musical analyses can be suggested through a theoretical group approach. In the second section, we consider the neo-Riemannian theory, and suggest a ...
PERMUTATION GROUPS AND CHORD TESSELATIONS Franck Jedrzejewski Commissariat a` l’Energie Atomique (CEA) INSTN - 91191 Gif-Sur-Yvette FRANCE ABSTRACT
2. CHORD TESSELATIONS
Analytical approaches to musical analysis have generally attempted to provide some kind of justification, and, in the last decades, to develop analytical elements based on tesselations of pitch class sets. When applied to atonal music, analytical approaches frequently fail to find internal coherence. Most of the time, tilings of chords are based on a torus representation, which follows the traditional principles of functional harmonic progression. A possible solution for determining a nice coherence of atonal chords progression is dependent on the topology of the tesselation involved. In the first section, we show that some tilings of chords could be related to the Klein bottle or to the projective plane. It is well known that tesselations are connected with groups. Consequently, possibilities of musical analyses can be suggested through a theoretical group approach. In the second section, we consider the neo-Riemannian theory, and suggest a number of revisions. We substitute the contextual transformations by three pointwise operations based on permutations, in order to link harmonic hierarchical or categorized structures with permutation groups. In the last section, we draw upon Olivier Messiaen’s works in order to show that some groups are more suitable than others for analytical approaches, but in any case, are related to some permutation groups.
According to the traditional theory of the French theorist Jean-Philippe Rameau (see [3]), harmonic chords are built from a succession of major and minor thirds. In the mathematical framework, the set of the twelve notes is usually identified with the set Z12 = {0, 1, ..., 11} (0 = C, 1 = Db, 2 = D, etc.) and the chains of major thirds are generated by an application T 4 : x → x + 4 mod 12. In the same way, chains of minor thirds are generated by T 3 : x → x + 3 mod 12. The cartesian product Z 12 Z3 × Z4 clearly defines a tiling of the plane in which the minor thirds are represented by the horizontal axis and the major thirds by the vertical axis. The tiling is a representation of the direct sum Z12 = 4Z3 ⊕ 3Z4 . A triangulation of each square leads to two chords : one major triad and one minor triad, constructed on the vertices of each triangle. The dual graph which is the graph built on interior points of each triangle is the starting point of the neo-Riemannian approaches dealing with Klumpenhouwer networks (see [4]). But the torus is not the only possibility of chord tesselations and another instructive example is given by the tiling of the projective plane P R 2 with Steiner triple system. Let X be the finite set {1, 2, .., n}. A Steiner system S(p, k, n) of order n is the set of k-subsets A in X called blocks such that every p-subset from X is contained in exactly one block of A (p < k). The Steiner triplet S(2, 3, 7) is the set of 7 triplets {1,2,4}, {2,3,5}, {3,4,6}, {4,5,7}, {1,5,6}, {2,6,7} and {1,3,7}. The unique Steiner triple system of order 7 is known as the Fano plane. The Steiner triple system S(2, s+ 1, s2 + s+ 1) is, for each integer s, a finite projective plane. Suppose now that we have a set of six musical objects, denoted by the first positive numbers X = {1, 2, ..., 6}. The set of triplets such that all 2-sets of numbers in X occur exactly in two triplets is not a Steiner triple system, but is related to S(2, 3, 7). It is a set of 10 triplets : T1 = {1, 2, 3}, T2 = {1, 3, 5}, T3 = {1, 5, 6}, T4 = {1, 2, 4}, T5 = {1, 4, 6}, T6 = {2, 3, 6}, T7 = {2, 4, 5}, T8 = {2, 5, 6}, T9 = {3, 4, 5}, T10 = {3, 4, 6}. As shown in figure 1, it is a triangulation of the projective plane. Two adjacent triplets have two points in common. Thus it is a way to move in a network of chord tesselations. Another example of chord tesselations is the Klein bottle, which has been studied by Robert Peck [8]. The direct sum Z12 = 6Z2 ⊕Z6 defines a tiling in which the horizontal axis is generated by the chain of fifths (T 7 : x → x + 7
1. INTRODUCTION Departing from the tesselations of pitch class sets, we examine the induced topological structures. Relationships between tilings and groups have been studied in [1] and [3]. The toroidal Tonnetz of the neo-Riemannian is one of the classical manifolds used in analytical approaches to tonality. Chromatic music based on triadic progressions is analyzed with Riemannian contextual transformations, even the tonality is not well established. In the post-tonal practice of the late nineteenth century, the music of Wagner or Liszt is directly mapped on a subnet of consonant triads. But the mapping is not always possible as in the case of triadic atonality. In order to get more flexible networks, we suggest to redefine the contextual transformations P, L, R, first introduced by Hugo Riemann (see [9]). We consider three new pointwise operations which are suitable to be used with any pitch class sets. From the algebraic point of view, they form a group which is isomorphic to a dihedral group, and then preserve pitch class sets.
