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Contents

Music in the Late Twentieth Century

“TOTAL SERIALISM”

Chapter:
CHAPTER 1 Starting from Scratch
Source:
MUSIC IN THE LATE TWENTIETH CENTURY
Author(s):
Richard Taruskin

That is how the Darmstadt “class of 1951” chose to interpret Messiaen's purpose, at any rate, when he played them his recording of the piece. Boulez in particular found the work inspiring, not only for the way in which it seemed to integrate “the four constituents of sound” as he listed them (surely under the influence of Messiaen's “divisions”) in “Schoenberg Is Dead,” but also for the way in which the whole piece arose out of a set of axioms, or what Messiaen scholar Peter Hill called its “fantastically detailed set of a priori rules.”33 It promised a new utopia: “total” or “integral” serialism.

All one had to do was introduce strict serial ordering into the four Messiaenic domains. And that is just what Boulez did in Structures for two pianos (1951). He paid tribute to Messiaen's example, and declared his intention to realize explicitly what Messiaen had been content merely to imply, by adopting the pitch succession of Division I in Mode de valeurs as an actual tone row in Structures, turning what had been for Messiaen a quarry of “stones” for a mosaic into a rigorously ordered pitch and intervallic sequence. Next, Messiaen's twelve chromatically graded durations were likewise put in a definite and rigorously maintained order, derived from the pitch order but operating independently. The method of derivation is what mathematicians call “mapping,” that is, a system of one-to-one correspondences. The starting point, as before, is Messiaen's Division I of note values, in which each successive pitch is assigned the next successive “degree” of the “chromatic scale of thirty-second notes.” Thus E♭ (pitch 1) is associated with the thirty-second note, D (pitch 2) with the sixteenth note, A (3) with the dotted sixteenth, A♭ (4) with the eighth note, and so on.

But where Messiaen maintained this pitch/duration association as a constant throughout Mode de valeurs, Boulez related the pitch-classes and durations independently to the order positions (the numbers in parentheses). This allowed him to create twelve “permutations” of the rhythm series that corresponded demonstrably (if only numerically) to the twelve possible transpositions of the pitch series, and to deploy the pitch and rhythm series independently of one another, like the color and talea in a medieval “isorhythmic” motet. The result is a truly fantastical set of a priori rules — fantastical in that the principle of correspondence is purely conceptual, devoid of any aurally perceivable relationship to the principle of pitch transposition on which it was based.

Here is why. As we know, when any twelve-tone pitch series is transposed— down, say, by one semitone — the result is a new ordering of pitch classes that preserves the same intervallic relationships as the original one. (One of the most basic aspects of twelve-tone technique, then, is that a “tone row” is really an “interval row” since it is the succession of intervals — the all-important motivic Grundgestalt—that remains constant when the row is subjected to its various transformations.) In Ex. 1-7a, the stated transformation of the Structures row (transposition down a semitone) is set down twice, each time both in terms of letter names and in terms of the reordering of the pitch numbers:

“Total Serialism”

ex.1-7a Transformation of Pierre Boulez's Structures row

In Ex. 1-7b, the same mapping operation is shown for the durational “scale,” each time maintaining the original assignment of durations to pitch numbers:

“Total Serialism”

ex. 1-7b Transformation of Pierre Boulez's Structures durational “scale”

Boulez's whole “precompositional strategy” can be represented as a pair of “magic squares” (Ex. 1-8). Running across the top and down the left side of the first square are the numbers corresponding to the original pitch/duration order of Messiaen's “Division I” as set forth in the preceding example. Note that in these squares, the number 1 always refers to E♭ (pitch 1 of the original series) and to a thirty-second note (the first “degree” of the chromatic scale of durations); the number 2 always refers to D (pitch 2 of the original series) and to a sixteenth note (the second “degree” of the durational scale); the number 3 to A (pitch 3) and a dotted sixteenth (the third “degree”), and so on. Thus the “transpositions,” reading down the left-hand column, are not by successive semitones (as in the explanatory example) but by the actual order of intervals in the row.

“Total Serialism”

ex. 1-8 Pierre Boulez, Structures Ia, “O” and “I” matrices

If the twelve columns in the first square, reading from left to right across or down from top to bottom, represent the twelve possible transpositions of the original row, then the same columns read from right to left or bottom to top represent the twelve possible retrogrades. The columns of numbers in the second square (from left to right or top to bottom) represent the twelve inverted row forms, and (from right to left or bottom to top) the retrograde-inversions. The relationship between these pitch rows and their associated durational rows is again mediated by the numbers. Under these rules a given series of durations can be called the “inversion” of another only by this arbitrary set of numerical correspondences—that is, only by an ad hoc definition or convention.

The remaining “constituents of sound” for which Messiaen had provided unordered “modes” are serialized in Boulez's composition according to a procedure that is even more arcane (so arcane, in fact, that it was not detected until 1958). Messiaen's collection of seven degrees of loudness was easily expanded to twelve, simply by making the gradations finer:

1

2

3

4

5

6

7

8

9

10

11

12

pppp

ppp

pp

p

quasi p

mp

mf

quasi f

f

ff

fff

ffff

And Messiaen's collection of twelve attacks could also be taken over (albeit with slight modifications for reasons that will soon become apparent).

