Klein-Bottle Tonnetze


KEYWORDS: transformation, generalized interval systems, Klumpenhouwer networks, Lewin, group theory, neo-Riemannian theory

ABSTRACT: Departing from the toroidal Tonnetz of neo-Riemannian theory, we construct a generalized Klein-bottle Tonnetz. Further, we examine associated transformational graphs and analytical contexts, using operators from the cyclic T group, the dihedral T/I group, and a generalized quaternion T/M subgroup. In the T/I example, corresponding regions within a Tonnetz are related by various Klumpenhouwer network isographies. Finally, we consider relationships among entire Klein-bottle Tonnetze, and place them into recursive supernetworks.

Received June 2003
Volume 9, Number 3, August 2003
Copyright © 2003 Society for Music Theory


[1.1] Our point of departure is the “Table of Tonal Relations,” or Tonnetz, used by Hugo Riemann (1902, 479) and other nineteenth-century German music theorists to model relationships among triads and keys. A rendering of this table appears in Figure 1.

More recently, neo-Riemannian theorists have resurrected the Tonnetz as a network on which to illustrate certain transformational relationships. It has been modularized to accommodate pitchclass space, essentially forming a grid on the surface of a torus, and has been further generalized using techniques from graph theory and abstract algebra. In the present study, we consider the implications of a particular algebraic relation on a class of related Tonnetze, using various groups of GIS-intervals. The fundamental regions underlying their graphs describe Klein bottles; therefore, we will refer to these constructs as “Klein-bottle Tonnetze.”

[1.2] Klein-bottle Tonnetze that incorporate both Tn and In operators are related to Perle cycles, and we may use both types of networks to study Klumpenhouwer networks. This relationship is an anti-isomorphism; hence, a T/I Klein-bottle Tonnetz and the Perle cycle to which it is antiisomorphic have the same algebraic structure. However, they have very different surface features. Corresponding regions within Perle cycles demonstrate strong Klumpenhouwer-network isographies, while the pitch-class contents of these segments belong to varying set-classes. In contrast, corresponding regions in T/I Klein-bottle Tonnetze display weak isographies, but the segments are members of the same set-class.

[1.3] Lewin (2002, 197) points out that Perle cycles are useful in Klumpenhouwer-network analysis, as they provide a method for modeling strong isographies. Their weakness, then, resides in the fact that they cannot be used to demonstrate recursive structures. Furthermore, neither the Perle-cyclic nor the recursive analytical method addresses specifically the set-class content of the pitch-class sets they interpret. Whereas this information may be of secondary importance in a transformational analysis, it is often musically salient, especially in passages with restricted setclass content. Klein-bottle Tonnetze are capable of modeling recursion, particularly with regard to musical contexts in which the set-class content is circumscribed. Indeed, such modeling is useful, as the musical literature contains many instances of limited-set-class passages that suggest Klumpenhouwer-network interpretations.

[1.4] Example 1a shows one such excerpt, the opening canon of no. 8, “Nacht,” from Arnold Schoenberg’s Pierrot Lunaire, op. 21. Example 1b demonstrates how the passage may be interpreted as a cycle of imbricated 3-3[014] trichords. Figure 2 presents a network of the passage, wherein these trichords appear as adjacent triangular subnetworks, labeled g1 through g9. As we will see later, this network is a Klein-bottle Tonnetz. Whereas the interpretation of 3-3[014] trichords elsewhere in the piece might suggest the exclusive use of Tn arrows, In arrows are appropriate here if we wish to show some degree of recursion. The inversional relation between adjacent trichords suggeststheir inclusion.

[1.5] Figure 3 places the above trichordal subnetworks into a supernetwork, in which nodes represent underlying graphs, and edges represent hyper-operators. It is also a type of Klein-bottle network. The recursion between the two networks is evident when we compare the operators of Figure 2’s edges with the hyper-operators of the edges of Figure 3. Regardless of node content—hence, taken only as graphs—we find a direct correspondence among these edges. The same operators which relate pitch-classes in Figure 2 also relate, via conjugation, the operators that interpret its trichords. Thus, we find a remarkable degree of consistency among the various levels of the example.