3.1. Permutations
5 T2
3
1
T3
T1
6 T5
T4 T6
6
2
4 T7
T8
T10
T9
3
5 Figure 1. Tesselations of the Projective Plane mod 12) and the vertical axis by the tritones (T 6 : x → x + 6 mod 12).
3. NEO-RIEMANNIAN TRANSFORMATIONS Hyer [2] and Lewin [5] investigated a group of transformations on triads from relations based on Hugo Riemann’s work. The three principal transformations originate in Riemann, and are called Parallel, Relative and Leittonwechsel. The relationship between the major triad C major (C, E, G) and the minor triad C minor (C, Eb, G) is interpreted as parallel and is defined in neo-Riemannian theory as a transformation that maintains the two pitches that form a fifth, and moves the remaining pitch to form another triad. The relative transformation exchanges major triad and relative minor triad. It maintains the two pitches that form a major third, and moves the remaining pitch a whole tone so as to form another triad. The major triad C major (C, E, G) is sent to the relative minor triad A minor (A, C, E). The third transformation called Leittonwechsel connects a major triad with a minor triad, transposed by a major third up. The C major triad is transformed to the E minor (E, G, B) triad. The Leittonwechsel transformation is defined by Lewin, as a transformation on triads that maintains the two pitches that form a minor third and moves the remaining pitch one semitone so as to form another triad. These transformations act directly on consonant triads and could not be defined pointwise. For instance, the Leittonwechsel exchanges the C major triad [0,4,7] and the minor triad E minor [4, 7, 11], so the pitch 4 (E) is exchanged with the pitch 7 (G), but the same transformation exchanges the E major chord [4, 8,11] and the Ab minor triad [8, 11, 3], so the pitch {4} could not be transformed in the pitch {7}. These transformations are said to be contextual.
The purpose of this section is to redefine the neo-Riemannian contextual transformations by pointwise transformations that globally exchange the set of major triads into the set of minor triads. Each transformation P, L, R (Parallel, Leittonwechsel and Relative) is an involution (P 2 = L2 = R2 = 1, where 1 is the identity). They could also be interpreted as actions on bundles similar to the local symmetries that Mazzola used in the context of his couterpoint theory (see for example [6]). The contextual transformations transform only triads, while the pointwise transformations could be used to transform any k−chords into k−chords. The new pointwise transformations involve the following permutations given in the usual and in the cycle notation. The Parallel transformation is defined by the permutation in cycle notation P = (0, 7)(1, 6)(2, 5)(3, 4)(8, 11)(9, 10) It transforms a major triad based on the pitch n into a minor triad based on the pitch (12 − n) mod 12. This transformation is the inversion I 7 (x) = −x + 7 mod 12. The Relative transformation R = (0, 4)(1, 3)(5, 11)(6, 10)(7, 9)(2)(8) It exchanges a major triad based on the pitch n with a minor triad based on the pitch (9 − n) mod 12. This transformation is the inversion I 4 (x) = −x + 4 mod 12. The Leittonwechsel transformation L = (0, 11)(1, 10)(2, 9)(3, 8)(4, 7)(5, 6) transforms a major triad based on the pitch n into a minor triad based on the pitch (4 − n) mod 12. This transformation is the inversion I11 (x) = −x + 11 mod 12. 3.2. The Hexatonic Group The hexatonic group is the group generated by the transformations P and L. It has the presentation : H = P, L | P 2 = L2 = 1, P LP = LP L Let T0 be the identity of the group, the group H is composed of six elements H = {T0 , I7 , T4 , I3 , T8 , I11 } where In is the inversion of order n (x → −x+n mod 12) and Tn is the transposition of order n (x → x+n mod 12). The group is represented by a graph, in which the vertices are the elements of the group, drawing an hexagon and the edges are alternatively the transformations P and L. Let A be a pitch class set of k notes, the vertices of the graph of H represent the pitch class sets, given by the action of the group H on A. For example, the atonal triad A = [0, 6, 11] gives five others triads P (A) = [7, 8, 1], LP (A) = [4, 3, 10] (apply first the application P and then L), P LP (A) = [3, 4, 9], (LP )2 (A) = [8, 7, 2] and P (LP )2 (A) = [11, 0, 5]. Remark that all triads are the same pitch class set (3-5 in the Forte’s classification).
