Structures Recomposed · 2014-07-05 · 25 Boulez: Structures Recomposed In [19], Pierre Boulez...

25

Boulez: Structures Recomposed

In [19], Pierre Boulez describes a compositional strategy called analyse creatri-ce, creative analysis, which is opposed to what he calls “sterile academic” anal-ysis in that the analytical results are used as germs to create new compositions.Before discussing Boulez’s ideas in detail, we should stress that his proceduretranscends the purely analytical or compositional activities: He proposes a co-herent double activity that includes both analysis and composition. This meansthat our own discourse in this chapter will deal with both, analysis and com-position, the latter more specifically realized by use of the music compositionsoftware RUBATO! [75].

25.1 Boulez’s Idea of a Creative Analysis

Let us explain the practical consequences of Boulez’s strategy for the analyt-ical and compositional e!orts1. Anne Boissiere [15] has given a concise sum-mary of Boulez’s ideas on creative analysis, which comprise these core items:The analysis focuses more on the limits of the given composition than on thehistorical adequacy. These limits open up what has not been said, what wasomitted or overlooked by that composer. This hermeneutic work is not drivento deduce a new composition as a special case of what has been recognized(deduction), nor is it meant to help build the new composition by a passagefrom the particular to the general (induction). Referring to Gilbert Simondon’sphilosophical reflections [106], the creative movement consists of the openingof a topological neighborhood of the given analysis within a space of analyticalparameters. In such a space, analytical structures similar to the given one areselected and eventually used as germs for the construction of new compositions.This ‘horizontal’ movement is called “transduction” by Simondon.

In this transduction process, what Boulez calls the composer’s gesture, isthe movement toward the creation of new compositions, which share precisely1 For a more philosophical discussion of this approach, we refer to [82, ch. 7].

G. Mazzola et al., Musical Creativity,Computational Music Science, DOI 10.1007/978-3-642-24517-6 25,© Springer-Verlag Berlin Heidelberg 2011

279

280 25 Boulez: Structures Recomposed

those analytical structures reflecting the given analysis. More concretely, wetake the analysis of the given work and make a number of “small” value changesto the analytical parameters. For example, if we have exhibited a set of pitchinversion symmetries that govern the given work, we may extend that set andinclude also time inversion symmetries (retrograde). Or if we have recognizedthat the voices of the di!erent instruments are derived by some systematicprocedure, e.g. time expansions (dilations) plus transpositions from a leadingvoice, then we may add more instruments and apply the same procedure, e.g.further time expansions (dilations) plus transpositions to define these addedvoices.

This creative gesture—building new works from the transgression of theanalytical structure discovered in the given work—is what Boissiere calls adetonation. It is precisely this act of breaking the given structures and steppinginto unknown neighborhoods that characterizes Boulez’s concept of an openwork.

Boulez’s approach is visibly akin to our concept of creativity. Boulez’screative analysis takes the given work as the critical concept in our theory andthen inspects its walls with the analytical e!orts. The open work is exactly thisanalytical search for walls where the given work might be limited. The creativeact then would consist in the action of opening those walls and stepping to newcompositional paths.

To our knowledge, these ambitious claims have not been backed by con-crete examples: How should and would such a strategy work in detail? This iswhat we have accomplished in a formal (mathematical) setup and on the levelof computer-aided composition and what we want to discuss in this chapter. Inview of Boulez’s poetical text, such an enterprise cannot be more than a firstproposal. But we believe that it could open a fruitful discourse on the role ofcreativity in the dialectic between analysis and composition. In this sense, ourapproach is not a thesis but a detailed experiment following Boulez’s ideas. Itis, therefore, completely logical to pursue the trajectory to its completion: tothe construction of a full-fledged composition2.

Our choice of Boulez’s Structures is not random; it relates to the promi-nent role that this composition has played in the development of serialism. Thisis also confirmed by the fact that Gyorgy Ligeti has published a very carefulanalysis of Structures, part Ia. Ligeti’s investigation [67] is neutral and pre-cise, but it abounds with strong judgments on the work’s compositional andaesthetic qualities. Therefore, our experimental application of creative anal-ysis to Structures is not by chance. The very success (or failure) of the se-rial method has been related to this composition, which was not only one ofBoulez’s sucesses, but also a turning point in his compositional development.2 It should however be noted that such a creative analysis had been applied in the

case of Beethoven’s op. 106 [71] before we knew about Boulez’s idea. The presentapproach is somewhat more dramatic, since we shall now apply Boulez’s idea totwo of his own works, namely Structures [17, 18].

