and Chronomchromie - Messiaen frequently indicates that rhythm, time, and even eternity are important musical concerns. Messiaen saw his conception of musical time as radically different from that of the musical mainstream, particularly with regard to the use of regular rhythmic divisions implying a predictable forward movement; these rhythms he chose to avoid in favor of "non-retrogradable," "added-value," and irrational rhythms. "[R]hythmic music is music that scorns repetition, squareness, and equal divisions, and that is inspired by the movements of nature, movements of free and unequal durations."3 Messiaen's music thus offers a case study for how a composer might deliberately subvert the expectation of rational time and of forward motion in music.
This essay will discuss excerpts from several pieces for organ and for piano from the years immediately following the writing of the Turangalîla Symphony, a time when Messiaen was most intensely preoccupied with rhythm. I will examine how, in these pieces, Messiaen employs retrograde, interversion, and other systematic order permutations to suggest forward, backward, and non-contiguous motion through time. These examples challenges composers, analysts, and listeners alike to develop different, multilayered conceptions of how time is composed in, and composes, music.
A number of theorists and writers about Messiaen's music have commented upon its apparent avoidance of directed motion and linear time. Paul Griffiths writes: "[Messiaen's] music has remained sublimely indifferent to [the Western tradition's] axioms of diatonic harmony, meter, forward motion expressed as development, and interdependence of rhythm and pitch structure." Griffiths cites the symmetries of harmony and of scales, dependence on the pulse rather than on the measure as the fundamental unit of time, the use of repetitive or static forms, and the avoidance of any counterpoint that "impl[ies] a control of time's unfolding." He concludes, "It is... the denial of forward-moving time that is the generative and fundamental substance of Messiaen's music[.]"4
Jonathan Kramer views many of Messiaen's middle-period works, including some discussed in this essay, as anticipating or exemplifying moment time,5 described by Karlheinz Stockhausen as a time in which "[w]hat counts is the here and now;... for long periods of time you have no thought for the past or future, because there's nothing but the present moment."6 In moment time, there can be no directedness, no leading up to or away from a central event, no cause and effect. Moment form creates a timelessness that leads Stockhausen to dub it "die unendliche Form."7
Messiaen's own view of how time passes in his music is perhaps the most intriguing. Griffiths quotes his comment about the rhythmic interversions8 in the Livre d'orgue, which, the composer says, allow him "like the hero of Wells's Time Machine to go back in time, and also to chop its course, alternately following it and returning."9 This paper will take the view that time is neither absent nor irrelevant in Messiaen's music, but rather that multiple and contradictory unfoldings of time are among its most noteworthy features. I will examine the techniques Messiaen employs to induce a feeling of simultaneous motion and stasis, or simultaneous forward and backward temporal movement, and, in conclusion, I will suggest the implications for composers.
Retrograde is one of the techniques that Messiaen undoubtedly had in mind when making his extraordinary claim about being able to go back in time. Of the three canonical operations of serialism - transposition, retrograde, and inversion - retrograde is the one with the most lasting interest for Messiaen, and he uses it systematically throughout his entire oeuvre. Retrograde does not depend, as transposition and inversion do, on any particular relationship (indeed, on any relationship at all ) between adjacent elements: one need not consider pitch or time intervals, direction, or any other comparison between elements, to be able to construct or perceive a retrograde.10 And, unlike transposition or inversion, retrograde applies easily to non-pitch elements - to durations, dynamics, or articulations - and can apply as easily to groups of notes as to single notes. However, recognizing a retrograde relationship does require a type of out-of-time listening: we must know the end of a series before we can identify the beginning of its retrograde.
