Yale University Department of Music Duke University PressYale University Department of Music http://www.jstor.org/stable/843873 . Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Duke University Press and Yale University Department of Music are collaborating with JSTOR to digitize, preserve and extend access to Journal of Music Theory. http://www.jstor.org 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: 195 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 197 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 ACE c-axis 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 198 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.