The deployment of loudnesses and attacks was geared not to the individual notes, as in Messiaen's Mode de valeurs, but to the overall “structure” of Structures, which consisted of a single complete traversal of the pitch and durational “matrices” contained within his magic squares. Each section would contain forty-eight row statements, twenty-four for each piano. Boulez derived a series of twenty-four loudnesses for each piano by taking the right-to-left diagonals of each square. The diagonal from the square based on Messiaen's original series (the “O” matrix) governed the loudnesses for piano I, and the diagonal from the other square (the one based on inversions, or the “I” matrix) governed the loudnesses for piano II.

These diagonals formed “Webernian” palindromes, which must have been Boulez's reason for selecting them. To arrive at a full forty-eight elements Boulez constructed additional palindromes by taking the identical left-to-right diagonals from the seventh order position both at the top and along the side of each square, and running them back-to-front then front-to-back. Thus (for the “O” matrix):

Right-to-left diagonal across:

12

7

7

11

11

5

5

11

11

7

7

12

ffff

mf

mf

fff

fff

quasi p

quasi p

fff

fff

mf

mf

ffff

Left-to-right diagonals from 7th place:

2

3

1

6

9

7

7

9

6

1

3

2

ppp

pp

pppp

mp

f

mf

mf

f

mp

pppp

pp

ppp

The series of attacks is coordinated with the pitch/duration series by taking the opposite diagonals (full across from upper left to lower right and, from the sixth position, from upper right to lower left). What Boulez apparently did not foresee is that the numbers 4 and 10 happen to be absent from all the diagonals he selected. In the case of the dynamic series he fudged a bit to incorporate the levels in question (p and ff). In the case of the attack series he simply left them out, resulting in a “row” of only ten members.

The only decisions that remained concerned the order of presentation of the concurrent (but independent) pitch and durational series. With the magic squares in hand these choices could be planned in fairly mechanical fashion, after which the composer could sit back, as it were, and let the music write itself. The real work, in short, was all “precompositional.” Thus, at the beginning of Structures (Ex. 1-9), Piano I simply goes through the twelve transpositions of the basic pitch series in the order given by the numbers read from left to right across the top of the “I” matrix as they match up with Messiaen's old Division I: 1 (= E♭) 7(= E) 3(= A) 10(= B♭) 12(= B) 9(= C) 2(= D) 11(= F) 6(= F♯) 4(= A♭G♯) 8(= C♯) 5(= G). (The pitches in parentheses are the starting pitches for Piano I's row forms through m. 64; Ex. 1-9 shows only the first two row forms.) Then, in similar mechanical fashion not shown in the example, Piano I goes through the twelve transpositions of the retrograde in an order determined by the numbers read from right to left across the bottom of the same magic square: 12(= B) 11(= F) 9(= C) 10(= B♭), and so on.

Piano II performs exactly the “opposite” (that is, reciprocal) set of operations. First it goes through the twelve inversions in an order determined by the numbers read from left to right across the top of the “O” matrix (i.e., Messiaen's original series), and then (from m. 65) it traverses the twelve retrograde inversions in an order determined by the numbers read from right to left across the bottom of the same magic square: 5 (= G) 8 (= C♯) 6(F♯) 4(G♯A♭), and so on.

As for the durational series, their order is determined, first, by taking the rows or columns in sequence according to their positions in the squares, and then (from m. 65) in an order determined by the size of their first components. Again each piano reciprocates the other's operations. The first durational series in Piano I, as shown in Ex. 1-9, is , which corresponds to the numbers either read from bottom to top along the right edge of the “I” matrix or from right to left across the bottom; the second, , corresponds to the next column to the left (or the next row from the bottom); the third (no longer shown in Ex. 1-9 but easily predictable, i.e., , corresponds to the next column or row in the same direction and so on. (When the articulation is staccato, the durations are measured from attack to attack rather than in sustained sound.)

“Total Serialism”“Total Serialism”

ex. 1-9 Pierre Boulez, Structures Ia, mm. 1–15

At m. 65 (Ex. 1-10), Piano I shifts over to a contrapuntal combination of three row forms. The one marked ppp uses the “I” matrix series that begins with 12 (the fifth column from the left or the fifth row from the top); the one marked pp uses the series that begins with 11 (eighth column from the left or row from the top); the one marked pppp uses the series that begins with 10 (fourth column from the left or row from the top). At m. 73, the single line played by Piano I uses the series that begins with 9 (sixth column from the left or row from the top), and so it goes, all the way down to 1.

