Dodecatonic Cycles and Parsimonious Voice-Leading in the Mystic-Wozzeck Genus

Do decatonic Cycles and P arsimonious V oice-Leading in the Mystic-W ozzec k Gen us ∗ V aibha v Mohan ty † Quincy House, Harvar d University, Cambridge, MA 02138 This pap er dev elops a unified voice-leading model for the genus of m ystic and W ozzeck c hords. These v oice-leading regions are constructed by perturbing symmetric partitions of the o cta ve, and new Neo-Riemannian transformations b et ween nearly symmetric hexachords are defined. The b e- ha viors of these transformations are shown within visual represen tations of the voice-leading regions for the mystic-W ozzeck genus. I. INTR ODUCTION [1.1] In a fo otnote in his 1996 article, Richard Cohn men tions that it should be p ossible to understand v oice- leading parsimon y of chords of the pitch-class sets 4-27 and 6-34 just as he has shown for 3-11 (the consonant triads). Adrian Childs’ 1998 pap er full describ es the the- ory for 4-27 in his pap er tw o y ears later. In Audacious Euphony: Chr omatic Harmony and the T riad’s Se c ond Natur e , Cohn (2012) reviews b oth the n = 3 and n = 4 cases, and he briefly discusses voice-leading parsimon y in the m ystic and W ozzec k c hords for the n = 6 case, but he do es not fully develop a unified v oice-leading model. The cen tral focus of this paper is to formalize the n = 6 v oice- leading regions, developing a “centipede” region analo- gous to the W eitzmann waterbug and Boretz spider as w ell as a “do decatonic” region—in analogy with the hex- atonic and o ctatonic regions—for the T n /T n I set class of nearly symmetric hexachords. [1.2] In Audacious Euphony , Richard Cohn (2012) uni- fies hexatonic cycles and W eitzmann regions/w aterbugs in to a single mo del of v oice-leading for all 24 ma jor and minor triads ( i.e. , nearly symmetric chords of cardinalit y n = 3). By analogy , he also constructs a unified geomet- ric represen tation for dominant seven th and fully dimin- ished seven th chords (the n = 4 case), combining Childs’ (1998) o ctatonic regions of voice-leading with the Boretz regions/spiders. Cohn also describ es generalized parsi- monious voice-leading mo dels for c hords with arbitrary cardinalit y n in a tonal system with m a v ailable tones in an o cta v e such that n is a divisor of m . Within the standard 12-tone system, he discusses in particular the existence of a voice-leading mo del similar to the W eitz- mann/hexatonic and Boretz/o ctatonic systems but for the n = 6 case inv olving the nearly symmetric mystic and W ozzeck c hords. [1.3] In this paper, I develop a visual representation for v oice-leading parsimon y in the m ystic-W ozzec k gen us, constructing a unified mo del of do decatonic cycles and cen tip ede voice-leading regions. The mystic-W ozzeck ∗ I thank Professor Suzannah Clark for discussions during the preparation of this paper. † E-mail: mohanty@college.harv ard.edu gen us and its voice-leading regions can b e generated us- ing a p erturbativ e method similar to what Childs (1998) uses for sev en th c hords whic h Cohn (2012) repurposes for ma jor and minor triads. In section II, I discuss symmet- ric partitions of the o ctav e and p erturbations of c hords generated from such symmetries. In section I I I, I re- capitulate Cohn’s work on the generation of a unified mo del for voice-leading parsimony in the minor-ma jor gen us, starting with the symmetric chords from section I I. I also consider a v ailable Neo-Riemannian transforma- tions that are used to v oice-lead within W eitzmann re- gions/w aterbugs and within hexatonic cycles. Section IV extends the approach of the previous section to the n = 4 case, and I walk through the p erturbative construction of the Boretz regions/spiders and the o ctatonic cycles as they are represen ted in Child’s (1998) article and Cohn’s (2012) Audacious Euphony . I also prop ose a pro cedure for reducing the dimension of Childs’ (1998) cubic dia- gram of the o ctatonic cycle. [1.4] While the voice-leading regions for the n = 3 and n = 4 nearly symmetric c hords are w ell-kno wn, I walk through their deriv ations in this pap er for the sak e of highligh ting the inheren t similarities and self- consistencies b etw een the n = 3, n = 4, and n = 6 cases. The hexatonic and o ctatonic v oice-leading regions are relatively easy to visualize as in 2 and 3 dimensions, resp ectiv ely . But, this is not the case with my prop osed do decatonic region, whic h w ould require 5 dimensions to represen t the region as a conv ex polyhedron. My pro- p osed metho d to reduce the dimension of Childs’ o cta- tonic region can b e directly applied to the n = 6 case to allo w for easy visualization of the dodecatonic region in 2 dimensions. W alking through the n = 4 case makes this dimension-reduction process esp ecially transparen t. The extension of the n = 3 and n = 4 deriv ations to the n = 6 case is carried out in Section V, forming the central p or- tion of this pap er. I construct what I refer to as c entip e de regions and do de c atonic cycles for voice-leading betw een m ystic and W ozzeck c hords. Section VI discusses the set-theoretic prop erties of the hexatonic, o ctatonic, and do decatonic regions and prop oses a future direction for researc h. 2 FIG. 1. Possible symmetric partitions of the o cta ve for the (a) n = 3, (b) n = 4, and (c) n = 6 cases. I I. SYMMETRIC P AR TITIONS OF THE OCT A VE [2.1] The splitting of an o cta v e into 12 tones allows for the generation of interesting symmetries that often translate into musically relev an t structures. F or exam- ple, in mathematics, the cyclic group Z 12 consists of the in tegers from 0 to 11, and elements of the group are re- lated to each other by addition mo dulo 12. Of utmost relev ance to music theory is the fact that Z 12 has gen- erators 1, 5, 7, and 11. Using the generator 1, one ma y construct all of the elements of the group by starting with one element—sa y , 0—and adding (mo dulo 12) the generator rep eatedly: 0 ≡ 0 mo d 12 1 ≡ (0 + 1) mo d 12 2 ≡ (1 + 1) mo d 12 . . . 