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This content downloaded from 143.107.252.35 on Wed, 12 Jun 2013 17:10:53 PM All use subject to JSTOR Terms and Conditions SOME ASPECTS OF
THREE-DIMENSIONAL TONNETZE
EdwardGollin
This paper explores how a three-dimensional(3-D) Tonnetzenables an interesting spatial representationof tetrachordsand the contextual transformationsamong them. The geometryof the Tonnetz,a function of its set-class structure,emphasizes certain operations within a larger group of transformationsbased on their common-toneretention proper- ties. After reviewing some features of the traditionaltwo-dimensional (2- D) Tonnetz,we will explore featuresof a 3-D Tonnetzbased on the famil- iar dominant-seventh/Tristantetrachord [0258]. We will then generalize the 3-D Tonnetzto accommodaterelations among membersof any tetra- chord class. Lastly, we will investigatethe relationof a Tonnetz'sgeom- etry to the group structureof the transformsrelating its elements.
The Two-dimensional Tonnetz
The traditionalTonnetz (as manifest in the nineteenth-centurywrit- ings and theories of Oettingen,Riemann, et al.) is an arrayof pitches on a (potentiallyinfinite) Euclideanplane.' Figure la illustratesa region of a traditional2-D Tonnetz.We may summarize several of its essential aspects as follows:
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This content downloaded from 143.107.252.35 on Wed, 12 Jun 2013 17:10:53 PM All use subject to JSTOR Terms and Conditions 1) Pitches are arrangedalong two independentaxes (not necessarily orthogonal), arrangedintervallically by perfect fifth on one axis and by majorthird on the other (See Figure lb).2 2) The triangularregions whose vertices are defined by some given pitch and those either a perfect fifth and majorthird above or a per- fect fifth and majorthird below correspondto some triad,either the Ober- or Unterklangrespectively of that given pitch. 3) The edges of the triangularregions correspond to the intervals within the triad. Only two of these lie along axes of the array;the thirdedge may be definedas a secondaryfeature resulting from the combinationof the primaryaxes. 4) The dual structureof major and minor triads is visually manifest within the Tonnetz.Triangles representing major triads are oriented oppositely from those representingminor triads. 5) Triangles/triadsadjacent to any given triangle/triadshare two com- mon tones if they share a common edge, or one common tone if they share a common vertex. Three particularoperations acting on triads, L, P and R (for Leitton- wechsel or Leading-toneexchange, Parallel, and Relative) which maxi- mize common-tone retention,occupy a privileged position on the Ton- netz: they are representableas 'edge-flips' among adjacent triangles/ triads.3Arrows in Figure la illustratethe mappingsof L, P and R about a C-majortriad. Severalnineteenth-century assumptions underlying the traditional2-D Tonnetzare not obligatory.For instance, if one assumes an arrangement of equally-temperedpitch classes (ratherthan just-intoned pitches), the Tonnetzwould be situatednot in an infinite Cartesianplane, but on the closed, unboundedsurface of a torus.4 RichardCohn (1997) and David Lewin (1996) have demonstratedthat the Tonnetzneed not representrela- tions merely among triads, but may do so among members of any tri- chordalset class; Cohn (1997) has furthershown the extensionof the Ton- netz in chromaticcardinalities other than c= 12.5 Further,one need not adhereto the restrictionof representingtrichords in two dimensions,but may extend the Tonnetzin three dimensions to accommodaterelations among tetrachords.6
An [0258] Tonnetz
Figure2 illustratesa portionof a 3-D Tonnetz,consisting not of points on a plane but of points (representingpitch classes) in a space lattice. Whereastwo axes were sufficientto locate points in the 2-D Tonnetz,the additionaldegree of freedom in three dimensions requiresan additional axis to describe the location of all points. I have chosen axes of a non- orthogonalcoordinate system in Figure 2 (interaxialangles are all 60?), 196
This content downloaded from 143.107.252.35 on Wed, 12 Jun 2013 17:10:53 PM All use subject to JSTOR Terms and Conditions B F# C# G# D#
D A E B
r 1R L.VL
Bb F C G D
Db Ab Eb Bb