25.1 Boulez’s Idea of a Creative Analysis 281

In view of Boulez’s principle of creative analysis, when applied to his composi-tional turning point in the Structures, one is immediately led to the question:Would it be possible to write a world of new music on the principle of serialismor was it just a radical experiment without much long-range e!ect? This isan important question when taking seriously the idea of creative analysis, andnot only as a recipe for fabricating yet another work. And it is also an impor-tant question relating to a more systematic and demystifying understanding ofmusical creativity using analytical activity.

In our case, the Structures, the Boulezian gesture of opening a work’slimits is a doubly critical and di"cult one: On the one hand, it should helpdetermine whether the huge calculations that led to the composition are worthbeing reused with aesthetic success. On the other hand, the method of seri-alism also marks the computational limits of humans to compose music. Thislatter fact will lead to the question of using music technology and in particularcomputers in a creative context.

We must understand here how to integrate computational power into cre-ative works of music, and on what level of creation this can or should be done.Boulez’s Structures is an excellent testbed to learn this lesson. It teaches usthat the control of laborious computational processes cannot be systematicallydelegated to very limited human calculation power and that there is a life be-yond strictly human composition. To paraphrase Schoenberg, “Somebody hadto be Boulez.”

Of course, computers are widely used by modern composers, but it is acommon belief that creativity is separated from such procedures; it terminateswhen the big ideas are set. And computers are just doing the mean calculations.Apart from being classically wrong, we shall see that this is not realistic. Infact, no composer would contest the creative contribution of trying out a newcomposition on the piano—playing it on the keys and listening to its acousticalrealization, which may give a strong feedback for the creative dynamics, even onthe gestural level of one’s hands, as is testified by Ligeti and other composers,see [68]. We have to contradict Marshall McLuhan: The medium is not alwaysthe message. But it gives the message’s germ the necessary mold and resonanceto grow into a full-fledged composition.

Before delving into the technical details, we should address the questionof whether not only computational computer power is necessary or advanta-geous for modern compositions, but also conceptual mathematical power. Isn’tmusical composition anyway su"ciently controlled by plain combinatorial de-vices: permutations, recombinations, enumerations, and the like? The questionis in some sense parallel to the question of whether it is su"cient to control acomputer’s behavior on the level of binary chains, or machine language. Or elsethe question of whether it is not completely su"cient to perform a compositionfor piano by simply controlling the mechanical finger movements and forgettingabout all those psycho-physicological ‘illusions’ such as gestures.

The parallelism lies in the fact that all of these activities are shaped byhigh-level concepts that create the coherence of low-level tokens in order to


express thoughts and not just juxtapose myriad atomic units. Of course canone write a computer program in machine language, but only after having un-derstood the high-level architecture of one’s ideas. The artistic performance ofa complex composition only succeeds when it is shaped on the high mentallevel of powerful gestures. And the composition of computationally complexmusical works needs comprehensive and structurally powerful concepts. Com-binatorics is just a machine language of mathematical thinking. We shall seein the following analysis that it was precisely Ligeti’s combinatorial limitationthat hindered his understanding of the real yoga of Boulez’s creative construc-tions. You can do combinatorics, but only if you know what is the steeringidea—much as you can write the single notes of Beethoven’s HammerklavierSonata if you know the high-level ideas. The mathematics deployed in the mod-ern mathematical music theory is precisely the tool for such an enterprise. Itis not by chance that traditional music analysis is so poor for the compositionof advanced music: Its conceptual power is far too weak for precise complexconstructions, let alone for their computer-aided implementation.

This lesson is interesting in the creativity process when it comes to suc-cessfully inspecting those walls. The critical point in wall inspection is torecognize walls, but this might be very di"cult for many reasons. One rea-son is that one might stand too close to a wall and not be able to see thatthere is a wall exactly where one stands. One then needs tools to recognizethe bigger architecture of the critical concept, to conceive entire walls insteadof staring at an unrecognized surface detail. Higher-level languages are exactlywhat is needed to conceive higher architectures of critical concepts in the cre-ative process. Why is this needed? Because the abstraction provided by higherlanguages reduces phenomena to their essential characteristics. Modern math-ematics are such higher languages; they enable insights into what is essential.The great mathematician Alexander Grothendieck had exactly this ingeniouscapability to construct those powerful abstractions that gave access to solutionsof di"cult problems (such as the Weil conjectures or Fermat’s conjecture). Ifyou want to open a wall, you have to understand where it is fixed and howit is connected to other parts of that critical concept. We shall now see thatBoulez had exactly this capability to understand the higher architecture ofserial approaches to musical composition to solve some of its basic problems.