11 ≡ (10 + 1) mo d 12 . If the generator is 7, the group can b e constructed with the same technique, but the elements start to app ear in a different order: 0 ≡ 0 mo d 12 7 ≡ (0 + 7) mo d 12 2 ≡ (7 + 7) mo d 12 9 ≡ (2 + 7) mo d 12 . . . 5 ≡ (10 + 7) mo d 12 . As commonly done in musical set theory , if one assigns a pitch class to each integer ( i.e. , C = 0, C ] = 1, . . . , B = 11), the order in which the elements are generated starts to form either the chromatic circle or the circle of fifths. Generators 5 and 11, likewise, form the circle of fourths and the descending c hromatic circle, respectively . Hence, the mathematics can serv e as a to ol for the rigor- ous construction of well-kno wn m usical phenomena. [2.2] The n umber 12 has divisors 1, 2, 3, 4, 6, and 12. The divisor n indicates how the notes in the o ctav e ma y b e partitioned symmetrically . F or the case where n = 3, one starts with a particular note—p erhaps C — and selects every 12 / 3 = 4th note that app ears in the c hromatic circle 2 . Connecting these notes with lines, one sees that an equilateral triangle is formed. F urthermore, there are 4 individual equilateral triangles that can b e formed, and no vertices intersect. Figure 1(a) shows that symmetrically partitioning the octav e for the n = 3 case yields the 4 augmented triads with nonintersecting sets of pitc h classes: { C, E , G] } , { D [, F , A } , { D , F ], A] } , and { E [, G, B } . [2.3] F or the n = 4 case, one finds the symmetric c hords in the same manner. This time, the cardinality of the c hord is also n = 4, and one mu st select ev ery 12 / 4 = 3rd note in the c hromatic circle to complete the chord, whic h is a fully diminished sev en th chord. Figure 1(b) clearly illustrates that connecting the notes in a c hord generates a square, and there are 3 indep endent squares that share no pitch classes with each other. These c hords are com- prised of the collections { C , E [, G[, A } , { C ], E , G, B [ } , and { D , F, A[, B } . [2.4] The n = 6 case is determined using the same metho d. By choosing ev ery other note (12 / 6 = 2), one constructs a whole-tone scale, whic h geometrically forms 2 Note here that any circle generated by 1, 5, 7, and 11 may b e used for this purpose. 3 FIG. 2. Perturbations of the A[ augmen ted triad, generating a W eitzmann region of ma jor and minor c hords. a regular hexagon sup erimposed onto the c hromatic cir- cle. Figure 1(c) sho ws the 2 indep enden t whole-tone scales, which are comprised of { C, D , E , F ], A[, B [ } and { D [, E [, F , G, A, B } . [2.5] The n = 1, 2, and 12 cases function similarly , but v oice-leading b etw een classes of single note, dyads, and nearly-full chromatic sets is relativ ely unin teresting. Nearly Symmetric Chords [2.6] The correspondence b et w een these symmetric ge- ometries and well-kno wn pitc h collections demonstrates the m usical relev ance of certain mathematical structures. Eac h of the follo wing three sections b egins with a prelim- inary discussion on p erturbation of the symmetric c hords previously discussed. As Dmitri Tymoczko (2011) heav- ily emphasizes in his b o ok A Ge ometry of Music , many of fundamental triads, sev enth chords, and scales com- monly used W estern m usic are nearly ev en c hords; that is to say , they are not quite symmetric chords, but they are only a few semitone displacements a wa y from sym- metric chords. While Tymoczko has a relatively op en definition of “nearly ev en,” in this paper I specifically fo- cus on c hords that are exactly a single-semitone displace- men t (SSD) from one of the symmetric chords previously discussed. I shall refer to this sp ecific class of p erturb ed c hords as ne arly symmetric . Nearly symmetric c hords of a giv en cardinality n demonstrate a sp ecific, consistent pattern of voice-leading. I II. WEITZMANN W A TERBUGS AND HEXA TONIC CYCLES [3.1] The literature on parsimonious voice-leading b e- t ween ma jor and minor triads is extensive, as the field of Neo-Riemannian theory essentially developed from this cen tral topic. Richard Cohn’s (1996) article on hexa- tonic cycles thoroughly develops this topic, fo cusing on the set of maximally smo oth cycles that are generated from single-semitone displacemen ts of the ma jor and mi- nor triads. F or the collection of 24 ma jor and minor triads, 4 independent maximally smooth cycles—called hexatonic cycles—each consisting of 3 m a jor and 3 mi- nor c hords can b e constructed via a simple pro cedure: one starts with a consonan t triad and needs to v oice lead to another consonant triad only utilizing single-semitone displacemen ts. This results in a transformation