(a) A region of traditional2-D Tonnetzand three contextualinversions (sharinga common edge) with a C-majortriad
/M3 axis P5axis (b) The axis system of the traditional2-D Tonnetz
Figure 1 and have labeled these a, b, and c (avoiding labels x, y, and z to distin- guish my axes from those of the Cartesiancoordinate system). Axes are labeled accordingto a right-handedconvention (right thumbpoints in a positive directionalong the a-axis, right index finger in a positive direc- tion along the b-axis, middle finger in a positive direction along the c- axis). Points within the lattice are arrangedat unit distances in positive and negative directionsalong the axes from all other points. The regular arrangementof points in this Tonnetzconstitutes one of two uniform ways of filling space with spheres-crystallographers refer to this ar- rangementas cubic closest packing (ccp).7 Throughoutthis discussion, we will assume equal temperament.Do- ing so induces a modular geometry to the 3-D Tonnetz-the Tonnetz occupies the closed, unbounded volume of a hyper-torusin 4-dimen- sional space. Units in a positive directionalong the a-axis correspondto the musical intervalof 4 semitones,units in a positive directionalong the b-axis correspondto the intervalof 7 semitones, and units in a positive directionalong the c-axis correspondto the intervalof 10 semitones.Any
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b-axis
a-axDF#is a-axis
Figure 2. A region within an [0258] Tonnetz point in the lattice may be located in referenceto any otherpoint in terms of units traversedalong each of the axes, and consequently,if the pitch class of the first point is known, one can determinethe pitch class of the second. For instance, in Figure 2, a point one positive unit along the b- axis from C (=0) is 0+7 = G. A point one positive unit along the a-axis, one positive unit along the b-axis, and one negative unit along the c-axis from C is (1.4) + (1-7) + (-1.10) = 1 = C#. The 60* angle is chosen because it is the angle formed at the edges of a regulartetrahedron meeting at a vertex. Just as triangularareas of the 2-D Tonnetzcorrespond to triadicelements (or trichordalelements, in the case of a generalized2-D Tonnetz),tetrahedral volumes in our 3-D Ton- netz correspondto its tetrachordalelements. In Figure 2, one can observe that any point, along with the points lying one positive unit along the a-, b-, and c-axes, describe the vertices of an upward-pointingtetrahedron and that the pitch classes representedby these points constitutea 'domi- nant-seventh'chord. Similarly,any point and the points lying one nega- tive unit along a-, b-, and c-axes describe the vertices of a downward pointing tetrahedron,and the pitch classes representedby these vertices correspondto some Tristanor half-diminishedseventh chord. Analogous to the invertedtriangles/triads of the 2-D Tonnetz,the dual structuresof Tristanand dominant-seventhchords are visually manifest as oppositely orientedtetrahedra in the 3-D Tonnetz(i.e., pyramidswith peaks pointed upward,versus those with peaks pointed downward). Before I discuss relationsamong tetrachordal elements in our Tonnetz, it will be helpful to develop a contextualnotation for identifyingthe pitch elements of the tetrachords.For this, we can adapta contextualnotation
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This content downloaded from 143.107.252.35 on Wed, 12 Jun 2013 17:10:53 PM All use subject to JSTOR Terms and Conditions developed by Moritz Hauptmannto identify the tones of a triad (Haupt- mann [1853]1991). Figure 3a presentsHauptmann's designation for the tones of a major triad, and one of his versions for designatingthe tones of a minor triad. Roman numeral I designates the Einheit of a triad, that chord tone to which the others refer by means of 'directly intelligible intervals' (for Hauptmann,these are the octave, perfect fifth and major third). Roman numeralII refers to the tone in the triadthat lies a perfect fifth from the Einheit,roman numeral III to the tone thatlies a majorthird from the Ein- heit. The difference between major and minor for Hauptmannis that in the case of major,the Einheit lies a perfect fifth and a majorthird below the other chord tones (i.e. it has a perfect fifth and majorthird), whereas in minor the Einheit lies a perfect fifth and major third above