When music is retrograded, cause and effect are reversed. Example 1 shows two excerpts from the pedal part of the sixth movement of the Livre d'orgue, the opening and the conclusion of the movement. In listening to the first excerpt, if one were searching for aural clues as to its construction one might hear the ascent from A
flat 2 to F4 (mm. 4 - 5) as an echo of the rising contour of the first two notes, or the B3-F#2 (mm. 6 - 7) as a transposition of the same interval just before, F4 - C3 (mm. 5 - 6). One might retrospectively relate the E
flat 3 - A2 of mm. 8 - 9 to the E
3 - A flat 2 of mm. 3 - 4, in which each of the earlier pitches is displaced by a semitone. Certainly the reason that the low D
flat of m. 12 seems a satisfactory conclusion to the section is not only that it completes the twelve-note aggregate, nor merely that it is the third in an exponentially-increasing sequence of durations, but also that it balances the low D natural of m. 2, the next-longest and next-lowest note of the series. However, in listening to the concluding section, the D
flat (now two octaves higher) must serve, not as a terminus, but as an initiator - of the decreasing durations that follow in mm. 62-63, and of the new contour consisting of descending pairs of notes. Now if we hear any relation between A2 - E
flat 3 (mm. 64-65) and the transposed A flat 3 - E4 (mm. 69-70), it is the former that is the antecedent, the latter the consequent. And now the low D, the starting pitch in m. 2, must be understood as a final note. The teleological categories cause and effect, origin and goal, must change places.
A similar process occurs in the first movement of the Livre d'orgue, entitled "Reprises par interversion," in which the second half is an exact retrograde of the first. The movement consists of four clearly demarcated sections, whose relationships are detailed in Figure 1.
Messiaen's descriptions in the score11 make it clear that he conceives of each of the three later sections as independently deriving from the first - the second by an order permutation that starts with the first and last elements and works its way in to the middle, the third by a permutation that starts with the middle elements and works its way outward to the first and last, and the fourth by retrograde. For Messiaen, the first section is the cause, and sections 2 through 4 are effects.
It is easy enough to hear the second section, separated from the first by four beats of rest, as a consequence of the first - the different organ timbres, the widely separated registers, and the presence of repeated pitches, all help us to understand that elements present in section 1 are now reappearing, scrambled and overlaid, in section 2. Section 2 seems like a development and intensification of the first section. An alternate interpretation of sections 3 and 4, however, is equally or more plausible than the composer's: the retrograde relationship of section 3 to section 2 is the fact that really stands out on hearing them, because the point of symmetry announces itself so clearly. And, after the rests in m. 39, the fourth section gives the impression of coming from the previous section: it effects a stretching and thinning out of section 3 in the same way that section 2 compresses the first section. Thus the cause-and-effect relationship heard between the first section and the denser second section is reversed when this material appears in retrograde as sections 3 and 4.
Example 1: Excerpts from the pedal part of Livre d'Orgue VI. Mm. 61 - end present the "note - durations" of mm. 1 - 12 in retrograde.
Click to view Figure 1: A Summary of the Form of Livre d'Orgue mvt. I ("Reprise par interversion") Mm. 19, 29, and 39 consist only of rests.
Click to view Example 2: An excerpt from Cantéyodjayâ pp. 13 - 14, built from a "chromatic scale of durations forward [in the left hand] and retrograded [in the right]." The numbers show each note's durations in 32nd notes. The vertical line at the beginning of the third system shows the midpoint of the rhythmic palindrome.
When Messiaen uses a forward and a backward moving process at the same time, the effect is striking. Example 2 shows an except from the piano piece Cantéyodjayâ. The right hand repeats a seven-note gesture in durations that gradually increase in length from one to twenty-three thirty-second notes. At the same time, the left hand plays a quasi-chromatic ascending line, which eventually uses all twelve pitch-classes, in durations that decrease from twenty-three to one thirty-second note. The composite rhythm of this excerpt is palindromic, with the second half the retrograde of the first. However, because the notes are not likewise palindromic, there is less of a sense that the first half causes or engenders the second. Instead, the musical interest gradually passes from the right to the left hand as the written-out ritardando at the start gives way to an accelerando at the end.
Besides retrograde, Messiaen is also fond of exhausting all the possible order permutations of a small number of elements. (A collection of n elements has n! (n factorial) different orderings - that is, n x (n-1) x (n-2) ... x 2 x 1.) A typical example occurs at the start of the first movement of the Livre d'Orgue, discussed previously. The opening three measures present three rhythms that Messiaen identifies as Hindu rhythms: pratâpaçekhara, gajajhampa, and sârasa. In the remainder of section 1, mm. 4 - 18, Messiaen cycles through the remaining five orderings, as shown in Figure 2 below.
