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Olivier Messiaen's permutations symétriques in theory and practice Sawatzky, Grant Michael 2013

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OLIVIER MESSIAEN’S PERMUTATIONS SYMÉTRIQUES IN THEORY AND PRACTICE by GRANT MICHAEL SAWATZKY  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Music)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  January 2013 © Grant Michael Sawatzky, 2013  Abstract This study begins by looking at the three compositional techniques that exemplify what Olivier Messiaen called the “charm of impossibility.” Messiaen sought this aesthetic desideratum by way of techniques that in some way impose a limit on the generation of new musical material, while also implying some kind of symmetric structure. In the first chapter, using precise, formal language, I describe each technique as the application of a particular function to the elements of a particular domain. This approach exposes the similarities (as well as certain differences) between these techniques. My explanation of the more wellknown techniques—modes of limited transposition, and non-retrogradable rhythms—is somewhat atypical, in that I define them in terms of functions, objects, and cycles/orbits; but this approach is purposeful, because it fosters a clearer understanding of the often misunderstood permutations symétriques. The second chapter focuses exclusively on symmetric permutations (SPs), surveying some existing explanations of SPs before considering the ways that their applications manifest symmetry. Further, it goes into considerable mathematical detail in order to relate Messiaen’s SP orbits (consisting of m different orderings of n elements) to the larger context of the symmetric group Sn. The third and final chapter turns to real musical examples, starting with serial procedures that are precursors to the SP technique proper, and then examining ways in which Messiaen used SP to manipulate pitch-class and/or rhythmic series in his compositions between 1950 and 1992. Finally, the conclusion outlines what it is that a rigorous theoretical approach to Messiaen’s music might contribute to the existing Messiaen literature—namely, that successful analysis of this kind can better inform some of the more speculative, philosophical lines of inquiry into Messiaen’s music, which often allude to the mathematical nature of his techniques. ii  Table of Contents Abstract ....................................................................................................................................................... ii Table of Contents .................................................................................................................................... iii List of Tables ............................................................................................................................................ iv List of Figures............................................................................................................................................ v List of Examples ...................................................................................................................................... vi Acknowledgments ................................................................................................................................. vii Dedication .............................................................................................................................................. viii Chapter 1: A formalism for Messiaen’s “charm of impossibilities” ....................................... 1 1.0 – Introduction ............................................................................................................................... 1 1.1 – Modes of limited transposition ........................................................................................... 4 1.2 – Non-retrogradable rhythms ...............................................................................................11 1.3 – Symmetrical permutations .................................................................................................17 Chapter 2: Permutations symétriques in theory .........................................................................20 2.0 – Some existing descriptions of Messiaen’s SPs..............................................................20 2.1 – Permutation schemes ...........................................................................................................30 2.2 – Contextualizing orbit cardinalities with regard to Sn................................................46 Chapter 3: Permutations symétriques in practice ......................................................................61 3.0 – Hearing interversions: introduction to the musical examples ..............................61 3.1 – Musical examples: Messiaen’s “experimental period” (1949–1951) ..................66 3.2 – Musical examples: Catalogue d’oiseaux (1958) and after ..................................... 100 3.3 – Conclusion.............................................................................................................................. 148 Works Cited .......................................................................................................................................... 152  iii  List of Tables Table 1: Exhaustive list of TINV pitch class sets (adapted from Cohn 1991) ....................................................... 8 Table 2: Summary of “Two-by-two” family of permutations .................................................................................... 37 Table 3: “Two-by-two” family of permutations (part 1 of 4) .................................................................................... 39 Table 4: “Two-by-two” family of permutations (part 2 of 4) .................................................................................... 40 Table 5: “Two-by-two” family of permutations (part 3 of 4) .................................................................................... 41 Table 6: “Two-by-two” family of permutations (part 4 of 4) .................................................................................... 42 Table 7: Largest possible orbit defined by a permutation on n (A000793) ....................................................... 47 Table 8: Permutation types in S12.......................................................................................................................................... 51 Table 9: Summary of orbits in S12 ......................................................................................................................................... 52 Table 10: Summary of orbits in S32 ....................................................................................................................................... 52 Table 11: Twenty most common orbit cardinalities according to percentage (among all permutation types) of types whose functions are of a given period ...................................................................................... 53 Table 12: Twenty most common orbit cardinalities according to percentage (among all f in Sn) of functions with of a given period ................................................................................................................................. 54 Table 13: Ten most common permutation types among types whose period is 36 ....................................... 54 Table 14: Orderings of durations {2,3,5,7,11,13,17,19,23,29,31} in subsections 1-A through 24-A of Timbres-durées .................................................................................................................................................................... 71 Table 15: six orderings of 12 durations/PCs of the Pedal in Les yeux dans les roués ..................................... 74 Table 16: Superposed PC interversions and their resulting interval classes (Île de feu II) ......................... 91 Table 17: Interval class- and Rhythmic profiles of superposed interversions from Île de feu II .............. 95 Table 18: Statistics on “personnages rhythmiques” episode in Le Chocard des Alpes .................................. 104 Table 19: Statistics on “Mode de valeurs” episodes in La Chouette Hulotte ..................................................... 107 Table 20: Matrices for Catalogue d’oiseaux ‘s SP orbit of pc interversions...................................................... 108 Table 21: Superposed pc interversions and resultant ics in La Courlis Cendré ............................................. 113 Table 22: Superposed pc interversions and resultant ics in La Buse Variable ............................................... 114 Table 23: Usages of interversions from Chronochromie SP orbit ........................................................................ 115 Table 24: Matrices for Chronochromie SP orbit of IOI interversions.................................................................. 121 Table 25: Statistics on composite rhythms from superposed interversions of the Chronochromie SP orbit ...................................................................................................................................................................................... 125  iv  List of Figures Figure 1: "Rotation" (transposition) and "Mirror" (inversion) Symmetries of MLTs ................................... 10 Figure 2: Messiaen’s SP demonstration in TRCO-III (annotated) ........................................................................... 23 Figure 3: Sherlaw-Johnson’s figure of alleged SPs in Mode de valeurs et d’intensités .................................... 28 Figure 4: A different representation of Sherlaw-Johnson’s contour inversion symmetries in Mode de valeurs et d’intensités........................................................................................................................................................ 28 Figure 5: Functions of the Chronochromie permutation type (λ = 18, 6, 4, 3, 1) and other permutations with a period of 36 amongst S32 .................................................................................................................................. 56 Figure 6: Distribution of 248 unique orbit sizes among the 8439 permutation types.................................. 57 Figure 7: Distribution of functions among 248 unique orbit sizes ........................................................................ 57 Figure 8: Network of permutation functions among sections in Reprises par interversion ........................ 67 Figure 9: Graphic Score for Reprises par interversion – each section as a (re)ordering of the 54 elements of section 1........................................................................................................................................................ 70 Figure 10: Network of permutation functions among six different orderings (a-f) internal to Subsection A in Timbres-durées .......................................................................................................................................... 72 Figure 11: Network of permutation functions among six different orderings of Pedal material in Les yeux dans les roués ............................................................................................................................................................. 74 Figure 12: Orderings of 64 IOIs in Soixant quatre durées as members of SP orbits ....................................... 79 Figure 13: Two large-scale rhythms, and their non-retrogradable composite rhythm from Soixante Quatre Durées ...................................................................................................................................................................... 80 Figure 14: 24 orderings of four durations in Le merle noir ....................................................................................... 85 Figure 15: 12 unique composite rhythms obtained by superposition in Le merle Noir ............................... 87 Figure 16: Matrix for SP of 12 pitch classes in Île de feu II, mm. 70–75 ............................................................... 88 Figure 17: Matrices representing SP of series of durations in Île de feu II.......................................................... 93 Figure 18: Superposition and composite rhythms in Île de feu II ........................................................................... 94 Figure 19: Personnages Rythmiques and composite rhythms in Le Chocard des Alpes ............................... 104 Figure 20: graphic score – “mode de valeurs” episodes in La Chouette Hulotte ............................................. 106 Figure 21: Chronochromie SP orbit matrices to scale (duration proportional to length) ......................... 122 Figure 22: Superposition and composite rhythms of Chronochromie SP orbit.............................................. 123 Figure 23: Twelve composite rhythms of Chronochromie SP orbit ..................................................................... 124 Figure 24: Graphs – Number of attacks per (scrolling) timespan in composite rhythms 1–12 ............. 133 Figure 25: Graph – Number of attacks per (scrolling) timespan in composite rhythms 1, 6, 7, & 8 .... 136 Figure 26: Graphic score – Gagaku .................................................................................................................................... 142 Figure 27: Graph – Attacks per (scrolling) timespan in Gagaku, with formal boundaries ....................... 143 Figure 28: Fragments of interversions 31-33 in Saint François d’Assise........................................................... 147  v  List of Examples Example 1: Inversion Symmetry in Mode de valeurs et d’intensités ....................................................................... 26 Example 2: Open-fan permutation on invented tone row resulting in T11 transposition .......................... 44 Example 3: “Two-by-two, even↓, odd↑” permutation resulting in similar rhythm .................................... 44 Example 4: Livre d’Orgue – I. Reprises par interversion ............................................................................................... 69 Example 5: Livre d’Orgue – VI. Les yeux dans les roués ................................................................................................. 75 Example 6: Livre d’Orgue – VII. Soixante quatre durées .............................................................................................. 78 Example 7: Le merle noire (Coda) ......................................................................................................................................... 83 Example 8: Île de Feu II – interversions of 12 pitch-classes (mm. 70–75) .......................................................... 90 Example 9: Île de Feu II – SP of of 12 pitch-classes and durations (Interversions 1–4) ............................... 92 Example 10: Catalogue d’oiseaux (première livre) – I. Le Chocard des Alpes .................................................. 103 Example 11: Catalogue d’oiseaux (troisième livre) – V. La Chouette Hulotte .................................................. 105 Example 12: Catalogue d’oiseaux (septième livre) – XIII. La Courlis Cendré ................................................... 111 Example 13: Catalogue d’oiseaux (septième livre) – XI. La Buse Variable ........................................................ 112 Example 14: Chronochromie – II. Strophe I .................................................................................................................... 137 Example 15: Chronochromie – IV. Strophe II ................................................................................................................. 138 Example 16: Éclairs sur l’au delà… – IV. Les élus marqués du sceau .................................................................... 139 Example 17: Sept Haïkaï – II. Le parc de Nara et les lanterns de pierre .............................................................. 140 Example 18: Sept Haïkaï – IV. Gagaku .............................................................................................................................. 141 Example 19: Saint François d’Assise – Acte I, 1er tableau: La croix ....................................................................... 144 Example 20: Saint François d’Assise – Acte I, 3e tableau: Le baiser au lépreux................................................ 146  vi  Acknowledgments First, I would like to thank the Social Science and Humanities Research Council of Canada for generously providing financial support for this thesis project. I also thank Éditions Musicales Alphonse Leduc for granting permission to reproduce excerpts of Messiaen’s Le merle noir (1952), Livre d’orgue (1953), Catalogue d’oiseaux (1964), Chronochromie (1963), Sept Haïkaï (1966), Éclairs sur l’au-delà… (1998), and Saint François d’Assise (1992 & 2007). I am grateful to John Roeder, an extremely knowledgeable and helpful advisor who provided invaluable guidance and feedback on my research, even from the earliest stages of the project. I am also grateful to Richard Kurth for his encouragement, and for his insightful comments, suggestions; and thoughtful advice—musical or otherwise. Further, I would like to acknowledge and thank William Benjamin, Gyula Csapó, Edward Gollin, Gregory Marion, and Simon Rose, all of whom helped this project take shape in one way or another. I would also like to thank Markus Bandur, Marc Battier, and Julian Hook for their instructive, and kind responses to my email queries. Finally, heartfelt thanks to my wife Elizabeth for being ever patient and supportive while I worked on this project, and to my loving family for their encouragement.  vii  Dedication  Dedication  for Hans Georg Epp  Chapter 1: A formalism for Messiaen’s “charm of impossibilities” ...the imagination can function only spatially; without space the imagination is like a child who wants to build a palace and has no blocks. —Czesław Miłosz, If only this could be said  1.0 – Introduction Messiaen opens his 1944 treatise Technique de mon langage musical (TMLM) with a short chapter on his compositional techniques that demonstrate “the charm of impossibilities,” explaining: “This charm, at once voluptuous and contemplative, resides particularly in certain mathematical impossibilities of the modal and rhythmic domains” (Messiaen TMLM, 13).1 He goes on to show that the modes of limited transposition” (modes à transpositions limitées, hereafter MLT) and “non-retrogradable rhythms” (rythmes non rétrogradables, hereafter NRR) are two techniques that exemplify this quality, each in their own way (Messiaen TMLM, chapters XVI and V respectively). In the third tome of Traité de rythme, de couleur, et d’ornithologie, (hereafter TRCO) Messiaen reflects on his earlier writing, adding “symmetrical permutations” (permutations symétriques, hereafter SP) to the list, giving him a trinity of techniques that evidence the charm of impossibilities.2 He also gives account (in more detail than in TMLM) of exactly how each technique demonstrates this charm: The first chapter of my first book [TMLM] bore the title "[The] Charm of Impossibilities." If we take the word charm in its literal, etymological sense as ‘magic spell’, it means that certain mathematical impossibilities, certain closed circuits, possess a bewitching power, a magic force, a charm. This is English quotations from TMLM are from Satterfield’s translation, (Messiaen, 1944 [1956]); while translations of passages from Traité de rythme, de couleur, et d’ornithologie are my own. 2 Messiaen uses his term interversion to label a particular ordering of permuted elements—and uses permutation to refer to his technique. “Symmetrical permutation” has become the standard English term (though symmetric permutation is better, in my opinion). Interversion has generally been retained in English translations, though it has occasionally been translated as interchange or, misleadingly, inversion. I will adopt interversion to mean an ordered series, and reserve permutation to mean the action of reordering the elements in the series. 1  1  the case for the Modes of Limited Transposition, which cannot be transposed beyond a certain number of transpositions without falling onto the same notes, because they contain small transpositions within themselves. This is the case for the non-retrogradable rhythms, which cannot be read in retrograde order without yielding exactly the same order of durations as when read forwards, because they contain in themselves small retrogradations. Finally, it is the case for symmetrical permutations, which— as soon as a small number of interversions are produced—stop when they stumble onto the initial chromatic [i.e. ordering], recapturing the original [interversion]. This happens because we always read them in the same order.3 (Messiaen, TRCO-III, 7)  More recent commentators who have committed themselves to the task of unpacking Messiaen’s often enigmatic and symbolism-laden explanations of his own technical musical processes are in general agreement that the “certain mathematical impossibilities” common to MLTs, NRRs, and SPs are manifest in either some kind of symmetry, or in the imposing of some kind of limitation on the generation of musical material, or both.4 Roberto Fabbi, for instance, contends that each of the techniques evincing Messiaen’s “impossible charm” possess “specific internal constitutions [that] impose a mathematical limit on duration and/or pitch, forcing certain behavior patterns” (Fabbi 1998, 58), while Jean Marie Wu tells us that “the musical impossibilities are… established by creating structural symmetries in his musical language” (Wu 1998, 85). These observations do not go much further than Messiaen’s own perfunctory explanations in TRCO.  “Le premier chapitre de mon premier livre avait pour titre: « Charme des impossibilités ». Si l’on prend le mot charme dans son sens étymologique, réel: enchantement magique – cela veut dire que certaines impossibilités mathématiques, certains circuits fermés, possèdent une puissance d’envoûtement, une force magique, un charme. C’est le cas pour les « Modes à transpositions limitées », qui ne peuvent se transposer au-delà d’un certain nombre de transpositions sous peine de retomber dans les mêmes notes, parce qu’ils contiennent en eux-mêmes de petites transpositions. C’est le cas pour les « rythmes non rétrogradables », qui ne peuvent être lus en sens rétrograde sans que l’on retrouve exactement le même ordre de valeurs que dans le sens droit, parce qu’ils contiennent en eux-mêmes de petites rétrogradations. C’est le cas enfin pour les « permutations symétriques » qui s’arrêtent au bout d’un petit nombre d’interversions en butant sur un chromatisme de départ tôt retrouvé, parce qu’on les a lues toujours dans le même ordre de lecture.” (Messiaen, TRCO-III, 7) 4 e.g. Bauer 2007, 146; Cheong 2007, 111; Fabbi 1998, 58; Johnson 1975, 108-109; Wu 1998, 113. 3  2  There do exist formal treatments of some of Messiaen’s techniques, notably, Julian Hook’s (1998) algebraic study of NRRs, “rhythmic characters,” and composite rhythms in the Turangalîla Symphony; Eleanor Trawick’s (1991) treatment of permutation of tone rows in Livre d’Orgue; Christoph Neidhöfer’s (2005) theory of harmony and voice leading for Messiean’s music; and Franck Jedrzejewski’s (2006, 60ff.) mathematical account of Messiaen’s MLTs (in the more general context of Jedrzejewski’s “Limited Transposition Sets”).5 The present study will demonstrate that precise mathematical definitions of each of the three techniques (MLTs, NRRs, and SPs) can help clarify just how each of these very different musical processes embody the “charm of impossibilities,” that is, limitation and symmetry. In the case of MLTs and NRRs, these definitions are straightforward. But SPs have not been so well studied, and it takes special formalism to relate all three techniques. The first two chapters develop this formalism, which might be summarized by saying: the extent to which each technique does (or does not) exhibit symmetry and does (or does not) impose a limitation can be made explicit by considering it as a particular function acting on a particular domain. Toward the end of chapter 2 the thesis moves from the theoretical account of SPs toward the consideration of Messiaen’s applications of the technique, including questions of which attributes of the SP technique might be perceptible to the listener, and how it is that an understanding of SPs can meaningfully contribute to the analysis of pieces that feature the technique.  Jedrzejewski’s Limited Transposition Sets are Richard Cohn’s Transpositionally Invariant (TINV) sets (Cohn, 1988, 1991). I will adopt Cohn’s nomenclature hereafter. 