that is more sp ecifically an in volution: a ma jor triad can be transformed to only a minor triad via a single-semitone displacemen t, and a minor can b e transformed to only a ma jor triad via a single-semitone displacement. This pro cedure can be rep eated un til one returns to the start- ing c hord. A maximally smo oth cycle of ma jor and minor triads consists of exactly 6 chords, and there are 4 inde- p enden t maximally smo oth cycles for this collection. [3.2] There are multiple wa ys in whic h the elemen ts of the hexatonic cycle can b e constructed; the pro cedure presen ted ab ov e is one of them. I now describ e a pro- cedure that Cohn (2012) mo dels after Childs’ (1998) ap- proac h: namely , I exploit the near symmetric nature of the ma jor and minor chords to derive the voice-leading regions. [3.3] In the previous section, I show ed that for the n = 3 case, there are 4 indep endent augmented triads that symmetrically partition the o cta ve. Figure 2 shows the triad { A[, C , E } . If the A[ is p erturb ed down a semi- tone, a C ma jor triad is obtained, and it is written as C +. If the A[ is p erturb ed up a semitone, an A mi- nor triad is obtained, notated as A − . P erforming down- w ard and up ward p erturbations on the pitch C results in E + and C ] − , and p erturbing the pitch E generates A[ + and F − from the augmented triad. This collec- tion of chords { C + , A − , E + , C ] − , A[ + , F −} is known as the W eitzmann region (Cohn 2012). The 3 other inde- p enden t augmen ted triads similarly generate 3 separate W eitzmann regions, and no tw o W eitzmann regions share an y chords. [3.4] An well-kno wn visual represen tation of the W eitz- mann region is the W eitzmann w aterbug, shown in Fig- ure 3 (Cohn 2012). The legs on one side of the water- bug’s b o dy corresp ond to the (+) chords, while the legs on the other half corresp ond to the ( − ) chords. The fa- miliar Neo-Riemannian transformations that act within this region are R (relativ e), N ( Neb enverwandt ), and S (slide). Applying R to C + requires mov emen t of a sin- gle voice by 2 semitones to produce A − . Using Douthett and Steinbac h’s (1998) formal definition of P m,n -related c hords, one would sa y that C + and A − are P 0 , 1 related b ecause one can shift b etw een the t wo c hords only by mo ving 1 note by a whole tone and 0 v oices by semitone. The transformations N and S b oth require mo ving tw o v oices in parallel motion by a single semitone (each). S mo ves the p erfect 5th, shifting the pitches C and G to C ] and G] (so C + b ecomes C ] − ), while the pitch E re- mains inv ariant. N shifts the minor 3rd, E and G are transformed to F and A[ (so C + b ecomes F − ), leaving 4 FIG. 3. A W eitzmann w aterbug, adapted from Cohn (2012). FIG. 4. Repro duction of Cohn’s (2012) unified v oice-leading mo del for nearly symmetric triads. the ro ot in v arian t. Th us, chords related b y N or S are said to b e P 2 , 0 related. [3.5] The four W eitzmann w aterbugs that are gener- ated from the 4 augmented triads hav e no in tersection, so to represen t the relationships betw een the full system of waterbugs, one can use a diagram like Douthett and Stein bach’s Cube Dance (1998) or Cohn’s (2012) unified w aterbug/hexatonic figure in Audacious Euphony . F or the sak e of visual clarity , only Cohn’s diagram is recre- ated here in Figure 4 , ev en though Cube Dance contains additional voice-leading information. Cohn’s diagram is constructed simply by placing the waterbugs in a square suc h that the ro ot names match at the “bridge” regions of the w aterbugs. F or example, the left bridge region con- tains the chords { C + , E + , G + } from one waterbug and { C − , E − , G −} from the adjacent waterbug. Arranging the w aterbugs in this wa y rev eals that the chords within the bridge regions are exactly the ones in the hexatonic cycles (Cohn 2012). Moreov er, all 24 ma jor and minor c hords are represen ted in this figure, so Figure 4 is in- deed a unified mo del of voice-leading for the ma jor/minor collection. 5 FIG. 5. Hexatonic cycles shown as hexagons (Cohn 2000). FIG. 6. Av ailable voice-leading transformations in a hexatonic region (Cohn 1996). [3.6] The hexatonic regions (or cycles) themselves are constructed from the familiar Neo-Riemannian transfor- mations. The four hexatonic cycles, which Cohn (1996; 2000) names “Northern,” “Southern,” “Eastern,” and “W estern,” are shown as hexagons instead of circles in Figure 5 . In his 1996 paper, Cohn draws these hexa- tonic regions as circles and the 2000 article presents them as hexagons. F or the sake of geometric consistency with the n = 4 and n = 6 cases which I will present, I choose the hexagonal representation. This is particularly useful b ecause a “musical” meaning can b e attributed to each v ertex and edge of the hexagon: every v ertex corresponds to a ma jor or minor chord, and each straigh t line (form- ing the sides of the hexagon) corresp onds to the identical v oice leading distance. One can say that t wo chords X and Y connected by a straight line in a given hexatonic region are P 1 , 0 -related, since one can construct Y from X (and X from Y ) b y only moving 1 note by a semitone. The transformations P (parallel) and L ( L eittonweschel ) are resp onsible for transformations b etw een P 1 , 0 -related c hords. Starting with a chord of a