25.2 Ligeti’s Analysis

Ligeti’s analysis [67] of Structures Ia exhibits the totality of rows appearingin this section of the composition. It starts from the given serial rows SP forpitch classes and SD for durations (the primary parameters), as well as SL forloudness and SA for attack (the secondary parameters). It then presents andinvestigates that central 12! 12-matrix3 Q = (Qi,j) which gives rise to all row3 Ligeti names it R, but we change the symbol since R is reserved for retrograde in

our notation.

25.3 A First Creative Analysis of Structure Ia from Ligeti’s Perspective 283

permutations for all four parameters. Whereas the construction of Q is rela-tively natural, the subsequent permutations thereof for the primary parametersseem to be completely combinatorial, and even more radically those for the sec-ondary parameters. Ligeti attributes to these constructions the qualification ofcombinatorial fetishism4. This is even worse when it comes to the secondaryparameters, where Boulez applies what Ligeti calls chessboard knight paths,a procedure that in Ligeti’s understanding qualifies as purely numerical gamewithout any musical signification.

This (dis)qualification remains valid in Ligeti’s final remarks on the newways of hearing, which are enforced by this new compositional technique. Hecompares the result to the flashing neon lights of a big city, which, althoughbeing driven by precise machines, generate an overall e!ect of statistical soundswarms. He concludes that with this radical elimination of expressivity, stillpresent in Webern’s compositions, the composition finds its beauty in theopening of pure structures. And Boulez—we follow Ligeti’s wording—in sucha “nearly obsessive-compulsive neurosis, strains himself at the leash and willonly be freed by his colored sensual feline world of ‘Marteau.’ ”

Ligeti’s main objection to Boulez’s approach is that he makes abstrac-tion from the parameters and plays an empty game of numbers instead. Wenow want to contradict this verdict and show that in the language of modernmathematics—topos theory to be precise—Boulez’s strategy is perfectly nat-ural, and in fact, only reasonable when dealing with such diverse parametersas pitch classes, durations, loudnesses, and attacks. When we say “natural,”we mean mathematically natural, but the fact that a musical constructionis only understood by advanced mathematical conceptualization, and not bynaive combinatorial music theory, proves that mathematical naturality e!ec-tively hits the musical point. This fact will be confirmed later in this chapterby our ability to implement our findings in the music software Rubato in orderto comply with the creative part of Boulez’s principle. Music theorists have tolearn that from time to time, conceptual innovations may even enlighten theirossified domains. It is not the music’s fault if they are “dark to themselves”5.

25.3 A First Creative Analysis of Structure Ia fromLigeti’s Perspective

We observe right from the beginning that there is no intrinsic reason to transferthe twelve-pitch-class framework to the other parameters. If the number twelveis natural in pitch classes, its transfer to other parameters is a tricky business.How can this be performed without artificial constructs?4 “. . . schliesslich die Tabellen fetischartig als Mass fur Dauernqualitaten angewandt

. . . ”5 Title of a Cecil Taylor LP.


To understand Boulez’s procedure, let us first analyze the matrix Q con-struction. It yields one pitch class row for every row. The ideas run as fol-lows. We start with a modern interpretation of what is a dodecaphonic pitchclass series SP . Naively speaking, SP is a sequence of twelve pitch classes:SP = (SP,1, SP,2, . . . SP,12). More mathematically speaking, it is an a"ne mor-phism6 SP : Z11 " Z12, whose values are determined by the twelve values one1 = 0 and the eleven basis vectors ei = (0, . . . 1, . . . 0), i = 2, 3, . . . 12, wherethe single 1 stands on position i # 1 of that sequence. This reinterpretationyields SP,i = SP (ei). The condition that a series hits all twelve pitch classesmeans that the images SP (ei), SP (ej) are di!erent for i $= j.

This reinterpretation of a dodecaphonic series means that it is viewed asa Z11-addressed point of the pitch class space Z12 in the language of topostheory of music [78]. This language views the series as a point in the spaceZ12, but just from the perspective of a particular domain, or address, namelyZ11. In topos theory of music, a space Z12 is replaced by its functor @Z12,which at any given address B, i.e. module over a specific ring, evaluates tothe set B@Z12 of a"ne module morphisms f : B " Z12. This means that theaddress B is a variable and that our dodecaphonic series is just a point at aspecific address among all possible addresses. In other words, the change ofaddress is completely natural in this context. What does this mean? Supposethat we have a module morphism g : C " B between address modules. Thenwe obtain a natural map B@Z12 " C@Z12 that maps f : B " Z12 to thecomposed arrow f · g : C " Z12. For example, if we take B = C = Z11, and ifg(ei) = e12!i+1, then the new series SP · g is the retrograde R(SP ) of theoriginal series.