[1.6] In the following sections, we generalize the theory of Klein-bottle Tonnetze. First, we arrive at an algebraic and graph-theoretical abstraction, and consider the relation of Klein-bottle Tonnetze to the more familiar toroidal models. Then, we examine various pitch-class networks that possess these properties. Specifically, we focus on those Klein-bottle Tonnetze which incorporate the cyclic T group, the dihedral T/I group, and a generalized quaternion subgroup of the T/M group. We examine next the various isographies that relate entire Klein-bottle Tonnetze to one another. Finally, we make some connections between neo-Riemannian and Klein-bottle Tonnetz theories, and suggest some further areas for musical analysis.


[2.1] Theoretically, Riemann’s table in Figure 1 extends infinitely on a plane in two dimensions. Using its underlying graph, we may define these dimensions as follows: translation by x moves each node to the right by one, and translation by y moves each node upward by one. These translations correspond to musical transpositions by a just perfect fifth and a just major third, respectively. In other words, we map the figure’s interval content onto the infinite cyclic groups X and Y. By assigning finite orders to X and Y—musically, by accepting enharmonic and octave equivalence—we identify the figure’s two sets of parallel edges. Now each of the axes on the plane is circularized, and the figure forms a grid on the surface of a torus. The transformation group for the graph is given by the product set XY, and, because X and Y are both groups of translations, XY is also cyclic group, hence commutative. 18 Accounting for node content, this group is generated by T7 and T4, which yields the familiar cyclic T group.

[2.2] Given a particular relation, we observe another geometry which arises from the product of two cycles: the Klein bottle. Initially, a Klein bottle is constructed like a torus. If we start with a rectangle, and bend it to identify two parallel edges, we obtain an open cylinder. If we then bend the cylinder around to join its two ends, we get a torus. To make a Klein bottle, we need to return to the cylinder. Again, we join its two ends, but not by bending the cylinder around; rather, we thrust one end through the side of the cylinder, and connect it to the other end internally. The surface must not really intersect itself through where the bottle’s neck is thrust. Rather, a fourth dimension is needed to go around the surface instead of through it. The result is an unbound surface with no inside or outside.

[2.3] Figure 4 shows a pair of rectangles, and the identifications of their sides that yield respectively a torus and a Klein bottle.

In Figure 4a, we identify the edges indicated by the ordered pairs (e,y) and (x,yx) by taking e to x, and y to xy, forming an upright cylinder. Then we identify edges (e,x) and (y,xy) by taking e to y and x to xy, obtaining a torus. In Figure 4b, we identify (e,w) and (x,xw) by taking x to e and w to xw. Again, we have an upright cylinder. Now we identify (e,x) with (xw,w) by taking e to xw and x to w, thus forming a Klein bottle.

[2.4] To arrive at the relations necessary for a Klein bottle group, we start with two distinct cyclic groups W and Z, generated by w and z, respectively. On the underlying graphs of our Tonnetze, we define the operations w and z not as linear translations, like x and y, but rather as parallel glide reflections. A glide reflection is the product of two operations: first it reflects in an axis, then translates parallel to that axis. Figure 5 shows a fragment of a W-cycle of glide reflections.

From e, w reflects first across an axis parallel to Y, taking e to *. Then w translates upward, taking * to w. This combination completes a move by w. A subsequent move by w also reflects across the same axis, now taking w to +, and translates upward, taking + to w2, and so forth. Of course, it is important to keep in mind that, like translations, glide reflections act on the entire plane, and not just on a single point.

[2.5] Figure 6 shows a portion of another cycle of glide reflections, this one generated by z, together with our previous W-cycle. A move by z reflects across a different axis, parallel to Y, and then translates upward. Hence, both operations are sense-reversing with regard to a vector pointing in the direction of the (horizontal) X-axis. We note the following important relation,

          DEFINITION 2.5.1 w2 = z2,

which, for a finite w and z, incorporates into a presentation of a Klein-bottle group, G.