given mo dality , the only c hord in the hexatonic region of the opp osite mo dal- it y that is not P 1 , 0 -related to the starting chord is the hexatonic p ole, reached by the H transformation. The geometric functions of these 3 transformations on a sam- ple chord, C +, are shown in Figure 6 . I will show that in the next tw o sections, construction of the voice-leading regions for the n = 4 and n = 6 cases follow the same pro cedure as that for the n = 3 case, and the musical in terpretation of the geometric elements (e.g. straight lines and P m,n -relatedness) remains consistent. IV. BORETZ SPIDERS AND OCT A TONIC CYCLES [4.1] Adrian Childs’ (1998) pap er extends the previ- ously describ ed unified voice-leading model to the n = 4 case. In this section, I arriv e at Childs’ results using the same metho d outlined in the previous section, starting from symmetric partitions of the o ctav e 3 . In section I I, I show ed that for the n = 4 case, the o ctav e is symmet- rically partitioned indep enden tly b y 3 fully diminished 3 In fact, Childs first prop osed the p erturbative approac h used to generate the nearly-symmetric seven th chords in his 1998 pa- per. Cohn adapted this approach to the triadic case, which he presented in the first edition of Audacious Euphony , which was published in 2000. In this pap er, though I present the deriv a- tion of the n = 3 voice-leading regions first, the p erturbative approach to the n = 4 indeed was published first. 6 FIG. 7. P erturbations of the F ] fully diminished seven th c hord, generating a Boretz region of dominant seven th and half- diminished seven th c hords. FIG. 8. A Boretz spider (Cohn 2012). c hords. The p erturbations of one of these p ossible fully diminished c hords are sho wn in Figure 7 . Notationally , the symbols (+) and ( − ) are used to represen t the domi- nan t seven th and half-diminished sev en th chords, respec- tiv ely . [4.2] Starting with C ] fully diminished sev enth, if the bottom C ] is perturb ed do wnw ard, the resulting c hord is C dominant seven th, written as C +. Perturb- ing the C ] up ward results in E half-diminished sev- en th, written as E − . P erturbing the E down ward re- sults in E [ +, and shifting the same note up ward re- sults in G − . In general, a down w ard p erturbation re- sults in a dominant seven th c hord, and an upw ard p er- turbation results in a half-diminished seven th c hord. Con tinuing these p erturbations for the remaining tw o notes in the initial C ] fully diminished seven th chord, one can generate a Boretz region, given by the collec- tion of chords { C + , E − , E [ + , G − , F ] + , A] − , A + , C −} (Cohn 2012). There are 3 indep enden t Boretz regions, and the union of these 3 sets gives the full collection of all dominant seven th and half-diminished sev enth chords. [4.3] As the W eitzmann waterbug conv enien tly repre- sen ts the W eitzmann region, the Boretz spider, shown in Figure 8 , is the n = 4 visual representation of the Boretz region (Cohn 2012). In eac h of the 3 Boretz spiders, a dominan t sev en th c hord corresp onds to a leg on one half of the spider’s bo dy , and the half-diminished sev enth c hords are placed on the other half of the b o dy . W ell- defined inv olutions can b e applied to any giv en chord in order to transform one chord to an y other c hord on the other side of the Boretz spider’s b ody . These particu- lar transformations are R ∗ , S 3 ( 4 ) , S 6 , and S 3 ( 2 ) (Childs 1998; Cohn 2012). [4.4] The R ∗ transformation is the n = 4 analogy to the triadic R transformation. Though there is no formal definition of a “relative” half-diminished seven th chord for a giv en dominant seven th chord or vice versa, the R ∗ transformation mo ves 1 v oice b y 2 semitones, just lik e R . Th us, C + and E − are P 0 , 1 -related. [4.5] The remaining S -t yp e transformations for the Boretz spiders hav e functions that are analogous to the triadic S and N transformations. In S 3 ( 4 ) , S 6 , and S 3 ( 2 ) , one “slides” 2 v oices by 1 semitone in parallel motion. Th us, c hords that are related by S 3 ( 4 ) , S 6 , and S 3 ( 2 ) are said to b e P 2 , 0 -related, just like in the n = 3 case. The first num b er in the sup erscript ( e.g. the “3” in S 3 ( 4 ) ) refers to the interv al within the 4-note chord that is held in v arian t. The num ber in parentheses denotes the inter- v al that “slides” due to the transformation. F or example, if one applies S 3 ( 4 ) to F +, then the set-theoretic inter- v al 3, which is a minor 3rd, is held in v arian t. There are 2 minor 3rds—pitch class E to G and G to B [ —so one lo oks at the n umber 4 to determine the interv al that is shifted. The interv al 4 is a ma jor 3rd, so pitch classes 7 FIG. 9. Recreation of Cohn’s (2012) unified v oice-leading model for nearly symmetric seven th c hords. C and E m ust b e shifted. A down w ard shift do es not result in a dominant seven th or half-diminished seven th c hord, so S 3 ( 4 ) sp ecifically transforms C + to G − and vice v ersa. As a comparison, S 3 ( 2 ) lea ves the minor 3rd E to G in v arian t while shifting D and B [ , transforming C + to C ] − . Cohn (2012) writes S 6 ( 5 ) as S 6 b ecause it is im- plied that inv ariance of the interv al 6 requires shifting of the perfect 4th, as there is only 1 possible tritone within a dominant sev en th or half-diminsihed seven