Our claim is that all of Boulez’s constructions are simply address changemaps and as such follow a very systematic construction. So the combinatorialityis viewed as a particular technique from topos theory. Of course, Boulez did notknow this, since topos theory was not even invented at that time, and Yoneda’slemma, which is the password to all these mathematical constructions, was onlypublished in 1954, one year after the publication of Structure I. But this makeshis approach even more remarkable; one could even state that in view of thistemporal coincidence, Boulez’s Structures are the Yoneda lemma in music.

25.3.1 Address Change Instead of Parameter Transformations

The trick that enables Boulez to get rid of the unnatural association of di!erentparameters with the serial setup stemming from pitch classes is this: For any(invertible) transformation T : Z12 " Z12, we have a new pitch class series,namely the composition T · SP . For a transposition T = Tn, we get the n-fold6 An a!ne morphism f : M ! N between modules M, N over a commutative ring R

is by definition the composition f = T t ·g of a R-linear homomorphism g : M ! Nand a translation T t : N ! N : n "! t + n. A!ne morphisms are well known inmusic theory, see [78].


transposed series. For an inversion U(x) = u#x, we get the inverted series, etc.Now, it is evident that one can also obtain this e!ect by an address change,more precisely, if T : Z12 " Z12 is any a"ne transformation, then there isprecisely one address change C(T ) : Z11 " Z11 by a base vector permutationsuch that the diagram

Z11 C(T )####" Z11

SP

!!"!!"SP

Z12T####" Z12

(25.1)

commutes. Instead of performing a parameter transformation on the codomainof the pitch class row, we may perform an address change on the domain Z11.Note, however, that the address change C(T ) is also a function of the underlyingseries SP .

What is the advantage of such a restatement of transformations? We nowhave simulated the parameter-specific transformation on the level of the univer-sal domain Z11, which is common to all parameter-specific series. This enablesa transfer of the transformation actions on one parameter space (the pitchclasses in the above case) to all other parameter spaces, just by prependingfor any series the corresponding address change. So we take the transformationT on Z12, replace it with the address change C(T ) on Z11, and then applythis one to all other series, i.e. building SD · C(T ), SL · C(T ), SA · C(T ). Thismeans that we now have a completely natural understanding of the derivationof parameter series from address changes, which act as mediators between pitchclass transformations and transformations on other parameter spaces. This isthe only natural way of carrying over these operations between intrinsically in-compatible parameter spaces. We replace the spaces by their functors and acton the common addresses. This is quite the opposite of purely combinatorialgaming. It is functoriality at its best. For a deeper understanding of Boulez’sselection of his duration series, referring to Webern’s compositions, please see[35].

25.3.2 The System of Address Changes for the Primary Parameters

Now, nearly everything in Boulez’s construction of part Ia is canonical. Themost important address change is the matrix Q. It is constructed as follows. Itsith row Q(i,#) is the base change C(TSP (i)!SP (1)) associated with the transpo-sition by the di!erence of the pitch class series at position i and 1. The natural7number Q(i, j) in the matrix is therefore Q(i, j) = SP (i) + SP (j) # SP (1), asymmetrical expression in i and j. Moreover, we now see immediately from thedefinition of the operator C in the above commutative diagram that the compo-sition of two permutations (rows) of the matrix is again such a permutation row;in fact, the transpositions they represent are the group of all transpositions.7 We represent elements x # Z12 by natural numbers 0 $ x $ 11.


We may now view Q as an address change Q : Z11 ! Z11 " Z11 on the a"netensor product Z11 ! Z11, see [78, E.3.3], defined on the a"ne basis (ei ! ej)by Q(ei ! ej) = eQ(i,j). For any such address change X : Z11 ! Z11 " Z11, andany parameter Z series SZ " ParamSpace with values in parameter spaceParamSpace, we obtain twelve series in that space by address change SZ ·X ofthe series, and then restricted to the ith rows of X, or equivalently, prependingthe address change (!) rowi : Z11 " Z11 ! Z11 defined by rowi(ej) = ei ! ej .