5  3  1.1 – Modes of limited transposition The function associated with MLTs is, as the name suggests, pitch-class transposition. As we have seen, Messiaen explains the transpositional invariance of his modes as the result of the modes “containing small transpositions within themselves.” Present-day analysts such as Fabbi (1998, 59) and Simpson-Litke (2010, 10) have explained MLTs in much the same way, MLTs being composed by dividing the octave into equal parts (whether 2, 3, 4 or 6), each segment being “filled in” with the same ordered intervallic pattern.6 While there is nothing wrong with this explanation of MLTs, a contrasting explanation (adapting and elaborating Starr 1978) will better suit the purposes of this project. Let f be the function that is pitch-class-set transposition by 1 semitone. Then the twelve discrete transpositions can be conceived as powers of f: f 2 is transposition by 2; f 3 is transposition by 3, and so on.7 (For the remainder of section 1.1 f indicates pc transposition T1, but later on, f will represent various permutation functions.) When one of these powers of f, f n for 0 ≤ n ≤ 11 is repeatedly applied to a “seed” pitch-class set A whose cardinality is 1 (that is, a set containing a single pc), it generates sets that are transpositions of A: f n(A), f 2n(A), f 3n(A),…, f mn(A), and so on. At a certain point in this iteration, when mn = n mod 12, f mn(A) = f n(A) and the series of sets becomes a repeating cycle, which (following standard mathematical terminology) we will call an orbit. For example, Let A = {0} and let n = 4. Then f n defines an orbit whose cardinality is 3:  This conception essentially explains Messiaen’s MLT’s as “transpositional combinations” of pc-sets (Cohn, 1988, 23). 7 We can say that f n is transposition by n mod 12. Because we are dealing with pitch-classes, all equalities cited in the context of MLTs and transpositionally invariant pc-sets are mod 12. 6  4  The union of the elements within an orbit defined by f n applied to a distinct monad {i} constitutes a pc set which we will refer to as Cni.8 The cardinality of an orbit (or alternatively, the cardinality of Cni, symbolized |Cni|) is  for n > 6 and n = 0, otherwise  . When n (or  12 – n for n > 6) and 12 are relatively prime, f n generates the aggregate. (We are interested in the size of the orbit because it is an indication of how many unique transpositions exist for a given MLT; later on, orbit sizes will be pertinent to the discussion of Messiaen’s SPs). For example: Let A = {0}, then f 0(A) generates {0}{0}, whose union is the set C00 = {0} f 1(A) generates the orbit {0}{1}{2}{3}{4}{5}{6}{7}{8}{9}{A}{B}{0}, whose union is the set C10 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}. f 2(A) generates the orbit {0}{2}{4}{6}{8}{A}{0}, whose union is the set C20 = {0, 2, 4, 6, 8, A}. f 3(A) generates the orbit {0}{3}{6}{9}{0}, whose union is the set C30 = {0, 3, 6, 9}. f 4(A) generates the orbit {0}{4}{8}{0}  , whose union is the set C40 = {0, 4, 8}.  f 5(A) generates the orbit {0}{1}{2}{3}{4}{5}{6}{7}{8}{9}{A}{B}{0}, whose union is the set C50 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B} f 6(A) generates the orbit {0}{6}{0} whose union is the set C60 = {0, 6} . Every set Cni is transpositionally invariant under f n (and f 12–n). Now let us consider using one of these Cni as a seed set itself. As above, when a power of f, f k for 0 ≤ k ≤ 12, is repeatedly applied to a seed pitch-class set Cni, it generates a sequence of pitch-class sets that are transpositions of Cni: f k(Cni), f 2k(Cni), f 3k(Cni),…, f mk(Cni), The label Cn was coined by George Perle (1984, 199) to signify sets whose elements are generated by a single function applied to a seed pitch-class. Cohn (1991) adopts this nomenclature in his discussion of transpositional invariance. 8  5  et cetera. At a certain point in this iteration, when mk = k mod 12, f mk(Cni) = f k(Cni) and the sequence of sets becomes an orbit. Each element in one of these orbits is a unique transposition of the set-class Cni. For example, consider the orbit of pc-sets defined by f 2 applied to the seed C30. C30 = {0, 3, 6, 9} f 2(C30) = {2, 5, 8, 11} f 2(f 2(C30)) = f 4(C30) = {4, 7, 10, 1} f 2 (f 2(f 2(C30))) = f 6(C30) = {6,9,0,3} = {0, 3, 6, 9} = C30 As a set class under transposition, Cn contains |  |  distinct members, each the union of  sets generated by applying f n repeatedly to a distinct monad {i}. Transposing any Cni repeatedly by f k produces an orbit of sets. When k and n are relatively prime, the sets in the f k orbit exhaust all possible transpositions of Cni, and equally, the aggregate. Daniel Starr’s Invariance Theorem states: (1) that if a pitch-class S is a Cni or if it is the union of a Cni and one or more of transpositions of Cni by k, then S must be transpositionally invariant under f n. The converse is also true. If a set is invariant under f n, then S must be either a Cni or the union of Cni and one or more of its transpositions by k: for if S contained only part of a certain Cni, the application of f n to S would generate at least one of the pcs missing from the incomplete cycle, and S would not be invariant under f n. Therefore (2) if S is invariant, S is the union of complete Cnis. In combining (1) and (2) we conclude that S is transpositionally invariant if and only if S is a Cni or the union of Cni and one or more of its transpositions by k.9  The theorem as it appears above is a direct paraphrase from Starr 1978, 17–18, replacing some of Starr’s terminology with terms used in this thesis. Here is Starr’s original formulation: “(1) if S comprises an F-cycle or a union of intact F-cycles, then S must be F-invariant. We will also show that the converse is true. If any set S which [sic] is invariant under a TTO F, S must be the union of one or more intact F-cycles, for if S contained only part of a certain F-cycle, the application of F to S would generate at least one of the PCs missing from the incomplete cycle, and S would not be Finvariant. Therefore (2) if S is F-invariant, S is the union of complete F-cycles. In combining (1) and (2) we conclude that S is F-invariant if and only if S is the union of complete F-cycles.” 9  6  Hereafter, pitch-class sets with this property (transpositional invariance) will be referred to as TINV sets (after Cohn 1991). C1 and C5 are trivial cases, since the aggregate is invariant under any transposition. C0 is another trivial case, since every pc set is invariant under the identity f 0. But equipped with Starr’s Invariance Theorem, one can quickly exhaust the list of all possible TINV sets by considering all the possible unions of Cni and transpositions of Cni for any given n. C2i, the whole-tone scale, is Messiaen’s mode 1—the only Cni that he calls an MLT in and of itself. Composing C2i  f 1(Ci) produces the aggregate, so no new TINV set is obtained. A set C3i can be composed with either f 1 or f 2 of itself to produce an octatonic collection—Messiaen’s mode 2. Continuing in this manner, the cumulative list of TINV (transpositional) set classes is obtained. Table 1 (below) lists every TINV pc-set, and is organized in ascending order of n.  7  Table 1: Exhaustive list of TINV pitch class sets (adapted from Cohn 1991)  8  We take the domain S of the transposition function f to be the set of all possible pitch-class sets (including the null set) with 12 or fewer elements, whose cardinality is 212 = 4096. Oftentimes, for analytic purposes, these 4096 are reduced to the 224 set classes under transposition and inversion. The list might be “reduced” in a different way to emphasize an aspect of S that is more germane to the discussion of MLTs and TINVs. The function f 1, transposition by 1, when applied to every element in S partitions S exhaustively into 351 disjunct orbits (provided the distinction between inversionally related pc-sets is maintained). These 351 pitch-class set orbits are the 351 transpositional set classes—each element of each orbit being a member of the same set-class. Of these, 335 orbits have a cardinality of 12 (that is, there are 335 transpositional set-classes that contain 12 distinct pc sets). The remaining orbits are the 17 TINV set classes: 9 orbits of cardinality 6, 3 orbits of cardinality 4, 2 orbits of cardinality 3, and 2 orbits of cardinality 1. We have already seen this structure in table 1 above—the orbit cardinalities being reflected in the rightmost column, labeled “# distinct pc sets” per set class. This is not the typical way of conceptualizing transpositional invariance, but it will help to clarify their connection to Messiaen’s SPs. Presumably Messiaen considered the TINV pc-sets with fewer than six notes as too small to constitute a “mode”, and so did not include them in his list of MLTs. The only remaining TINV set class (other than the aggregate) is [013679]. It has the same transpositionally invariant property as Messiaen’s MLTs, but was nevertheless not included in Messiaen’s list of them, and it bears the distinction of being the only TINV set class that is not also inversionally invariant (Cohn 1991, 14; Simpson-Litke 2010, 10). When Messiaen’s modes are displayed on circular pitch-class diagrams, as in figure 1, below, transpositional invariance manifests as rotational symmetry. All but [013679] exhibit mirror symmetry on various axes as well, manifesting their inversional symmetry. Messiaen’s MLTs then, can be  9  formally described as the subset of TINV sets that are also inversionally invariant, and whose cardinality is greater than 4 and less than 12. The property of mirror symmetry helps link Messiaen’s MLTs to other “charming” features, as will be explained below.  Figure 1: "Rotation" (transposition) and "Mirror" (inversion) Symmetries of MLTs  10  1.2 – Non-retrogradable rhythms Non-retrogradable rhythms (hereafter NRR) are likewise affiliated with a certain function, retrograde. This function has a different domain, and a different effect than the transposition function I called f in the previous section, which acted on sets or sequences of pitch classes. Retrograde is a permutation, which, strictly speaking, acts upon neither pitches nor durations, but on the positions in a sequence. Any rhythm, for instance, may be regarded as a sequence in which the content of each position in the sequence is a particular duration or interonset interval (IOI). Symbolically the series (1, 2, 3, 4, 5, … , n) can represent the successive individual durations that compose a rhythm if we take the integers 1 through n to represent the positions in the sequence, each of which (in this example) corresponds to a duration or IOI. Nota bene, in this thesis, the context of ordered sets or sequences, integers between round brackets will always refer to the positions themselves, not to the durations associated with those positions. When integers are meant to represent the durations of positions in a sequence, the integers will appear within angle brackets. For example, consider the integer sequence <3,1,2,2,4,4>. If these integers within angle brackets are interpreted as durations measured in sixteenth-notes, then the rhythm would be transcribed as:  .  For our purposes, permutation functions are best represented in cycle notation wherein “we map each element to the one on its right, except for the elements that are last within their parentheses. Those are mapped to the first element within their parentheses” (Bóna 2004, 74). For example, consider a sequence A = <2,3,4,5>. Applying the permutation f = (1 4)(2 3), maps the content in first position to fourth position and vice versa, and maps the content of the second position to third position and vice versa. That is: f (A) = <5,4,3,2>, the retrograde of A.  11  There are n! different permutation functions of n elements, one of which is the retrograde function. These functions form a group (in the formal algebraic sense) called the symmetric group Sn (Fraleigh 1999, 96).10 (Later on we will consider this group in further detail in the context of SPs.) Likewise, there are n! different orderings of n things. This group is called ORDn. These groups are, in fact, two different ways of describing one and the same structure, but the distinction between function and the results of applying those functions proves helpful for explaining NRRs and SPs. Just as we applied transposition to a seed pc set, we can apply a permutation function to a “seed” sequence  . When such an f is repeatedly applied to A, it generates  an orbit of sequences that are permutations of A: f 0(A), f 1(A), f 2(A), f 3(A),… etc.. At a certain point in this iteration f m(A) = (A) and the sequence of sets becomes an orbit. This occurs when m is the least common multiple (LCM) of the discrete cycle lengths of f expressed in cycle notation, Each element in such an orbit is a unique ordering of the seed sequence A = <a1, a2, a3, …, an>, so long as in A, no ai = aj for i ≠ j. When a function recursively applied to a seed series generates an orbit with a cardinality m we say that the function is of order m, or that the function has a period of m. For example, consider the orbit of orderings of 4 elements defined by f = (1)(2 3 4) applied to the seed A = <2,3,4,5>. Since the lengths of the discrete cycles of f are 1 and 3, f defines an orbit whose cardinality is equal to LCM(1, 3) = 3. Accordingly, f 3 = f 0, the identity function, as shown by this demonstration: A = <2,3,4,5> f (A) = <2,5,3,4> f (f (A)) = f 2(A) = <2,4,5,3> f (f (f (A))) = f 3(A) = <2,3,4,5> = f 3(A) = A  “Definition (Symmetric Group): Let A be the finite set {1,2,3,…,n}. The group of all permutations of A is the symmetric group on n letters, and is denoted by Sn. Note that Sn has n! elements, where n! = n(n – 1)(n – 2)…(3)(2)(1)” (Fraleigh 1999, 96). 10  12  Let us now consider a special permutation, the retrograde function. It simply exchanges the contents of the first position with the contents of the last position, the contents of second position with the contents of the second-to-last position, and so on. Symbolically, it is expressed in cycle notation thusly: where n is odd, and where n is even,  (  )(  )( (  )(  )( )(  ) )(  (  ) )  (  ).  In either case, the LCM of the cycle lengths is 2, so the retrograde defines an orbit of 2 elements. This is to say the retrograde function is always an involution, that is, a function that is its own inverse: apply it once, obtain a new ordering; apply it again, obtain the ordering you started with. Thus for most sequences the retrograde function is of order 2. But when it is applied to a special sequence ANRR wherein the value in the first position is equal to the value in the last position, the value in the second position is equal to the value in the second-to-last position, and so on, the retrograde of the rhythm is indistinguishable from the original, and so the retrograde function is indistinguishable from the identity function. In this sense, an artificial limit is imposed on the retrograde function by a special property of the series to which it is applied. Non-retrogradable rhythms figuratively evidence mirror symmetry in the sense that retrograde function exchanges the values in corresponding positions around a central axis. We can say a few things about the special properties necessary for producing NRRs. First of all, it must be the case that (except for one-duration NRRs) some of the durations in an NRR must be equal, or more accurately, for each duration in the sequence (excepting the middle duration if the number of positions is odd), there must be a corresponding position of the same duration. If we think of a rhythm as drawing its durations from every member of a source set, then the equality of some durations demands that the source be a multiset, which is an unordered collection wherein some elements are not unique, and where 13  “multiplicity is explicitly significant” (Weisstein, N.D.). For any multiset M = {m1, m2, m3, …, mn} where |M| = n, and k is the number of unique elements in M, M is the union of k subsets, each of the form λi = {mi1, mi2, …, mij} where mi1 = mi2 = mij, and the cardinality of each λi , |λi|, is >0. Below, I will write |λi| as CARDi. When the n elements of a sequence A are all distinct, then the number of distinct orderings of A, |ORDn| = n!. But when the n elements (from which A is composed) constitute a multiset, then |  |  (  )(  )  (  )  . (Note that, regardless of whether or not  the resource set for a series A is a multiset, |Sn| = n!. That is, there are always n! functions that rearrange n elements, even if the performing every of those n! rearrangements yields fewer than n! distinguishable orderings.) For a multiset M to afford one or more NRRs: if |M| is odd, then CARDi, the cardinality of each equal-member subset, must be even for all but one i (1 ≤ i ≤ k) (equivalently, CARDi must be odd for only one i). if |M| is even, then CARDi, the cardinality of each equal-member subset, must be even for all i (1 ≤ i ≤ k) (equivalently, no CARDi is odd). We will use the label MNRR to denote any multiset that fulfills these criteria. The number of possible MNRRs of a given cardinality n is the number of partitions of n with at most one odd part (this number is found in the nth position of Sloane’s integer sequence A100824 (Jovovic 2005)). For example: Let n = 6, then the number of partitions with at most one odd part is 3, the 3 partitions being 6, 4+2, and 2+2+2 (each having no odd parts, in this case). Respectively, these correspond to MNRR1 = {a,a,a,a,a,a}, MNRR2 = {a,a,a,a,b,b}, and MNRR3 = {a,a,b,b,c,c}. For n = 5, there are 4 MNRRs: they are the partitions 5, 4+1, 3+2, and 2+2+1, corresponding to MNRR1 = {a,a,a,a,a}, MNRR2 = {a,a,a,a,b}, MNRR3 = {a,a,a,b,b}, and MNRR4 = {a,a,b,b,c}. Every MNRR affords a certain number of NRR-types. For example, MNRR = {a,a,a,b,b} affords two NRR-types: <a,b,a,b,a> and <b,a,a,a,b>. Every  14  individual NRR-type, of course, itself describes an infinite number of NRRs, since the durational values a, b, c, ..., n can be any n unique rational numbers > 0. The number of NRR-types for all sets whose cardinality is n is given by the nth position in Sloane’s integer sequence A152536 (Jovovic 2008).11 Specifically, the number of NRRs, x, afforded by a given MNRR is determined as follows. Where n is even  and where n (and CARD1) are odd  ( ) (  )(  )( (  (  )(  )  (  )  ,  ) )(  )  (  )  .  This is to say, the n elements of an MNRR can be arranged into x different ordered series ANRR such that the first and the nth elements are equal, the second and the (n-1)th are equal, the third and (n-2)th are equal, etc.. In such cases—as previously mentioned—the retrograde function f = (1 n)(2, n–1)(3 n–2)... produces a result indistinguishable from the application of the identity function to the same seed series A. That is, f (ANRR) = ANRR. Having thus formalized two of the three techniques that evidence the “charm of impossibilities,” we can now assess what is so “impossible” about them. In the case of MLTs, we found that the function f 1 (i.e. transposition by 1 semitone) acts on the domain of all 212 possible subsets of {0123456789AB}. The f function parses this group into 351 orbits, each comprising a distinct transpositional set class. Most of these orbits contain 12 elements. But when acting on the 17 elements of the TINV subset of S, the f function behaves atypically, defining orbits of 6 or fewer elements. Analogously, the retrograde function f parses the group ORDn of every possible n-sequence into  inversely related pairs  An integer sequence is a function that maps elements from the set of non-negative natural numbers to the set of integers, symbolically, g : N0 → Z. If g(n) = an for each a ∈ Z, then g ={(0, a1), (1, a2), (2, a3), …}. So the integer sequence defined by g is a1, a2, a3, … and we call an the nth term of the sequence. (This definition of sequence adapted from Chartrand, Polimeni & Zhang 2008, 267). 11  15  (that is,  orbits of 2 elements each). But when the contents of positions 1 through n are  chosen from a special n-element multiset MNRR , the paired sequences internal to a few among the  orbits within Ordn become indistinguishable from one another—and thus (in  Messiaen’s terms) a few of the orbits are “short-circuited.” Commenting on the “charm of impossibilities” as evidenced in the two techniques reviewed thus far, Messiaen says: Immediately one notices the analogy of these two impossibilities and how they complement one another, the rhythms realizing in the horizontal direction (retrogradation) what the modes realize in the vertical direction (transposition). (Messiaen 1944, 13) Strictly speaking, this statement isn’t quite true, since the mirror symmetry of palindrome rhythms, if adapted to the “vertical” domain of pitches would bring about inversional symmetry. And although each of Messiaen’s modes exhibits mirror symmetries—the reason for Messiaen’s confusion, perhaps—the germane symmetry-metaphor for transposition is rotational symmetry. More strictly, however, we can say that the affinity between the two techniques is this: that both MLTs and NRRs involve special subdomains, that when acted upon by a “run-of-the-mill” function, will afford fewer unique results than is typical of that function.  16  1.3 – Symmetrical permutations Permutation symétrique is the remaining compositional device exemplifying Messiaen’s “charm of impossibility,” and is the least understood among them. The technique—though it does not appear as often as NRRs and MLTs—is incorporated into compositions from a significant span of Messiaen’s long career: from as early as Île de feu II (1950), to as late as Eclairs sur l’au-delà… (1992), completed in the year of his death. As was the case for NRRs, the pertinent function with regard to SPs is a permutation function, which acts upon the order positions of a given series. It is often unclear in both recent publications and Messiaen’s own treatise whether integers represent order positions, or the values that occupy those positions. The confusion occurs because in most of Messiaen’s examples the resource from which the content for positions 1 through n in a series is selected is the set of integer durational multipliers 1 through n: {1, 2, 3, 4,…,n}. When a series of IOIs is selected from such a resource, and is in either entirely ascending or descending order, Messiaen calls the series chromatique durées. Following suit, we call such a resource set a “set of chromatic durations.” For the time being, we are not concerned with content, but only order positions, because unlike NRRs, the content of the resource set (which for Messiean’s SPs is never a multiset) has nothing to do with how it is that SPs achieve their “impossible charm.” Messiaen commences his explanation of SPs with an attempt to catalogue the myriad ways in which a given series of n different elements can be (re)ordered. In our formal terms, every ordering of n different elements is an n-member series in which every position 1 through n is occupied by a different integer. Any reordering of such a series, obviously, also features a different integer in every position as well. By interpreting the integer in each position of a sequence as a duration (or IOI), Messiaen realizes each series musically as a  17  rhythm, and these rhythms he calls “interversions.”12 As mentioned in the discussion of NRRs, there are n! different ways to permute n distinct elements (i.e. the group Sn), and— since in the context of SPs we are talking specifically about the permutations of n distinct elements—exactly n! different orderings (or interversions, or rhythms) that can be composed from those n elements (namely, the group ORDn). Symbolically, |Sn| = |ORDn|. Messiaen explains: In order to understand what it is that [the technique of] Symmetrical Permutation achieves, we must first address the problem of permutation in general. Every given number has a certain number of corresponding interversions. The greater the number, the more its corresponding number of interversions is augmented—at a rate that seems disproportionate at first, [since] the number of interversions very quickly reaches astronomical figures [as n increases incrementally].13 He then writes out every possible ordering of 3 elements, of 4 elements, even the 120 orderings of 5 elements, each in their respective lexicographic order; he seems to enjoy compiling such “complete sets” (TRCO-III, 8–10). But for n ≥ 5, producing tables of all possible orderings become too arduous a task (“Je ne continue pas: on voit jusqu’où un tel travail nous entraînerait” (TRCO-III, 11)). Messiaen goes on to demonstrate that by choosing some particular permutation function (ordre de lecture, in his terms) and recursively applying that function to some seed series (his seed is typically <1,2,3,4,…,n>), one obtains a relatively small subset of ORDn (TRCO-III, 11–38). This should sound familiar. Just as with NRRs, we can apply a permutation function  to a “seed” sequence  . In the case of NRRs, f was the  retrograde function. In the case of SPs, f could (theoretically) be any  . When such an f  Chapter 3 includes a few instances of SPs where the contents of each order position is something other than a duration. e.g. Catalogue d’oiseaux and Livre d’orgue. 13 Pour comprendre l’utilité des « Permutations symétriques », il faut d’abord se pencher sur le problème des permutations tout court. Chaque nombre donné possède un nombre d’interversions correspondantes. Plus le nombre donné est grand, plus le nombre d’interversions augmente, avec une rapidité d’augmentation exacte et logique mais qui paraît disproportionnée au premier abord et attient très vite des chiffres astronomiques (TRCO-III, 7). 