          DEFINITION 2.5.2 G = {w,z | wm = zn = e, w2 = z2}.

[2.6] The product of any two glide reflections is a translation. Accordingly, we may now define the translations x and y above in terms of w and z.

          DEFINITION 2.6.1 x = z-1w.

          DEFINITION 2.6.2 y = w2 = z2.

Figure 7 is an illustration of e, w, x, y, and z, all originating from e.

Using 2.6.1-2, we note further that

          THEOREM 2.6.3 w = zx


          THEOREM 2.6.4 z = wx-1.

We may also give an alternative definition of z:

          COROLLARY 2.6.5 z = xw.

[2.7] Now we construct an appropriate fundamental region for our Klein-bottle group. Whereas the relation in 2.5.1 suggests Figure 8, its graph does not intuitively resemble a Klein bottle.

Therefore, we will construct a variant that is easier to visualize. In doing so, we will move the top half of the figure by some member of the group. Multiplying the nodes of the upper triangle, (w,z,y), on the left by z-1 gives (x,e,z).

Next, we reconstruct the diagram using the original bottom half of Figure 8 and this variant of its top half (Figure 9). This diagram is also a fundamental region.

Now we can more readily visualize the Klein bottle, using the following identifications of sides: (e,w)  (x,z) (by x), and (w,z)  (x,e) (by z-1).

[2.8] A Klein-bottle group may be generated equivalently by using either two glide reflections, as above, or by one glide reflection and one translation. Since the latter conforms more to our notions of the dihedral T/I group, we observe the following relations:

          DEFINITION 2.8.1 w2 = y; xy = yx; w(x)w-1 = x-1

that incorporate into a presentation of a group G in terms of its generators w and x.

          DEFINITION 2.8.2 G = ⟨w,x | w2 = y; xy = yx; w(x)w-1 = x-1; wm = xn = e⟩.

Given no further relations, we note that G is non-commutative.

[2.9] In the generalized group, only the members of subgroup Y commute always with every member of the group. Therefore,

          THEOREM 2.9.1 Y is in the center of G.

Accordingly, it is easy to show that Y is also a normal subgroup. Furthermore,

          THEOREM 2.9.2 X is a normal subgroup of G;


          THEOREM 2.9.3 if X has an even order k, then xk/2 is in the center of G.

Now we may define the center of G, CG.

          DEFINITION 2.9.4 If 2 | |X|, then CG = ⟨y,x|X|/2⟩; otherwise, CG = ⟨y⟩.

[2.10] Specifically, because X is normal in G, we may offer an alternate definition of G to 2.8.2,

          DEFINITION 2.10.1 G = XW,

using the product formula for the set XW. The order of G is thus determined:

          DEFINITION 2.10.2 |G| = |W||X| / |W ∩ X|.


          THEOREM 2.10.3 2 | |G|.

Figure 10 shows a larger segment of an abstract Klein-bottle Tonnetz. Given no further relations, we find 2|G| unique triangles formed by mutually adjacent nodes in such a graph.

[2.11] In certain circumstances, G may be commutative, and in any such Klein-bottle Tonnetz, x generates an involution.

          THEOREM 2.11.1 If ab = ba for any a,b in G, then |X| = 2.

For example, all Klein-bottle Tonnetze that use the commutative T group will have X = ⟨T6⟩, the only involution in that group.

[2.12] Finally, we note the condition under which the groups of two Klein-bottle Tonnetze, G and G′, are isomorphic. This condition will be of particular consequence in later sections. We define the isomorphism in terms of a bijective mapping, F, of G onto G′.

          DEFINITION 2.12.1 Let G and G′ be two groups, and let F be a bijective set mapping F: F(G) = G′. F is an isomorphism of G to G′ if F(h)F(g) = F(hg), for any g,h in G.