Given any such address change matrix X : Z11 ! Z11 " Z11, we thereforeget twelve series in every given parameter space. So we are now dealing withthe construction of specific matrix address changes, and the entire procedureis settled. The general idea is this: One gives two address changes g, h : Z11 "Z11 with g(ei) = eg(i), h(ei) = eh(i) and then deduces a canonical addresschange g!h : Z11!Z11 " Z11!Z11 by the formula g!h(ei!ej) = eg(i)!eh(j).So, when X is given, we obtain a new address change of the same type bybuilding the composed address change X · g ! h. For example, the retrogradematrix in Ligeti’s terminology is just the matrix Q · Id ! R deduced from Qby the address change Id ! R. And Ligeti’s U -matrix is deduced from Q byU ! U , where U is the address change associated with the inversion at e!, i.e.it is the composite Q · U ! U .

Now everything is easy: For the first piano, for the primary parameterspitch class P and duration D, and for parts A and B, Boulez creates one matrixQ1

P,A, Q1D,A, Q1

P,B , Q1D,B address change each, all deduced from Q by the above

composition with product address changes T 1P,A = U ! Id, T 1

D,A = U · R ! U ·R, T 1

P,B = U · R ! U · R, T 1D,B = R ! U via

Q1P,A = Q · T 1

P,A, Q1D,A = Q · T 1

D,A, Q1P,B = Q · T 1

P,B , Q1D,B = Q · T 1

D,B . (25.2)

This is quite systematic, but the second piano is now completely straight-forward, in fact the product address changes of this instrument di!er just byone single product address change, namely U ! U :

T 2P,A = U ! U · T 1

P,A, (25.3)

T 2D,A = U ! U · T 1

D,A, (25.4)

T 2P,B = U ! U · T 1

P,B , (25.5)

T 2D,B = U ! U · T 1

D,B . (25.6)

25.3.3 The System of Address Changes for the SecondaryParameters

For the secondary parameters, loudness and attack, Boulez takes one suchvalue per series—deduced from the given series SL, SA—that was derived forthe primary parameters. Intuitively, for each row in one of the above matrixes,we want to get one loudness and one attack value.


For loudness, we start with the Q matrix address change for piano 1 andwith the U matrix for piano 2. We then take an address change a : Z11 "Z11 ! Z11 for part A, and another c : Z11 " Z11 ! Z11 for part B. Theseaddress changes are very natural paths in the given matrix. Path a is just thecodiagonal of the matrix, i.e. a(ei) = e12!i ! ei, while path c is the path shownin Figure 25.1.

Fig. 25.1. The two paths a, c for loudness, part A and part B, in Ligeti’s Q matrixfor piano 1; same paths in the U matrix for piano 2.

Contradicting Ligeti’s verdict, these paths are by no means arbitrary.They are both closed paths if one identifies the boundaries of the matrix. Patha is a closed path on the torus deduced from Q by identifying the horizontaland vertical boundary lines, respectively. And path c is closed on the sphereobtained by identifying the adjacent left and upper, and right and lower bound-ary lines, respectively. The torus structure is completely natural, if one recallsthat pitch classes are identified exactly like the horizontal torus construction,while the vertical one is a periodicity in time, also a canonical identification.The sphere construction is obtained by the parameter exchange (diagonal re-flection!) and the identification of boundary lines induced by this exchange.

For the attack paths, one has a similar construction, only that the pathsa and c are rotated by 90 degrees clockwise and yield paths ! and ". Again,piano 1 takes its values on Q, while piano 2 takes its values on the U matrix.So apart from that rotation, everything is the same as for loudness.

Summarizing, we need just one product address change given by the Utransformation for the primary parameters in order to go from piano 1 topiano 2, while one rotation 90 degrees su"ces to switch between the secondaryparameter paths. Observe that this rotation is just the address change on thematrix space Z11 ! Z11 induced by a retrograde on each factor! It could not besimpler, and barely more beautiful.


25.3.4 The First Creative Analysis

A first transduction is now immediate. Of course, there are many ways to shiftfrom the given analytical data to neighboring data in the space of analyti-cal data. A first way is obvious, and it is also the one in which we urgentlyneed to remedy the evident imperfection of the given construction, namely thenumber of instruments. Why only two instruments? In order to obtain a moreintrinsically serialist construction, one should not work with two, but withtwelve instruments. This is achieved in the most obvious way: We had seenthat the second piano is derived from the first by taking the U matrix insteadof the Q matrix. This suggests that we may now take twelve address changesUi : Z11 " Z11, starting with the identity Id, and generate one instrumentalvariant for each such address change, starting with the structure for the firstpiano and then adding variants for each successive instrument.

