For the purposes of this investigation, a simple working knowledge of multiplication is sufficient in order to understand the constellation that includes this harmony-producing technique. The most straightforward approach to multiplication has been taken by Lev Koblyvakov so I take his work as my example in explicating this compositional process.. Boulez is able to generate additional harmonic materials from a single generalized series by re-partitioning it into new pitch-class sets that are then subject to the process of multiplication. Fig. 3.8 shows five different partitionings of the
Le Marteau series, referred to by Koblyakov as “Domains.”17 Fig. 3.9 verticalizes the partitions in each Domain (distinguished by a series of Greek letters), each labeled a
through e so that the process of multiplication can take place according to the schematic shown in Fig. 3.2.
Fig. 3.8. Five Domains as described by Koblyakov as they result from five different partitions of the Le Marteau series
Boulez creates a master lexicon of matrices, generated from a series of Domains, the totality of which is often erroneously referred to as a multiplication matrix, shown in
17
Fig. 3.10. Consequently, the term “multiplication table” will be used to refer to the totality of the matrices in Fig. 3.10, with individual matrices identified by their Domain name. The schematic shown in Fig. 3.2 is applied to each Domain, labeled Αλ, Βη, Γα,
∆ε, and Επ.
Fig. 3.9. Domains with verticalized partitions of the series; annotations provided by the author
Similarly to the matrix shown in Fig. 3.6 where the top line contains the series’ partitions, the top lines of each matrix in Fig. 3.10 are substituted with a new set of partitions of the same series that define the Domain, with each partition labeled a–e. The products, or
blocs sonores, in later rows are identified by their combination of partitions. The products that appear in the remaining rows are designated by another series of Greek letters, Μυ,
Fig. 3.10. Transcription of Boulez’s sketch of multiplication matrices for Le Marteau
Νυ, Ξ, Οµ, and Πι. Combining a Greek Domain-name with Greek row-name specifies a particular set of five blocs sonores within a given row of the matrix. Even in the use of Greek letters, Boulez follows an initial succession as he typically does whether or not the series of representative symbols involves the Greek or Latin alphabets, Roman or Arabic numerals, or another series of symbols.
The first use of this particular multiplication table was in the unpublished Oubli signal lapidé, in which Boulez primarily uses the table’s partitions (indicated by the letters a–e)for the local organization of pitches and gestures. The first page of this unpublished piece, provided by the Paul Sacher Foundation, is shown in Fig. 3.11 with analytic markings highlighting the individual Μυ partitions from the Γα and Επ
Domains. Further analysis of this work would be a useful endeavor to see if Boulez progressed to using the full potential of the multiplication table as well as his possible employment of any logical path to connect products from particular Domains.
The use of a particular multiplication matrix within the table provides Boulez with the opportunity to organize local structures and characterize them according to the properties of the harmonies contained within it. Boulez tends to observe the groupings of blocs sonores, keeping them distinct in order to highlight the features of a particular matrix. He uses these blocs sonores to organize materials from the various movements of Le
Marteau, most straightforwardly in the first movement, “avant l’artisanat furieux.” Fig. 3.12 shows a multiplication matrix that Boulez has rearranged to generate new
trajectories through the products. By following the arrows, beginning with product ea in the upper left-hand corner, one can trace the route through the matrix that Boulez uses in
Fig. 3.11. Analysis of the unpublished Oubli signal lapidé; reprinted with permission from the Paul Sacher Foundation
his employment of Domain Βη in mm. 42–52 of “avant l’artisanat furieux.” Note that Boulez alternates between paths that maintain or avoid what he refers to as “partially isomorphic objects,” or those that have an “original object in common; all the objects in b, for example will have the common structure b, while the other structure will be variable: ab, bb, cb, db, eb.”18 He refers to products as “non-isomorphic” if they are “pure objects” in that they are multiplied by themselves, e.g., dd, cc, bb, etc.
Fig. 3.12. Boulez’s path through a matrix derived from Domain Βη for “avant l’artisanat furieux,” mm. 42-52, as described by Koblyakov19
In the score and analysis, shown in Fig. 3.13, Boulez’s path through the matrix can be found by following the boxed products according to the following scheme: m. 42:
ea (vibraphone, guitar and viola); m. 43: cb (viola), bc (flute in G); mm. 43-44: cd
(vibraphone); m. 45: de and dd (guitar and viola); m. 46: cc (vibraphone), bb (flute and vibraphone), ca (guitar and viola), although the meaning of the T5 transposition here is not apparent; m. 47, ba (vibraphone and guitar); m. 48: cb (viola); m. 48/49: dc
(vibraphone); m. 49: db (guitar and viola); m. 50/51: ca (vibraphone and guitar); m. 52:
da (vibraphone and guitar). 18
Boulez, On Music Today, 80. 19
In a similar fashion to the example in Fig. 3.13, Boulez chooses a path through the
Επ Domain in “Don,” shown in Fig. 3.14, in which each product shares a specific
partition in common, in this case a. The succession of products are as follows: ea and da
(vibraphones), ca (harps), ba (mandolin and guitar), and a (celeste and tube bells).
While Boulez uses all matrices from the multiplication table in “Don” and “Tombeau,” the framing movements of Pli selon Pli, the ramifications of their use for large-scale form will be discussed in greater detail in Chapters Five.
Fig. 3.14. Blocs sonores from Domain Επ in “Don”