Certain groups are isomorphic to themselves (by a map other than the identity), and such internal relationships define (nontrivial) automorphisms.

          DEFINITION 2.12.2 Let G be a Klein-bottle group, and let F be a bijective set mapping F: F(G) = G. F is a group automorphism if, for any g,h in G, F(h)F(g) = F(hg).

[2.13] The opening of Witold Lutosławski’s Funeral Music (Example 2) provides an illustration of how we can map musical materialsto the nodes of a Klein-bottle graph. The piece begins with a twelvetone row, P0, presented in the Cello I solo. It consists of a fragment of a cycle of alternating tritones and descending minor seconds. This row is followed immediately in the same voice by a statement of I6, then another statement of P0, and so forth. The Cello II solo enters in strict imitation at the half note with a P6 form of the row. It continues correspondingly with I0, and again P6. Because of the row’s intervallic construction, every other harmonic interval in this canon is either a unison or an octave.

[2.14] The full cycle of alternating T11’s and T6’s, in that order, is twenty-four units long. We can map its operators onto the elements of an order 24 cyclic group W, as shown in Figure 11.

Next, we can produce another order 24 cycle by alternating T6’s and T11’s, in that order. We map its elements onto a cyclic group Z. (See Figure 12.)

Under this mapping, w2 = z2, as T6T11 = T11T6 = T5. Furthermore, w24 = z24 = e. Thus, all the relations for a Klein-bottle group from 2.5.2 are satisfied.

[2.15] Successive application of the members of W and Z to the pitch-class 6 yields the following two cycles of pitch-classes:

          W(6) = (6,5,11,10,4,3,9,8,2,1,7,6,0,11,5,4,10,9,3,2,8,7,1,0) and,
          Z(6) = (6,0,11,5,4,10,9,3,2,8,7,1,0,6,5,11,10,4,3,9,8,2,1,7).

Alternating members of these respective cycles belong to the same pitch-class. The Cello I solo presents a twelve-member fragment of W(6), beginning on the pitch-class 5 in the cycle’s second position: (5,11,10,4,3,9,8,2,1,7,6,0). This line continues with a twelve-member fragment from the retrograde of Z(6): (11,5,6,0,1,7,8,2,3,9,10,4). The Cello II solo imitates the Cello I solo with a (non-retrograded) fragment of Z(6), beginning on the pitch-class 11 in the cycle’s third position: (11,5,4,10,9,3,2,8,7,1,0,6). It continues with a fragment from the retrograde of W(6): (5,11,0,6,7,1,2,8,9,3,4,10).

[2.16] These twelve-tone rows can be modeled entirely using a closed segment from the Kleinbottle Tonnetz generated from this W(6) and Z(6). Figure 13 presents this fragment. The Cello I solo begins with the pitch-class 5 in the figure’s lower left-hand corner, and proceeds upward with pitch-classes 11 and 10 in the the figure’s middle and leftmost columns, respectively. The line continues with pitch-classes 4, 3, 9, 8, 2, 1, 7, 6, and 0, maintaining the alternation between these two columns. Next, it presents pitch-classes 11, 5, and 6, rounding the uppermost portion of the figure, and it concludes with pitch-classes 0, 1, 7, 8, 2, 3, 9, 10, and 4, descending and alternating between the rightmost two columns. At this point, it reaches pitch-class 5 again, and then repeats the entire cycle, and so forth. In short, it describes a jagged clockwise path around the figure. The Cello II solo begins with the pitch-class 11 at the bottom of the middle column, and it proceeds around the diagram in a jagged counterclockwise path.


[3.1] We will now formalize the previous example. We will take G as a group of operations, and will examine the network obtained by applying all the members of G to a common object, p. For example, if p is a pitch-class integer, we obtain a Klein-bottle Tonnetz. We start with the concept of a P-set.

          DEFINITION 3.1.1 P = G(p).

In other words, P is the orbit of p under G. We determine the order of P by extension of 2.10.2.