This yields a total of twelve instruments and for each a succession oftwelve series for part A and twelve series for part B, according to the twelverows of the matrix address changes as discussed above. For the ith series, thisgives us twelve instruments playing their row simultaneously. Boulez has, ofcourse, not realized such a military arrangement of series. We hence propose acompletion of the serial idea in the selection of the numbers of simultaneouslyplaying series. Observe that the series SP of pitch classes has a unique innersymmetry that exchanges the first and second hexachord, namely the inversionI = T 7.#1 between e and e!, i.e. the series defines the strong dichotomy No. 71in the sense of mathematical counterpoint theory [78, chapter 30]. In part A, wenow select the instrument SP (i) from below and then take I(SP (i)) successiveinstruments in ascending order (and using the circle identification for excessiveinstrument numbers). For part B we take the I-transformed sequence of initialinstrumental numbers and attach the original serial numbers as successivelyascending occupancies of instruments. Figure 25.2 shows the result.

The next step will be to transform this scheme into a computer programin order to realize such compositions and to test their quality. It is now evidentthat such a calculation cannot be executed by a human without excessive e!ortsand a high risk of making errors. Moreover, it is also not clear whether suchcreative reconstructions will yield interesting results, or perhaps only for specialtransformational sequences U1, U2, . . . U12. We come back to this issue after theanalytical discussion of the second part of Structures.

25.4 Implementing Creative Analysis on RUBATO!

As already mentioned in the introduction, the concrete realization of a varietyof creative analyses in terms of notes is beyond human calculation power, orat least beyond the patience of the artistic creator. Therefore, we have imple-mented the above mathematical procedure on the music software RUBATO!

[75]. This involves seven new rubettes (Rubato PlugIns), specifically pro-grammed for our procedure (see also Figure 25.3): BoulezInput, BoulezMartix,

25.4 Implementing Creative Analysis on RUBATO! 289

Fig. 25.2. The instrumental occupancies in part A, B, following the autocomplemen-tarity symmetry I = T 7.% 1 of the original pitch class series. The lowest instrumentsare taken according to the series, while the occupancies are chosen according to theI-transformed values. For example for the first column, we have the serial value 3,and its I transformed is 4, so we add 4 increasingly positioned instruments.

Transformation, BaseChange, Chess, SerialSystem, and Boulez2Macro. We callthem boulettes to distinguish them from general-purpose rubettes.

In order to understand the data flow of this network of rubettes, we needto briefly sketch the data format that is used; details are found in the docu-mentation [75]. Rubettes communicate exclusively via transfer of denotators.These are instances of forms, a type of generalized mathematical space com-prising universal constructions, such as powersets, limits, and colimits, that arederived from mathematical modules.

The outputs A and B of boulette Boulez2Macro create one zero-addresseddenotator for each part A, B of Boulez’s score. These two denotators, MA, MB ,are not just sets of notes, but are more refined in that they include hierarchiesof notes. This is the form where MA, MB live: It is a circular form, namely

MacroScore:.Power(Node)Node:.Limit(Note, MacroScore)Note:.Simple(Onset, P itch, Loudness, Duration, V oice)Onset:.Simple(R), P itch:.Simple(Z), Loudness:.Simple(Q)Duration:.Simple(R), V oice:.Simple(Z)


So the formal notation of these denotators is

MA:0@MacroScore(MA,0, MA,1, . . . MA,m),MB :0@MacroScore(MB,0, MB,1, . . . MB,n)

with the nodes MA,i, MB,j , respectively. Each node has a note, its anchor note,and satellites, its MacroScore set denotator. Observe that the concept of ananchor with satellites is grano cum salis also the approach taken by Boulez inhis multiplication of chords, where the anchor is the distinguished note, andwhere the satellites are represented by the intervals of the other notes withrespect to the anchor. This output A, B is then united in the Set rubette andits output C is sent to the AllFlatten rubette, which recursively “opens” allthe nodes’ satellite MacroScore. How is this performed? Given a node withempty satellite set, one just cuts o! the set. Else, one supposes that its satelliteMacroScore has already recursively performed the flattening process, resultingin a set of notes. Then one adds these notes (coordinate-wise) to the node’sanchor note. This means that the satellites are given a relative position withrespect to their anchor note. A trill is a typical example of such a structure: Thetrill’s main note is the anchor, while the trill notes are the satellites, denotedby their relative position with respect to the anchor note.

The output from the Boulez2Macro boulette is given as a MacroScoredenotator for strong reasons: We want to work on the output and take itas primary material for further creative processing in the spirit of Boulez, aprocessing that, as we shall see, requires a hierarchical representation. Themultiplication of chords used in Structure II implicitly also uses a hierarchicalconstruction of the above type. Therefore, the chosen MacroScore form actsas a unifier of conceptual architectures in parts I and II of this composition.