th c hord. [4.6] Cohn (2012) joins the 3 independent Boretz spi- ders in a unified v oice-leading model for the n = 4 case, and Douthett and Steinbac h (1998) ha ve a simi- lar figure—P o wer T ow ers—whic h sho ws additional voice- leading capabilities within the bridge region. F or visual simplicit y , Cohn’s figure is repro duced in Figure 9 . As with the triads, the (+) chord root names on one spider m ust b e “bridged” with the ( − ) c hords with the same ro ot names on another spider. F or example, the b ottom bridge region of Figure 9 unites { C ] + , E + , G + , B [ + } on one spider with { C ] − , E − , G − , A] −} on the adjacen t spider. The 3 bridge regions that are generated are—in analogy with the n = 3 case—referred to as o ctatonic regions. Childs (1998) shows that eac h o ctatonic region can b e displa yed as a cubic net w ork, and one of the three cub es is constructed in Figure 10(a) . Geometrically , a cub e is a desirable structure for describing this t yp e of c hord collection b ecause it is not only a conv ex p olytope, but it is also p ossible to assign its v ertices to (+) and ( − ) chords suc h that no (+) vertex has an y edge shared with another (+) v ertex, and no ( − ) v ertex has an y edge shared with another ( − ) vertex. [4.7] Although a cub e is conv enien t for visualizing the n = 4 voice-leading region, visualizing conv ex p olyhe- dra in n > 3 spatial dimensions b ecomes an imp ossible task. A metho d for reducing the dimension of higher- dimensional geometric structures will surely prov e useful when trying to visualize the n = 6 case, since 5 spatial dimensions w ould be required. Thus, I propose a metho d for flattening such geometric represen tations of bridge re- gions to 2-dimensions; this becomes an esp ecially pow er- ful and useful to ol when dealing with the n = 6 case, and it can also b e applied to n = 4. (The n = 3 hexatonic cycles already ha v e a 2-dimensional represen tation.) The geometric structure sho wn in Figure 10(b) is m y alter- nativ e to Childs’ (1998) cubic netw ork for the o ctatonic region. Since there are 8 chords in the o ctatonic col- lection, an o ctagon provides the neatest “frame” for the structure, just like a hexagon do es for the n = 3 case. I arrange the (+) and ( − ) chords around the o ctagon in c hromatically sequential order, alternating b et ween the (+) and ( − ) chords, just lik e Cohn do es for the hexa- tonic cycles. How ev er, unlike the hexatonic cycles, each v ertex in the o ctagon is connected to more than 2 other v ertices. Examining Childs’ cubic netw ork, I dra w lines inside the perimeter of the o ctagon to connect chords that also hav e connections in the cub e. As a result, an y 8 FIG. 10. A octatonic region represented as a (a) cube (Childs 1998) and (b) a 2-dimensional netw ork. FIG. 11. Octatonic cycles (Childs 1998), shown as 2-dimensional graphs. (+) chord is connected to all other ( − ) c hords except its own o ctatonic p ole, and an y giv en ( − ) chord is con- nected to all other (+) c hords except its o wn o ctatonic p ole. The 3 indep endent o ctatonic regions are shown in the 2-dimensional form in Figure 11 . [4.8] The Neo-Riemannian transformations that act within this o ctatonic region are shown in Figure 12 . Of the 4 inv olutions that are allow ed, 3 are “slide” trans- formations: S 2 , S 4 , and S 5 . These 3 chords follo w the same notational scheme as the S -t yp e transformations that acted within the Boretz region, and they are ab- breviations of S 2 ( 3 ) , S 4 ( 3 ) , and S 5 ( 6 ) . The S 2 trans- formation holds the ma jor 2nd (set-theoretic interv al 2) in v arian t while sliding the minor 3rd, so the chord C + w ould b e transformed to C − . S 4 w ould hold the ma jor 3rd inv ariant while shifting the minor 3rd, so C + trans- forms to F ] − . Lik ewise, S 5 w ould transform C + to A − , since the perfect 4th is held in v arian t while the tritone is shifted. In eac h of the ab ov e cases, the chords related b y these S -type transformations are P 2 , 0 -related since tw o v oices are shifted by 1 semitone eac h, and 0 voices are shifted by a whole tone. [4.9] Within the 2 geometric representations of the o ctatonic region in Figure 10 , a solid line b etw een t wo c hords represen ts an iden tical v oice-leading distance. That is to sa y , any tw o chords connected by a solid line are P 2 , 0 -related. This fact may seem obvious from Childs’ (1998) cubic diagram; but in the 2-dimensional reduc- tion, the corresp ondence b et ween length of a connecting line and voice-leading distance is lost. While this sac- rifice of information must b e made in the reduction of dimension, the flattening of geometric structures to 2 dimensions provides a pow erful and, most importantly , geometrically consistent metho d for treating the n = 3, n = 4, and—as I will show in the next section— n = 6 case on equal fo oting, visually . [4.10] Nonetheless, it is easy to see that given a par- ticular chord in an o ctatonic region, there is only one c hord of the opposite mo dalit y that is not P 2 , 0 -related to the starting chord. This is kno wn as the octatonic p ole (Childs 1998), and I refer to it as the O transfor- mation 7 . On Childs’ cubic representation, the o ctatonic p ole is presen t on the vertex that is farthest aw a y from 7 Neither Childs (1998) nor Cohn (2012) explicitly asso