12  18  is repeatedly applied to a series A, it generates an orbit of sequences that are reorderings of A: f 0(A), f 1(A), f 2(A), f 3(A),…, f m(A), and so forth. Each element in an orbit is a unique reordering of the seed sequence A = <a1, a2, a3, …, an>, since—because the resource set for SPs is not a multiset—in A no ai = aj for i ≠ j. Essentially an NRR is an SP that uses one particular  whose period is 2. The only difference being NRRs require the resource set  to be a special kind of multiset. Messiaen, and many commentators after him, position the SP technique as a way of selecting a coherent and relatively small number of interrelated orderings of n elements from what would have otherwise been an enormous collection of all possible orderings. Wai-Ling Cheong, for instance, emphasizes the limitation imposed by symmetrical permutations: “Just as the modes of limited transposition enjoy fewer than twelve transpositions and the non-retrogradable rhythm retains the original duration pattern when played backward, a symmetrical permutation scheme can also drastically cut down the number of permutations derivable from any given series” (Cheong, 111). Amy Bauer credits Messiaen’s SP technique as being the selection of a particular permutation function that “tames the chromatic cycle…which otherwise could produce [32!] permutations” (Bauer 2007, 149).14 Such observations do not truly capture the essence of SPs. In the next chapter, we will demonstrate how SPs do not impose the same kind of limitation as do MLTs and NRRs; and neither do they manifest a symmetry in the manner of MLTs and NRRs.  Bauer’s use of the term “cycle” is somewhat ambiguous. It seems to mean permutation function, e.g.: “The cycle in Ile de feu II generates only 10 individual duration series” (Bauer 2007, 148). 14  19  Chapter 2: Permutations symétriques in theory Any sufficiently advanced technology is indistinguishable from magic. —Arthur C. Clarke, Profiles of the Future  2.0 – Some existing descriptions of Messiaen’s SPs One of the first published explanations of Messiaen’s SPs (and almost certainly the first English-language explanation) is the “Introduction by the Composer” written for the liner notes of the premiere recording of Chronochromie: In the case of a chromatic scale of durations ranging (as here) from the [32nd note] to the [whole note], the number of possible permutations is so vast that it would take half a human lifetime just to write them down—and several years to play them. Therefore, it is necessary to make a selection— and to select in a way that will give the maximum opportunity for dissimilarity between one permutation and another. 15 (Messiaen 1965) In a continuation of this explanation elsewhere, he elaborates: To arrive at this, I read my scale of chromatic values in a certain order, then having written down the result, I number from 1 to 32 the succession of note-values obtained, then I read my result thus numbered in the same order as the first time; I write down this second result and again number from 1 to 32 the succession of numbers obtained. Then I read my second result in the same order as the first time, which gives yet a third result, then I read my third result in the same order as the first time: I do the same thing for the fourth result, and so on until I arrive again at the chromatic scale of durations with which I began. This gives a reasonable number of permutations (not too far from the number of objects chosen), and also permutations sufficiently different to be juxtaposed and superimposed. 16 (Samuel [1967] 1976, 90) In this account, Messiaen explains what he believes the SP technique achieves, and narrates how each subsequent interversion is produced. (Shortly we will consider his claims that SPs  The translator of these notes is unknown. This first recording of Chronochromie appeared in 1965, and was conducted by Antal Dorati, who, the liner notes credit as having conducted the world premiere in Paris three years prior. The list of Messiaen’s compositions in New Grove cites 1959– 1960 as the period in which Chronochromie was composed (Griffiths 2001). 15  20  select a number of interversions “not too far from the number of objects chosen,” and later, that they are “sufficiently different [enough] to be juxtaposed and superimposed.”) It is clear that he is at least intuitively aware that some permutations have smaller orbits than others. But he does not explain how an “order of reading” imposes a certain limit on the number of permutations in the orbit. By reconceiving of Messiaen’s “order of reading” as a permutation function in cycle notation, we can determine this limit from the least common multiple of each of the cycle lengths. Without this knowledge, though, one would have to discover the size of the orbit by brute force, writing each subsequent permutation until the initial series is reproduced. This more arduous method is precisely what Messiaen does in TRCO. In this way, Messiaen’s own publications on the topic of SPs are more demonstration than explanation. Figure 2, below, reproduces a portion of Messiaen’s demonstration of SPs from TRCOIII (15-36), with my annotations in blue. Messiaen begins by describing a series of “chromatic durations” in ascending order, shortest to longest (see series “0” in figure 2). Such a series is what was described in the previous chapter as a seed series (so it follows that the content of the series is not the object of focus for the time being). Messiaen then describes an “order of reading” by which he obtains the next interversion (see “l’ordre de lecture 3 | 28 | 5 | …” in figure 2). This reading is what we have described in the previous chapter as a permutation function, although Messiaen’s description and orthography do not of course adopt the current mathematical conventions. The integers both above and below the noteheads represent order positions, and not the content of those positions. (The correlation between the upper row and the durational value measured in sixteenth notes holds only for the first interversion.) Reading the integers above and below the durations in any of the interversions column by column—top to bottom then left to right—we can interpret Messiaen’s figures as saying: the content from position 3 in the prior interversion  21  is mapped to position 1 in the present interversion; the content from position 28 from the prior interversion is mapped to position 2 in the present interversion; and so on. This orthography for permutation functions is very similar to what mathematicians call two-line notation, wherein there are two rows of order positions one above the other, and the content of an order position in the top row maps to the order position represented by the integer immediately below (Bóna 2004, 73). For example: The permutation function that turns the series <4, 3, 2, 1> into <4, 1, 3, 2> is written in two-line notation as  . The difference between two-line notation and Messiaen’s  notation is that two-line notation is organized in ascending order according to the top row (arguments), while Messiaen’s is organized in ascending order according to the bottom row (values). The result is that Messiaen’s notation is a kind of “passive” expression of a permutation—emphasizing the way in which an interversion was obtained from the previous one, instead of emphasizing the permutation function that turns one series into the next. His one-line expression of the “ordre de lecture” (written 3|28|5|… ) is also confusing because it is the same sequence of integers as would represent the durations in interversion 1 if the seed set were <1,2,3,4,…,n>. Since that is not always the case, it is better not to name an SP orbit simply by its first interversion.  22  Figure 2: Messiaen’s SP demonstration in TRCO-III (annotated)  23  Another point of clarification regarding best practice for labeling SP orbits: it is preferable to label the seed series (chromatic durations in ascending order in this example) as “interversion 0” despite the fact that TRCO lists it as the last interversion. This way, we can do simple addition and subtraction with the exponents of f and the interversion numbers, mod m (where m is the LCM of the discrete cycle lengths of f expressed in cycle notation), to describe relationships between series in an orbit. For example, we can say that if m = 12, interversion 7 = f 3(interversion 4) = f 2(f 4(interversion 1) = f -1(interversion 8) = f 8(f 11(interversion 0). Hereafter, I will always label the seed series as interversion 0, though Messiaen lists it last, as interversion m. In figure 2 Messiaen’s notation is translated both into standard two-line notation and also cycle notation. Again, I prefer cycle notation because it is the most compact expression of permutation functions, and because it makes evident the discrete cycles within a permutation function, which determine the size of the orbit defined by that function. Robert Sherlaw-Johnson’s analysis of SPs in Chronochromie is likely the first published account of Messiaen’s SPs that correctly observes that the size of the orbit defined by a given permutation function is equal to the least common multiple of the cycle lengths of that permutation function (Sherlaw-Johnson, 1975, 160). In this respect, his understanding of how it is that SPs impose a limit is on the right track; but it is not entirely without fault. In a discussion of Chronochromie, he describes SPs as a way to systematically limit the number of permutations from an “astronomical” number [presumably he is referring to 32!] to a more manageable size, 36 (Sherlaw-Johnson 1975, 160), yet in his  24  analysis of Île de feu II, he asserts that the permutation of 12 objects produces only 10 interversions, rather than the “complete cyclic series of twelve permutations (which one might expect when twelve objects are permutated cyclically amongst themselves)” (Sherlaw-Johnson 1975, 109). So there seems to be confusion here about whether the limit that SPs impose is less than n or less than n!. Regardless, his observations implicitly raise a general question that he does not address: what does the cardinality of an orbit need to be in order to impose a satisfactory “limit”? This question will be addressed in section 2.2 below. Sherlaw-Johnson’s explanation suffers further from a misunderstanding of how SPs manifest symmetry. He identifies this property first in Mode de valeurs et d’intensités, illustrating its "use of symmetrical permutations of the twelve-note series” with a diagram soon to be reproduced below as figure 3 (Sherlaw-Johnson 1975, 109). On eight separate occasions in the piece, he explains, one of the three “voices” states the twelve members of its “mode” in succession (in some order), and that “the notes of each group are permutated in a type of symmetrical order at each appearance” (Sherlaw-Johnson 1975, 107). At first glance it is difficult to know what the integers in Sherlaw-Johnson's figure are meant to represent; but it turns out they indicate the relative pitch height (lowest integer being highest pitch) and also the relative duration (lowest integer being shortest duration) with respect to all the pitches and durations in each voice’s respective mode. In his figure, each pitch is connected to the next by a line that is horizontal if the second is a step higher/longer or lower/shorter than the second, or that slopes diagonally downward or upward if the changes are greater than that. (Further, he tries to represent non-adjacent pitches relatable by stepwise motion on the same horizontal plane.) In other words, the symmetries he identifies here involve contour inversion, not an inversion (or otherwise “symmetric” transformation) of the order positions themselves.  25  Take for instance the fourth statement of an entire mode, from measures 53–57 (see example 1, which also shows the initial statement of the mode, with strictly descending pitches, in a non-periodic and asymmetric sequence of intervals, and strictly increasing durations that are successive multiples of the 32nd note). In the fourth statement, every successive pair of attacks is inversionally symmetrical with respect to exact duration, in the specific sense that their durations sum to 13: 6+7, 12+1, 5+8, 11+2, 4+9, and 10+3. This also means, because the original mode is strictly descending, and the pitches remain attached to their original durations, that the fourth statement is also symmetrical with respect to relative pitch height, around the axis between contour pitches 6 and 7: each successive pair involves symmetrical pitch elements from the original descending contour, and the contour of the fourth statement is symmetrical, pairwise, although it does not show exact pitch symmetry because the original pitch sequence is neither periodic nor symmetric.  Example 1: Inversion Symmetry in Mode de valeurs et d’intensités  Sherlaw-Johnsons figure (figure 3) does not communicate these symmetries as effectively as it might have. My figure 4 makes the inversion symmetries of these eight series more evident by representing the pitches from high to low as "contour pitches" 1–12 arranged vertically (Morris 1987, 26); and indicating the order positions for each series by the 26  smaller integers in rounded boxes.17 (Conceiving of the symmetry of these series in terms of inversion of rhythmic contour is also possible, but less perceptible, and certainly less intuitive than pitch-height contour inversion).18 These symmetries, while interesting in their own right, do not arise from a symmetrical permutation process. Certainly each of the eight statements of twelve-note series is some “permutation” of the mode, and one could relate each statement to the next one (or any other) by some permutation function. But there is no permutation function that relates more than one series to another among the eight twelve-note series Sherlaw-Johnson identifies in Mode de valeurs et d’intensités, and so we can say that the piece is not an instance of SPs in practice. This is because—to define SPs formally in a way that is consistent with what Messiaen himself labels as SPs—we will say that a composition is an instance of the SP technique if and only if that composition contains n different series—of pitches, of IOIs, or of both, presented either simultaneously or in succession— where n > 2, and where the n series are n adjacencies in a single orbit defined by a single permutation function. (Additionally, we know that the contour inversion symmetry Sherlaw-Johnson identifies in Mode de valeurs et d’intensités cannot be the defining characteristic of SPs, since it is not present in later instances of SPs that the composer himself identifies and labels as “interversions”.) By this definition, the first instance of SPs is Île de feu II (1950)—the fourth of the Quatre Études de rythme—which will be considered further in chapter 3.  Sherlaw-Johnson asserts that “in three cases [namely, series 2,3, and 8] the symmetry is slightly disturbed, apparently in order to avoid sounding a note at the same time as [it sounds in] another part” (1975, 107). In his figure, these disturbances to the symmetry are indicated by the integers in parentheses. Correspondingly, asterisks in my figure mark the order positions that that I have altered to illustrate Sherlaw-Johnson’s “ideal” symmetrical form of the series. 18 A possible exception, re imperceptibility of rhythmic contour symmetries: in the fourth statement, the periodic IOI of 13 32nds between every pair of attacks might be noticed. 17  27  Figure 3: Sherlaw-Johnson’s figure of alleged SPs in Mode de valeurs et d’intensités  Figure 4: A different representation of Sherlaw-Johnson’s contour inversion symmetries in Mode de valeurs et d’intensités  28  Recent analyses by Amy Bauer (2007) and Vincent Benitez (2009) expand on Sherlaw-Johnson’s work, offering different proposals for how symmetry is manifest in Messiaen’s SPs. They both describe abstract mirror symmetries based on the way certain rhythms may be taken to suggest time flowing simultaneously backward and forward, and based on the symmetrical “movements” of specific durations (around various axes) between interversions (Bauer 2007, 149–150; Benitez 2009, 282–284). These claims about SP symmetries, and the significant perceptual challenges they entail, will be addressed later this chapter, and early in the next. Benitez, along with Wai-Ling Cheong (2007), employs some concept of “permutation schemes”—in other words, categories of permutation functions that share some common characteristics. Cheong refers to these categories as “permutation schemes”; going so far as saying the SP of 12 durations in Île de feu II and the SP of 32 durations in Chronochromie are “essentially the same permutation operation” (Cheong, 111). Though this assertion is not sufficiently qualified, it implies that there exist certain permutation schemes that can be scaled to manipulate series of differing lengths, and we shall see that this is true of Messiaen's procedures. (Confusingly though, Cheong elsewhere uses “permutation scheme” to mean individual permutation functions.) Benitez describes certain “permutation techniques,” or schemes (taken from Messiaen’s Appendix to TRCO-III), which, like the “retrograde function,” are permutation functions that can be scaled to series of different sizes while displaying some unifying characteristics with regard to symmetry or limitation of the SP orbit, and/or the interversions in that orbit.  29  2.1 – Permutation schemes The simplest example of what we will call a “permutation scheme” is the retrograde function, which we have already covered. Regardless of the length of the series to which the retrograde function is applied, we can write it in cycle notation as: where n is odd,  (  )(  )(  )(  )  (  and where n is even,  (  )(  )(  )(  )  (  ) ).  Intuitively, we can conceptualize such a permutation as “the same function” applied to series of differing lengths, though technically speaking, every retrograde function is unique—or at very least, there are two different retrograde functions, one that reorders an even number of elements, and one odd. Nevertheless, it is easy to conceptualize the (infinitely many) retrograde functions as belonging to a single permutation scheme because, as we have seen, the LCM of its cycle lengths behaves extremely predictably as the length of the series to which the retrograde function is applied is increased—the LCM always being two, for a series of any length. More generally, a permutation scheme is a set of all permutations that may be expressed by a single functional notation involving a variable n. Usually the notation will specify two closely related functions for odd and even n. For the retrograde permutation scheme, it was possible to do this using cycle notation, since every cycle length is either 2 or 1. In other cases, we may need to express permutation functions in two-line notation in order to demonstrate the consistencies of the scheme. In order for a permutation scheme to be of potential value as a music-theoretical construct (whether used in analysis, or as a compositional tool) we should be able to take advantage of the notation, which enables us to scale the permutation f to series of different lengths. Then, we want to observe some correlation between the length of the series, n, and  30  the size of the orbit defined by f recursively applied to that series. There are a few particular permutation schemes that we will assess by this criterion: the “double-line” scheme, the “open-fan” (or “open scissors”) scheme; the “closed-fan” (also called “closed scissors”) scheme; and the “two-by-two” scheme. These schemes may be inferred from some of the examples of SPs given by Messiaen in the Appendix to TRCO-III. There it is not clear whether Messiaen considered these names to denote functions or orderings. Evidence in favor of the latter includes the fact that he labeled each element of each series with a integer describing its duration and not the order position from whence it came, along with the fact that he lists in many cases only what would be “interversion 1.” But he later used these names to indicate functions on order positions (e.g., in his analysis of Reprises par interversion, (TRCO-III 178–180) or his description of the “3 en 3” ordre de lecture (TRCOIII 12)). Benitez’s (2009) recent analysis of SPs considers schemes to be functions when he describes certain permutation “technique[s] applied to…[a] durational series” (284) and when he asserts that a series is reordered by the “open-fan algorithm” (286); but later, he names individual interversions within an orbit generated by the recursive application of an “open-fan” function as being the 4x4 permutation pattern, and 2x2 permutation pattern (287). These names can be appropriately used for either functions or orderings so long as we are precise about what we mean. Messiaen, for example, describes the rhythm <1,3,5,7,9,11,13,15,16,14,12,10,8,6,4,2> (the integers representing durations measured in 16th notes) as “Interversion de 2 en 2 (movement droit, en allant de 1 à 15) – puis de 2 en 2 (movement retrograde, en allant de 16 à 2)” or, “two-by-two interversion (movement from the right [ascending] from [odds] 1 through 15, and then two-by-two interversion (from left to right [descending] from [evens] 16 through 2) (TRCO-III, 321). (Hereafter I adopt the shorthand “Two-by-Two: odds↑, evens↓” for this scheme, and analogous shorthand  31  descriptions for other schemes that can similarly described as a “two-by-two” of some variety.) As a name for a type of rhythm, this describes a certain ordering of chromatic durations 1 through n wherein the “odd” durations in ascending order are followed by the “even” durations in descending order. As a name of permutation function, it can mean “the permutation function that transforms <1,2,3,4,…,n> into <1,3,5,…n–1,n,n–2,n–4,…,4,2> for even n (or into <1,3,5,…,n,n–1,n–3,…,4,2> if n is odd). In this way, a certain rhythm is representative of a certain permutation, since we can always read a rhythm as being an “interversion 1” of an orbit whose seed “interversion 0” is the some particular, predetermined ordering. This is to say, we can read every series Ai ∈ Ordn as a unique function f ∈ Sn if we interpret it as being derived from a certain Aj ∈ Ordn. This works out conveniently enough for Messiaen whenever his seed series, or “interversion 0,” is <1,2,3,4,5,…,n>. But describing permutation functions this way is akin to describing how to shuffle a deck of cards by listing the order of the cards in the deck after the shuffle: it is meaningless unless the prior order of the cards is known, and even then it makes for a cumbersome notation. A better approach is to consistently describe permutation functions by using only order positions, since in this way, we can use the same notation regardless of the specific content of the order positions (this will be especially important in chapter 3 when we examine the permutations of durations used in Île de feu II, Timbres-durées, and Livre d’orgue). The scalable “double-line” permutation scheme implied by Messiaen’s example rhythm <1,9,2,10,3,11,4,12,5,13,6,14,7,15,8,16> (TRCO-III 322-323) is the family of permutations each of which—in every subsequent interversion of its respective orbit— divides the prior series into two segments as equally sized as possible (in this case, equal halves), and then interleaves the contents of the first segment with the contents of the second. Hereafter, this particular permutation scheme is written as “Double-line, 1↑, ↑n”, 32  since it interleaves the values in order positions 1 and up with the values from the “middle” of the series up to order position n. Mathematicians refer to this permutation function as the Faro shuffle, or the riffle shuffle. Given a deck of cards (even in number) this shuffle is performed by cutting the deck in half (one half constituted of order positions 1 through , and the other half constituted of order positions  through n) and then interleaving the  cards from the two piles. There are two different ways to perform this shuffle; either with the card in order position 1 mapping to 1 and n to n, or order position 1 mapping to 2 and n to n – 1. The former is often dubbed the riffle “out” shuffle, the latter “in” (presumably referring to whether the cards on the extremes stay out, or mix in to the deck). If the number of cards in the deck is odd, the shuffle is performed by splitting the deck into piles of  and  (either 1 and  then interleaving the piles—the first and last cards from the larger pile , or  and n) always mapping to positions 1 and n. There is no “in”  riffle shuffle on an odd number of cards, only two different “out” shuffles, depending on which is the larger portion of the deck. Writing out the Double-line 1↑, ↑n (riffle-out) function in cycle notation for the first few even n’s, it quickly becomes evident that both the cycle-lengths and number of discrete cycles per function change somewhat erratically as n increases incrementally, so it is not possible to write a general expression of the scheme in cycle notation. f = Double-line 1↑, ↑n (or riffle-out): n = 4, f (<A,B,C,D>) = <A,C,B,D>  f = (1)(2 3)(4)  n = 6, f (<A,B,C,D,E,F>) = <A,D,B,E,C,F>  f = (1)(2 3 5 4)(6)  n = 8, f (<A,B,C,D,E,F,G,H>) = <A,E,B,F,C,G,D,H>  f = (1)(2 3 5)(4 7 6)(8)  Here, the “disadvantage” of two-line notation—that it doesn’t make explicit the discrete cycles—is an advantage, allowing for a general expression of the permutation scheme. For  33  example, the riffle-out shuffle that corresponds to Messiaen’s “Double-line, 1↑, ↑n” can written in two-line notation as: when n is odd,  (  and when n is even,  ( )  ( (  )  ( (  ) ( ) ( ) ( ) (  ) ( ) ( ) ( ) (  ) ) ( ) )  )  .  