**1. “Tombeau”: Serial Harmony**

(Note: Due to copyright restrictions, examples from the score of Pierre Boulez’s “Tombeau” have not been included. The full orchestral score is available from Universal Edition.)

Pierre Boulez’s

*Pli selon pli*“(Fold by fold”), composed in 1957–62 for soprano and orchestra, is a masterpiece in associations between text and music. Planned as a large-scale portrait of the French symbolist poet Stéphane Mallarmé (1842–1898), the piece’s five movements approach and solve the problems of text setting in different ways. Composed immediately after the Sonata No. 3 for piano, Boulez’s first essay in formal mobility, the piece is also a masterpiece in the establishment of a system of harmonic order that contributes both to a correspondence of melody and harmony, and to formal coherence. “Serial harmony”—the term is not Boulez’s and has no reference to principles of tonal harmony—is a convenient means for describing his approach at this time: A twelve-note set serves as a “parent set,” from which other sets are derived and from which an array of harmonic units is established. The approach focuses less on the linear control offered by serial operations and more on the coherence that can be achieved with intervals and harmonic units, which are subjected to serial techniques.

“Tombeau,” the fifth and final movement of

*Pli selon pli*, is an audibly explicit demonstration of the efficacy of serial harmony and the ways in which it can contribute to form. Boulez’s concerns with serial harmony may be traced to his critiques of the inadequacy of serialism to relate vertical and horizontal structures, as in the way in which “accidental harmonies” could result from the emphasis on horizontal aspects. He has been harshly critical of the absence of control over vertical structure in much of Schönberg’s twelve-tone music and his own early works, yet has credited Webern’s achievement in the fifth movement of the Cantata No. 2, Op. 31 with composing “…the four melodic lines such that they meet at the same point to form a specific harmony. The counterpoint becomes convincing because the vertical, horizontal and diagonal aspects are controlled by the same laws.” (Célestin Deliège.

*Pierre Boulez: Conversations with Célestin Deliège*, p. 94) Regarding his own music, he noted, “I have often been absorbed by a concern to discover clear harmonic relationships. If we can write harmony with melodic line under laws common to both, then we can begin to find a solution that will considerably enrich the musical vocabulary.” (Célestin Deliège.

*Pierre Boulez: Conversations with Célestin Deliège*, p. 91)

His solution in the early 1950s, coincident with the composition of

*Le marteau sans maître*, was a technique for generating an array of harmonies by means of operations on a twelve-note set. In an interview in the 1970s, he observed, somewhat obliquely, that:

In my most recent works, practically all the pitches are deduced from each other by means of harmonic systems such as those which can be multiplied by each other. I believe it is impossible to write in two different dimensions following two different sets of rules, and that one must in fact follow rules that apply reciprocally to the horizontal and the vertical.

(Célestin Deliège.

*Pierre Boulez: Conversations with Célestin Deliège*, p. 90)

Introduced and discussed in his article

*Èventuellement…*from 1952, the technique to realize serial harmony has come to be known, somewhat erroneously and perhaps from a misguided translation from the French, as “frequency multiplication.” (Pierre Boulez,

*Boulez on Music Today*, pp. 39–40) One advantage of this technique as a means for systematizing pitch is that the total pitch content of a work, both the horizontal and vertical dimensions, is based on a single twelve-note set; another is that it provides a cogent system for non-tonal harmonies. Of particular note is that the same array of harmonies in “Tombeau” was used in a number of subsequent compositions, up to

*cummings ist der dichter*of 1970; succeeding works would explore different resources for pitch organization. Although Boulez has never explained why he has worked with the same array over a period of approximately twenty years—and then disregarded it—it is unquestionably this recycling of material that contributes to what one could designate as the “Boulezian” sound.

From a theoretical standpoint—arguably, from an audible one as well—“Tombeau” unfolds its process of serial harmony in an exceptionally clear manner (one might even say “fold by fold”) over the course of its fifteen minutes. The first eight measures of the piece introduce a twelve-note set (the “parent set”) in the piano, which is identified in the score as a principal instrument (

*piano principal*). In these opening measures the twelve pitches of the parent set are grouped into five harmonic units of two, four, two, one and three notes. Measures 9–19 present a second statement of this set, though varied by harmonic units of four, two, one, three and two notes. Measures 20–39 continue with a third statement of this set, whose harmonic units comprise two, one, three, two and four notes. This is followed by a fourth statement in mm. 40–48 with harmonic units of one, three, two, four and two notes, and a fifth statement in mm. 53–61 of three, two, four, two and one notes. The five statements of the set in these measures are:

The linear rendition of the parent set, though not given in this work in any instance, can be deduced from Example 1 as:

**Example 1.**The linear rendition of the parent set, though not given in this work in any instance, can be deduced from Example 1 as:

**Example 2.**

The four sets in mm. 9–61 are created by varying the harmonic groupings of the parent set, though not the pitches themselves; for this reason they could be designated as “derivative sets.” The number of notes in each harmonic unit, but not the notes themselves, is modified by rotation, whereby the number of notes in the first harmonic unit of a set is moved (or rotated) to the end of the following derivative set. The “magic square” in Example 3 illustrates this, in which each number represents the number of notes in each of the five harmonic units:

Parent Set: 24213

Derivative Set 1: 42132 [“2” of the Parent Set is moved to the end of this set]

Derivative Set 2: 21324 [“4” of Derivative Set 1 is moved to the end of this set]

Derivative Set 3: 13242 [“2” of Derivative Set 2 is moved to the end of this set]

Derivative Set 4: 32421 [“1” of Derivative Set 3 is moved to the end of this set]

**Example 3.**

Continuing past the fourth derivative set would reproduce the parent set. The emphasis is placed on the ordering of the harmonic units in these sets, rather than on the ordering of individual pitches; hence, the serial aspect of the system.

Each of these five sets serves as source material to produce frequency-multiplication sets. In this process, the intervals of one harmonic unit are transposed to each of the pitches of a second harmonic unit; thus, units of greater harmonic density are created from units of lesser harmonic density. Clearly, the technique works more successfully with harmonic units, rather than with single pitches, which explains why the pitches of the five sets in Example 1 are grouped into five units. Presumably, its nomenclature can be attributed to the nature of the technique. If a unit of three pitches is multiplied with another unit of three pitches, then the result is nine pitches (3x3), duplicate pitches being excluded. This is illustrated in Example 4, where the intervals of one three-note unit are multiplied with (or transposed to) the pitches of another three-note unit, producing nine pitches:

**Example 4.**

D is the only pitch that is duplicated in the three steps of multiplication; a harmonic unit comprising eighth unique pitches is the result.

Because of the importance of interval associations in this technique, set-theory analysis can be instrumental in determining the presence of the harmonic units in the music and in revealing the interval structure of these units. To illustrate the latter, the two harmonic units from Example 4 will be used. Firstly, determine the prime-form designation of each unit (see Alan Forte,

*The Structure of Atonal Music*, pp. 3–7):

**Example 5.**

Then, add, from left to right, each number of the first unit to each number of the second unit (any resultant numbers greater than twelve are reduced by twelve):

0 1 3

+ 0 2 6

_____

0 2 6 (0 + 0 2 6)

1 3 7 (1 + 0 2 6)

3 5 9 (3 + 0 2 6)

**Example 6.**

Lastly, arrange the numbers in arithmetical order, with duplications eliminated. The result is the pitch-class (pc) set [0,1,2,3,5,6,7,9] (or 8-Z29, according to Forte’s classification), and this, as Example 7 shows, is the prime-form designation of the definitive harmonic unit in Example 4:

**Example 7.**

Note that the process is commutative in terms of interval content; which is to say, switching the order of the units in Example 4 will produce the same pc set:

0 2 6

+ 0 1 3

_____

0 1 3 (0 + 0 1 3)

2 3 5 (2 + 0 1 3) = [0,1,2,3,5,6,7,9]

6 7 9 (6 + 0 1 3)

**Example 8.**

Unquestionably, the most practical aspect of frequency multiplication from a compositional standpoint is the relationship that is established amongst various harmonic units, as Boulez has emphasized. The process, in fact, centers primarily on intervals, rather than frequencies; in essence, the intervals of one harmonic unit are “mapped” onto the pitches of another unit. In Example 4, the definitive eight-note harmonic unit is created by mapping the intervals of the first unit on the pitches of the second unit. As a result, the definitive unit will have a superstructural association with both units; that is, each of the initial units will be embedded in multiples instances in the definitive unit. For example, the definitive unit [0,1,2,3,5,6,7,9] comprises five [0,1,3] pc sets (the first unit) and five [0,2,6] pc sets (the second unit). Interval associations between harmonic units are thus the essence of this technique and of Boulez’s serial harmony, in general; for this reason, the term “interval mapping” might be a more appropriate designation, though one that is not in current usage.

Boulez realized an array of frequency-multiplication sets by applying the process in Example 4 to each of the five sets (the parent set and the four derivative sets of Example 1). Each of the five harmonic units of a set was multiplied with every other unit of the set including itself, in which each unit was used as a “fixed multiplicative factor”. For example, unit 2, as the fixed multiplicative factor, was multiplied with units 1, 2, 3, 4 and 5; unit 3, as the fixed multiplicative factor, was multiplied with units 1, 2, 3, 4 and 5, and so forth. Boulez provided a schematic diagram to illustrate this process, as well as to illustrate the resultant associations in the array, what he called “isomorphic relations.” In the following diagram, the letters “a,” “b,” “c,” “d” and “e” represent the five harmonic units of the parent set, and the letters “m,” “n,” “p,” “q” and “r” are the number of notes per unit.

