j + 10 k + ` + 6 m ) (5) where mo d 12 : Z → Z 12 returns the remainder of the argument divided by 12. F rom this definition, one ma y see that, starting at the origin, the notes along the ˆ q 1 in the p ositiiv e direction are C , E , G] , etc. The notes along the ˆ q 2 in the p ositive direction are C , A[ , E , etc. Similar logic can b e applied to the other three axes. Notes that are not along an y axis are determined simply by linearity . 3 I I I. NEARL Y SYMMETRIC HEXA CHORDS No w that the coordinate space has b een constructed precisely , I now introduce the geometric definitions of the m ystic and W ozzeck c hords. As describ ed in section 1.3, the m ystic and W ozzeck c hords are in v ersionally related nearly symmetric hexac hords, and I will sho w that the particular definition of ϕ in the previous section has been pro vided so that each mystic c hord and eac h W ozzec k chord forms a 5-simplex in R 5 . A. W ozzeck chords A W ozzeck c hord is obtained from the down ward perturbation of any tone in a whole-tone scale and will be denoted with a (+) sym b ol suc h that “ C W ozzeck” can b e written as C +. By the conv en tion presen ted in an earlier work (Mohan ty 2018), a W ozzeck chord will b e lab eled b y the lo w er of the tw o tones comprising a minor 2nd. This is to sa y that C + is the collection of pitc h classes { C, D [, E , F ], G], B [ } . In the coordinate space defined in the previous section, a W ozzeck chord which has its ro ot at the p oint ( i, j, k , `, m ) is given b y the collection of tones { ( i, j, k, `, m ) , ( i + 1 , j, k , `, m ) , ( i, j + 1 , k , `, m ) , ( i, j, k + 1 , `, m ) , ( i, j, k , ` + 1 , m ) , ( i, j, k , `, m + 1) } . This corresp onds to the collection of vertices of a 5-simplex in R 5 with orientation we will refer to as (+). B. Mystic chords A mystic chord is given b y a up ward perturbation of a tone within the whole-tone scale; these c hords are denoted with the ( − ) symbol. Thus, C − refers to the “ C mystic” c hord and is comprised of the pitch classes { C, D [, E [, F , G, A } . A mystic chord is giv en b y the collection of 6 tones { ( i, j, k , `, m ) , ( i + 1 , j, k, `, m ) , ( i + 1 , j − 1 , k , `, m ) , ( i + 1 , j, k − 1 , `, m ) , ( i + 1 , j, k , ` − 1 , m ) , ( i + 1 , j, k , `, m − 1) } . These points directly correspond to the set of vertices of a 5-simplex with orien tation opp osite to that of the W ozzec k chords—the ( − ) orientation. C. Dualit y in the m ystic-W ozzeck genus As ma jor and minor triads—represented b y triangles (or 2-simplices) in the 2D T onnetz —hav e opp osite graphical orien tations, the dominant seven th and half-diminished seven th chords in Gollin’s (1998) 3D T onnetz also are “upside do wn” images of eac h other. This notion of orien tation is w ell-describ ed mathematically and can easily be obtained b y comparing the set of R 5 co ordinates with the general forms of the W ozzeck collection from section 3.1 and the m ystic collection from section 3.2. A nearly symmetric hexachord can only b e represented in one of the tw o inv ersionally related forms, so the notions of (+) or ( − ) orientation holds for the mystic-W ozzeck genus as it does for the ma jor and minor triads as w ell as the T ristan gen us. The mathematical notion of orien tation, whic h is a signed quantit y , preserv es this analogy as well. IV. NEIGHBORS IN THE 5D TONNETZ As described by Cohn (2012), the nearly symmetric hexac hords exhibit parsimonious v oice leading, and small- displacemen t c hord transitions are fully described b y a set of Neo-Riemannian transformations, whic h are defined in the author’s previous work (Mohant y 2018). The table of these Neo-Riemannian transformations and the result of applyinng these transformations to C + are display ed in T able 1 . Starting with some arbitary chord on the 2D T onnetz , it is easy to see that applying the standard triadic Neo-Riemannian transformations R , P , L , S , and N to the starting chord result in a c hord that shares either an edge or a corner with the starting c hord. In the 2D and 3D T onnetze , not all of the neigh b oring corner or edge chords are represented b y the ab ov e transformations, but for the 5D T onnetz ev ery neigh b or has an associated transformation. The p olar relation H do es not—and should—not share an y common tones with the starting chord, so it is not a neigh b or. Examining Gollin’s (1998) diagrams, it is clear that the same rule holds for the 3D T onnetz ; all of the w ell-defined Neo-Riemannian transformations except the o ctatonic p ole O transformation corresp ond either to an edge-preserving or corner-preserving neighbor chord. One ma y exp ect the rule to hold for the 5D T onnetz , and indeed it do es, as I will sho w. Since the full 5-dimensional space cannot be directly visualized