This is the only permutation scheme to which Messiaen directly gives the name “double-line,” though, the riffle-in shuffle could also be described as a double-line permutation scheme, namely Double-line ↑n, 1↑. We could also imagine performing a riffle shuffle (of either sort) on a deck with the cards turned upside down (that is, with the order positions of both halves of the series retrograded). In this way, the same mechanisms of the riffle-out and riffle-in shuffles would result two additional double-line permutation functions, which we could describe as “Double-line ↓1, n↓” and “Double-line n↓, ↓1” respectively. There are two ways of choosing which half of the “deck” will map to the odd order positions, and there are four ways of choosing which deck-halves to “flip” or retrograde (both, neither, one, or the other), and so there are 4*2 = 8 different permutation schemes that could be described as “double-line” schemes. The four I have not yet described are those that result when only one half of the deck is turned-upside down (i.e. retrograded). (This would be impractical if actually dealing with playing cards, since half the deck would be face up, but poses no such problems when shuffling order positions of a series). These “half flipped” riffle shuffles are the permutation schemes Messiaen calls "open- and closedfans," but we can see that they could just as easily be described as double-lines: the “openfan” being either Double-line ↓1, ↑n, or Double-line ↑n, ↓1; and a “closed-fan” being either Double-line 1↑, n↓, or Double-line n↓, 1↑. All eight of the double-line permutation schemes  34  can be written as a general function in two-line notation, and all define orbits whose cardinality changes erratically as n increases incrementally. Next let us consider the scalable “two-by-two” permutation scheme suggested by Messiaen’s example rhythm <1,3,5,7,9,11,13,15,16,14,12,10,8,6,4,2> as a permutation of <1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16> (TRCO-III 322-323). It is the family of permutations each of which—in every subsequent interversion of its respective orbit—presents the contents from the odd order positions (from the prior interversion) in ascending order, before presenting the contents from even order positions (from the former interversion) in descending order. Hereafter, this permutation scheme will be called “Two-by-two, odd↑, even↓”. It is evident that other schemes are also “two-by-twos.” If a two-by-two permutation scheme is one that presents all the contents of one parity (in either ascending or descending order) and then the other (in either order), it follows that there are eight different two-bytwo schemes, since there are two ways of choosing which parity of order positions will appear first and four ways of choosing whether or not the values from order positions of one or the other parity will be reversed: Two-by-two, odd↑, even↓ Two-by-two, odd↑, even↑ Two-by-two, odd↓, even↑ Two-by-two, odd↓, even↓ Two-by-two, even↑, odd↓ Two-by-two, even↑, odd↑ Two-by-two, even↓, odd↑ Two-by-two, even↓, odd↓ Again, these functions are easily conceptualized if we imagine them as ways to shuffle a deck of n cards. Every two-by-two permutation is isomorphic to a shuffle that deals the  35  cards into two piles, starting from the top of the deck—first card to the left pile, second card to the right pile, third card left, fourth card right, and so on—then flips either or neither pile or both, then places one pile on top of the other. Whether the cards from odd positions come first or second is a matter of which of the two piles is placed atop the other (two choices). Whether the cards from odd order positions of either parity are running in the same (ascending) or reversed (descending) order (relative to their order in before the shuffle) depends on whether the piles get “flipped” upside down (four choices). Writing out the function “two-by-two odd↑, even↑” in cycle notation for the first few even n, it becomes evident that, as with the double-line schemes, the cycle lengths and number of discrete cycles per two-by-two function change erratically as n increases incrementally, and so it is not possible to give a general expression of these functions in cycle notation. f = Two-by-two odd↑, even↑ (or two-pile shuffle, flip both): n = 4, f (<A,B,C,D>) = <A,C,B,D>  f = (1)(2 3)(4)  n = 6, f (<A,B,C,D,E,F>) = <A,C,E,B,D,F>  f = (1)(2 4 5 3)(6)  n = 8, f (<A,B,C,D,E,F,G,H>) = <A,C,E,G,B,D,F,H>  f = (1)(2 5 3)(4 6 7)(8)  The Two-by-two odd↑, even↑ permutation scheme can be expressed for every n as: when n is odd,  and when n is even,  (  )  (  (  )  )  (  (  )  )  (  (  )  )  (  (  )  )  (  )  (  )  .  Comparing the notation for the “Two-by-two odd↑, even↑” and “Double-line 1↑, ↑n” permutations above, we can see that one is the inverse function of the other. (Inverses of functions written in cycle notation can simply be read right-to-left, or when written in twoline notation, read bottom-to-top). As it turns out, each of the eight double-line schemes is the inverse of one of the two-by-two schemes (see table 2), and thus, are all part of a single 36  family of permutation schemes. It is possible to say that all two-by-two schemes are of the same family because they are all a variation on a basic shuffling mechanism, and because the functions from each of the schemes—though not identical—have periods related to one another, that effect a similar upper limit on orbit size.  Table 2: Summary of “Two-by-two” family of permutations  Thus far there is no reason to consider an orbit of interversions as having directionality, so there is no difference (for the time being) between the orbits defined by inversely related functions. Therefore it is sufficient to consider only the “two-by-two” functions. Tables 3-6 provide information about each of the two-by-two permutation schemes. Each table contains two sub-tables for two different permutation schemes. Above every sub-table appears the shorthand name of each scheme, along with the shorthand name of its double-line inverse. Immediately below the name are the two-by-two functions (for odd and even n) expressed in two-line notation. Below these, I provide a reference to the integer sequence (as catalogued in Sloane’s online database oeis.org) that corresponds to the periodicities of the permutation function f on n. The sub-tables are made up of three columns: the leftmost listing the series length n, the middle column listing the function in cycle notation for the given series length, and the leftmost column giving the periodicity of the function when applied to a seed series of length n. The periodicities of f—that is the orbit cardinalities of that f recursively applied to 37  some seed series of length n—correspond to existing integer sequences catalogued in Sloane’s database (oeis.org),19 but Sloane’s n (which is the (n+1)th term of the respective sequence) is not always equal to our n (which is the series length). For this reason, each of my references to a Sloane integer sequence replaces the respective “Sloane n” with a value r, and provides a formula by which r is derived from the series length n. (The definition of r appears in the reference above each sub-table—the variable r is then used in formula in the Orbit cardinality heading of the leftmost column.) Looking at the orbit sizes for n’s 3 through 16 on tables 3–6 we can see that Messiaen’s claim that the number of interversions is near n is not quite true. In certain cases Messiaen’s claim is incorrect, because the period of f is considerably less than n. For instance the Two-by-two, odd↑, even↓ function (or its “closed-fan” inverse) on a series whose length is 16 has a period of 5. And the “same” function (that is, the function of the same permutation scheme) applied to a series whose length is 32 has a period of 6. What can be said about all of the functions of schemes in the two-by-two family is that they have a period less than or equal to n (see tables 3 through 6, and the continuation of the corresponding integer sequences A003558, A002326, and A065457). It is possible to describe a "three-by-three" family (of 6*4*2=48 schemes) with corresponding “triple-line” inverses; and a “four-by-four” family (of 8*6*4*2=384 schemes) with corresponding “quadruple-line” inverses, but it is not necessary to do so here, since Messiaen’s examples of them make up only a very small percentage of schemes from those larger families (whereas he describes at least 4 of the 8 orbits of the two-by-two family).20  Two-by‐two, odd↓, even↓ (Inverse: Double--‐line, ↓1, n↓) on even-length series is the only exception. 20 Interestingly, some “four-by-four” orbits are reordered versions of two-by-two orbits. This is because if f is a two-by-two scheme, then f i is a 2i-by-2i scheme (mod n), and because whenever k and the period of f are relatively prime, f k and f generate orbits comprising the same interversions (in a different order). 19  38  Table 3: “Two-by-two” family of permutations (part 1 of 4)  39  Table 4: “Two-by-two” family of permutations (part 2 of 4)  40  Table 5: “Two-by-two” family of permutations (part 3 of 4)  41  Table 6: “Two-by-two” family of permutations (part 4 of 4)  42  Recall that Messiaen says he chose reading orders (permutations) that produced satisfactorily different adjacent interversions (reorderings). But the question of whether any of these different permutation schemes just discussed achieve satisfactorily different adjacent interversions is difficult to answer. Without referring to a specific instance, the contents of the order positions are unknown. Further, it is not known exactly what Messiaen’s criteria for “satisfactory difference” are. The question of whether certain shuffles can achieve satisfactory randomization of n elements is of interest to mathematicians and computer scientists (not to mention casinos and online gambling services), and has been expertly addressed by Golomb (1961) and Diaconis, Graham & Kantor (1983). However, these types of questions have nothing to do with the content in the n order positions. In contrast, Messiaen’s criteria—that SPs should generate adjacent interversions sufficiently different one from the other to be suitable for juxtaposition or superposition—do require knowledge of the content of every order position. It is relatively easy to thwart the differentiation that reordering can provide by inventing a series which, when permuted by one of the “two-by-two” or “double-line” permutation schemes, becomes a highly similar interversion. Two examples will illustrate. Example 2 shows an invented twelve-tone row, which when reordered according to Messiaen’s oft-used “open-fan” permutation yields a pitch-class transposition of the original row (this “sleight of hand” is only possible when the period of f is 12, but other functions with different periods could conceivably produce similar adjacent interversions as well).21 Certainly it would be difficult, if not impossible to hear the two rows back to back and pick up on the implied mirror symmetry (which asks the listener to trace an imaginary path connecting the content in each order position and its order position from the prior interversion) when the transpositional relationship is so apparent. Similarly, example 3 Babbitt explains transpositions as permutations on order positions in his 1960 essay “Twelve-tone Invariants as Compositional Determinants” (Babbitt 2003, 57ff.). 21  43  invents a rhythm (composed from a resource set of “chromatic durations”: a sixteenththrough a half-note) that, when permuted by the “Two-by-two, evens↓, odds ↑” function, yields a rhythm that might be deemed unsuitable for juxtaposition with the original, since the durations in their respective order positions i (for 1 through 11) differ by only a single sixteenth-note value. For example, it would be difficult to hear the series segments <9,10,8,5> and <8,9,7,4> (from respective order positions 4-7) as contrasting rhythms (example 3). (If superimposed, their composite rhythms would be interesting enough, since the attacks in the two streams would diverge from each other significantly).  Example 2: Open-fan permutation on invented tone row resulting in T11 transposition  Example 3: “Two-by-two, even↓, odd↑” permutation resulting in similar rhythm  44  Not much else can be said about the way different permutations manipulate certain series, without knowing something about the contents of those series.22 It would be possible (if time consuming) to tabulate and exhaustive list of each of the orbits defined by each of the “two-by-two” functions acting on Messiaen’s favorite seed series <1,2,3,4,….,n> or <n, n–1, n–2,…,1> of a certain length, but that would be unnecessary, since the majority of those orbits are not used in Messiaen’s compositions. What has been demonstrated here is that it is possible to calculate the size of the orbit for any of these functions without so much as writing down one’s seed series and first interversion. Had Messiaen been privy to such information, he certainly could have selected a particular function on a series of a certain length so to find an orbit of a certain desired size. It seems that, ironically, part of the “charm” of SPs for Messiaen was that they defined “unexpectedly small” orbits. For the magician executing a card trick, the permutations are magic for the opposite reason—that the properties of a particular shuffle can calculated ahead of time, and exploited.  One exception: the “two-by-two evens↑, odds↑” and its inverse—whether on a series of odd, or even length—defines an orbit of cardinality 2k wherein every interversion i is the retrograde of interversion i+k mod 2k. 22  45  2.2 – Contextualizing orbit cardinalities with regard to Sn In section 2.1 it was demonstrated that none of the permutation schemes in the twoby-two family affords an orbit of interversions whose cardinality is greater than n. Not all permutation functions, however, are limited to a period of n or less. For example, consider the one used by Messiaen in Chronochromie:  Since the cycle lengths of this function are 6, 4, 18, 3 and 1, the period of this function is LCM(6,4,18,3) = 36. While the cardinality of this orbit of interversions still falls in the range of being relatively near n, the permutation seems to have been chosen arbitrarily (as opposed to the systematic nature of the permutations schemes) so it certainly could have been otherwise. If we want to determine whether Messiaen’s SP achieves an orbit whose cardinality is legitimately “small”, we need to have something to measure it against. As mentioned previously, Messiaen, as well as recent commentators paraphrasing his explanation, marvel at how small an SP's orbit (whose seed series contains n elements) is compared to n!. In fact, an SP’s orbit can never be as large as n!, nor larger, since for n > 2, there is no member of the symmetric group Sn that can generate the entire group. Indeed, as we know, an SP’s orbit will be the LCM of its component cycle lengths, that is, the LCM of a limited set of numbers that sum to n. That number will always be much smaller than n!, which is the product (not LCM) of n integers. A better measurement of the relative size of a particular orbit would be to compare its size to the size of largest possible orbit that can be generated by a permutation of n  46  elements. The question of finding this number was first addressed in the early 20th century (Landau 1903). (Messiaen, in all likelihood, was unaware of any research in this field, though one imagines he would no doubt be delighted to learn that mathematicians find this problem “reclusive, eccentric, and charming” with a “penchant for disguise…[i.e.] masquerading in equivalent forms” (Miller 1987, 497).) The largest order of a permutation of n elements (that is, the largest possible LCM of the partitions of n) is given by “Landau’s function,” specified in table 7 (from oeis.org/A000793).  Table 7: Largest possible orbit defined by a permutation on n (A000793)  Simply comparing the orbit of Messiaen’s choosing with the largest possible orbit does not provide a complete picture of its relative size, however. This is because many permutations have the same period (with the exception of the identity function: the only element whose period is 1). To accurately assess the relative size of an orbit, the distribution of periodicities across all permutations must be determined. To do this, we need to be able to find the number of permutations f ∈ Sn with a given period x. It is a twopart question. First, we need to determine which partitions of n have an LCM of x. A partition “is a representation of n as a sum of positive integers, called summands, or parts of 47  the partition. The order of the summands is irrelevant” (Andrews 1994, 149).23 We are interested in the partitions of n because they represent the number of unique permutation types amongst which the n! elements of Sn are distributed. (The number of partitions of n is provided by the integer sequence A000041). By permutation type, I mean a category that comprises all permutation functions that have identical cycle lengths. For example, f1 = (1)(2)(3 4)(5 6)(7 8)(9 10 11 12) and f2 = (1 2 3 4)(5 6)(7 8)(9 10)(11)(12) represent the same permutation type. Symbolically we write partitions as λ ⊢ n : λ = {λ1, λ2, λ3,…, λk} (which we read as: lambda is a partition of n, such that…), where k is the number of summands of the partition. Second, having found all of the different permutations types whose cycle lengths have the desired LCM, we need to calculate the number of permutation functions for every one of those types. To find the number of permutation functions for a given permutation type, we begin by dividing the total number of permutations, n!, by the product of summands of the given partition. Doing this divides out all duplicate functions that would result from different rotations of the same function in cycle notation (e.g. (1 2 3 4)(5 6 7 8) = (2 3 4 1)(6 7 8 5), so we do not want count it twice). Next we find the cardinality of the set of automorphisms of λ for a given partition of n, where AUT(λ) is the subgroup of Sk which fixes λ. The cardinality of AUT(λ) is calculated as follows. Given a partition λ = {λ1, λ2, λ3,…, λk}, construct a set L = {l1, l2, l3, … , lq} each of whose members contain all summands of the  Partition is used differently in the twelve-tone literature, where it is concerned with the “contents” of each summand. I use it here in the mathematical sense: the number of partitions of n is the number of different (unordered) ways to sum to n. 23  48  same size, where all lj are disjunct. Then |  Dividing ((  )( )( ) ( )  ( )|  ∏| |  ) by |AUT(λ)| divides out all duplicate functions that are the same  function with its cycles written in different-but-equivalent orderings. Thus we arrive at the formula for the number of functions of a certain permutation type, r. Symbolically: (  ( )( )( )  )(  ( )  |  ( )|  )  For example, let n = 12, and let the permutation type λ = (1, 1, 2, 2, 2, 4). We want to find the number of permutation functions on 12 elements that, when written in cycle notation, have cycle lengths λ1 = 1, λ2 = 1, λ3 = 2, λ4 = 2, λ5 = 2, λ6 = 4. Then since λ1 = λ2 ; λ3 = λ4 = λ5 ; and λ6 is unique, |AUT(λ)| = (2!)(3!)(1!) meaning there are (  )( |  ) ( )|  (  )(  )  permutation functions of this type.24 Equipped with this information, a lot more can be said about the structure of a particular Sn and about the relative size and commonness of particular orbits relative to the group. For example, functions of the permutation type λ = (1, 1, 2, 2, 2, 4), from the example above, account for 0.26% of the 479,001,600 different permutations on 12 elements. Table 8 lists all 77 permutation types in S12 (Column 1), the number of functions of each of those types (Column 4), and the respective percentage of each of those functions among S12 (Column 5). The information in a table such as this shows that the largest possible orbit size is far smaller than n!; allows for the comparison of a given orbit size with the mean, median  Here I would like to credit Simon Rose, without whose assistance and guidance I would not have come to this formulation. Simon Rose obtained a Ph.D. in mathematics from the University of British Columbia in 2012, and is currently conducting post-doctoral research at Queens University in Kingston Ontario. 24  49  and mode orbit size of all of Sn; and further, allows for the calculation of the likelihood that a randomly chosen function will be of a certain type, or have a certain period (Table 8, Column 3). Such a blind, statistical view of the likelihood of Messiaen’s choosing one function over another is perhaps inappropriate when n = 12, since in such cases, Messiaen tended to choose permutations from schemes that achieve certain types of shuffles (i.e. interleaved/separating odds from evens)—the period of those functions being restricted to values related to the multiplicative order 2 (mod 2n+1). Other statistical information, such as mode, median and mean orbit size, is nevertheless useful, at least in so far as proving a good measure of the size of some particular orbit (see table 9).  50  Table 8: Permutation types in S12  51  Table 9: Summary of orbits in S12  Messiaen’s Chronochromie permutation, on the other hand, as mentioned before, appears to be an arbitrarily composed permutation function, so calculating the likelihood of Messiaen’s choosing a function of this permutation type (or, a function of a different type with the same period) is a meaningful statistic. I have composed a table that describes S32 in the same way that table 8 describes S12, which allows for the calculation of such values. Because it lists all the data for all 8439 permutation types in S32 it is too large to include here, but relevant data from the table can be summarized in a more compact fashion. Table 10 lists the number of permutations types, and the number of unique orbit cardinalities, as well as the mean (139.9), median (58) and mode (60) orbit cardinalities of all permutations in S32. In light of this data, it we can more precisely identify the Chronochromie permutation’s orbit of 36 as being relatively small. Figure 5 shows that 77% of all functions in Sn have a period greater than 36, while only 21% have a period less than 36.  Table 10: Summary of orbits in S32  Figure 6 summarizes the likelihood of choosing at random a function with a certain period by showing the distribution of the distribution of the 8349 permutation types among the 248 unique orbit cardinalities. The corresponding table 11 lists the twenty most common orbit cardinalities according to the number of permutation types with a given periodicity among the 8349 unique permutation types in S32. By this measure, permutation types whose periodicity is 36 are 9th most common among types, with 5 of the 8 more  52  common periodicities being larger than 36. But this measurement is somewhat skewed because of the uneven distribution of 32! elements amongst the 8349 permutation types.  Table 11: Twenty most common orbit cardinalities according to percentage (among all permutation types) of types whose functions are of a given period  Figure 6 shows the distribution of the 8349 permutation types amongst the 248 unique periodicities, while figure 7 shows the distribution of the 32! permutation functions of S32 among the same 248 unique orbit cardinalities types. Table 12 lists the twenty most common orbit cardinalities according to the percentage of functions with a given period among the 32! elements of Sn, while table 13 lists the ten most common permutation types among permutation types whose period is 36. These more accurate measures position functions whose period is 36 as the 18th most common periodicity among all functions in Sn, and the position functions of the Chronochromie permutation type as fourth most common among permutations whose period is 36.  53  Table 12: Twenty most common orbit cardinalities according to percentage (among all f in Sn) of functions with of a given period  Table 13: Ten most common permutation types among types whose period is 36  The likelihood of randomly selecting a permutation whose period is 36 (of which there are 240 permutation types) is 1.32%, and the likelihood of choosing a permutation of the Chronochromie permutation type in particular, is 0.08%.25 Superficially, these odds seem low; but bearing in mind the distribution of permutation functions among the 248 unique orbit cardinalities, it becomes apparent that they are actually relatively likely choices, since There are only three permutation types whose period is also 36 that are more probable the Chronochromie permutation type. Together, the four most common permutation types whose orbit cardinality is 36 constitute 52% of all permutation functions with that period. The remaining 48% are spread among 236 different permutation types (see figure 6). 25  54  only 17 periodicities are more common, and 230 less common. This seems to position Messiaen’s choice of permutation as rather unremarkable, except that of the 17 more common orbit cardinalities, 12 are greater than 36. According to Messiaen’s criterion of the orbit cardinality being near n, the only suitable orbit cardinalities among the twenty most common are 30, 31, 32, 28, 36 (and perhaps 24 if near enough). These account for approximately one third of the 43% of functions in Sn constituted by the functions with one of the twenty most common periods. (Meanwhile, the likelihood of selecting a permutation whose function is the element of maximal order, 5460 (of which there is a single permutation type, λ = (13,7,5,4,3) is 0.02%.) Erdös and Turán (1965) provides a generalized, formal treatment of related problems (concerning the distribution of functions with certain periods within Sn). Miller (1987, 498) summarizes two pertinent aspect of their research: “that [in any Sn] very few permutations on n elements have orders as large as the maximum order” and that “’most’ (in a sense that Erdös and Turán made precise) permutations on n elements have an order whose logarithm is about (  ).  