ciated a letter with the o ctatonic p ole transformation. In analogy with H for hexatonic pole, I coin O for o ctatonic p ole. 9 FIG. 12. Av ailable v oice-leading transformations in an octatonic region (Childs 1998), shown in the 2-dimensional represen ta- tion. FIG. 13. Perturbations of the whole-tone scale including C , generating a collection of W ozzeck and m ystic c hords. the starting chord. In the 2-dimensional representation, the O transformation connects tw o chords of the opp o- site mo dality that are not joined b y a solid line. Like the hexatonic p ole for triads, a chord and its o ctatonic p ole share no pitch classes. O applied to C + results in E [ − . [4.11] Discussion of an octatonic region also motiv ates a search for maximally smo oth cycles. In a single hex- atonic region, there is only one allow ed hexatonic cycle: the alternating P - L cycle. Hexatonic cycles are aptly named b ecause all of the chords in a hexatonic region are utilized in a maximally smo oth cycle, a definition whic h w as formalized by Cohn (1996). In the o ctatonic region, one will notice that an y closed path with no loops is indeed a maximally smo oth cycle, according to Cohn’s definition. The union of all pitc h classes contained in the c hords of any one of these maximally smo oth cycles is an o ctatonic collection given by set class 8-28, so these maximally smo oth cycles can b e called o ctatonic cycles. V. CENTIPEDES AND DODECA TONIC CYCLES [5.1] This section forms the crux of this pap er. Here, I presen t the p erturbative deriv ation of nearly symmet- ric hexac hords that comprise the mystic-W ozzeck gen us. These chords exhibit voice-leading prop erties similar to the ma jor/minor triads and dominant/half-diminished sev enth c hords, and their v oice-leading “arthropo d” and bridge regions can b e visually represented just as in the n = 3 and n = 4 cases. [5.2] Firstly , I m ust return to the symmetric p er- turbation of the octav e discussed in section I I. In the n = 6 case, there are only tw o wa ys to partition the o c- ta ve, and these are the tw o non-intersecting whole-tone scales. Figure 13 shows the p erturbations of one of these whole-tone scales, given by the pitc h class collec- tion { C, D , E , F ], G], B [ } . As with the triads and sev- en th c hords, if one perturbs any note in the whole-tone scale down w ard by a half step, the resulting chord is as- signed a mo dalit y , and it is given the (+) symbol. In this case, a down w ard p erturbation of a note in the whole- tone scale results in a W ozzeck chord, where the name is taken from Alan Berg’s op era Wozze ck (Cohn 2012). An upw ard p erturbation of any note in the whole-tone scale results in a m ystic chord, which was brought to the atten tion of theorists primarily due to Alexander Scri- abin’s use of the chord in his compositions. The upw ard p erturbation results in the opp osite mo dalit y , so I notate m ystic chords with the ( − ) sym b ol. [5.3] I assign an arbitrary naming scheme for the “ro ot” of a mystic or W ozzeck chord so that the chords can be discussed simply by naming a letter name and mo dalit y sym b ol. In general, the ro ot is the lo wer of the tw o notes in the minor 2nd interv al in the m ystic or W ozzec k c hord. F or example, the W ozzeck chord { C, D [, E , F ], G], B [ } is 10 FIG. 14. A cen tipede generated from p erturbations of a whole-tone scale. notated as C +, and the mystic c hord { C, D [, E [, F, G, A } is notated as C − . [5.4] Returning to Figure 13 , one sees that individ- ually p erturbing the six notes of whole-tone scale b oth up ward and down w ard results in six m ystic chords and six W ozzeck chords. F or the whole-tone scale b egin- ning on C , the collection of nearly symmetric hexa- c hords chords is given b y { A] + , C ] − , D + , F − , F ] + , A − , G] + , B − , C + , D ] − , E + , G −} . I propose that this col- lection can b e represented visually as a centipede 9 , in analogy with the W eitzmann waterbug and Boretz spi- der. [5.5] The “centipede” for mystic and W ozzeck chords is shown in Figure 14 . As with the triads and sev enth c hords, the legs on one half of the centipede’s b o dy are all of the (+) mo dality , and the ( − ) chords are assigned to the legs on the other side of the b o dy . It is easy to see that, given a starting chord, 5 of the 6 chords of the opp osite modality are P 2 , 0 -related to the starting chord, and the remaining chord of the opposite mo dalit y is P 0 , 1 - related to the starting chord. This is directly analogous to the n = 3 and n = 4 cases. Thus, I can define Neo- Riemannian transformations that act on chords in the cen tip ede that are analogous to the “relative” and “slide” transformations that act within the W eitzmann w aterbug and Boretz spider. [5.6] I define the R ∗∗ transformation as the “relative” transformation that connects tw o P 0 , 1 -related chords 9 T rue centipedes do not typically hav e 12 legs, but a newb orn garden symphylan ( Scutigerella immaculata ) rep ortedly is in- deed b orn with 6 pairs of legs but grows more ov er the course of its lifetime (Michelbacher 1938). The garden symphylan is com- monly referred to as the garden cen tip ede, whic h is why I hav e chosen “centipede” as a name for this voice-leading region. of opp osite mo dality . As with dominant sev enth/half- diminished seven th c