Figure 2: Livre d'Orgue I. ("Reprises par interversion"), mm. 1 - 18. Systematic reordering of the three constituent rhythms.
One way of understanding Messiaen's system for reordering the elements is to note that the initial one generates the next five: pratâpaçekhara starts the first and second orderings; gajajhampa, the second element, starts the next two; sârasa, the final element of the initial ordering, starts the last two orderings. The right-hand column shows another model, giving the reordering operation needed to transform each ordering into the next one. (The last row shows that another rotation to the right would transform the last ordering back into the initial one.) Exactly this same ordering algorithm is used with three-element groups elsewhere in this piece, for example in movements III and V.
The passage in question has the paradoxical effect of remaining static and yet, at the same time, progressing and developing. The constant cycling-through of the same elements can give the music a sense of immobility. On the other hand, since Messiaen employs a clearly-defined process for the reordering, the section is directed forwards - at least from the composer's point of view, if not from that of the proverbial naive listener. The interpretation of time in this excerpt however, is made more complicated because two of the three rhythms undergo transformation each time they are repeated. The pratâpaçekhara is gradually augmented in duration - a forward-moving process. The gajajhampa starts out augmented and undergoes gradual rhythmic diminution, ending with what Messiaen thinks of as the "original" form - this is a backward-moving process. The sâ rasa does not change; it is static.
Example 3 shows another excerpt from Cantéyodjayâ, a somewhat more complicated example of the exhaustive permutation of three elements. The right hand constantly repeats a sequence of two, three, four, and eight repeated pitches. The left hand permutes four pitches, all of which, as their order is changed, retain fixed durations of two, three, four, and eight sixteenth notes. (The apparent canon between the two parts does not last beyond the first two measures of the excerpt.) The eight-note D2 always precedes the dotted-eighth E1, so that the pair constitute a single durational element labeled "2 + 3" in the summary table, shown in Figure 3. This summary table indicates how the three elements - 2 + 3 sixteenths, 4 sixteenths, and 8 sixteenths duration - are permuted through all the possible orderings, returning at the end to the starting order.
Click to view Example 3: Cantéyodjayâ, pp. 19 - 20. The right hand constantly repeats a sequences of 2, 3, 4, and 8 repeated pitches. The left hand plays fixed pitches of 2, 3, 4, and 8 sixteenth notes' duration, permuted as shown in Figure 2. (The score uses an 8va notation in this section: both lines are in extreme registers, starting five octaves apart).
Figure 3: Permutation of left-hand durations. Cantéyodjayâ pp. 19 - 20. The measure numbers refer to the first through fifteenth measures of Example 3, not to those of the entire score.
Messiaen elsewhere systematically permutes sequences of four and more durations: in Le Merle Noir, (for flute and piano) the piano part in the final section cycles through all 24 orderings of four durations - in both hands simultaneously. And in the "Offertoire" of the Messe de la Pentecôte, Messiaen starts, but does not complete, a cycle through all 120 orderings of five durations. Sections such as these are static, with a homogeneous texture of circulating short durations and self-similar pitch materials; and yet they are also not static - they are not mere repetitions, but present a gradually-shifting kaleidoscope of predefined elements, evolving and developing in accordance with a predefined logic.
Messiaen uses the term interversion it its general sense to refer to any sort of reordering, systematic or otherwise, such as those I have just described. (The French word is the equivalent of the English "permutation.") He also uses the term in a more specific and idiosyncratic sense to refer to a chiasmic, or crosswise, retrograde operation, in which a sequence of elements is reordered by taking the last, the first, the next-to-last, the second, ... and so on, until reaching the middle. Or one might take the middle two elements then the preceding and succeeding elements, ... and so on, until reaching the first and the last. (The two sorts of interversion are illustrated, respectively, in the tables from Figure 1 (p. 67) that show the derivation of a second and third section by interversion of the first.) Messiaen sometimes refers poetically to these operations, respectively, as an éventail fermé - a closed fan - and éventail ouvert - an open fan. It is to interversion in this restricted sense that Example 4 and Figure 4 refer.