55  Figure 5: Functions of the Chronochromie permutation type (λ = 18, 6, 4, 3, 1) and other permutations with a period of 36 amongst S32  56  Figure 6: Distribution of 248 unique orbit sizes among the 8439 permutation types  Figure 7: Distribution of functions among 248 unique orbit sizes  57  The formulas for calculating the number of functions of each permutation type provide a more accurate measure of relative orbit size. In this way, we can better quantify how it is that different permutation functions—specifically, the functions Messiaen deems SPs— effect limitation. How it is that Messiaen’s SPs are symmetrical is less obvious. For the permutation schemes such as those of the two-by-two family (which can be scaled to reorder series of different lengths in a like manner), we can intuit the “same” function precisely because there is some symmetry—whether “translational” or “mirror”—inherent in the instructions for which order positions are to map to positions 1, 2, 3,… in the subsequent series. But since the Chronochromie permutation does not manifest either of these types of symmetries, this explanation is not satisfactory. Another possibility is that the symmetry lies in the composite image of all pathways traced from each order position to the order position to which its contents map in the subsequent interversion (as shown in example 2 above). These conceptual images create perfect bilateral symmetries in the case of Messiaen’s open and closed-fan permutations. Symmetries of this kind between interversions in the Chronochromie orbit would less perfect. But more importantly, these symmetries—while they might be evident on paper— are extremely difficult, if not impossible to hear in real time, since they ask a listener to form two (or more) accurate spatial representations of the series in which every attack forms a sound-object assigned to a particular order position, and then assess symmetry by cross-referencing the respective order positions of each sound-object between the different series.26 Alternatively, we could say that there is an implicit mirror symmetry whenever a listener can hear “stepwise” motion between contents in non-adjacent order positions,  The challenge (or impossibility) of hearing serial procedures acting on durations is addressed in Peter Westergaard’s analysis of Milton Babbitt’s Composition for twelve instruments (Westergaard 1965). 26  58  which according to Messiaen and Benitez is suggestive of time flowing in both directions (Messiaen TRCO-I, 54–57; Benitez 2009, 282–283). This conception of symmetry in SPs is faced with two challenges. First, hearing symmetry in this way would be contingent on a listener’s a priori association of acceleration, deceleration, ascent and descent with a certain direction of time-flow. Second, and more importantly, even supposing a listener did make such associations, the heard “symmetry” occurs completely and utterly independently of the mechanics of Messiaen’s SPs and their orbits. Perhaps the best explanation of symmetry in SPs is simply this: that every f ∈ Sn applied recursively to any seed series A ∈ ORDn is symmetrical in so far as it is a cyclic subgroup of the Symmetric group of n, which is, after all, a group of symmetries. Of course, by this definition every permutation is a SP—only, some are better suited for Messiaen’s purposes than others. This makes SPs different from MLTs and NRRs in two respects. At the end of previous chapter I said that the common trait between MLTs and NRRs is that they both involve special subdomains, that when acted upon by a “run-of-the-mill” function, will afford fewer unique results than is typical of that function. The same observation does not hold for SPs since there is no particular function associated with the technique; and also because—since Messiaen permutes series whose contents are all unique—whichever function is chosen only imposes the “limit” that is its period, meaning the limitation is independent of characteristics of the domain. As we turn to looking at actual musical instances where Messiaen employed permutations of series of durations and his SP technique, we are now well equipped to evaluate Messiaen’s choice of permutation functions (and the ensuing orbit of interversions, where applicable) from two differing perspectives: in terms of properties of permutation  59  functions themselves (as elements of Sn), and then in terms of musical materials that result from the particular way Messiaen implements certain interversions in a composition.  60  Chapter 3: Permutations symétriques in practice As for the appearance of the wheels and their construction: their appearance was like the gleaming of beryl. And the four had the same likeness, their appearance and construction being as it were a wheel within a wheel. —Ezekiel 1:16 (English Standard Version)  3.0 – Hearing interversions: introduction to the musical examples It was demonstrated in chapter 2 that, in theory, the SP technique does not favor one permutation over another. Every permutation function, when recursively applied to some seed series, defines an orbit whose cardinality is equal to the LCM of the cycle lengths of that function. So what determined Messiaen's actual choices of permutations? It must have to do with his stipulation that each of the interversions in one of his SP orbits ought to be a “satisfactory” shuffle of the elements in the series, suitable for juxtaposition and/or superposition (Samuel [1967] 1976, 90; Messiaen, 1965). But since this quality depends on the content of the order positions as much as on the mechanism of the SP technique itself, it is best appreciated on a case-by-case basis.27 Thus, we turn now to SP in practice, examining specific pieces in an attempt to determine its musical effects. If the SP technique is to be “heard” in actual musical settings, then either a listener must cognize the permutation function itself by perceiving the relation of adjacent interversions in an orbit, or hear something in the resulting rhythm or series that is  To clarify: this it is not to say that the mechanism of a given SP is unimportant, only that consideration of content is necessary in order to test Messiaen’s claims (that SPs are a “satisfactory” shuffle whose interversions are suitable of superposition and juxtaposition). Assessing the musical effectiveness of a given SP, one should certainly consider its component cycles, which produce independent processes, working in polyphony, so to speak. For instance, if an SP has some very long component cycle(s), then in general the transformations through their orbit(s) will be harder to notice. On the other hand, when the cycles are mostly of shorter length, the chances of noticing their periodicities will increase. 27  61  characteristic of their combination.28 If the former, the listener needs first to be able to discriminate between very small differences between long durations, since oftentimes the content of order positions 1 through n constitute a set of “chromatic durations”: IOI values one through n. In the (rather extreme) case of the set of chromatic durations used in Messiaen’s Soixante quatre durées from Livre d’Orgue this means observing as small a difference as between a whole note and a whole note less one 32nd note.29 (In Messiaen’s own performance of this piece (Messiaen, 1992) the tempo is approximately eighth-note = 50 MM, which makes the difference between sixty-three and sixty-four 32nds approximately the difference between 19.7 and 20 seconds.30) Other compositions, especially those of Messiaen’s experimental period (1949–1951 (Sherlaw-Johnson 1975, 101ff.)), present similar perceptual challenges with regard to discriminating between durations of very similar lengths. Messiaen forthrightly claimed to have “ventured to bring us a step forward in [being able to hear the difference between two long note-lengths]” with his use of chromatic durations; saying of the sixty-four durations in the sixth movement of Livre d’Orgue: “if I can think them, read them, play them, if I can hear all sixty-four of them, with all their differences, then so can others, too” (Messiaen [1958] 1986). Even if certain gifted listeners can learn to do this, however, the task still remains to compare position of each unique duration to its position in the “neighbouring” interversion(s) of the given orbit if one is to “hear” an interversion as a permutation. And since, as we will soon see, Messiaen typically superposed two or three orbit-adjacent  Recent analysts have wondered to what extent SPs can be heard. Wu suggests that SPs “are probably better understood mentally and felt subliminally than consciously perceived aurally” (Wu 1998,108). Trawick is not bothered by the fact that SPs are only discoverable via analysis, saying “to dismiss detailed analysis of [a piece containing SPs] on the grounds that the serial relations are not really perceptible would demonstrate a naiveté verging on philistinism” (Trawick 1991, 33-34). 29 A portion of the score is reproduced below as example 6. 30 Messiaen only tempo indication on the score is “Modéré.” My rough calculation of the tempo is based on IOIs between attacks in the RH part for the first page, which I measured using Sonic Visualizer software, which was developed by Chris Cannam, Queen Mary, University of London. 28  62  interversions, the listener would have to isolate two or more simultaneous streams to hear the SP technique as such. This is almost certainly beyond perception. If the permutation functions themselves can not be heard, there remains the implicit question of what, if anything, distinguishes an instance of Messiaen’s SPs from a randomized, unpredictable series of pitches and/or durations.31 I do not mean to dismiss all musical structures that are beyond perception as being irrelevant; because certainly analysis of SPs accounts for the events in many of Messiaen’s pieces.32 But neither do I want to prematurely retreat to the position that SPs are entirely imperceptible at the expense of failing to appreciate certain perceptible differences between the unpredictable rhythms generated by SPs, and other series that are likewise unpredictable—but unpredictable in a different way.33 One can only hope to identify SP of durations-series (aurally) by listening for some distinctive properties in the resulting rhythms, in at least one of the component cycles of the SP, possibly in more than one cycle. Musical examples to follow will identify a few measures by which certain distinctive properties might be made evident, and will contrast the SP technique with some of Messiaen’s experiments with other systematic  There is also a less precise way that a listener might “hear” SPs. Upon learning about Messiaen’s charm of impossibility and SPs, and upon discovering through score study that a certain piece contains a few interversions, one might be able to appreciate the aesthetic quality of the technique to a certain extent, even if the technique itself is not directly perceived. Bauer addresses this, citing Messiaen’s conception of his musical writing as being akin to the “symbolic language to the secret language forged by the early Christians, … [whereby he] implies that only the initiate can truly grasp his meaning” (Bauer 2007, 161). Paul Griffiths sees it in much the same way, saying: “it cannot be that [Messiaen] intends his operations of interversion to be followed by his audience” (Griffiths 1985, 156). He continues with an apt, if irreverent, analogy that likens Messiaen to “a pre-Vatican II priest with his back turned on the faithful, … bringing about transformations of which they can be aware only by its atmosphere and by its result.… The process by which that result is achieved must remain mysterious, except for those who examine the score by eye as well as by ear, and except of course for the composer” (ibid.). 32 Messiaen called score reading a “supplementary pleasure” that allows keen listeners to better “grasp the construction of the work” (Samuel [1967] 1976, 89). 33 This idea of different passages sounding “unpredictable, but in different ways” comes from a composer Darren Miller, who told me that, after having spent hours upon hours listening through “random” passages (realized on DisKlavier), he was able to differentiate with some accuracy which of several different random number generating scripts was used to composed the passage—even when the parameters were the same. 31  63  compositional methods (other uses of permutation, the “mode de valeurs” technique, and superposition of “personages rhythmique”). The musical examples in this chapter are organized more or less chronologically, though certain exceptions are made in order to present instances of serial manipulation of series (of durations and/or pitches) other than SPs, before discussing SPs proper. Section 3.1 outlines the development of Messiaen’s approach to using permutation as a compositional tool. It begins with analysis of three movements from Messiaen’s Livre d’Orgue as well as a brief consideration of Messiaen’s withdrawn electronic piece Timbresdurée. In each case, permutations (though not SPs per se) are used to manipulate pitches, durations, and rhythms, and they have some bearing on the sectional design of the piece. Next is an examination of Le Merle Noir which, while still not a true instance of the SP technique, represents a more systematic approach to permutation. Finally, 3.1 concludes with examples of SP proper in the piano composition Île de feu II, the last of Messiaen’s Quatre études de rythme. Section 3.2 begins with examples drawn from Catalogue d’Oiseaux. Some involve SPs proper, while others serve as counterexamples in which unpredicatble rhythms or series are generated by some method other than SP. The remainder of the musical examples come from compositions that employ the Chronochromie orbit of interversions in some manner: the “strophes” (movements II and IV) from Chronochromie; movements II and IV from Sept Haïkaï; short excerpts from the opera Saint François d’Assise; and the fourth movement of Éclairs sur l’au-delà….34 While surveying the usage of SPs and other techniques in certain of Messiaen’s compositions and commenting on which function (if any) is used in each case, Evidently, the composer considered his Chronochromie SP orbit to be a substantial resource for rhythmic material, and found its interversions interesting enough to have incorporated them into considerably more music than just a single work or movement. To my knowledge, there is no instance of the SP technique in his music (post 1960) other than those that draw their interversions from this same orbit. 34  64  the figures and tables in this chapter demonstrate a method of analyzing SPs that allows for meaningful assessment of particular interversions (i.e., rhythms or tone rows) and their superpositions.  65  3.1 – Musical examples: Messiaen’s “experimental period” (1949–1951) Reprises par interversion, the first piece in Messiaen’s Livre d’Orgue, represents one of Messiaen’s early experiments in composition with permutations of musical materials. Although it bears “interversion” in its title, it is not an instance of SP as defined in chapter 2. The piece is nevertheless of particular interest, because its permutation of rhythms and durations determines its four-part form. Example 4 below shows the first two sections. Section 1 (up through the end of the first system on second page) features three different rhythms (each of which Messiaen labels with Hindu names). As they repeat, Messiaen permutes the order in which they appear until they have been sounded in every of the 3! = 6 possible orderings.35 On the score, Messiaen labels the second section “Même musique, en éventail fermé, des extremes au centre”—“the same music, a closed fan, from the extremes to the centre.” ”Closed fan,” as presented in section 2.1, is a function that reorders a series of elements. The series here, consists of 54 elements, most of which are a single duration and pitch. A few, however, are a “parcel” of more than one discrete attack, which together occupy a single order position. In all, there are a total of 72 discrete attacks that are parceled into 54 order positions in the first section. The second section of the piece (example 4, systems 5–7) permutes, using a closed-fan scheme, this series of 54 elements, interleaving the first half of series with the retrograde of its second half. This can easily be seen by comparing the beginning of the fifth system with the beginning of the first system and the end of the fourth system. The third section is likewise a permutation of the 54 elements of the first, bearing the heading: “same music, an open fan, from the centre to the extremes.” Based on the  Each of the rhythms is slightly altered each time they appear: every IOI in each is either uniformly lengthened or shortened. This is a technique Messiaen called personnages rythmiques. Refer to Hook’s algebra of rhythm for a formal definition of this technique (Hook 1998, 103ff.). 35  66  information about closed- and open-fan schemes provided in chapter 2, the orbit sizes for these permutations on 54 elements could be determined easily enough. But it is unnecessary to do so, since each of sections 2, 3, and 4 is generated by a different permutation of the event series of section 1. In other words, the sections are not four adjacent interversions of a single orbit. Nevertheless, as figure 8 shows, the functions in this piece constitute a “looser” family of functions and orderings in the following sense: although the closed-fan and open-fan functions—recursively applied to a single seed series—define two different orbits, when they are each applied (once) to a single seed series they produce interversions that are retrogrades of one another (or nearly so, as we will see). This family is not closed under these operations, that is, applying every function to every element would yield elements not on the diagram (and not used in the composition).  Figure 8: Network of permutation functions among sections in Reprises par interversion  Likewise, the fourth and final section is the “retrograde” of the first; so the composition as a whole is an NRR. This overarching design contains a few irregularities that result from the fact that some among the 54 elements in the series contain more than one attack, and  67  that the attacks within a single order position are impervious to retrogradation. For example, attacks 8–10 constitute the ninth element of the series, and attacks 11–12 the tenth, so the sequence <Eb,G,E>, appears in the same order in the third measure of system 1 as it does in the third measure of system of system 5; and <F#,F> appears in the same order in the third measure of system 1 as it does toward the end of the third measure of system 5 (example 4). Thus neither sections 2 and 3, nor 1 and 4 are exact retrogrades of one another. Figure 9 is a graphic score of the piece in its entirety, where pitch class is indicated by vertical position of the coloured cells, and where duration is indicated (proportionally) by the horizontal length and by colouration of cells (the spectrum orange-yellow-green corresponding to shortest to longest IOIs). The figure serves to illustrate the approximate NRR across the piece as a whole, with annotations included to highlight certain of the nonretrograding multiple-attack-durations.  68  Example 4: Livre d’Orgue – I. Reprises par interversion  69  Figure 9: Graphic Score for Reprises par interversion – each section as a (re)ordering of the 54 elements of section 1  70  There are at least two other compositions from the same period that exhibit a similar approach to formal design, in that subsequent sections are generated by some permutation of the elements of a prior section. Messiaen’s withdrawn electronic piece Timbres-durées, for instance, comprises subsections of one of four types of musical materials which, over the course of fifteen minutes, are presented in every one of their 4! = 24 possible orderings, thereby defining twenty-four sections.36 Additionally, permutations manipulate the materials within subsection A, which is defined as a series of eleven unique durations (whose lengths are the first eleven primes, measured in sixteenth notes). Table 14 shows that this series appears in one of six different orderings (denoted by lower case letters a-e) in each of the 24 iterations (Messiaen omits durations from several series; omissions are greyed out in the table).  Table 14: Orderings of durations {2,3,5,7,11,13,17,19,23,29,31} in subsections 1-A through 24-A of Timbres-durées  Like Reprises par interversion, Timbres-durées permutes its elements in certain predetermined ways. The durations appear either in ascending or descending order, or in  See Battier [forthcoming] and Battier 2010, 17ff. for a summary of the permutation of the four subsections, as well as studies (including reproductions) of the unpublished manuscript copies of various drafts of the score. 36  71  an “open-fan” ordering (one of two), or in a “closed-fan” ordering (one of two).37 Figure 10 diagrams the relationships amongst the six orderings of this family.  Figure 10: Network of permutation functions among six different orderings (a-f) internal to Sub-section A in Timbres-durées  Again—as with the permutations among sections in Reprises par interversion— although there is a strong and symmetric pattern of retrograde relationships between the orderings, the relationships between all A subsections in Timbres-durées do not define a closed family. Neither do the functions that relate one subsection A to the next follow any consistent pattern as they would in an SP piece. In fact, more than the five functions shown in figure 10 are needed to describe the relationship between each subsequent iteration of A subsections in Timbres-durées (for example, the succession from b to e from 17A to 18A is a composition of f3 and f4). This is to say that although permutations of some kind are playing an important role in compositions such as these, they do not exhibit the uniformity of permutation function, and the resulting closed orbit, that is a defining characteristic of the SP technique proper. The process is perhaps more obvious here in that the “normal order” of the first statement of subsection A is the 11 durations in increasing order, where as the “normal order” in Reprises par interversion was an freely composed series, whose resource set is a multiset. 37  72  Les yeux dans les roués—the sixth piece from Messiaen’s Livre d’orgue—is a third example of a composition that uses permutations of some initial material to generate new subsections of the piece. Here, a series of twelve distinct pitches/durations in the pedal part is presented six times over the course of the piece, each time in a unique ordering. Example 5—a reproduction of the first two pages of the score—shows the first two iterations of the pedal series (a rest separates the iterations). Table 15 lists the six orderings of the pedal series, and figure 11 diagrams the network of permutation functions that relate one section to another. Figure 11 is, perhaps, more tight-knit than very similar network of orderings for timbres-durées in that it uses only a single “swap” function (there is only way to swap the contents of adjacent order positions when n is even). Nevertheless, it too is an “open” network, insofar as new orderings of the 12 elements could be obtained if the permutation functions were applied to every ordering. Although this analysis shows, abstractly, that Les yeux dans les roués is composed in a similar way as Reprises par interversion and Timbresdurées, it is unique, and more complex than those prior examples because permutation of a series of pitches/durations makes up only one layer of the composition.: the permutation of the pedal series is presented at the same time as two furious streams of straight sixteenthnotes in the right- and left-hand parts.38 In Reprises par interversion and Timbres-durées, in contrast, permutations of a series account for the pieces in their entirety. In all musical examples discussed below Messiaen superposes two or more orderings at a time; often gilding these structures with additional layers of unrelated, improvisatory musical materials.  