a{m b{n c{p d{q e{r

m x [aa ab ac ad ae]

n x [ba bb bc bd be]

p x [ca cb cc cd ce]

q x [da db dc dd de]

r x [ea eb ec ed ee]

**Example 9.**

(Pierre Boulez,

*Boulez on Music Today,*p. 79)

The diagram shows that every harmonic unit appears twice in the array (i.e., “ab” is the same as “ba,” as part of the commutative process), except for units multiplied by themselves (i.e., “aa,” “bb,” “cc,” “dd” and “ee,” read diagonally). This example also shows the relationships between multiplied units of a single frequency-multiplied set. For example, all units on line “m” share the common unit “a” (each of which is duplicated in column “a”), yet each is distinguished by unique groupings with other units of the set. The significance of the letters “m” through “r” will be discussed subsequently.

Following the diagram in Example 9, the parent set, the four derivative sets and the array of frequency-multiplication sets in “Tombeau” are reproduced in Example 10.

**Example 10.**

*Every*pitch in “Tombeau” can be explained in reference to this example, an astonishing instance of adherence to a pre-compositional plan and the significance of the plan.

The numbers in Example 10 are means to designate specific sets and harmonic units, and to designate the ways in which they are formed. Sets in the same column (read vertically) are derived from the twelve-note set on the uppermost staff. For example, set 1-22 (column 1, line 2) is derived from set 1-00 by using the second unit of set 1-00 as the fixed multiplicative factor, where this unit is multiplied with every unit of the same set, including itself: 2x1, 2x2, 2x3, 2x4 and 2x5, or “ba,” “bb,” “bc,” “bd” and “be,” according to Example 9. Similarly, sets on the third staff (-33) use the third unit of each twelve-note set as the fixed multiplicative factor and so forth. Note from Example 10 that Boulez disregarded using the initial unit of each set as a fixed multiplicative factor, which would result in sets x-11 from line “m” of Example 9, for reasons that are not apparent.

A crucial aspect of this technique is that the registral distribution of notes in the harmonic units of the twelve-note sets is decisive in determining the pitch transpositions, though not the interval content, of the definitive units. Boulez has noted, “Each of the superimpositions of frequencies is evidently susceptible to modification each time it is reproduced, all the notes being novel as to

*tessitura*and susceptible to permutations along a vertical line in relation to the intervals of definitions.” (Pierre Boulez,

*Notes of an Apprenticeship*, pp. 167–68) This is illustrated in Example 9, where the letters “m,” “n,” “p,” “q” and “r” indicate the number of different transpositions available for each multiplied unit based on the number of notes in each of the original harmonic units. For example, the first unit of set 1-00, if multiplied with the second unit, will produce two different transpositions of the definitive harmonic unit based on the registral distribution of notes in the first unit (F/E-flat vs. E-flat/F):

**Example 11**.

However, the definitive units in Example 11 are transpositionally equivalent, inasmuch as each preserves the same interval content. Consequently, from a theoretical standpoint, the pitches of sets x-22 through x-55 in Example 10 are relative; however, these pitches can be justified as absolute in “Tombeau,” for these are the transpositions that Boulez used. Ultimately, this would signify that Boulez was more concerned with the interval associations produced with this technique, rather than with the actual pitch transpositions, suggesting, again, that the term “frequency multiplication” is a misnomer.

One final step remains in the production of the harmonic units in Example 10. Boulez transposed units that are multiplied with a fixed multiplicative factor in relation to the interval between the lowest pitch of the fixed multiplicative factor and the lowest pitch of the first unit of the set. This can best be illustrated when one pitch is used as a fixed multiplicative factor, resulting in a transposition of the twelve-note set (e.g., set 1-44 of Example 10, an interval-class (ic) 3 transposition of set 1-00). Prior to multiplying the fourth and first units, Boulez transposed the first unit ic3 to A-flat/G-flat, ic3 being the interval between the fourth unit (A-flat) and the lowest note of the first unit (F):

The remaining units were treated similarly, thereby resulting in an ic3 transposition of set 1-00 (

**Example 12.**The remaining units were treated similarly, thereby resulting in an ic3 transposition of set 1-00 (

*cf*. Example 2):**Example 13.**

This transposition is applied to each process of multiplication in Example 10, presumably to establish pitch commutativity (in addition to interval commutativity) between harmonic units in the array. For example, if the order of units in Example 11 is reversed, the same harmonic unit will be produced, yet at a different transposition (i.e., at ic5):

**Example 14.**

Transposing the second unit ic5 in relation to the interval between the lowest notes of units one and two will yield the identical unit:

**Example 15.**

Seemingly, Boulez applied this transposition specifically for this reason and applied it in every instance for purposes of consistency.

Such is the methodology of serial harmony in Boulez’s “Tombeau.” It is unquestionably a cogent strategy for establishing interval associations between simple and complex harmonic units, in which “progressions” between these units are governed by a serial order. The next entry will address how this array of serial harmony is instrumental in defining the form of “Tombeau.”