“Total Serialism”“Total Serialism”“Total Serialism”“Total Serialism”

ex. 1-10 Pierre Boulez, Structures Ia, mm. 65–81

Piano II, meanwhile, has started with the sequence of durational series beginning with the right-hand column (read bottom to top) or bottom row (read right to left) in the “O” matrix, and progressing thereafter across to the left or up to the top. At m. 65 (Ex. 1-10), the pattern reverses: now Piano II's durational series are chosen from the left-to-right rows or top-to-bottom columns. But the order of selection is no longer governed by a simple predetermined rule (or “algorithm,” to use the mathematical word). Here, and only here, in other words, Boulez seems to have chosen the order of presentation “freely”—that is, spontaneously, in the act of composing. (For the record, the order, counting the rows from top down, is 5/8, 6/4, 2/11, 12/9, 10, 7/1/3, with numbers grouped by slashes representing row forms that are played simultaneously, in counterpoint.) Boulez never acknowledged this spontaneous choice, but he did acknowledge others. One was “density,” as he called it, meaning the number of row forms deployed simultaneously in any given subsection of the piece. The number of contrapuntal lines varies from one to three in each piano (which means, potentially, four to six in toto). Even here, Boulez seems to have followed “rules” where he could. Consecutive row forms that are assigned by the “diagonals” to the same dynamic level are often played together. But not always; and the inconsistency must count as a “liberty.” The other conspicuous “liberty” is registral distribution. Although the score seems to be notated like conventional piano music, there is no a priori assignment of the players’ right and left hands to the top and bottom staves, nor does the assignment of a note to the upper or lower staff imply anything about its register. Instead, the contrapuntal lines so frequently cross, and leap so capriciously from register to register (often extreme ones) that it is impossible to hear the texture as consistently linear, the way one can hear the texture of Messiaen's Mode de valeurs. Thus Boulez's composition is not at all like Messiaen's, its putative model, in aural effect. There is no consistent “hypostatization” to which the ear can grow accustomed. Since Boulez treats his “four constituents of sound” as independent variables, any pitch can occur in any register at any loudness and with any duration or attack.

Not only that, but whereas Messiaen clearly imagined his musical “atoms” or “particles” as sounds, and took acoustical factors (like the greater loudness and sustaining power of the low end of the keyboard) into account in devising his algorithms, Boulez's are entirely abstract or “conceptual.” Boulez's series of twelve dynamic gradations, in particular, is entirely utopian, both in its assumption that the twelve levels can be manipulated as discrete entities on a par with pitches and durations, and also in the way levels are assigned to pitches regardless of register.

When the texture becomes dense, moreover, so much interference is caused by the mixing of registers that there can be no hope of grouping the individual notes into contrapuntal lines by ear, hence no way of perceiving by listening the relationship between the sounds heard on the “surface” of the music and the axioms that motivated their choice. In bluntest terms, then, the paradox created by “total serialism” is this: once the algorithms governing a composition are known (or have been determined), it is possible to demonstrate the correctness of the score (that is, of its component notes) more decisively and objectively than is possible for any other kind of music; but in the act of listening to the composition, one has no way of knowing (and, no matter how many times one listens, one will never have a way of knowing) that the notes one is hearing are the right notes, or (more precisely) that they are not wrong notes.

Indeed, by excluding beams from the notation, Boulez makes it difficult to gain this knowledge even by eye. (Hence, too, not only the arduousness but also the tediousness of the foregoing explanation; the reader is forgiven for skimming.) The extreme fragmentation of the texture into atomic particles insures that, paradoxically, all the meticulous “precompositional” planning—the music's basic theoretical justification—is lost on the listener, and even on the score reader. The music yields its secrets—that is, its governing algorithms or a priori rules—to nobody's senses, only to the mind of a determined analyst (which is why so much of it remained secret for so long).

The value of technical analysis as a separate musical activity, therefore, experienced an unprecedented boom. (Boulez was overheard at Darmstadt to say that the age of the concert had passed; scores need no longer be played, just “read”—i.e., analyzed.) Along with the growth of integral serialism, then, there grew up a new musicological specialization, that of music analyst (sometimes loosely identified with the much older and broader calling of music theorist), and a number of outlets for the practice in the form of specialized journals. The first to appear was Die Reihe (“The row”), widely regarded as the unofficial Darmstadt house organ, issued between 1955 and 1962 by the Vienna publisher Universal Edition, and coedited by Stockhausen and the equally intransigent Herbert Eimert, who as early as 1924had published a little practical method for twelve-tone music, the first of its kind. An English translation of Die Reihe, published in the United States between 1958 and 1968, retained the German title (though it might have been called The Row or The Series).

Notes:

(33) Peter Hill, “Piano Music II,” in The Messiaen Companion, ed. P. Hill (Portland, Ore.: Amadeus Press, 1995), p. 319.

Citation (MLA):
Richard Taruskin. "Chapter 1 Starting from Scratch." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 21 Oct. 2014. <http://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-001010.xml>.
Citation (APA):
Taruskin, R. (n.d.). Chapter 1 Starting from Scratch. In Oxford University Press, Music in the Late Twentieth Century. New York, USA. Retrieved 21 Oct. 2014, from http://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-001010.xml
Citation (Chicago):
Richard Taruskin. "Chapter 1 Starting from Scratch." In Music in the Late Twentieth Century, Oxford University Press. (New York, USA, n.d.). Retrieved 21 Oct. 2014, from http://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-001010.xml