The right- and left-hand parts each sound their own twelve-tone row, which themselves are permuted according to various (arbitrarily selected) permutation schemes. It is difficult to trace these rows since Messiaen occasionally omits pitches from the row (as he did the series of durations in Timbres-durées), or elides beginnings and endings of row forms (Trawick 1991, 25). The row forms and their permutations will not be investigated further here. 38  73  Table 15: six orderings of 12 durations/PCs of the Pedal in Les yeux dans les roués  Figure 11: Network of permutation functions among six different orderings of Pedal material in Les yeux dans les roués  74  Example 5: Livre d’Orgue – VI. Les yeux dans les roués  75  Soixante quatre durées, the last of six pieces in Messiaen's Livre d’orgue, also superposes three textural layers (see example 6, which reproduces the first two pages of the score). One layer is restricted to the pedals, where sixty-four dyads are sounded such that the IOI between each attack is a unique duration between one and sixty-four thirty-second notes (hence the title of the piece). A second layer also consists of sixty-four attacks, the IOIs between which are the sixty-four unique durations between one and sixty-four thirtysecond notes. This second layer is sounded by either the right or left hand at different points in the piece, and is predominately dyads, although the simultaneities of this layer expand into triads in mm. 90–107, and a few sonorities with four or more notes occur thereafter. A third layer (also shared between the right and left hands) projects freer, improvisatory outbursts, some of which are stylized birdsong transcriptions (TRCO-III 226).39 It is important to recognize that the sixty-four durées in this piece are not durations in the strict sense, but rather IOIs: they measure the interval between attacks, rather than time spans from onsets to offsets. For example, in example 6 the right-hand attack in measure 17 is labeled 57, which is the time-span—measured in 32nd notes—until the left-hand attack (labeled 58) three measures later. The IOI is 57, but the duration of the E-B dyad itself is 14. In the intervening span the right hand assumes the role of the improvisatory layer, with the left hand taking over what was the uppermost stream on the next attack. In TRCO, Messiaen begins his analysis of Soixante quatre durées by writing out a series of chromatic durations, one through sixty-four 32nd notes, in ascending order (TRCO-III 218–225). Next, he narrates his method of choosing a new ordering of those elements: first taking the last (i.e. longest) four durations in ascending order for positions 1 through 4; then choosing the first (i.e. shortest) four durations in descending order for positions 5 The bird-song layer is an instance of what Messiaen called monnayage or “minting”: Messiaen thought that hearing short notes would assist a listener in being able to measure long durations because the short values would imply the basic durational unit of which all long durations are multiples (TRCO-III, 84, 226, 320ff.; Bauer 2007, 148). 39  76  through 8 in the new ordering. Continuing this process, Messiaen composes a new ordering of all sixty-four elements, which he describes as being an order of reading “from the extremes to the centre,” and thus symmetrical: <61,62,63,64,4,3,2,1,57,58,59,60,8,7,6,5,…,32,31,30,29>.  This language is familiar from Messiaen’s description of SPs, and especially the “closed-fan” scheme, but the “ordre de lecture” employed here is not exactly that (wherein n maps to 1, and n–1 maps to 3, etc.) unless the series is interpreted as being composed of 16 rather than 64 elements, each element being a “parcel” of four successive integers in either ascending or descending order. (“Chunking” the series into 16 parcels would certainly make it easier to try and hear the series as permutations of one another, but Messiaen is quite deliberate about describing the piece in terms of its 64 durations.) In either case, neither of the permutation functions by which Messiaen derives these orderings from the (ascending) series of chromatic durations defines an orbit that constitutes an instance of an SP as defined in chapter 2. This is because neither function describes the relationship between the actual series in the piece. The function that does describe the relationship between the two layers of the piece is the retrograde function. Figure 12, below, illustrates Messiaen’s explanation for the derivation of these two orderings, from the seed series <1,2,3,…,32>, showing the two larger implied orbits, in blue and red, of cardinalities 5 and 10, respectively (where f1 derives the series which is sounded by in the right hand, and f2 the series sounded in the pedal (as of m. 1)). In the centre of the figure, the black arcs represent the retrograde function. The use of a series with its retrograde could be considered an instance of SP (an orbit of 2 orderings), but it is something of a trivial case.  77  Example 6: Livre d’Orgue – VII. Soixante quatre durées  78  Figure 12: Orderings of 64 IOIs in Soixant quatre durées as members of SP orbits  79  Figure 13: Two large-scale rhythms, and their non-retrogradable composite rhythm from Soixante Quatre Durées  Segment 1  Segment 2  Segment 3  Segment 4  Segment 5  Segment 6  Segment 7  Segment 8  61 29  30  62 31  32  63 36  35  29  30  2 29  32  35  28  57 25  26  27  58  25  26  6  53 21  22  23  21  22  10  21  40  28  9  18  19  17  18  14  24  44  24  17  50  5  45 13  14  15  16  13  14  15  3  48  20  25  42 56  43  9  10  11  11  41  25  60  5  6  7  8  11  33 2 3  4  64  2 3  4  23  28  34  35  7  28  38  11  40  9  6  5  8  7  6  5  10  9  15  14  13  15  14  13  45 13  3  19  18  17  18  17  49 20  5  14  23  22  21  22  21  53 24  28  31  7  11  13  10  27  58  28  8  11  24  17  2  41  25  27  3  9  54  28  4  10  20  44  2  11  33  23  3  12 41  50  28  26  25  26  25  57 28  21  6  36  32 62  31  30 61  29  35  32  29  2 30  29  63 34  37  16  19  59 11  38  4  42  48  39  38  23  46  55  38  34  52  19  12  8  25  51  11  7  28  47  33  33  56  27  52  10  6  11  43  23  41 9  37 5  28  47  46  13  31  51 20  34  60 39  55  13  7  59 28  54  49 17  64  80  Julian Hook, in his seminal analysis of Messiaen’s Turangalîla, remarks that “when several [parts] play rhythms that clearly function [i.e. act] together, the composite rhythm can be considerably more revealing than any of the individual parts” (Hook 1998, 103). Such is the case with the two large-scale rhythms in this piece. A composite is “the rhythm heard when [two or more rhythms] are presented simultaneously; its points of articulation are all points of articulation of [every one of the rhythms]” (Hook 1998, 103).40 Hook has shown that composite rhythm of every (finite) rhythm superposed with its retrograde is always an NRR (Hook 1998, 103-105). A simple example of this is the opening passage of the eighteenth of the Vingt Regards sur l’enfant Jesu (Regard de l’Onction terrible), where the right hand performs sixteen straight sixteenth notes, followed by a series chromatic durations one through sixteen (measured in sixteenths) in ascending order; while the left hand performs the retrograde version of this same rhythmic series. The approximate NRR in Soixante quatre durées is a more sophisticated manifestation of the same principle: two of its layers combine to approximate a single large-scale NRR that serves as a gentle, if unpredictable, backdrop against which melodic outbursts are contrasted. Figure 13, above, illustrates this process in an immediately visible manner, proportionally representing durations as lengths (red-yellow-green corresponding respectively to short-medium-long). In each segment, the “open/closed fan”-like rhythms are displayed one above the other, with the composite rhythm slightly below. The colour 40Writing  a formal, symbolic definition of composite rhythm is possible, but not often necessary, since, as Hook observed, “musical notation…provides an intuitive method for calculating a composite rhythm.” He nevertheless offers a precise algorithm for calculating the composite rhythm, Z, for two rhythms X and Y. (I have made small adjustments to the formula to reflect the notation used elsewhere in this thesis.) Let X = <x1, x2,…, xm> and Y = <y1, y2,…, yn>, where each xi and yi is a single duration or rest. Let z = min{x1, y1}; assume that z is defined to be a rest if both x1 and y1 are rests and a duration otherwise. Then either z = x1 or z = y1 or both. If z = x1 < y1, define y1’ = y1 – z, X’ = <x2, x3,…, xm> and Y’ = <y1’, y2,…, yn>. If z = y1 < x1, define x1’ = x1 – z, X’ = <x1’, x2,…, xm>, and Y’ = <y2, y3,…, yn>. Finally, if z = x1 = y1, define X’ = <x2, x3,…, xm> and Y’ = <y2, y3,…, yn>. Then, in every case, X * Y = <z, (X’ * Y’)>. This formula reduces the calculation of the composite rhythm of X and Y to the calculation of a composite of two rhythms X’ and Y’ of smaller total duration, and the method may be repeated recursively until one of the rhythms is empty. (Hook 1998, 103 fn.)  81  coding makes the retrograde symmetry immediately visible (left-right and top-bottom on the graphical layout). Interestingly, as a byproduct of the particular way that Messiaen parceled together groups of four consecutive chromatic durations, the piece manifests a simultaneous attack between the streams every 260 thirty-second notes, and at no other time. This is because the durations are parceled such that IOIs in positions (8i+1)+n, and (8i+1)+(n+4) will always sum to 65 (where n<4). On paper, this structure divides the composition into eight equal sections. But this division is not supported in any perceptible way in the music: at times Messiaen seems to emphasize moments when there are simultaneous attacks between streams (e.g. “Section 7”, bottom system p. 40 (Messiaen, 1953)), but elsewhere he obscures them (e.g. “Section 6”, second system, p. 39 (ibid)). When it comes to other rhythms that are related one to another by some permutation function other than retrograde, the composite rhythm will not typically be an NRR; but in keeping with Hook’s observation, it seems worthwhile to consider such rhythms as “functioning together” to some extent. As such, subsequent analyses in this chapter take composite rhythms into consideration as much as (or even more than) they interpret the superposed series as polyrhythmic streams.41 The piano accompaniment from the coda to the 1951 composition Le Merle Noir (for flute and piano, reproduced as example 7) provides a good example of superposed streams combining (more or less) into a single entity.42 The coda uses the permutations of a series as  Messiaen’s attitude toward his own rhythmic serial procedures was inconsistent. Despite elsewhere imploring listeners to discriminate intervals between durations in discrete (and perhaps simultaneous) streams, he offers what might be taken as an endorsement for hearing complex polyrhythmic music in terms of composite rhythms; saying, “[one ought to] realize that … in [a] lack of rhythm, there are thousands of superimposed rhythms which combine in a great rhythm and in blocks of duration” (Samuel [1967] 1976, 88). 42 In his introduction to The Messiaen Companion Peter Hill describes Messiaen’s Le merle noir for flute and piano as “the first work entirely based on birdsong.” He then positions the work as Messiaen’s “way forward”—an escape from the “cul-de-sac” that was his period of radical experimentation (Hill 1995, 8). Not only is this a gross oversimplification (given the serial 41  82  compositional material in a more exhaustive way than does any work from the “experimental period.” It consists of free bird-song flourishes in the flute, supported by a piano accompaniment that exhausts the 24 distinct orderings of four durations (1, 2, 3, and 4 sixteenth-notes) twice over—once in each hand.43 Further, the sequence in which the 24 orderings are presented is combinatorial: the 24 orderings are exhausted between the two hands as soon as each has sounded its first 12 orderings; and exhausted again as soon as each has sounded its remaining orderings (see figure 14). It so happens that the sequence in which they appear in one hand is a rotation of the sequence in which they appear in the  Example 7: Le merle noire (Coda)  procedures found Messiaen’s quintessential birdsong opus, Catalogue d’oiseaux (Cheong (2007)); it is not even true of this piece. I point this out to demonstrate that Messiaen’s “experiments” with permutation, though not always at the fore, remained part of this compositional language long after 1951, as Section 3.2 shows. Manifestly, Messiaen was not seeking a “way forward.” 43 In Messe de la pentecôte – II. Offertoire (Les choses visibles et invisibles) Messiaen presents many of the orderings of five elements, but he curtails the episodes before exhausting every one of one of the 5! = 120 orderings, despite superposing two, or even three orderings at a time. This illustrates the “need” for the SP technique—to be able to work with permutations of greater than four durations, while maintaining some sense of completeness or closure.  83  Example 7 continued: Le merle noire (Coda)  84  other (beat-class transpositions of +120 sixteenth notes, out of a total cycle of 240), meaning the composite rhythm of the first half is identical to the composite rhythm of the second half (see figure 15).44  Figure 14: 24 orderings of four durations in Le merle noir  Messiaen’s particular use of permutation of durations in this piece involves a paradox. Compositionally, or at least theoretically, such a technique (exhausting all possible permutations, and superposing them combinatorially) espouses an aesthetic that values completion, order, and cohesion. What the technique yields is quite the opposite: in this case a protracted composite rhythm of unpredictable alternation between durations as short as one and as long as four sixteenth-notes, that oscillates rather unpredictably between monads and dyads. While there is a simultaneous attack exactly every ten  There ways to superpose two of 24 orderings (assuming their initial attacks are alligned), but I have calculated that among these there are only 168 unique composite rhythms (though some may have a different profile as to simulteneous attacks between streams). The retrograde-related rhythms <1,3,1,2,1,2> and <2,1,2,1,3,1> (where integers represent durations in sixteenth-notes) can each be obtained six different ways. 44  85  sixteenths, this regularity is obfuscated in at least two ways. First, as previously mentioned, there are other dyads that appear occasionally in the interim (see the accent symbols beneath the composite rhythms in figure 15). Second, since the pedal is depressed throughout the episode, the occasional attacks in the lower register (approx. C4–C#5) sustain longer than the notes in higher register, and tend to stand out both as contour accents, and at times, even as a separate melodic stream (e.g., the left-hand <Bb4, C#5, B4, A4,…> in the first two systems of example 7). The paradox—that Messiaen’s compositional approach seems to value “completeness” and “order” while in effect producing a “randomization” of elements (within certain parameters)—is common to many of the musical examples that follow.  86  Figure 15: 12 unique composite rhythms obtained by superposition in Le merle Noir  1  r.h. 1 l.h. 2 composite 1 >  2  4  6  7  3 3  4  1  2  4  3  4  2  1  1  4 2  1  4  2  1  4  3  1  3  18  composite  3  1  1  1  2  1  2  3  3  1  3  2  2  1  r.h. l.h.  1 4  3  1  >  10  r.h. l.h.  1  1  3  r.h. l.h.  1  20 1  3  2  1  3  4  3  4  1  4 3  1  3  1  3  4  1  2 2  1  4  r.h. l.h.  1  3 3  1  2  3  4  2 2  1  4  1  1  4 1  1  3 3  2 3  1  1  4 2  1  1  2 2  1  3  1 >  3  1  2  3  4 1  2  1  4  1  3  1  2  1 >  2  3  2  4  3  r.h. l.h.  1  2 2  1  1  2  1  22  1  4  2  3  1  2  1 3 1  1  1 1  2 2  1  4  3  1  3  1  1  1 1  1  r.h. l.h.  4  composite  23  1 >  1  2  1  2  1  2  1  3 4 3  2  3  1  3  1  3  1  4  2  2  2  1  4  3 2  3  1  2  1 1  2  1 >  4 2  1  >  24  r.h. l.h.  4 3 3  composite  1  >  3  composite  4 1  r.h. l.h.  2 2  >  >  3 1  4  composite  2  2  3  >  >  3  composite  >  3  composite  21  3  2  1  4  r.h. l.h.  3  >  >  >  12  3 3  4  1  4  composite  4  2  3  composite  2 2  1 4  >  11  1  3  > 3  composite  4 4  > 3  composite  1  r.h. l.h.  >  9  4 >  > 2  1 1  19 2  1 >  4 4  2 3  3  2  r.h. 2 l.h. 1 composite 1 >  4 4  >  r.h. 2 l.h. 1 composite 1 >  3  1  4 4  composite  1  3 3  1  17  >  r.h. l.h.  r.h. l.h.  1  2 4  >  8  1  3 3  1  r.h. 2 l.h. 1 composite 1 >  2 3  3  1  2  3 4  1  16  1  r.h. 2 l.h. 1 composite 1 >  2  1  2  15  1  4 3  1  3  1  3  1  1 2  r.h. 2 l.h. 1 composite 1 >  >  3  1  14  3  >  r.h. 1 l.h. 2 composite 1 >  4 4  1  r.h. 2 l.h. 1 composite 1 >  >  2  1  13  4  >  r.h. 1 l.h. 2 composite 1 >  4 4  3  1  r.h. 1 l.h. 2 composite 1 >  5  1  r.h. 1 l.h. 2 composite 1 >  3 3  1  r.h. 1 l.h. 2 composite 1 >  3  2  3 1  2 1  1  1  1  2 4  1  1  2  >  87  Île de feu II proves an interesting example of permutation symétrique, since it contains two separate applications of the technique—one acting on pitch-classes of uniform duration (example 8), the other on pitch-classes each with a unique duration (example 9). Toward the end of the piece, a brisk passage in straight sixteenth notes articulates consecutive 12tone rows (six in each hand). (The rows are not labeled as “interversions” in the published score, as they often are in later compositions. Nor is the passage identified as an instance of the SP technique in TRCO.) The permutation function f = (1 11 2 5 3 7 12 10 8 4 9 6) relates the first row of the right hand to the first row of the left hand; the first row of the left hand to the second row of the right hand, and so on (see figure 16 and example 8).  Figure 16: Matrix for SP of 12 pitch classes in Île de feu II, mm. 70–75 f = (1 11 2 5 3 7 12 10 8 4 9 6) [order positions] or f = (F C Bb Db E D F# B A G Eb Ab) [pitch classes]  According to Schoenbergian criteria, the selection of rows here may seem rather haphazard, since the rows are not related by standard TTOs, and are devoid of the rich motivic connections common in twelve-tone music. Here Messiaen simply uses the aggregate as a “mode” that contrasts the other modal flavors elsewhere in the piece. But that the row 88  forms seem arbitrary should not be altogether surprising, considering the fact that Messiaen did not so much choose the rows, as he chose a row, which—by way of its derivation from the ordered chromatic scale (ascending from C)—defines a permutation function that generates the other rows.45 Nonetheless, there is a compelling intervallic property that emerges in this passage which results from the superposition of the interversions. Since the durations for all order positions are uniform, it is appropriate to display the interversions on a matrix. Table 16 pairs together the interversions that are presented simultaneously; vertically-aligned pitch-classes represent pitches that sound as dyads in the actual music. Beneath each vertical pair of pitch-classes the table lists the interval class between the two. The rightmost portion of the table (in dotted lines) tallies the multiplicity of each such vertical interval class. Of course, since the duration for each pc is fixed, the superposition of any two adjacent interversions will afford an ordering of the same set of twelve dyads, so the multiplicity of each vertical interval class will be the same for all superpositions. But it shows an intervallic consistency between all superpositions of adjacent interversions: within this SP orbit, is it not possible to superpose two adjacent interversions and obtain a dyad of interval classes 0, 1 or 6. This is because, when superposing two interversions at time, it is only possible to obtain the interval classes that are found between the pitch-classes that are adjacent in the cycle notation (expressed in terms of pcs rather than order positions; see figure 16 and table 16). That three of the possible seven interval classes (0–6) do not occur makes the five-measure passage sound  Messiaen said something very similar himself regarding the SPs in Chronochromie. From an interview with Claude Samuel: Samuel: You choose the [orderings] that interest you? Messiaen: I’m not the one who chooses, it’s the procedure. Samuel: But you control the procedure. Messiaen: Exactly. (Samuel [1986] 1994, 80–81) 45  89  quite uniform: one continuous episode, rather than six consecutive double statements of the aggregate.  Example 8: Île de Feu II – interversions of 12 pitch-classes (mm. 70–75) f = (1 11 2 5 3 7 12 10 8 4 9 6) [arrows trace paths of pcs 5 and 8]  90  Table 16: Superposed PC interversions and their resulting interval classes (Île de feu II)  f = (F C Bb Db E D F# B A G Eb Ab) IC: 5 2 3 3 2 4 5 2 2 4 5 3  91  Example 9: Île de Feu II – SP of of 12 pitch-classes and durations (Interversions 1–4)  92  Figure 17: Matrices representing SP of series of durations in Île de feu II  f = (1 12 11 9 5 4 6 2 10 7)(3 8) A)  B)  C)  93  Figure 18: Superposition and composite rhythms in Île de feu II  94  Table 17: Interval class- and Rhythmic profiles of superposed interversions from Île de feu II  95  The other SP orbit in Île de feu II is made up of orderings wherein each of the twelve order positions is occupied by a unique pitch class, which is assigned a unique duration and a unique dynamic marking/articulation (see example 9).46 Again Messiaen superposes two interversions at a time (one in each hand), and presents all 10 interversions of this SP orbit over the course of three separate episodes: interversions 1 and 2 immediately followed by interversions 3 and 4 (mm. 8–27), interversions 5 and 6 immediately followed by interversion 7 and 8 (mm. 35–54), and finally, interversions 9 and 0 (or 10, in Messiaen’s labeling) (mm. 76–85). The permutation function that relates one interversion to the next is f = (1 12 11 9 5 4 6 2 10 7)(3 8) which, as mentioned in chapter 2, is of the “open-fan [2]” scheme, with cardinality 10. We know (from his explanation in TRCO) that Messiaen chose his permutation function by way of selecting an interversion 1, then deriving the subsequent interversions in the same manner that interversion 1 was derived from a seed series. In other pieces the seed series was <1,2,3,4,…n>.47 Had Messiaen used that seed series here, however, the permutation function would be different, and its orbit would have been of cardinality 11 rather than 10. Were this the case, Messiaen could not have “evenly” superposed the interversions two at a time as to use each only once. It seems then that while he did not possess a complete understanding of the mechanics of the SP technique, Messiaen knew it well enough by this point to be able to produce an orbit cardinality that suited a particular compositional purpose.  Because of this, Peter Hill mistakes the “interversion” episodes in Île de feu II for being instances of “the mode de valeurs” compositional technique (Hill 1995, 320). I will show at the outset of section 3.2 how the two techniques are decidedly different, and how they might be distinguished aurally. 47 As discussed in chapter 2, names like “open-fan 2” have been used to describe certain orderings, as well as certain permutation functions. Note well, the seed series, interversion 0, for this SP orbit is <12,11,10,9,8,7,6,5,4,3,2,1>. Jedrejewski (2006, 189) and Benson (2006, 341–342) have both remarked on the implication of the Mathieu group M12 in Île de feu II, which can be generated by the functions f1=(1 12 11 9 5 4 6 2 10 7)(3 8) and f2 = (1 11 10 8 4 5 3 7 2 9 6)(12). The latter is the function that would derive the same interversion 1 = <6,7,5,8,4,9,3,10,2,11,1,12> from the seed series interversion 0 = <1,2,3,4,5,6,7,8,9,10,11,12>. But that series is not present in the music, and neither does that function relate any two interversions. 46  96  Figure 17 shows how it is possible to display a complete SP orbit (such as the orbit for the series of durations in Île de feu II) on a matrix, as was done for SP of series of pitch classes in figure 16. In this case, vertical alignment is an indication of values (be they durations or pitches) occurring in the same order position. The columns also illustrate the discrete cycles of the given permutation function (see figure 17A, where the 10-cycle is coloured in gradations of purple, and the 2-cycle in green). Reading such a matrix horizontally indicates the distribution of different values throughout a given series (see figure 17B, where the spectrum orange-yellow-green denotes shortest-medium-longest IOIs, respectively, in a given series). But these matrices suffer from serious deficiencies when their values represent durations. This is because when realizing the series as rhythms, order position is not a particularly salient feature, since the moment when the attack from a given order position occurs can vary greatly from interversion to interversion. For example, in the Île de feu episodes, order position 8 can occur as early as timepoint 48 and as late as timepoint 72 (relative to the initial attack of the interversion rhythm, 0, measured in sixteenth notes). Given the total timespan of the rhythm (78 sixteenth notes), this means order position 8 might occur as early as 61.5%, and as late as 92.3% of the way through the total timespan. A more meaningful metric is: which timespans are occupied by which IOIs. Figure 17C accomplishes this by translating the information from the prior two matrices so that the integers represent timepoints when an attack occurs (all series starting with an attack at timepoint zero). Figures D and E are a better visualization still, representing durational values proportionally as lengths (D, like A, is coloured according to cycle, while E, like B, is coloured according to IOI). They illustrate the difficulty of perceiving (aurally) the orderposition cycles when the contents of the order positions are non-uniform durational values that seem obvious when displayed on a nonproportional matrix. 