Quatre Études de rythme

Baining firedancers, Papua New Guinea

The title refers to Papua New Guinea, and the thematic material of this movement has "all the violence of the magic rites of this country".[4] The piece consists of five sections, in alternating pairs of musical ideas. The first part of each section is a melodic theme with accompaniment, the second is a departure which Messiaen calls a trait ("episode"). The first section consists of a thematic statement and trait. The first half of the next three develop the initial theme. The fifth and final section introduces a new, longer theme in two periods which are immediately repeated before the etude is concluded by a final trait amounting to a short coda.[5]

This movement is the most-discussed of the four, as the first work by a European composer to apply numerical organisation to pitch, duration, dynamics, and mode of attack (timbre).[2] Because the treatment of the parameters is modal and not serial (that is, the elements are treated simply as a scale, without any implications for how they are to be ordered), there is no question of the material determining the work's form.[6] According to the composer's own description, there are separate modes composed of 36 pitches, 24 durations, 12 attacks, and 7 dynamics. The duration scale is separated into three overlapping scales, called "tempi" by the composer, which correspond to the high, middle, and low registers of the piano, and occur in simultaneous superimposition. The first tempo uses 12 chromatic durations applied to the demisemiquaver, the second does the same with the semiquaver unit, and the third with the quaver.[7]

According to Messiaen:

The durations, intensities and attacks operate on the same plane as the pitches; the combination of modes reveals colors of durations and intensity; each pitch of the same name has a different duration, attack and intensity for each register in which it appears; the influence of register upon the quantitative, phonetic, and dynamic soundscape, and the division into three temporal regions imbues the passage with the spirit of the sounds that traverse them, creating the potential for new variations of colors.[4]

The first of the four to be composed, this étude alternates two separate sets of refrains with longer strophes given over to the rhythmic neumes of the title. The first set of refrains is marked "rythme en ligne triple: 1 à 5, 6 à 10, 11 à 15", which means there are three durations, short, medium, and long, which are progressively expanded, upon each repetition, by the addition of a semiquaver unit.

The second set of refrains is marked "Nombre premier en rythme non rétrogradable" (A prime number in non-retrogradable rhythm) and, like the first set of refrains with which they alternate, are expanded upon repetition—in this case by a series of progressively larger prime numbers: 41, 43, 47, and 53 semiquavers.

The strophes which occur between the refrains are marked "neumes rythmiques, avec résonances et intensités fixes" (rhythmic neumes, with fixed resonances and intensities). From his studies of the neumatic notation symbols of plainchant, Messiaen had formed the idea of exploring the rhythms corresponding to them. "In an interplay of transposition, the neumatic symbol as an indication of a sinuous melodic entity is now applied to a rhythmic motive. Each rhythmic neume is assigned a fixed dynamic and resonances of shimmering colours, more or less bright or somber, always contrasting."[4]

The collage-like rhythmic structure grows from the iambic rhythm found at the beginning of the first strophe. As in the refrains, there is a process of augmentation upon repetition, in which new duration units are added in order to form more complex rhythmic cells. The pitches grow out of the major seventh and the tritone found at the beginning of the first strophe and in its third bar, respectively. Five basic dynamic-envelope patterns are applied to different groups, while at the same time the melodic motives constantly re-emerge in different rhythms.[8]

Although Messiaen's programme note for this étude says that it is, like the first étude, "also dedicated to Papua or New Guinea", his posthumously published analysis emphatically states that it refers only to the former, not the latter.[9] "[T]he principal theme, ferocious and violent, has the same character as the themes of the first étude; the variations on this theme alternate with permutations, successively permuted according to the same process and superimposed two by two; the piece closes with a cross-handed perpetual motion in the depths of the keyboard".[4]

The opening strophe from "Île de feu I" is modified to become the cyclic theme of this étude. As in the other pieces, this alternates with episodes, in this case based on a mode of twelve durations, twelve sounds, four attacks and five intensities.[10] The rhythmic focus in this piece concerns a symmetrical permutation scheme for a duration scale initially given in descending order of durations. The elements are reordered according to a wedge-shaped process, which Messiaen elsewhere called "permutation in the form of an open fan". The first permutation produces a new ordering which Messiaen called "Interversion I". The same permutation scheme applied to this first interversion produces "Interversion II", and so on:[11]

"Île de feu II": duration-row permutations

Initial series Interversion	12	11	10	9	8	7	6	5	4	3	2	1
I	6	7	5	8	4	9	3	10	2	11	1	12
II	3	9	10	4	2	8	11	5	1	7	12	6
III	11	8	5	2	1	4	7	10	12	9	6	3
IV	7	4	10	1	12	2	9	5	6	8	3	11
V	9	2	5	12	6	1	8	10	3	4	11	7
VI	8	1	10	6	3	12	4	5	11	2	7	9
VII	4	12	5	3	11	6	2	10	7	1	9	8
VIII	2	6	10	11	7	3	1	5	9	12	8	4
IX	1	3	5	7	9	11	12	10	8	6	4	2
X	12	11	10	9	8	7	6	5	4	3	2	1

A peculiarity is that the successive applications of this permutation do not yield the twelve permutations which might be expected. This is because the values 10 and 5 map into each other under this operation—which can be seen by scanning down columns 3 and 8 in the table. Consequently, the original order returns once the remaining ten values have circulated. This unexpected restriction has been cited as an example of what Messiaen called the "charm of impossibilities".[11]

Messiaen later applied such permutation schemes in works of larger scale, such as the orchestral work Chronochromie, completed in 1960.[12]