3.3. The Octatonic Group The Octatonic group is generated by the two permutations P and R. K = P, R | P 2 = R2 = 1, (RP )2 = (P R)2 The group has eight elements. With the previous notation of transpositions and inversions, we get K = {T0 , I7 , T9 , I10 , T6 , I1 , T3 , I4 } The graph is an octagon, whose edges are alternatively the transformations P and R. As in the case of the hexatonic group, to fill the graph with pitch class sets, one has to choose a vertex and to associate a pitch class set A to this vertex. By applying alternatively the P and R permutations on the set A, we describe all the pitch class sets associated with the vertices of the graph. For example, the interior octagon of the figure 2 is filled starting from the set A = {0, 1, 6}. Applying the R permutation, we get the set R(A) = {4, 3, 10} which is associated with the neighboring vertex. Recall that the main advantage of the pointwise transformation is that we could use a pitch class set of k notes. For example, the set A could be a five notes set, namely A = {0, 1, 6, 9, 11}. Thus the neghboring vertex will be associated with the pitch class set R(A) = {4, 3, 10, 7, 5}. 3.4. The PLR Group The group PLR is generated by the three permutations. It has been calculated by the GAP software (http://www.gapsystem.org), a free system for computational discrete algebra, especially computational group theory. It has the presentation P, L, R | P 2 = L2 = R2 = (P L)3 G= = (P RL)2 = (P R)4 = 1 Substituting P , L, R by the inversions I 7 , I4 , I11 , one verifies that G is the dihedral group. The elements of G are drawn on the figure 2. The interior octogone is rotated by 3π/4 and identified with the exterior octogone. The figure is then a torus on which three octagons are drawn in the meridian plane and four hexagons, based on the succession PLPLPLP, lie on the transversal planes of the torus. Because of the planar representation, these hexagons are not easily seen. Starting from a point situated on the interior octogone, the path PLPLPLP leads to a point situated on the exterior octogone which is exactly the same as the starting point. The identification of these two points yields the hexagonal figure. As G is the dihedral group, the figure is a tesselation of the transpositions and inversions of a pitch-class set. In the tonal music context, one chord, for example a major triad [C, E, G] (but it could also be a chord of four notes or more), is placed on a vertex of the figure and propagated with the P, L, R transformations. From the figure obtained, the consonant triads are connected with the transformation LR, when the base pitch moves a fifth up.
Figure 2. Atonal triads in the PLR group
LR
LR
LR
LR
D −−−−→ A −−−−→ E −−−−→ B −−−−→ F# We understand why isographies could easily be drawn with this representation. (For musical applications of isographies, see [4]). But this representation does not concern only consonant triads, and our scheme could be filled with any pitch class set, for example, with the atonal triad [0, 1, 6] (see figure 2). In the piano piece of George Crumb entitled Gargoyles (Makrocosmos vol. II), the right hand moves
PR
LR
PL
(P R)2
PR
[2, 3, 8] −−−−→ [5, 6, 11] −−−−→ [0, 1, 6] −−−−→ [3, 4, 9] −−−−→ [7, 8, 1] −−−−→ [1, 2, 7] in the same manner, the left hand moves : PR
LR
PL
(RP )2
PR
[1, 0, 7] −−−−→ [10, 3, 4] −−−−→ [11, 10, 5] −−−−→ [1, 2, 8] −−−−→ [0, 5, 6] −−−−→ [6, 11, 0] The same transformations are used in both hands, except in the last triads of our example, in which the reverse transformation is used (PRPR on the right hand, and RPRP on the left hand). In the last chord, the triads are played together. 4. MESSIAEN’S WORK In this final section, we present two simple examples of groups drawn upon Olivier Messiaen’s work. On the basis of suitable permutations, it seems that some groups are more adapted than others for the analysis of modal
or atonal pieces.In the PLR group, the graph is a mapping of all the transpositions and the inversions of a given pitch class set. Sometimes, it is convenient to get in a given group a transformation between two distinct pitch class sets.