Click to view Example 4: Livre d'Orgue, I ("Reprises par interversions"), mm. 1 - 3, 16 - 18, and 20 - 22. The excepts illustrate the use of "interversion" - simultaneous forward and backward permutation. Mm. 20 - 28 are the interversion of mm. 1 - 18, and the correspondence between the elements of mm. 20 - 22 and those of previous measures is indicated.
Click to view Figure 4: Interversions of Île de Feu II with notes shown at time points from pp. 1 - 2, 3 - 4, and 6 - 7 of the score. (Octave doublings, dynamics and articulations ommitted.)
From the point of view of time and goal-directedness, this reordering is the most interesting. The material from mm. 1 - 9 repeats in mm. 20 - 28 with minimal alterations, while at the same time mm. 10 - 18 appear in retrograde - with the before and the after, the cause and effect, reversed. As one's attention is drawn now to one voice and then to the other, the course of time is indeed fragmented, alternately moving ahead and going backwards.
One of the most elaborate and comprehensive examples of interversion from this period occurs in the piano work Île de feu II. The piece as a whole has a verse-refrain form - a recursive structure which itself simultaneously suggests progressing forwards and doubling back. Ten successive interversions of a twelve-note series always occurring in pairs, constitute the refrains, shown in Figure 4.12 The simplified example shows the notes, each of which has a fixed duration, as time points, with the durations indicated above; it omits the octave doublings, and also the fixed dynamics and articulations that are attached to each of these "note-durations."
"Interversion II" is the interversion of the treble line, "Interversion I" -- the interversion moving from the middle out, as an "open fan." "Interversion III" is the group: "Interversion X," a chromatic ascending line, could intervert to generate "Interversion I," and the whole process could continue.13 In Ile de deu II, the first system of Example 4 (interversions I and II) leads directly into the second (interversions III and IV) to constitute the second refrain. Interversions IX and X come as the final refrain, but with the music of a new "verse" underlaid in the bass.
The various temporal and teleological directions implied by this intricate structure boggle the mind. These refrains occur three times, and they are similar enough in character, sounding a but like a random twelve-tone wind chime, that they become static reference points. Yet there are also clear forwards and backwards progressions implied: from the simultaneous backwards and forwards reiteration of the series, to the mutual forwards and backwards reference of each refrain to the others. And, since the last refrain (and in particular its bass line, the final interversion) is the simplest and least random-sounding, one tends to hear it as the source for the entire process, as well as the goal of that process. Interversions I - IX would thus be extrapolations backwards from the compositionally anterior last refrain.
Messiaen's music provides a particularly rich illustration of new musical approaches to time in the mid-twentieth century. Despite the proliferation of musical styles and philosophies in the past fifty years, all music shares a concern, whether implicit or explicit, with time and listeners' experience of time. In fact, in the most abstract or the most minimal compositions, time may be the only shared reference between the concert hall and the mundane business of the outside world. Different ways that music can exist in time, or time in music, ought to be of tremendous interest to composers. Even if we do not all set out watches in the same way to the multiple and contradictory clocks of Messiaen's music, his experiments are surely as suggestive as they are challenging.
Many of the analytical terms associated with music of the common-practice era and the early twentieth century assume a continuous, forward-directed, linear time - whether in discussing harmony (chord progression, prolongation), melody (imitation, sequence, pickup), or formal structure (introduction and coda, antecedent and consequent, recapitulation). Messiaen and his students view time as one of many musical parameters or spaces to be explored,14 and in doing they reflect the changing scientific view of time. Messiaen refers to his rhythmic experiments in the Quatre Études de rythme as "an extension of Einstein's ideas on the effect of sleep."15 Messiaen's metaphors for the exploration of time in his music, suggesting that time is relative, and that a composer can control the flow of time, reverse its course, and even jump from one instant to another further away, provide a compelling and provocative insight into his music.