hords, there is no formal definition of a “relativ e” mystic and W ozzeck c hord, but the action of R ∗∗ is nonetheless well-defined: 1 voice m ust be mo v ed b y 1 step. As an example, C + is transformed to D] − b y R ∗∗ . [5.7] The 5 remaining Neo-Riemannian transforma- tions in this region are “slide” transformations that in- v olute b etw een P 2 , 0 -related chords: 2 voices are shifted in parallel motion by 1 semitone each. This means that 4 v oices are in v arian t in the transformation. I define the follo wing transformations following the notational con- v ention of slide transformations in the n = 4 case: S W ( 1 ) , S A ( 3 ) , S F , S A ( 5 ) , and S W ( 3 ) . The following abbrevi- ations denote the collection of 4 pitch classes that are held inv ariant: W = [0 , 2 , 4 , 6] (4-21, or the W hole tone tetramirror), A = [0 , 2 , 4 , 8] (4-24, or the A ugmented sev enth chord), and F = [0 , 2 , 6 , 8] (4-25, or the F rench sixth set). The letter that app ears in the sup erscript (outside of the paren theses) of the S -type transforma- tion denotes the set of pitches within the m ystic/W ozzeck c hord that do es not change when the transformation is applied. The num b er within the paren theses, as with the n = 4 case, denotes the interv al that is shifted. [5.8] As an example, supp ose one applies S A ( 3 ) to the W ozzeck chord C +, which is the pitch class col- lection { C, D [, E , F ], G], B [ } . The 2 augmented sev- en th chords which are subsets of this collection are { C, E , G], B [ } and { A[, C , E , G[ } . Since the interv al 3— a minor 3rd—is b eing shifted, the relev an t augmented sev enth chord that remains in v arian t during the trans- formation is { A[, C, E , G[ } , and the minor 3rd of B [ and D [ is the interv al that slides up w ard to B and D . The new chord is formed b y the pitc h class collection { C, D , E , F ], G], B } , whic h is B − , or the B mystic chord. 11 FIG. 15. A unified voice-leading mo del for nearly symmetric hexachords. The remaining Neo-Riemannian transformations in this region function the same w ay , and S F is thereby an ab- breviation of S F ( 5 ) . [5.9] In Figure 15 , I prop ose a unified voice-leading mo del a nalogous to Cohn’s (2012) diagrams for connect- ing the W eitzmann w aterbugs and Boretz spiders. By matc hing the ro ot names of one centipede with the cor- resp onding ro ot names on the other centipede, 2 bridge regions arise b etw een the centipedes. In analogy with hexatonic and o ctatonic regions, I refer to the b o xes in Figure 15 as do decatonic regions. In Figure 16 , I pro- p ose a visual represen tation of the voice-leading p ossibil- ities within the 2 do decatonic regions, in analogy with Figure 5 and Figure 11 . Presumably , 5 spatial di- mensions would b e required to construct a conv ex p oly- top e that presen ts all voice-leading p ossibilities within a do decatonic region. Thus, Figure 16 shows only a 2-dimensional reductive representation, constructed us- ing the same logical pro cedure as I used in the previous section to “flatten” Childs’ (1998) cubic netw ork for the o ctatonic region. [5.10] An y tw o chords that are connected by a solid line in the Figure 16 are P 4 , 0 -related. Thus, there are 5 av ailable Neo-Riemannian transformations that can be used to transform a starting chord to the 5 P 4 , 0 -related c hords of the opp osite mo dality . Using the naming con- v ention introduced for the cen tip ede’s Neo-Riemannian transformations, these 5 “slide” transformations are: S 1 , S 3 ( A ) , S 3 ( W ) , S 5 ( A ) , and S 5 ( F ) . The num b er that ap- p ears first in the superscript denotes the in terv al betw een the 2 notes that are held inv ariant. The letter within the paren theses denotes the 4-note collection that is shifted. The letters follow the naming sc heme described earlier in this section for the cen tip ede transformations. F or exam- ple, the transformation S 1 —whic h is an abbreviation of S 1 ( W ) —w ould transform C + to C − , since the minor 2nd of pitc h classes C to D [ would b e held inv ariant while the whole-tone tetramirror { E , F ], A[, B [ } would b e shifted do wnw ard a semitone. [5.11] Given a starting c hord (supp ose C +), there is one c hord (in this case, D − ) of the opp osite mo dalit y that is not accessible via one of the 5 S -t ype transforma- tions; thus, it is not P 4 , 0 -related to the starting c hord. This chord shares no pitch classes with the starting chord, so functionally it is similar to the hexatonic p ole for tri- ads and the o ctatonic p ole for sev en th chords; th us, it can b e referred to as the do decatonic p ole. The do decatonic p ole is accessible by the Z transformation 8 . The geomet- ric result of applying the Z transformation as w ell as the functions of the 5 S -type transformations are shown in Figure 17 . [5.12] Within the do decatonic region, one can construct v arious maximally smo oth cycles. Sequences of S -type transformations that generate closed paths with no lo ops along the solid lines in Figure 16 are, indeed, maximally smo oth cycles. The union of all pitch classes contained in the chords of any one of these maximally smooth cy- 8 Neo-Riemannian theorists often use D to denote motion to/from the “dominant,” so I use Z , which stands for the German zw¨ olf , or “tw elv e.” 12 FIG. 16. Do decatonic regions, sho wn as 