5. CONCLUSION The neo-Riemannian transformations (Parallel, Relative and Leittonweschel) defined by Hyer and Lewin are contextual transformations, and concern only the transformations of major and minor triads. There are not one to one transformations of the twelve pitch classes. Consequently, they could not be extended to defined global transformations of any pitch class set. We propose a switch of the musical semantics by introducing one to one applications, which are very closed to the contextual transformations. We redefine Parallel, Relative and Leittonweschel transformations by introducing permutations which are pointwise transformations. Main properties are maintained, despite the change of perspectives. We recover the dihedral group which preserves pitch-class sets. We show that permutation groups are well adapted to describe isographies or chord tesselations in various contexts : tonal, posttonal or atonal music. The new transformations are used to build the isographic net of atonal triads in a piece of Georg Crumb. Two pieces of Messiaen show the analytical power of permutation groups. In short, deconstruction of the contextual transformations, substituted by pointwise transformations in the permutational context, reveals alternate paths and opens new perspectives
4.1. Transpositions of Mode 2 Our first example deals with the mode 2 of Messiaen (C, C#, Eb, E, F#, G, A, Bb). which is a limited transposition mode. Trying to find affine transformations (see [7]), we got two permutations closely related to mode 2. The following permutation a = (0, 1, 6, 7)(2, 11, 8, 5)(3, 4, 9, 10) is the affine transformation x → 5x + 1 mod 12 and the permutation b a = (0, 3, 6, 9)(1, 8, 7, 2)(4, 11, 10, 5) is the affine transformation x → 5x + 3 mod 12. The group M is generated by the two permutations and has the presentation M=
a, b | a4 = (a−1 ba−2 )3 = 1, aab = baa, bab−1 a3 = ab
6. REFERENCES [1] Andreatta, M. “On Group Theoretical Methods Applied to Music : Some Compositional and Implementational Aspects”, Perspectives in Mathematical and Computational Music Theory, Epos Music, Osnabr¨uck, 2004, 169-193.
The group has only twelve elements and appears to be well adapted to the three transpositions of mode 2. 4.2. Ile de feu and the Mathieu Groups
[2] Hyer, B. Tonal Intuitions in ’Tristan und Isolde’, Ph. D. diss. Yale University, 1989.
Ile de feu 2 is a piece for piano composed in 1949-50 with three other pieces in Quatre e´ tudes de rythme. It is based on various inversions of two series of twelve notes, which is interpreted by two permutations. The first permutation is in cycle notation
[3] Jedrzejewski, F. Math´ematiques des syst`emes acoustiques. Temp´eraments et mod`eles contemporains, L’Harmattan, Paris, 2002. [4] Klumpenhouwer, H. A Generalized Model of Voice Leading for Atonal Music, Ph. D. diss., Harvard University, 1991.
a = (0, 6, 9, 1, 5, 3, 4, 8, 10, 11)(2, 7) The second permutation is
[5] Lewin, D. Generalized Musical Intervals and Transformations, Yale University Press, New Haven, 1987.
b = (0, 5, 8, 1, 6, 2, 4, 3, 7, 9, 10)(11) The GAP software shows that the group generated by the two permutations a and b is one of the Mathieu groups. The group is the Mathieu group M 12 , and has the following presentation. M12 =
a, b | a10 = b11 = (ab−1 )2 = aba−2 b2 a−1 b−1 = bab−2 a−1 b2 ab−3 a = (a2 bab)3 = 1
This group has 95040 elements and, of course, only some of them are used by Messiaen. However, it induces a real potential of combinations, which maybe charges the music of inner perspectives.
[6] Mazzola, G. The Topos of Music, Birkh¨auser, Basel, 2002.
[7] Noll, Th., Brand, M. “Morphology of Chords”, Perspectives in Mathematical and Computational Music Theory, Epos Music, Osnabr¨uck, 2004, 366-398. [8] Peck, R. “Klein Bottle Tonnetze”, Music Theory Online, 9 (3), 2003. [9] Riemann, H. Skizze einer Neuen Methode der Harmonielehre, Breitkopf und Hartel, Leipzig, 1880.