          THEOREM 3.1.2 |P| = |W(p)||X(p)| / |W(p) ∩ X(p)|.

In all cases, |X(p)| = |X|. Being a group of translations, X is fixed-point-free. However, in certain circumstances, |W(p)| < |W|, since W, as a group of glide reflections, may or may not be fixedpoint-free. Furthermore, sometimes |W(p) ∩ X(p)| > |W ∩ X|. This situation may occur in contexts in which certain images of p under both W and X are not unique. Moreover, because of the closure of G under its group operation, applying G to any member of P produces the same P-set.

          REMARK 3.1.3 G(pi) = G(pj) for any pi,pj in P.

[3.2] In all events, our Tonnetz is now a true GIS in the sense of Lewin (1987, 26–30). We may represent it by the ordered triple (S,IVLS,int), in which the space S of the GIS consists of P; the group of intervals IVLS acting on P is G; and int is a function which assigns a value g from G to any pair (pi,pj) in P × P. In other words, int(pi,pj) = g, for some g in G. Our Tonnetz is, moreover, transitive, and also satisfies Lewin’s Condition (B) for uniqueness: for every pi in P and g in G, there is a unique pj in P which lies the interval g from pi.

[3.3] We will now examine a segment from an abstract operational Tonnetz (Figure 14).

Following Lewin, Figure 14 is a network. Its space, which consists of the set of points S = {p1,p2,p3,p4}, is a subset of P. The intervals between these points, {e,w,w-1,x,x-1,y,y-1,z,z-1}, form a subset of IVLS = G. Table 3.3.1, then, gives the specific mapping of S × S into G.

[3.4] Next, for any pi in P, we observe two classes of triangles, t+(pi) and t-(pi), formed by mutually adjacent nodes on the graph of G(p); t+ triangles point upward and t- triangles point downward. We define the set of all such triangles,

          DEFINITION 3.4.1 TRI = {t+(pi),t-(pi) | pi is a member of P}

and assign only one triangle of each type to any node pi in the graph.

          DEFINITION 3.4.2 |TRI| = 2|P|.

The node pi of origin serves as the “Einheit” of the triangle, which we abbreviate “h.” We describe the Einheit as being either even or odd, based on the parity of the least power of w (from e) that determines it.

          DEFINITION 3.4.3 h is even if it is determined by an even least power of w.

          DEFINITION 3.4.4 h is odd if it is determined by an odd least power of w.

Therefore, we may define an equivalence relation which partitions all such triangles into four equivalence classes.

          DEFINITION 3.4.5 Let TRI be the relation: “points in the same direction as, and has the same Einheit parity as.”

We label the four classes of triangles as follows:

          DEFINITION 3.4.6 The four TRI-classes of triangles: t+/e, t+/o, t-/e, t-/o.

Figures 15a-b present these triangles for even and odd Einheiten, respectively; all even Einheiten appear on the left-hand side of the triangles, and all odd ones on the right.

[3.5] In general, each of the four types of triangles is determined by a unique mod 3 function.

          DEFINITION 3.5.1 t+/e = (x,x2(j+1)w,x2(j+1)-1w-1) : j = the power of x that determines h.

          DEFINITION 3.5.2 t+/o = (x-1,x2(j-1)w,x2(j-1)+1w-1) : j = the power of x that determines h.

          DEFINITION 3.5.3 t-/e = (x,x2(j+1)w-1,x2(j+1)-1w) : j = the power of x that determines h.

          DEFINITION 3.5.4 t-/o = (x-1,x2(j-1)w-1,x2(j-1)+1w) : j = the power of x that determines h.

Furthermore, the interval series of any triangle determines its total GIS-interval content.

          DEFINITION 3.5.5 Let (a,b,c) be the interval series of a triangle.
          Then, (e,a,a-1,b,b-1,c,c-1) is the total GIS-interval content for that triangle.

It is then easily demonstrated that any pair of t+ and t- triangles that share the same Einheit possess the same GIS-interval content.

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