2-dimensional graphs. FIG. 17. Av ailable voice-leading transformations in a dodecatonic region. cles giv es the full c hromatic set 12-1, so these maximally smo oth cycles can b e called do decatonic cycles. VI. DISCUSSION AND SUMMAR Y [6.1] In the previous 3 sections, I ha v e shown that voice- leading mo dels for nearly symmetric c hords of cardinali- ties n = 3, n = 4, and n = 6 follo w a set of patterns. In deriving the av ailable Neo-Riemannian transformations for each t yp e of chord, a symmetric chord is p erturb ed b oth down w ard and upw ard. Thus, the tw o c hords that are generated from the perturbation of the same note are in versionally related. [6.2] As I hav e mentioned b efore, the (+) and ( − ) la- b els assigned to ma jor and minor, dominant sev en th and half-diminished seven th, and W ozzeck and mystic chords signify the same direction of p erturbation with regard to the symmetric chord from which each collection is gener- ated. All 24 (+) and ( − ) c hords can b e organized in to dif- feren t groups, and one sees that the “arthrop od” regions as well as the “bridge” regions in the unified voice-leading mo dels are simply differen t w ays of grouping these 24 c hords in order to optimize voice-leading. [6.3] If one c hooses a particular chord from W eitzmann w aterbug, Boretz spider, or cen tip ede, one will alwa ys b e able to find exactly 1 P 0 , 1 -related chord of the opp osite mo dalit y , and n − 1 P 2 , 0 -related chords of the opp osite mo dalit y within the same arthropo d (Cohn 2012). If one c ho oses a particular chord from the hexatonic region, oc- tatonic region, or dodecatonic region, one will alwa ys be able to find exactly n − 1 P n − 2 , 0 -related chords of the opp osite mo dality within the same region, and there will b e exactly 1 “p olar” c hord which is the complement of the starting chord with resp ect to the union of all pitch classes within the region (Cohn 1996; Cohn 2012). [6.4] The union of all pitc h classes within a particu- lar hexatonic, o ctatonic, or do decatonic region is also of set-theoretic interest. As Cohn (1996) mentions, within a given hexatonic region, there are only 6 unique pitch 13 T ABLE 1. Summary of transformations, chord t ypes, and voice-leading regions for nearly symmetric chords; adaptation and extension of Cohn (2012). Some row names from Cohn (2012) hav e b een ommitted or mo dified, and I hav e added the n = 6 column. Gen us Sp ecies, n = 3 Sp ecies, n = 4 Sp ecies, n = 6 1 Symmetric partition Augmen ted triad F ully-diminished sev en th Whole-tone scale 2 Do wnw ard SSD Ma jor triad Dominan t sev enth W ozzeck chord 3 Up ward SSD Minor triad Half-diminished seven th Mystic c hord 4 Union of (2) and (3) Consonan t triads T ristan gen us Mystic-W ozzeck genus 5 arthropo d region W eitzmann w aterbug Boretz spider Centipede (this pap er) 6 Bridge regions betw een (5)’s Hexatonic region Octatonic region Do decatonic region 7 T ransformations within (5) ∼∼∼∼∼∼∼∼∼∼∼∼∼ • F or P 0 , 1 -related chords R R ∗ R ∗∗ • F or P 2 , 0 -related chords S , N S 3 ( 4 ) , S 3 ( 2 ) , S 6 S A ( 3 ) , S A ( 5 ) , S F , S W ( 1 ) , S W ( 3 ) 8 T ransformations within (6) ∼∼∼∼∼∼∼∼∼∼∼∼∼ • F or P n − 2 , 0 -related chords P , L S 2 , S 4 , and S 5 S 1 , S 3 ( A ) , S 3 ( W ) , S 5 ( A ) , S 5 ( F ) • F or p olar relation H O Z 9 Union of pitc hes in (6) 6-20 8-28 12-1 classes whic h are found; this collection is one of 4 distinct pitc h-class sets generated from the set class 6-20. Each of the 4 pitch-class sets corresp onds to a hexatonic region. Similar analysis sho ws that the o ctatonic regions corre- sp ond to set class 8-28 (Douthett and Stein bach, 1998). The dodecatonic regions correspond to the full c hromatic set, 12-1. [6.5] It is also of interest to note that the Neo- Riemannian transformations that are a v ailable in the arthrop od regions are complementary to the transforma- tions a v ailable in the bridge regions: the collections that are held in v arian t in one type of region are shifted in the other. F or example, for the n = 3 case, the S holds a single note inv ariant while shifting the p erfect 5th; the P transformation holds the perfect 5th in v arian t while shifting a single note. This relationship is easier to see in n = 4 and n = 6, as the names suggest a complemen tary relationship. As an example, one sees that for n = 4, the S 3 ( 4 ) transformation acts within the arthrop o d re- gion while S 4 = S 4 ( 3 ) acts in the bridge region (Childs 1998). In general, one of the arthrop od region trans- formations that connects P 2 , 0 -related c hords will hav e a complemen t in the bridge region, where transformations connects P n − 2 , 0 -related chords. [6.6] The Neo-Riemannian transformations, voice- leading regions, and set-theoretic prop erties I hav e dis- cussed throughout this pap er are summarized in T able 1 , which is an extended version of Cohn’s (2012) table co vering n = 3 and n = 4. A further extension of the the- ory for n = 6 presented in this pap er includes the rigorous dev elopment of a 5-dimensional T onnetz for voice-leading b et w een any m ystic and W ozzeck c hords. REFERENCES Childs, Adrian. 1998. “Mo ving b eyond Neo-Riemannian T ri- ads: Exploring a T ransformational Model for Sev en th Chords.” Journal of Music The ory 42, no. 2: 181-193. 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