the other chordtones (i.e., it is a perfect fifth and a majorthird to the tones below). One need not accept Hauptmann'sacoustical biases nor the dialecti- cal connotationsinherent in his reference to II and III as Zweiheit and Verbindungin order to adopt a similar system of tokens that identifies tones based on their intervallicenvironment within tetrachordsof some Tn/TnI class. In Figure 3b, I illustrate a neo-Hauptmanniansystem designating two forms of set class [0258]: a 'dominant-seventh'chord [C,E,G,Bb],and a 'Tristan'chord [C#,E,G,B].I designate C and B as 'Einheiten' of their respective chords-altering Hauptmann'ssymbol- ogy, I label these with a lower case romannumeral i. 'Einheit'here des- ignates a chord tone that lies four semitones, seven semitones, and ten semitones from the other chord tones. In (C,E,G,Bb), 'Einheit' is the familiarroot of the chord;in (C#,E,G,B),B is the 'dual root'-the upper tone in a traditionalthird-stacking. Roman numeralsii, iii and iv refer to those pcs thatlie four, seven and ten semitonesrespectively from the Ein- heit, either above (in the nominally 'major'dominant-seventh) or below (in the nominally 'minor'Tristan chord). Our initial choice of 'Einheit' was arbitrary-any tone could be the basis to which the others refer. What is importantis that the system of tokens offers a dual system of naming elements of any asymmetricaltetrachord class.8 Our neo-Hauptmannianlabels offer a means of identifying the con- textual transformsamong tetrachordswithin our lattice, in particular, inversions between tetrachordssharing common tones. Figure 4 illus- trates the common-tone mappings about a nexus tetrahedronrepresent- ing the dominantseventh chord [C, E, G, Bb]. Each mappingself-inverts: the directionof mapping within the Tonnetzis dependentupon whether the operation acts on a 'major' dominant seventh or 'minor' Tristan chord. Thus each mappingmay be viewed either as a mappingfrom the dominant seventh chord [C, E, G, Bb] to an adjacentTristan chord, or conversely a mappingfrom some Tristanchord to [C, E, G, Bb].
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This content downloaded from 143.107.252.35 on Wed, 12 Jun 2013 17:10:53 PM All use subject to JSTOR Terms and Conditions I IIl II II II I C G F ab C ~ffY P5 tl? ? (a) Hauptmann'sdesignation of triadicchord tones i ii iii iv iv iii ii i C E G Bb C# E G B 10I0 (b) a neo-Hauptmanniansystem of tokens identifying chord tones in set class [0258]
i ii iii iv iv iii ii i (u,v,w)(u+l,v,w) (u,v+l,w) (u,v,w+l) (u,v,w-1)(u,v-l,w) (u-l,v,w) (u,v,w) (c) neo-Hauptmanniantokens appliedto generalizedtetrachords in the 3-D Tonnetz Figure 3
Figure 4a isolates the six transformsthat map or exchange tetrahedra sharinga common edge. The exchange is representedspatially as a 'flip' of the two tetrahedraabout that common edge. For instance, the opera- tion Ii exchanges tetrahedraabout their common edge bounded by chordalelements i and ii. In Figure 4a, the Iii operationflips the central, upward-pointing[C, E, G, Bb] tetrachordabout the C-E edge, and exchanges it with the downward-pointing[E, C, A, F#] tetrachordjust below and to the front of the illustration.The operation Ii similarly 'flips' two tetrahedraabout their common elements ii and iii. In Figure4, Ilii exchanges the central [C, E, G, Bb] tetrachord(flipping it aboutthe G- E edge) with the downward-pointing[B, G, E, C#]tetrachord to the lower right in the illustration.Each 'edge-flip' maintainsat least the two com- mon tones that constitutethe common tetrahedraledge. In the case of Iii in this Tonnetz,mapped [0258] tetrachordsshare three common tones. Figure 4b isolates the four transformsthat map or exchange tetrahe- dra sharinga common vertex. The exchange is representedspatially by a 'flip' of the two tetrahedraabout that common vertex. For example, the operationII exchanges two tetrachordsthat sharea common 'Einheit.'In Figure 4b, Ii exchanges the central [C, E, G, Bb] tetrachordwith the downward-pointing[C, Ab, F, D] tetrachordbelow and to the forwardleft in the illustration.Each 'vertex-flip'maintains at least one common tone (the common vertex). However,I"l and I'vmap [0258] tetrachordssharing two common tones, and maps [0258] tetrachordssharing three. We can observe the followingIii degeneracy among transformsin this equal-temperedTonnetz: Il = Iv and Iv. The degeneracyarises from the symmetries inherent in our chosenIi= set class [0258], and from our imposition of equal temperament.Specifically, the equivalenceof I" and