97  Because the durational series from Île de feu II are superposed two at a time, it is appropriate to consider the composite rhythms that they afford. This tells us something about the rhythmic profiles of each episode (which I discuss shortly), but it is also a necessary step in determining the intervallic/harmonic profile of each superposition. Figure 18 displays all five superposed interversions, each as a series of rectangles whose lengths are proportional to the durations of their respective order position; the composite rhythm of each is shown likewise, directly below them. (Again the information is presented in two colourations, one representing the cycles, the other IOIs). As with the representation of composite rhythms from Le merle noir (figure 15), accent symbols below the composite rhythms indicate a simultaneous attack between the two interversions. The composite rhythms beneath the superposed interversions of figure 18A (purple/green) also display the interval classes between all simultaneously sounding pitch-classes. Table 17A summarizes this information, indicating that while the interval content of each interversion is distinct, all follow roughly the same profile: ICs 0 and 6 being least common, ICs 1, 2 and 4 most common. Likewise, table 17B summarizes the data from figure 18B. It tallies the number of simultaneous attacks in each superposition, and it also indicates median, average and mode IOIs composites (trivial for the interversions themselves), and assigns each interversion and composite a “smoothness” rating. The smoothness rating is the sum of the differences between each IOI and the IOI in the next order position. Given a series of durations A = <a1, a2, a3,…, an> , “smoothness” = ∑  |  |. The “smoothest” possible ordering  for a set of chromatic durations {1,2,3,…,n} is the series in either ascending or descending order, where the smoothness rating is equal to n–1. (The table also shows the percentage increase of each rhythm from the smoothest possible ordering of those same elements, and the average difference of successive IOIs, i.e. smoothness divided by one less than the 98  number of elements in the series; =  ∑  |  |  . These measures are helpful in making  more meaningful comparisons between rhythms of different lengths.) A few of statistics from table 17 stand out. It so happens that the two smoothest interversions (9 and 0) combine into the second-most “jagged” of composite rhythms. This, perhaps, is an example of what Messiaen meant when he claimed that his SPs afforded adjacent interversions that were suitably different from one another (though I still maintain that this is not a necessary property of the technique itself, it is only true of the particular ways Messiaen employed the technique). Additionally, table 17 reveals a certain consistency in the way Messiaen grouped the superposed interversions into the three separate episodes. The smoothness ratings of the composite rhythms in the first episode are very similar to one another. Likewise, the ratings of the composites internal to the second episode are highly similar to each other, while differing considerably from those in the first. Further, Messiaen groups the superposed interversions such that both the first and second episodes sound exactly five simultaneous attacks between their streams: the first episode involves three such simultaneities from the superposition of interversions 1 and 2 plus two from the superposition of interversions 3 and 4, and the second episode involves four from interversions 5 and 6 plus one from interversions 7 and 8. In section 3.2, similar metrics and visual representations will be used to investigate more complex musical examples that involve the permutation of 32 durations.  99  3.2 – Musical examples: Catalogue d’oiseaux (1958) and after The most well-known application of SPs is in Chronochromie, where Messiaen superposes three adjacent interversions from an SP orbit of a series of 32 durations. This very same orbit is the resource from which Messiaen selects material for a number of compositions written between 1960 and 1992 (see table 23). Before assessing these instances of the SP technique, a few other musical examples from Catalogue d’oiseaux show Messiaen’s continued use of “serial” techniques after 1951; whether SP (of pitch-classes); or other techniques for generating complex rhythms, such as mode de valeurs, or personnages rythmiques. Le Chocard des Alpes, the first piece of Messiaen’s large collection Catalogue d’oiseaux, opens with a passage that is somewhat similar in rhythm and texture to the “interversion” episodes from Île de feu II. The technique used to generate this rhythm, however, is not SP of durations, but rather, Messiaen’s personnages rythmiques.48 Here, the right hand sounds four repetitions of a series of three rhythmic characters, <A, B, C> while the left hand sounds three repetitions of its series of four rhythmic characters, <D,E,F,G>. Upon each repetition, each duration internal to rhythms B, D, and F is increased by one sixteenth note. The durations internal to rhythms C, and G remain constant, while the durations internal to rhythms A, and E are decreased by one sixteenth upon each repetition (figure 19 illustrates). (The overall length of the episode is determined by the procedure itself, terminating as soon as it finishes the first cycle wherein a duration internal to a rhythmic character has been decreased to IOI = 0 (Character “A” in figure 19).)  The idea is that the different rhythms, or “characters”, are differently modified by some value(s): the IOIs internal to one rhythm being increased, while in another they are decreased, and while still another remains unchanged upon each repetition. See Hook (1998, 103ff.) for a formal explanation of the technique. 48  100  On paper, this application of the technique affords slightly more than double the number of simultaneous attacks (per timespan) than did the superposed interversions from the SP orbit in Île de feu II, which manifested twelve simultaneous attacks across a span of 390 sixteenths (= 0.030 attacks/sixteenth), where the episode in Le Chocard des Alpes registers a total fourteen across a span of 208 sixteenths (= 0.067 attacks/sixteenth, see table 18). But this potentially distinctive attribute is difficult to perceive, since it is difficult (as a listener) to separate the right and left hand parts into discrete streams, and further, because Messiaen here grants himself greater freedom with regard to pitch materials than in Île de feu II, where all attacks were a single pc, sounded in perfect octaves. In Le Chocard des Alpes, the right- and left-hand sonorities range across the episode from single pitches to pentachords. As a result, it is not always possible for a listener to ascertain whether a large sonority is from a single stream, or the result of a simultaneous attack between streams (e.g. left hand {F4} with right hand {G4,Eb5}, fourth system, example 10), whereas simultaneous attacks between streams in Île de feu II have some phenomenological presence—being a struck dyad rather than a monad (doubled at the octave). The fifth piece in Catalogue d’oiseaux, La Chouette Hulotte, likewise opens with a serial episode; this time a “mode de valeurs et d’intensités” (example 11). This technique, famously introduced in the second of Messiaen’s Quatre études de rythme, associates a unique IOI (and articulation and loudness) to each pitch height. In La Chouette Hulotte the range is from A0 to A4, the highest pitch being a single sixteenth-note in length. Every lower pitch is assigned an IOI (measured in sixteenths) equal to one more than its distance (in semitones) below A4. Table 19 summarizes the mode, and provides a tally of the number of times each of the pitch/duration complexes is used across the two “mode de valeurs” episodes in the piece. Figure 20 is a graphic score of these episodes, assigning a colour to each pitch class (according to table 19) and representing durations proportionally as horizontal distances, 101  which also includes the composite rhythm below each coloured system. The lower portion of table 19 summarizes the data of these composite rhythms. Looking at figure 20 and table 19, it becomes apparent that Messiaen (when using the “mode de valeurs” technique) preferred to avoid two pitches of the same pitch class sounding at the same time. Beyond this, the episodes seem to be freely composed. As far as the composite rhythm of a given mode de valeurs episode, Messiaen is free to create or avoid simultaneous attacks (that is, struck dyads, or trichords) between two or three of the streams as he chooses.49 This is quite a different situation than in the case of superposed interversions that comprise durations of varied length, where the odds of a simultaneous attack between two streams or more streams is fixed, and where there is nothing prohibiting octaves or unisons being sounded together, or equal durations occurring over exactly the same time span. One could tally the frequency with which the different trichords occur over the course of the mode de valeurs episodes (just as was done for the interval classes for dyads across the interversion episodes from Île de feu II) but doing so is outside the scope of this project since the trichords that appear in La Chouette Hulotte’s mode de valeurs episodes are freely chosen by the composer, as opposed to being the direct byproducts of the procedure itself.  With one important exception: if the amount of time (in sixteenths) remaining in any sustained pitch q (at the moment of an offset in another stream) is equal to q+12n, for n>0, then the next possible simultaneous attack is prohibited according to the preference rule that there is no two pitches of the same pitch class sounding at the same time. 49  102  Example 10: Catalogue d’oiseaux (première livre) – I. Le Chocard des Alpes  103  Figure 19: Personnages Rythmiques and composite rhythms in Le Chocard des Alpes  Table 18: Statistics on “personnages rhythmiques” episode in Le Chocard des Alpes  104  Example 11: Catalogue d’oiseaux (troisième livre) – V. La Chouette Hulotte  105  Figure 20: graphic score – “mode de valeurs” episodes in La Chouette Hulotte  106  Table 19: Statistics on “Mode de valeurs” episodes in La Chouette Hulotte  107  The SP technique proper is used sporadically across Catalogue d’oiseaux (as shown by Cheong 2007), but in every instance it manipulates orderings of unique pitch classes, rather than pitch class/duration complexes (where durations too are varied). This makes the assessment of the superposed interversions more straightforward than in the well-known interversion episodes of Île de feu II. Short passages from La Courlis Cendré (example 12) and La Buse Variable (example 13) are particularly clear examples. In both cases, each event in the series of twelve unique pitch classes is assigned a uniform durational value (a sixteenth note and an eighth note, respectively); and both implement 35 unique orderings of the twelve pitch classes from one and the same SP orbit (see table 20).  Table 20: Matrices for Catalogue d’oiseaux ‘s SP orbit of pc interversions  108  The serial episode in La Courlis cendré (which Messiaen labels “l’eau” in the score, example 12) superposes two adjacent interversions (in right and left hands, respectively) creating a rushing stream of dyads in straight sixteenth notes, commencing with the superposition of interversions 1 and 2. Because the period of this permutation function is odd (its cycle lengths being 7 and 5) the last new ordering, interversion 0, is sounded only as it is superposed with the first repeated ordering, interversion 1. The odd parity of the orbit, and the distribution of the paired interversions into right and left hands, vaguely implies that the episode only represents half of its periodicity: it would take another pass through the cycle before interversions 1 and 2 would return in their original configuration. Table 21 shows the way that the superposed interversions of this orbit are paired in right and left hands in the SP episode from La Courlis cendré, and summarizes the interval-class content of the superpositions. In each  measure of this episode the permutation function  f = (1 2 6 12 3 8 7)(4 10 9 11 5) relates a twelve-note series in the right hand to the left hand’s twelve-note, f also relates the left hand’s twelve-note series to the right hand’s of the subsequent measure, and so on. Because the durational values across these series are uniform, and because there is no summand of this particular permutation function whose length is one, we know that there will be no interval class 0 dyads sounded at any point in the episode. Again, for pc interversions where duration is fixed it is the case that every superposition of interversions i and i+x ( mod m) produces an ordering of the same twelve dyads. In La Courlis cendré—as in the pc interversion episode in Île de feu II—the collection of dyads produced by the superposition of interversions is quite distinctive for the considerably uneven multiplicities of each of vertical interval class. Here, the episode exhibits a rather dissonant quality overall, since each pair of superposed interversions manifests four IC 1’s, and four IC 6’s, to only one each of ICs 2, 3, and 4. In La Buse Variable, 109  Messiaen differently superposes interversion from the same orbit to produces quite different interval content (see example 13 and table 22). Here, in every two  measure  span of the episode, the right hand concatenates interversions i and i+1, while the left hand sounds interversions i+2 and then i+4 (mod 35). Put another way, the permutation function f 2 relates all superposed interversions, while consecutive interversions in the independent hand are related, alternately, by f and f 3. Table 22 shows the f 2 permutation function in cycle notation, and lists the f 2 superpositions and the resulting interval content. This method of superposition achieves decidedly more consonant interval content across the twelve dyads between the superposed pc series: there are no tritones, but there is a preponderance of IC 4 and 5s (table 22). Only a single piece in the Catalogue—Le merle de roche—uses what might be considered SP of a series of IOIs. It contains an episode wherein chords in a series are held for oddly specific, unpredictable durations. As it turns out, this rhythmic series is the particular ordering of 32 chromatic durations that will come to be known as interversion 1 of the Chronochromie SP orbit.50 But one ordered series by itself cannot constitute an instance of SPs—the SP technique being a method of producing a number of orderings (≥2) all of which are related one to another by a single function.  The compositional history of this passage is curious. Messiaen obviously had worked out this ordering of the 32 chromatic durations by 1958, but later, when questioned about the episode, seems to have forgotten he used the series in Catalogue d’oiseaux (Holloday 1974, 432; Benitez 2011, 269). It is difficult to believe that Messiaen produced this ordering twice by accident. Either he was already working out the Chronochromie SP orbit while composing Catalogue d’oiseaux (c. 1957-8), or stumbled across a sketch of the rhythm while composing Chronochromie (c. 1960). 50  110  Example 12: Catalogue d’oiseaux (septième livre) – XIII. La Courlis Cendré  111  Example 13: Catalogue d’oiseaux (septième livre) – XI. La Buse Variable  112  Table 21: Superposed pc interversions and resultant ics in La Courlis Cendré f = (1 2 6 12 3 8 7)(4 10 9 11 5) f = (C Db F B D G F# C)(Eb A Ab Bb E Eb) IC: 1 4 6 3 5 1 6 6 1 2 6 1  113  Table 22: Superposed pc interversions and resultant ics in La Buse Variable f = (1 2 6 12 3 8 7)(4 10 9 11 5) f = (C Db F B D G F# C)(Eb A Ab Bb E Eb) IC: 1 4 6 3 5 1 6 6 1 2 6 1  f 2 = (1 6 3 7 2 12 8)(4 9 5 10 11) f = (C F D F# Db B G C)(Eb Ab E A Bb Eb) IC: 5 3 4 5 2 4 5 5 4 5 1 5 2  114  Table 23: Usages of interversions from Chronochromie SP orbit  Most analyses of pieces that employ rhythmic series from the Chronochromie SP orbit will reproduce some version of a 32 by 36 matrix listing all interversions of the orbit (e.g. Holloway 1974, 203; Sherlaw-Johnson 1975, 177; Keym 2002, 478; Cheong 2007, 119; Benitez, 2009, 291).51 It is important to be absolutely clear about what the matrix actually represents. The columns represent the order positions 1 through 32; the rows, the 36 interversions; and the integer in each cell represents an IOI. As mentioned with regard to displaying the information about the Île de feu SP orbit, such a matrix can be misleading, since unlike what we saw in the pitch-class SPs from Île de feu II and Catalogue d’oiseaux, the vertical alignment of values in this matrix (table 24) does not indicate that those IOIs sound together, or even overlap at all (save for the values in the first and last order positions). And since, unlike the Île de feu II SP orbit, Messiaen does not associate any Holloway’s table is, by my investigation, the first of its kind, with Sherlaw-Johnson’s published soon after. Holloway explains that he constructed the table from the three interversions from Chronochromie’s Strophe I, and that the completed table was “approved by Messiaen” (Holloway 1974, 202). 51  115  particular pitch-content with the IOI values in each of the order positions in this SP orbit it is not possible to make any general observations on the kinds of pitch materials that result from the superposition of adjacent interversions. The matrices in table 24 add value to the typical matrix representation of the Chronochromie SP orbit in two ways: the first is coloured to represent the different cycle lengths of the controlling permutation function; the second is coloured with an orangeyellow-green spectrum which corresponds to relative length of IOI from shortest to longest. Figure 21 adopts the same colour schemes, but stretches out each cell, representing the IOI value within each cell proportionally as a horizontal length. Messiaen typically uses the interversions from this orbit superposed three at a time, so figure 22 separates the 36 interversions into the twelve superpositions, and shows the twelve composite rhythms that result. Table 25 presents the statistical data that helps differentiate the composite rhythms according to the measures developed thus far in this chapter. I will refer back to figures 21-22 and tables 24-25 periodically throughout the remainder of the chapter, as most of the remaining musical examples make use of these composite rhythms. These tables and figures will help us to assess the relevance and accuracy of existing analytical statements regarding rhythms from the Chronochromie SP orbit. There are five movements from various pieces—Chronochromie II - Strophe I & IV. Strophe II; Sept Haïkaï - II. Le parc de Nara et les lanterns de pierre, & IV. Gagaku; and Éclairs sur l’au-delà – IV. Les élus marqués du sceau—for which the length is determined by a single iteration of superposed interversions from the Chronochromie SP orbit.52 Strophes I and II  Le parc de Nara et les lanterns de pierre is unique in that features only two simultaneous interversions, where the others superpose three (cf. example 17, examples 14-16). As a result the composite rhythm is significantly less dense. Further, the interversions are, to my ear, rather inconsequential feature of this particular composition. For these reasons, I focus my discussion around the other musical examples. 52  116  (examples 14 and 15), and Les élus marqués du sceau (example 16) seem to have been composed in the very same way (despite there being about 30 years between the composition of Chronochromie and Éclairs sur l’au-delà). Each contains a rhythm that is the superposition of three adjacent interversions from the Chronochromie SP orbit; composites 1 and 8, and 6, respectively (table 23). In all three cases the three interversions are sounded in three low metallic percussion parts, with every percussion attack being doubled by strings issuing seven or eight note chords. Atop this already dense texture Messiaen adds a flurry of birdsong melodies, in wind instruments and pitched percussion, which spans the entire movement (see examples 13, 14, 15). Unlike the SPs encountered in Île de feu II, the interversions in these compositions are ordered series of unique IOIs which are not assigned fixed pitch content. Messiaen governed the distribution of his chords across the series of IOIs in a strict way, but his method for doing so is entirely independent of the SP technique, and thus, beyond the scope of this paper (see Bauer 2007, 151-159, and Cheong 2003 for an introduction to the chord types used in Strophes I & II). Bauer (2007) and Benitez (2009) have recently presented partial analyses of Strophe I and Les élus marqués du sceau, respectively. The analytic methods introduced thus far help us evaluate certain analytical statements posited in these recent publications; and posit more efficacious strategies for future analyses of these and similar compositions. Bauer’s analysis of Strophe I includes a description of some of the distinctive properties she discerns across the 36 interversions of the Chronochronie SP orbit; namely, that the durations 19, 20, and 21, always occupy order positions 19, 20, and 21, and that duration 27, which is restricted to order position 27, is a “quilting point” which, occurring approximately fourfifths of the way through the series, “serves as a constant through all possible permutations of the duration series” (Bauer 2007, 150, 150fn). Comparing the matrices of table 24 with my duration/length proportional matrices in figure 21 calls this assertion into question. As 117  mentioned earlier in the discussion of Île de feu II, hearing musical materials in terms of their order positions is not a viable listening strategy for ordered series of IOIs. For example, IOI 27—which Bauer says “assigns meaning to the permutational algorithm” on account of its “universal recurrence…as the twenty-seventh member of the series” (Bauer 2007, 150fn)—occurs as early as 66% (interversion 0), and as late as 87% (interversion 6) of the way through the entire timespan of the series of IOIs. Conceding that the charm of these kind of features “remains largely impossible to hear,” she goes on to observe that “when three different [interversions] are juxtaposed [i.e. superposed], the combined effect generates a slower-moving rhythm, marked off by those points where two of three cycles attack simultaneously” (Bauer 2007, 150). This analytical observation requires some conception of separate and distinct rhythmic streams. Benitez’s brief analysis of Les élus marqués du sceau takes much the same approach. At first, he seems to imply a view similar to my own hearing of superposed interversions as constituting as single composite rhythm, when he asserts that “the three interversions generate a single, slowly moving durational complex.” But then he reverts to considering the interversions as independent streams, claiming that the superposed rhythms “partition that complex into smaller temporal segments whenever two or more of their durations [i.e. attack points] coincide.” Benitez goes on to suggest that Messiaen’s particular choice of interversions for Les élus marqués du sceau is one that manifests “simultaneous attacks [that] partition the music into two unequal parts, the second of which lasts approximately twice as long (355 [thirty-seconds]) as the first (173 [thirty-seconds]), suggesting an approximate golden-mean relationship.” (Benitez 2011, 290). There is a serious problem with this analytic observation. If Messiaen wanted to choose a composite rhythm of three superposed interversions from the Chronochromie SP orbit that would divide the movement into a golden-mean ratio by a simultaneous attack, he did not choose the best group of 118  superposed interversions. Given the total length of the movement—528 thirty-second note values—the golden mean ratio of roughly 1:1.618 would best be approximated by a partition after 201 thirty-second note values (i.e. timepoint 202), or a partition 201 thirtysecond note values before the end (i.e. timepoint 327). Composite rhythm 3 (with its simultaneous attack at timepoint 194), composite 4 (with either its simultaneous attack at timepoint 222 or its simultaneous attack at timepoint 344), composite 7 (with either its simultaneous attack at timepoint 208 or its simultaneous attack at timepoint 342), and composite 10 (with its simultaneous attack at timepoint 318) all yield a simultaneous attack between layers that is a closer approximation of the golden ratio than is the simultaneous attack at timepoint 174 in Composite 6. But there is a more fundamental problem here that extends also to Bauer’s reading of simultaneous attacks as accents that can define sectional boundaries or hyper-rhythms. It is tempting to frame the analysis around the simultaneous attacks between streams because it can feel as if there is not much else to say (apart from naming the interversions used, and showing their parent SP orbit). Indeed, looking at the number of simultaneous attacks between streams is one way that one might try to differentiate between various composite rhythms from the same orbit (as in table 25); or differentiate an instance of the SP technique (on a series of unique IOIs) from an otherwise generated unpredictable rhythmic series.53 It is true that for these three movements, Messiaen chose from among  We can calculate the likelihood of an attack occurring at a certain timepoint in a “random” rhythm composed of the n elements of the set {1,2,3,…,n}, which is to calculate the likelihood that a random ordering integers from the set {1,2,3,…,n} are arranged such that values in some consecutive order 53  positions (from 1) sum to t, where  ((  ) ( )). It becomes more likely for larger values t,  since it depends on the number of ordered partitions of t into distinct parts (given by A032020 (Bower, 1998)), and the number of summands for each of those partitions. (For a formal treatment of this mathematical problem, see Richmond and Knopfmacher 1995). For example, given a resource set {1,2,3,…,32}, there is only one way for there to be an attack at timepoint 2 (that is, if the value in order position 1 is <2>), so the likelihood of a randomly chosen permutation having one of these attacks is ( ) and the likelihood of a double, or triple  119  the composite rhythms of the SP orbit with the fewest possible simultaneous attacks between layers (see figure 22 and table 25), whereas for Gagaku (which will be considered shortly, example 18) he chose the superposition with the greatest number of simultaneous attacks; and this can make it seem as if simultaneous attacks were an important attribute to Messiaen.54 But these statistics, and especially the visual superpositions of interversions— whether in my figure 22, or Bauer’s annotated excerpts from Messiaen’s own rhythmic sketches (Bauer 2007, 150, from TRCO III; 39–66)—seriously overemphasize the importance of simultaneous attacks. Were the texture not so dense and complex, the simultaneities might indeed register a significant accent (whether in amplitude or in changing a small sonority into a larger one), as in the Île de feu II SP episodes, where a simultaneous attack between two interversions meant hearing a struck dyad, instead of the more commonly occurring single note attack. But in the strophes and Les élus marqués du sceau, a simultaneous attack between two layers means something like the difference between hearing an attack in a cymbal with a (pianissimo) seven-note chord in the strings, and hearing an attack on a cymbal and gong with a (pianissimo) fifteen-note chord in the strings. Given the overall texture of these movements, that difference is extremely difficult if not impossible to hear, and does not—in my opinion—give any significant phenomenological presence to the alignment of two simultaneous attacks.  