25.4.1 The System of Boulettes

But let us se first how the Boulez composition is calculated. We are given thefollowing input data:

Outlet 1 in the BoulezInput boulette contains the series for all param-eters as a denotator Series:@Z11BoulezSeries(SP , SD, SL, SA) of the formBoulezSeries:. lim(P, L, D, A) with the factor forms

P :.Simple(Z12), D:.Simple(R), L:.Simple(Z), A:.Simple(R3).

The attack form A has values in the real 3-space, where the first coordinatemeasures the fraction of increase of nominal loudness, the second the articula-tory fraction of increase in nominal duration, and the third the fraction of shiftin onset defined by the attack type. For example, a sforzato attack (sfz) wouldincrease nominal loudness by factor 1.3, shorten duration to a staccato by 0.6,and add to the nominal onset a delay of #0.2!nominal duration. As discussedin section 25.2, the address Z11 yields the parametrization by the twelve indices

25.4 Implementing Creative Analysis on RUBATO! 291

Fig. 25.3. The Rubato network generating MIDI files (played by the ScorePlay ru-bette) with arbitrary input from the creative analysis that is encoded in the Boulez-Input rubette.

required for a serial sequence of parameters. For example, the pitch class seriesis the factor denotator SP :Z11@P (3, 2, 9, 8, 7, 6, 4, 1, 0, 10, 5, 11).

Outlet 2 contains the two address changes for retrograde R and inversionU . They are encoded as denotators

R:Z11@Index(R1, R2, . . . R12), U :Z11@Index(U1, U2, . . . U12)

in the simple form Index:.Simple(Z), which indicates the indices Ri of thea"ne basis vectors, which are the images of the basis vectors ei C(R) andinversion C(U).

Outlet 3 encodes the above sequence U. = (Ui) of address changes forthe instrumental sequence, i.e. a denotator U.:Z11@Sequ(U1, U2, . . . U12) withUi:Z11@Index(Ui,1, Ui,2, . . . Ui,12), and the list type form Sequ:.List(Index).(This, in fact, also works for any number of instruments, but we restrict ourexample to twelve instruments as chosen above.)

Outlet 4 encodes the registers, which must be defined in order to transformthe pitch classes into real pitches. We give this information in the same formas the address change sequence, where the coordinates for the ith sequence arethe octave numbers where the pitches of the respective pitch class series in thecorresponding instrument are positioned. Octaves are numbered starting fromoctave 0 at pitch 60 in MIDI format. This information is also used to positionthe pitches according to an instrumental range.

The Split rubette takes the input series Series and sends its pitch classfactor SP to outlet 5. This denotator is taken as input of the BoulezMatrix


boulette and yields the famous matrix Q at outlet 6, which we interpret as adenotator Q:Z11 ! Z11@Index(Qi,j , i, j = 1, 2, . . . 12).

The boulette BaseChange is devoted to the calculation of the addresschanges on outlet 8 for the primary parameters described in the four formulagroups 25.2 and 25.3 !. for the primary parameters of the twelve instrumentsand takes as input the matrix Q from outlet 6 and the sequence U. of instru-mental address changes.

The boulette Chess is devoted to the calculation of the correspondingaddress changes on outlet 9 for the secondary parameters loudness and attack,as described by the chessboard paths.

Given the address change systems on input 8 and 9 of the SerialSystemboulette, and taking as a third input the total series from outlet 1, the totalsystem of series is calculated according to our formulas described in sections25.3.2 and 25.3.3. This yields outlet 10, which is finally added as input, to-gether with the input 4 of octaves to calculate the e!ective parameters. Thepitches, nominal durations, and loudnesses are now given, and the nominalonsets are calculated to produce the rectangular scheme shown in Figure 25.2.The attack data is used to transform the nominal values into the attack-specificdeformations, and we obtain the outputs A and B as required.

This output is a denotator of form MacroScore. Its nodes in part A andB are 144 series each. The anchor note of each serial node is taken to be thefirst note in the series. The satellites of this node are the remaining 11 noteswith their relative positions with respect to the anchor note. Moreover, theoutput denotators at A and B have one instrumental voice number for eachinstrument. Taking the union of these parts in outlet C, we obtain a largeMacroScore denotator MC = MA %MB . Selecting from this system, the seriesas shown in Figure 25.2 yields the final “raw” material, which will now be usedto generate more involved creative constructions in section 25.5. The system ascalculated by Rubato is shown in Figure 25.4. The graphical representation isrealized on the BigBang rubette for geometric composition. The input to thisrubette is the denotator MC , while the selection of the instruments accordingto the selection shown in Figure 25.2 is made by direct graphically interactiveediting. The functionality of the BigBang rubette is discussed in section 25.5.2.