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b-axisa-axis
a-axis a-axs AL -,?< LLal
Figure 4a. Six 'edge-flips' about a nexus tetrachord,(C,E,G,B?), within an [0258] Tonnetz
iv
c-axis D
b-axis
$ a-axis A
F E E C# A
A#
Figure 4b. Four 'vertex-flips'about a nexus tetrachord(C,E,G,Bb), within an [0258] Tonnetz 201
This content downloaded from 143.107.252.35 on Wed, 12 Jun 2013 17:10:53 PM All use subject to JSTOR Terms and Conditions (u,v,w+l) (u-,v,w+ (uv-
(u-1,v+1,w ( ufv+1w ,v+l,w) c-axis uv(u+l,v,w) XiS(u-,v-(u-1,v,w) 1,W)w u+l,v-l,w) b-axis (u,v-l,w) u+,v-,w) a-axis 1 uw-l +,w- (u,v,w-1) (u+1,v,w-1) Figure 5. A region of generalizedTonnetz about the point (u,v,w)
iv occurs because the intervalfrom ii to iv (6 semitones) spans half the cardinality of our chromatic universe. The equivalence of and results from the symmetry of the [036] trichordembedded Iii within anIiv [0258].9 Moreover,in the modulargeometry of the hyper-toroidalspace in which our equal-temperedTonnetz resides, lattice points identical in pitch are also identical in location. Thus, (X)Ii and (X)I', where X is some [0258] tetrachordin this Tonnetz,are the same tetrachordin our curvedmodular space.'0
A generalized 3-D Tonnetz
Figure 5 illustratesa region of lattice points in a generalized3-D Ton- netz adjacentto a point (u,v,w). The triple(u,v,w) indicatesa point u units (in a positive direction)along the a-axis, v units along the b-axis and w units along the c-axis from an origin point (0,0,0). An equivalentfamily of points could be constructedaround any point in the space. If we select coefficients a, 3, and y that correspondto intervalsin some universe of cardinalityN, and posit these as unit lengths along the a-, b-, and c-axes respectively,and if we fix the origin point as 0 in an integer notational system, then the identity of any point (u,v,w) is given by au + 3v + yw (mod N). For instance,fixing (0,0,0) = C = 0, and settinga, 3, andyequal to 1, 4 and 8 semitonesrespectively in a mod 12 pitch-classspace, a point (2,-3,5) would representthe pitch class (1.2)+(4--3)+(8-5) = 2-12+40 = 30 mod 12 = 6 = F#in an [0148] Tonnetz. Figure 3c illustrateshow our neo-Hauptmanniantokens may be af- fixed to points in the generalized3-D Tonnetz.If we let ii refer to a pitch class that lies one unit along the a-axis from some Einheit, let iii refer to a pitch class one unit along the b-axis, and let iv refer to a pitch class one unit along the c-axis (in eithera positive or negativedirection), then given 202
This content downloaded from 143.107.252.35 on Wed, 12 Jun 2013 17:10:53 PM All use subject to JSTOR Terms and Conditions some point in the lattice (u,v,w) as 'Einheit,'we may define a 'major' form of any tetrachordas the tetrahedroncomprising the four vertices (u,v,w), (u+1,v,w), (u,v+1,w), and (u,v,w+1). A tetrahedronrepresenting a 'minor' form of a tetrachordclass built from the same Einheit would consist of the four vertices (u,v,w), (u-1,v,w), (u,v-l,w) and (u,v,w-1). Analogously to the [0258] Tonnetz,contextual transformationsin the generalized 3-D Tonnetzmay be representedas edge- or vertex- flips among adjacenttetrahedrons. For instance, Iii is an edge-flip that inverts a tetrachord about its 'Einheit' and its element ii, mapping [(u,v,w),(u+1,v,w),(u,v+1,w),(u,v,w+1)] to [(u+1,v,w),(u,v,w),(u+1,v-1,w), (u+1,v,w-1)] and vice versa. I have been cautiousto specify the Tonnetzin which a particulartetra- hedron-invertingoperation occurs, since our neo-Hauptmannianlabels (and consequently our inversionlabels) are tied to the intervalliclayout of our axes. Were we to explore relationsin an [0258] Tonnetzwith dif- ferently arrangedaxes (for example positing units along the a-, b-, and c- axes of 3, 6 and 8 semitones respectively), the same operationsof our original [0258] Tonnetzwould obtain, but with different 'I' labels. The same is not the case if one considers operationswithin Tonnetzeof dif- ferent set classes: the family of common-tone preserving, contextual inversions of one set class is in general a differentfamily of contextual inversionsfrom any other.