in spatial coordinates, I hav e produced several reduced images in Figures 1 through 7 . In each figure 1-7, the central chord, which app ears as a hexagon with 4 T ABLE 1. Summary of Neo-Riemannian transformations for nearly symmetric hexachords and results of op erations on C +. Starting Chord T ransformation Resulting Chord F or P 0 , 1 -related chords C + R ∗∗ D ] − F or P 2 , 0 -related chords C + S A ( 3 ) B − C + S A ( 5 ) G − C + S F A − C + S W ( 1 ) C ] − C + S W ( 3 ) F − F or P n − 2 , 0 -related chords C + S 1 C − C + S 3 ( A ) A] − C + S 3 ( W ) E − C + S 5 ( A ) F ] − C + S 5 ( F ) G] − P olar Relation C + Z D − sev eral diagonal lines, is the arbitrarily chosen C +. This hexagon represents an orthographic pro jection of the C + 5-simplex onto 2 dimensions, and every solid line represents the edge of a 5-simplex in the R 5 T onnetz space. Despite v arying lengths of the solid lines in this pro jection, eac h line represen ts the same R 5 distance, whic h is precisely unit distance using the standard Euclidean metric. The permutation of the vertex lab els of a giv en c hord in differen t Figures 1 through 7 allo w for easy visualization of neigh b ors. A 5-simplex has 15 edges and 6 corners, so “true” picture of the 5D T onnetz is a simultaneous superp osition of all 7 panels sho wn in Figures 1 through 7 . In the figure, the Neo-Riemannian transformation relating C + and the neigh b oring chord—if such a transformation is w ell-defined—is giv en in b old next to the neighboring chord. The rules for the chord neigh b ors shown in Figures 1 through 7 generally hold for any W ozzeck c hord, and the neigh b ors for any mystic chord can b e quickly deduced by symmetry properties. A limitation of the 5D T onnetz is the inevitable fact that the entire T onnetz cannot be visualized with accurate represen tation of all spatial dimensions sim ultaneously . The orthographic pro jections used in this paper, moreo v er, cause the W ozzeck and mystic c hords to app ear geometrically as identical ob jects, whereas the 2D and 3D T onnetz clearly distinguish b et ween c hords of opp osite qualit y by clearly displaying orientation of simplices. F or the 5D T onnetz , the reader m ust activ ely examine the identities of the vertex pitch classes of a particular chord to parse whether the examined chord is a mystic or W ozzeck c hord. V. CONCLUSION In this article, I hav e presented an explicit construction of the 5D T onnetz for voice-leading b etw een nearly sym- metric hexachords. As discussed in previous work (Mohant y 2018; Cohn 2012), Mystic and W ozzeck chords ob ey v oice-leading rules similar to those for the ma jor-minor triadic complex as well as the T ristan genus. The 5D T on- netz presen ted here is intended as an analogy and extension of the 2D and 3D T onnetze to the remaining class of p erturbativ ely constructed c hords of cardinality n = 6. As Cohn’s (1996) hexatonic and W eitzmann waterbug regions define Neo-Riemannian transformations for ma jor and minor chords that can be represented on the 2D T onnetz , Childs’s (1998) o ctatonic and Boretz spider regions present Neo-Riemannian transformations that are used to v oice- lead b etw een dominan t sev enth and half-diminished sev en th chords that can be represented on Gollin’s (1998) 3D T onnetz . F or m ystic and W ozzeck chords, the do decatonic and cen tip ede regions (Mohan ty 2018) are comprised of Neo-Riemannian transformations that can b e depicted within the 5D T onnetz presen ted here. REFERENCES Childs, Adrian P . 1998. “Moving b eyond Neo-Riemannian T riads: Exploring a T ransformational Mo del for Sev enth Chords.” Journal of Music The ory 42, no. 2: 181-193. Cohn, Richard. 1996. “Maximally Smo oth Cycles, Hexatonic Systems, and the Analysis of Late-Roman tic T riadic Progres- sions.” Music Analysis 15, no. 1: 9-40. 5 Cohn, Richard. 2012. Audacious Euphony: Chr omatic Harmony and the T riad’s Se c ond Natur e . 2nd Edition. New Y ork: Oxford Universit y Press. Gollin, Edward. 1998. “Some Aspects of Three-Dimensional ‘ T onnetze ’.” Journal of Music The ory 42, no. 2: 195-206. Mohan ty , V aibhav. 2018. “Do decatonic Cycles and Parsimonious V oice-Leading in the Mystic-W ozzeck Genus.” Submitted for publication. Preprin t: APPENDIX: FIGURES FIG. 1. Edge-sharing c hords in the 5D T onnetz . The central chord is C +. 6 FIG. 2. Edge-sharing c hords in the 5D T onnetz . The central chord is C +. 7 FIG. 3. Edge-sharing c hords in the 5D T onnetz . The central chord is C +. 8 FIG. 4. Edge-sharing c hords in the 5D T onnetz . The central chord is C +. 9 FIG. 5. Edge-sharing c hords in the 5D T onnetz . The central chord is C +. 10 FIG. 6. Corner-sharing c hords in the 5D T onnetz . The central chord is C +. 11 FIG. 7. Corner-sharing c hords in the 5D T onnetz . The central chord is C +.