simultaneous attack, ( ) and ( ) respectively. An attack at timepoint 7 could result from either {7} in order position 1; <1,6>,<6,1>,<2,5>,<5,2>,<3,4>,<4,3> in positions 1-2; or any of the six orderings of {1,2,4} in order positions 1-3; so its likelihood is equal to ( ) ( ) ( ) . Because a simultaneous attack means there is one less IOI that gets divided, the greater the number of simultaneous attacks, the higher average IOI across the episode. One could just as easily use the same evidence to suggest that Messiaen was attuned to choosing composite rhythms with extreme average IOIs. 54  120  Table 24: Matrices for Chronochromie SP orbit of IOI interversions  121  Figure 21: Chronochromie SP orbit matrices to scale (duration proportional to length)  122  Figure 22: Superposition and composite rhythms of Chronochromie SP orbit  123  Figure 23: Twelve composite rhythms of Chronochromie SP orbit  124  Table 25: Statistics on composite rhythms from superposed interversions of the Chronochromie SP orbit  125  Aside from the number of simultaneous attacks per composite rhythm, the only other statistical characterizations of composite rhythm (from table 25) that can describe the distinctive qualities of the respective rhythms Messiaen chose (or might have chosen) are the “smoothness” metrics I defined above, which tally the total difference between all subsequent IOIs, and find the average difference between adjacent IOIs for each composite. The two strophes are quite similar to one another in this regard, being in the 42nd and 38th percentiles, respectively. Les élus marqués du sceau is considerably more “jagged,” being in the 84th percentile according to this measure. I leave it to the reader to decide on the extent to which one can learn to hear this quality, but to my ear, the 40-percentile-points between Chronochromie’s strophes and Les élus marqués du sceau are enough for noticeably different listening experiences—the strophes exhibit a certain steadiness, and a churning quality whereas the relative jaggedness of the superposed interversions in Les élus marqués du sceau manifests as a more turbulent, even aggressive character. Beyond the statistical information presented in table 25, what can be said about these composite rhythms, and their interaction with (or influence on) the other musical materials of the movements in which they appear? Benitez looks to moments in Les élus marqués du sceau where attacks in the composite rhythm align with the beginnings of birdsong motives in the melodic instruments to issue an accent (Benitez 2011, 292-293). This suggests that there is a certain degree of cooperation between the SP rhythm and Messiaen’s temporal placement of birdsong melodies. But then, considering the movement features a constant proliferation of birdsong across entirety of the 528 thirty-second note episode, and given that there is an attack in the composite rhythm on average every 5.87 thirty-seconds, it is not surprising that a significant number of percussive attacks align with beginnings of birdsong melodies. My own impression is that the shimmering string chords—whose attacks are emphasized by struck metallic percussion, and whose series of durations seems 126  unpredictable—are symbolic of a powerful Nature largely, if not entirely indifferent to the twittering birds, even while providing them the habitat in which to sing.55 There are other musical settings (for instance, episodes internal to Chronochromie’s Antistrophes, and musical interludes in the first act of Saint François) where birdsong melodies are supported by similar, shimmering metallic percussion sonorities, but where the attacks in the percussion all align perfectly with important moments in the birdsong melodies. Figure 28 is a graphic score of such an episode from Saint François (example 19) which shows the way the percussive attacks are coordinated with the birdsong gestures. It illustrates that Messiaen composed the episode such that the moments in the birdsong melodies that are “accented” by a simultaneous attack in the bells are almost always already an accented moment in and of themselves: either a durational- or contour accent of some sort. In this way, the slight accent resulting from the doubled attack is not as significant as the accent inherent in the melody. It is almost as if the moments of accent efface the bell attacks, leaving only a trace of bell resonance that so beautifully creates the glistening backdrop for the birdsong melodies. The point is that when Messiaen chose to write a metallic percussion part that was in full cooperation with the birdsong melody, he did so explicitly. Benitez’s idea to look for instances of alignment between the attacks of the interversion rhythms and important moments in the birdsong melodies has significant potential, but to develop it for the further analysis of Strophes I & II and Les élus marqués du sceau it would need to be integrated with a some method better equipped to identify the “important” moments across the many birdsong melodies. Assessing all of the birdsong gestures, and (perhaps their composites) in terms of melodic contour (as Schultz 2008 has  55  Messiaen likely would have rejected this metaphor on theological grounds.  127  expertly done for Réveil des Oiseaux) would be an important—even necessary—step toward this end. But, I leave such a task for the future. Another potentially valuable approach might be to consider the changing attack density of the bird song strata, tracking the number of notes-per-timespan across the episode. In this manner, one could look for cooperation between the SP rhythm(s) and the birdsong as manifested by the balance been the rapidity of certain birdsong outbursts with the density of composite rhythm of the superposed interversions. Such a detailed assessment of the birdsong densities is outside the scope of this project. However, we can employ this kind of measure as way to evaluate and differentiate the complex rhythms of the Chronochomie SP orbit. One of Bauer’s more casual analytical observations is based on by just such “attackper-timespan” data. (This observation is not critical to the argument, and hence, the data itself is not included in the article.) She asserts that “each [interversion] has a similar durational arc which peaks approximately two-thirds of the way through with a series of longer durations which in effect ‘slow down’ the [interversion]” (Bauer 2007, 150). This can be plainly seen in the “traditional” matrix representation of the SP orbit, which I have coloured (orange-yellow-green) according to the IOIs of each order position (table 24). On the proportional version of the matrix in figure 21, the “slowing down” towards the twothirds mark is less obvious, but still perceptible. The “slowing down” that Bauer observes is a feature of the individual interversions of the orbit, specifically; but it does extend (if in a limited sense) to the composite rhythms as well. Figure 23 (above)—which lists all 12 composite rhythms as proportional length coloured according to IOI—shows that the many of the “orangest” (i.e. shortest) IOIs fall in the first 33% of their respective composite rhythm; with somewhat fewer occurring near the last 33%; and relatively few in the middle, by comparison. 128  The analytic technology best suited to quantifying Bauer’s sense of the rhythms’ slowing down (or speeding up) is a precise tally of the number of discrete attacks per timespan for a given composite rhythm. Again, it is very important here to be absolutely clear that the superposed interversions are understood as a single composite rhythm. If one felt that the interversions were clearly audible and easily differentiated as separate streams, we might consider a simultaneous attack as two distinct attacks, which would increase the attack density count. Since there are three superposed interversions, each consisting of 32 durations (and thus, 32 discrete attacks) there are a total of 96 attacks across the whole. But then, since the beginnings are perfectly aligned, two of the attacks “disappear” at timepoint zero. Every subsequent simultaneous attack (between two of the three layers) subtracts 1 from the total number of attacks. In this way we can calculate the number of discrete attacks per composite rhythm, which, as table 25 shows, varies considerably across the twelve composite rhythms from the Chronochromie SP orbit. It follows, of course, that the number of discrete attacks, plus the number of simultaneous attacks per composite will always sum to 96. To reflect the listening experience—our sense that over the course of the piece the phenomenological “present” varies in content from containing relatively few, to relatively many discrete attacks—the referential timespan “scrolls” across the length of the episode, and thus, counts many of attacks more than once.56 The length of the timespan can easy be adjusted to whatever length (whether measured in seconds or note values) one feels best represents the length of the “now.” But for the purposes of this project (and because the tempos vary considerably across Messiaen’s different usages of the composite rhythms) we use a scrolling window 23 thirty-seconds notes in length, which is calibrated such that the  This is akin to Mustafa Bor’s method of assessing melodic contour in terms of a scrolling timespan (or fixed number of events); which treats heard melodies in the “way a landscape is experienced visually through the side window of a moving automobile or train” (Bor 2009, ii). 56  129  number of attacks per timespan falls to zero only during the longest IOI value (24) across all 12 composite rhythms. What is appealing about this analytical idea is that it reflects simultaneous attacks as “disappearances” of attacks, rather than accents, which is consistent with my own listening experience for IOI SPs post 1960. Plotting the scrolling attacks-per-timespan data for the composite rhythms on a graph (wherein the horizontal axis is timepoints (in 32nds) and the vertical accents is the number of discrete attacks per span) gives us the best answer as to whether the composite rhythms truly “slow down” toward the two-thirds mark; and further, give us another metric by which we can hope to differentiate between and identify unique characteristics of the composite rhythms. Figure 24 (multi-page) plots the data for each individual composite rhythm in this manner, and includes a trendline for each (based on a moving average for 12 values) which makes for easier comparison between profiles. It also includes a graph that similarly plots the data points and trendline for the average attacks-per-timespan across all twelve composite rhythms, and a graph that superposes all twelve trendlines onto a single graph. (Figure 25 superposes graphs in the same way, but only for those composite rhythms Messiaen uses as the basis for Strophes I & II (examples 14 and 15), Les élus marqués du sceau (example 16), and Gagaku (example 18) respectively.) Interestingly, these graphs all seem to converge to around 3-4 attacks per timespan about two-thirds of the way through the rhythm (c. timepoints 272-304): presumably, this results from of durations 19-21 being restricted to order positions 19-21, and duration 27 remaining fixed in position 27. The general “slowing down” is a little harder to observe in the graphs for the individual rhythms. But there is a slight “slowing down” across all the composites that can be seen in the graph showing the average number of attacks per span: the average number of attacks per span is more or less restricted to between 4.5-6 attacks per span through timepoint 144, and between 3 and 4 attacks per span through the 130  remainder of the length. Measuring and plotting attack density in this manner could be used in future analyses of these and similar pieces, and might help make stronger arguments for (or against) cooperation between the different strata in a composition. To illustrate, consider Gagaku, the fourth movement of Messiaens Sept Haïkaï (example 18). Unlike the other movements based on entire iterations of superposed interversions, this movement includes no birdsong. Here, the string chords (whose attacks were determined by the interversions in the strophes and Les élus marqués du sceau) are independent of the composite SP rhythm. In fact, it seems as if Messiaen made a conscious decision to place the majority of string attacks on timepoints when there is no attack in any of the interversions (see figure 26). This deliberate non-alignment is, in a contrary way, a type of cooperation between the now independent string and percussion layers. Further differentiating it from the other movements featuring complete statements of the superposed interversions, Gagaku features a lyric solo melody, performed in unison by the trumpet, oboe, and cor anglais. The melody seems to be a loose interpretation (or “stylized transcription” (Irlandini 2010, 197)) of traditional Japanese court music (TRCO V, 497). It is organized into a straightforward ABCAB scheme. Irlandini has shown that the three series of chords in the strings, a rather subtle element of the overall texture, are also organized according to the same ABCAB structure as the melody (Irlandini 2010, 202). (My figure 26 is a graphic score of the movement that summarizes the form of the piece, as described in Irlandini’s article.) Given the clear formal structure as defined by the melodic and harmonic aspects of the movement, one may well consider the superposed interversions (which were, after all, a pre-composed musical element) as a relatively incidental “add-on,” whose purpose is to provide an unpredictable rhythmic underpinning. But there may have been more to it than that. Adding the information about the formal design of the movement to the graph that 131  plots the number of attacks per timespan across the composite rhythm 7, we see that the sectional boundaries are almost perfectly aligned with significant peaks and troughs on the trendline (figure 27). This is to say, that the beginnings and ends of melodic and harmonic phrases are also those moments that signal an imminent and significant change in the attack density. Perhaps Messiaen organized the phrase rhythm of the unison melody according to the structure of the particular composite rhythm he chose, or intuitively chose a composite rhythm that cooperates remarkably well with the phrase rhythm of the melody. Applying these kind of analytical methods to similar compositions could prove an effective way to describe the kinds of relationships Messiaen achieved between different layers of his more heterophonic music.  132  Figure 24: Graphs – Number of attacks per (scrolling) timespan in composite rhythms 1–12  continued over page  133  Figure 24 continued  continued over page  134  Figure 24 continued  continued over page  135  Figure 24 continued  Figure 25: Graph – Number of attacks per (scrolling) timespan in composite rhythms 1, 6, 7, & 8  136  Example 14: Chronochromie – II. Strophe I  137  Example 15: Chronochromie – IV. Strophe II  138  Example 16: Éclairs sur l’au delà… – IV. Les élus marqués du sceau  139  Example 17: Sept Haïkaï – II. Le parc de Nara et les lanterns de pierre  140  Example 18: Sept Haïkaï – IV. Gagaku  141  Figure 26: Graphic score – Gagaku  142  Figure 27: Graph – Attacks per (scrolling) timespan in Gagaku, with formal boundaries  143  Example 19: Saint François d’Assise – Acte I, 1er tableau: La croix  144  Figure 28: Graphic Score – Saint François d’Assise – Acte I, 1er tableau: La croix (excerpt)  145  Example 20: Saint François d’Assise – Acte I, 3e tableau: Le baiser au lépreux  146  Figure 28: Fragments of interversions 31-33 in Saint François d’Assise  147  3.3 – Conclusion I would like to outline here a few open questions that might be addressed in the future. The first concerns the superposition of series of durations, each an  , where  the resource set for A is {1,2,3,…,n}. Having looked at the variety of ways in which two are superposed in the coda of La merle noire, I counted that there exist only 168 unique composite rhythms among the ( from  ) = 276 unique superpositions of two elements  . A more general investigation would seek how many unique composite rhythms  exist given the superposition of any m series  where the resource set is  {1,2,3,…,n}. Series of durations of this description are seldom used by Messiaen (to my knowledge the superposition of 4-permutations in Le merle noir and the superposition of 5permutations in the Offertoire from Messe de la Pentacôte are the only instances) but investigation is nevertheless warranted, if only because it seems an interesting mathematical problem. The second open question has to do with SPs proper, specifically, the interrelatedness among different composite rhythms that result from the superposition of adjacent interversions in an orbit. For example, given a permutation function f whose period is m; a seed series, A = <1,2,3,4,…,n> ; and given a composite rhythm compi that is the superposition of k successive interversions (where k | m, and k < m), it would be interesting to find a way to calculate composite rhythms comp2, comp3, … ,  (where compi results from the  superposition of rhythms f (i-1)(A), f (i)(A), f (i+1)(A),…, f (ik-1)(A)) and then generalize to describe the relationship between composite rhythms for other functions f , for other series lengths n, and for different numbers of superposed interversions, k. This problem, like the first, might be of limited use with regard to Messiaen’s compositions, but is perhaps of general mathematical interest.  148  I did not adopt a mathematical approach in this project for its own sake. Rather, I believe that a rigorous approach is important in attempting to move music-theoretical treatments of Messiaen’s music beyond analyses that are overly dependent on Messiaen’s own conception of his work. This is not to dismiss Messiaen’s own descriptions of his music outright: the primary sources (TMLM, TRCO, and published interviews: Samuel [1967] 1976, [1986] 1994) are substantial, and are valuable logical starting points for research. But Messiaen’s analysis of his own compositions tends toward a decidedly taxonomic approach (see Griffiths 1985, 93). They are something like a musical equivalent of a bird-watching guidebook—cataloguing the variety of musical techniques that one might expect to encounter during a particular work. Moreover, the mathematics of this project aimed to dispel certain misguided notions of how exactly SPs are similar to (and distinct from) to MLTs and NRRs; to quantify the relative “smallness” of Messiaen’s SP orbit(s); and to critically evaluate the extent to which symmetric permutations are symmetrical. To the extent it has succeeded, it can better inform some of the more speculative, philosophical lines of inquiry into Messiaen’s music, which allude to the mathematical nature of Messiaen’s techniques. By engaging in formal treatments of Messiaen’s techniques we can take “what he thought he was doing” under advisement, and try to focus instead on what he actually did. Such an approach does the music a service by encouraging analysis of Messiaen’s music on its own terms—rather than propping up Messiaen’s own readings of his music. My hope is that in subsequent musictheoretic and analytical writing on Messiaen’s compositions, trivial observations (e.g marveling that a certain permutation function is of this period rather than that; or narrating how Messiaen chose each subsequent integer of his ordre de lecture) can be avoided entirely. Future analyses, then, might better describe the characteristics of complex composite  149  rhythms, and more convincingly relate those rhythms to the other discrete strata of Messiaen’s heterophonic works. Another hope is that a rigorous analysis of the musical materials, and a more precise understanding of his compositional techniques will make Messiaen’s best metaphors and allusions all the better. Trawick, having analyzed Les yeux dans les roués, espouses a similar view, saying: “detailed—even occasionally tortuous—examination of the relations within the piece can help to reveal the composer’s own attitude towards it, and can thus help the listener to approach it from a different, although not necessarily a better, perspective” (Trawick 1991, 34). The strange creatures of Ezekiel’s biblical vision are directly invoked in the epigraph for Les yeux dans les roués, and might be evoked by the dazzling organ melodies in that virtuosic movement. But their mysterious “wheels within wheels” might well come flashing into mind whenever one senses symmetric permutation is afoot— knowing, as we do, that the individual cycles of a permutation independently circulate subsets of the series among themselves: reordering, while intimating size the orbit of rhythms the permutation defines. This metaphor suggests one final point. As much as some of Messiaen’s explanations of his techniques seem to betray a penchant for “completeness” and “closed circuits,” he also seems to be drawn to initiating musical processes, only a small part of which are revealed in the sounding music. We remember that the composite rhythms in the strophes are only a sliver of their larger orbit, which in turn is one small orbit within the astronomically large group S32. And then, more often than not, Messiaen gives us only a small portion of that composite rhythm. Two such tiny fragments of a composite rhythm are used to powerful effect as bookends to the third tableau of Messiaen’s opera (example 20, figure 29). Perhaps we should try to listen as Messiaen might have listened: while Saint Francis experiences moments which reveal to him the glimpses of the unfathomable nature of God, we 150  experience the overwhelming nature of the superposed interversions as we try to hold in mind the awesome scope of their parent orbit, and the astronomical group of which it is a small subset. Thus, we can try to appreciate these passages as a glimpse of a process that, even when revealed in its fullness, we know to be largely beyond perceptibility. It is as if a window is suddenly opened, through which one receives a ghostly impression of a cosmological order—and then, just as suddenly, the window is abruptly closed.57 (Keym 202, 275)  (Unknown Artist 1888, 163)  My own translation. Keym’s original reads: “Es ist, als würde plötzlich ein Fenster geöffnet, durch das man einen schemenhaften Eindruck von der… kosmologischen Ordnung erhält, und nach einiger Zeit ebenso abrupt wieder geschlossen.” 57  151  Works Cited Andrews, George E. 1994. Number theory. Mineola, New York: Dover. Babbitt, Milton. [1960] 2003. “Twelve-tone invariants as compositional determinants.” In The collected essays of Milton Babbitt, ed. Stephen Peles et al., 55–69. Princeton: Princeton University Press. Battier, Marc. 2010. “Messiaen and his collaborative musique concète rhythmic study”. In Olivier Messiaen: the centenary papers, ed. Judith Crispin. Newcastle: Cambridge Scholars Publishing. ______________. Forthcoming. “Timbres-durées” d’Olivier Messiaen: une oeuvre entre conception abstraite et materiaux concret. Paris: Groupe de recherches musicales et Institut national de l’audiovisuel. Bauer, Amy. 2007. “The impossible charm of Messiaen’s ‘Chronochromie.’” In Messiaen Studies, ed. 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