Group Structure of Transforms within the Tonnetz
If one exhaustivelycomposes the L, P, and R transformsin the triadic 2-D Tonnetzof Figure 1, the resulting closed group of operationsis a dihedralgroup known as the Schritt/Wechselgroup (S/W)." The group, acting on harmonic triads, consists of 12 mode-preservingoperations (Schritte)and 12 mode-invertingoperations (Wechsel). The symbols So, S1, S2, , will here indicate 12 Schrittethat map a triadto the mode- identical.... triadSIlwhose root lies 0, 1, 2,..., 11 semitones away in the direc- tion of chord 'generation'(up in the case of majortriads, down in the case of minor).12For example, (E major)S3=(G major),but (E minor)S3=(C# min.). Wo,W1, W2 etc. are Wechselthat map mode-invertedtriads whose dual roots lie 0, 1, 2, etc., semitones apartin the directionof chord 'gen- eration.'Thus in the triadic2-D Tonnetz,L equals WI1, P equals W7 and R equals W4.13 Analogously, if we exhaustively compose the edge- and vertex-flip- ping transformsin our [0258] Tonnetz,the resultinggroup of operations is identical in structure(or isomorphic)to the S/W group acting on har- monic triads. The isomorphic S/W group acting on [0258] tetrachords maps mode-identicaland mode-inverted tetrachords based on the directed intervalsamong their 'Einheiten.'Thus in our [0258] Tonnetz,II is equiv- 203
This content downloaded from 143.107.252.35 on Wed, 12 Jun 2013 17:10:53 PM All use subject to JSTOR Terms and Conditions alent to Wo, Iii is equivalentto W4, etc.14 Note that if we alter our system of labels (i.e., select some othertone as 'Einheit')the Wechselassociated with each particularcontextual inversion are all changed by some con- stant.The groupsresulting from the relabelingof tetrahedralelements in the Tonnetzare thereforeautomorphic images of one another. The exhaustivecomposition of the contextualinversions of any asym- metricaltetrachordal (or trichordal)set class will similarlyyield an S/W group isomorphic to the S/W groups acting on triads or [0258] tetra- chords (or else isomorphic to a dihedral subgroupthereof-contextual inversions among members of [0248] for instance yield only the even Schritteand Wechselof the largergroup). This remarkableresult, that the same group of operationsunderlies Tonnetzeof such drasticallydifferent geometries (and even of different dimensions), suggests that the Tonnetzis an entity independentof the S/W group. It also suggests an interestingrelation between the structure of the Tonnetzand the transformationsamong its members.Elements of mathematicalgroups are largely indifferentto the music-analyticalsitu- ations upon which they are called into service in transformationaltheo- ries. In the context of the S/W group qua group, any Wechselis basically like any other (all of order 2, etc.). Yet one could hardly claim that the musical effect of the Relativetransform (W4) acting on a triad(e.g., map- ping C min.) is at all the same as that of the Gegenkleinterz- wechselmaj.E--A (W3, e.g. mapping C min.). Thus we may view the Tonnetzas an independentframework maj.--Abthat allows one to make distinc- tions among the (otherwise indifferent)transforms that underlie its ele- ments. It allows one to posit analyticalmeaning or value to certainfam- ilies of transformsfrom the largergroup based on the distance between elements mappedby those transformswithin the Tonnetz,whose geome- try in turn is determinedcontextually by the intervallic structureof its chordal elements. Distance in the Tonnetz,moreover, is a function of common-toneretention. Hence, the function of the Tonnetzin neo-Rie- manniantheory is not unlike that of Lewin's INJ function:it provides a measure of the progressiveversus internalquality of a contextualfunc- tion acting on a 'Klang' of a trichordalor tetrachordalset class.'"
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1. For a historyof the Tonnetzand its evolutionfrom Eulerthrough Riemann, see Mooney1996. A studythat discusses the Tonnetzand its relationto Funktionsthe- orie in Riemannand his successorsis Imig 1970. 2. Thenineteenth-century basis of intervalselection was acoustical:the intervals cor- respondedto whatRiemann (along with numeroustheorists before him) believed to be the only intervals(along with the octave)given by nature. 3. RichardCohn (1997) has recentlyexplored the Tonnetzrepresentation of these operations,noting their potential for common-toneretention as well as theirpar- simoniousvoice-leading. The abbreviations derive from Brian Hyer 1989 and 1995 (101-38). 4. Cohn 1997cites Lubin1974 as the firstrecognition of the toroidalstructure of an equallytempered Tonnetz. Hyer's Figure 3 (1995, 119) presentsa Klangnetz,the geometricdual of a Tonnetz(see Douthettand Steinbach1998), in pitch-class space,which similarlyinduces a toroidalstructure. 5. Lewin'shexagonal graph of a 3-valuedCohn function (Lewin 1998; 1996, 189), is, like Hyer's,a dualof an [013] Tonnetz. 6. Otherwritings that extend the Tonnetzinto threedimensions include Vogel 1993 and Lindleyand Turner-Smith 1993. 7. It is so called becausethe unit cells of the lattice(the smallestunits that can fill space by translationalone) are face centeredcubes (latticepoints exist at the six verticesand at the centerof each cube face). 8. HenryKlumpenhouwer (1991, chap. 2) exploresa similarsystem of contextual labeling. 9. Theembedded [036] is also thereason that IIi! and IiR map tetrachords in our[0258] Tonnetzwith threecommon tones. It is interestingto observethat none of the 29 tetrachord-classesin a chromaticuniverse of cardinality12 arewithout degener- ate common-tone-preservingcontextual transforms: any tetrachord in c= 12 either (a) containsan ic 6 dyad,(b) is symmetricalor embedsa symmetricaltrichordal subset,or (c) satisfiesconditions (a) and(b). 10. This wouldnot be truein a just-intonedversion of our Tonnetz,where (C,E,G,B,) Iii= (G#,E,C#,A#)# (C,E,G,Bb)Itv = (Ab,Fb,Db,Bb).Such a Tonnetzwould, how- ever,occupy an infiniteEuclidean space. See Vogel 1993, 123 ff., andLindley and Turner-Smith1993, 66-68, for discussionsof unequallytempered [0258] Ton- netze. 11. Riemannexpounds his system of Schritteand Wechselas directedroot-interval relations among triads in Skizze einer neuen Methode der Harmonielehre (Rie- mann 1880). Klumpenhouwer1994 reinterpretsand extendsRiemann's Schritte andWechsel into a groupof operationsacting on the set of majorand minor triads. 12. Thisis a vestigeof Riemann'sbelief thatmajor triads were acoustically generated fromthe root up (viaovertones), and that minor triads were acoustically generated fromtheir upper, dual root down (via undertones). 13. Onecan derivethe completeS/W groupfrom the L andR transformsalone. Since LR = W11W4= S7 is an elementof order12, it may be combinedwith L or R or any Wechsel(all areof order2) to generatethe completegroup.
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This content downloaded from 143.107.252.35 on Wed, 12 Jun 2013 17:10:53 PM All use subject to JSTOR Terms and Conditions 14. Analogously,we canproduce generators for our[0258] S/W group.lifi Iii = WIIW4 = S7is an elementof order12, which may then be combinedwith any edge- or ver- tex-flipto generatethe completegroup. 15. INJ is discussedin Lewin 1987, 123 ff.
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