25.5 A Second More Creative Analysis andReconstruction

One of the most creative extensions of techniques in musical composition isthe opening of the transformational concept. This was already a crucial argu-ment in Boulez’s own construction of derived Q matrixes, where he inventedthat ingenious tool of address change in order to extend pitch class transforma-tions to parameters where such operations would not apply in a natural way.Our extension of Boulez’s approach was presented above and implemented inRubato’s boulettes, yielding the denotator MC :0@MacroScore().

25.5 A Second More Creative Analysis and Reconstruction 293

Fig. 25.4. The final “raw” material for twelve instruments. Instruments are distin-guished by colors. Satellites pertaining to a given anchor note are connected by raysto that note.

In this section, we shall add other extensions of the given transforma-tions and apply them to the construction of huge extensions starting from thepresent “raw material” MC . There are two threads of extensions: The first isthe conceptual extension, i.e. conceiving new types of transformations, whilethe second deals with the associated concrete manipulation of compositions onthe level of graphically interactive gestures.

The background of this double strategy is the following general idea: Theformulaic rendition of compositional tools, when implemented in software, per-tains to what is somewhat vaguely called algorithmic composition. This is whathappens in Rubato’s boulettes. The drawback of such an implementation isthat the result is “precooked” in the cuisine of the code and cannot be in-spected but as a res facta. A composer would prefer to be able to influencehis/her processes in the making, not only when it is (too) late.

This is why we have now realized a di!erent strategy: The transforma-tions, which are enabled by the the BigBang rubette8, are immediately visiblewhen being defined and can be heard without delay. The general idea behindthis approach is that any algorithm should be transmuted into a graphicallyinteractive gestural interface, where its processes would be managed on thefly, gesturally, and while they happen (!). Why should I wait until rotation ofmusical parameters is calculated? I want to generate it, and while I actuallyrotate the system by increasing angles, I would like to see the resulting rotatedset of note events and also hear how that sounds, and then decide upon thesuccess or failure of that rotation.8 The BigBang rubette is Florian Thalmann’s work.


25.5.1 The Conceptual Extensions

The conceptual extension of transformations has two components: the exten-sion of the transformations as such and the application of such transformationsas a function of the hierarchical structure of the MacroScore form. The se-rial transformations on the note parameters of a composition usually comprisethe a"ne transformations generated by inversion (pitch reflection), retrograde(onset reflection), transposition (pitch translation), and time shift (onset trans-lation). But they also include the construction of assemblies of iterated trans-formations, and not just one transformed note set but the union of successivelyapplied transformations. The latter is typically realized by regular patterns intime, where rhythmical structures are constructed. So we have these two con-structions: Given a set of notes M and a transformation f , one either considersone transformed set f(M) or else the union

#i=0,1,...k f i(M). The latter is well

known as a rhythmical frieze construction if f is a translation in time. If wegeneralize frieze constructions to two dimensions, using two translations f, g inthe plane, we obtain a wallpaper

#i=0,1,...k,j=0,1,...l f

igj(M).

25.5.1.1 Extensions of Single Transformations

The natural generalization of such transformational constructions is to includenot only those very special transformations, but also any n-dimensional non-singular a"ne transformation f in the group

#"GLn(R), whose elements are all

functions of shape f = T t · h, where h is an element of the group GLn(R)and where T t(x) = t + x is the translation by t & Rn. It is well known frommathematical music theory that any such transformation can be decomposedas a concatenation of musical standard transformations, which each involveonly one or two of the n dimensions. In view of this result, we have chosen thegeneralization of the above transformations to these special cases in 2-space:(1) translations T t, (2) reflections RefL at a line L, (3) rotations Rot" by angle!, (4) dilation DilL,# vertical to the line L by factor # > 0, (5) shearing ShL,"

along the line L and by angle !. These are operations on real vector spaces,while we have mixed coe"cients in the MacroScore form. The present (andquite brute) solution of this problem consists of first embedding all coe"centsin the real numbers, performing the transformations, and then recasting theresults to the subdomains, respectively.

Given the group#"GLn(R) of transformations (generated by the above two-

dimensional prototypes), we now have to deal with the hierarchical structureof denotators in the MacroScore form. How can transformations be appliedto such objects? To this end, recall that a MacroScore denotator is9 a setM of nodes N = (AN , SN ), which have two components: an anchor note AN

from the (essentially) five-dimensional form Note and a MacroScore-formedsatellite set SN . Common notes are represented by nodes having empty satellite9 All denotators in this discussion will be zero-addressed.


sets. Given a transformation f & #"GLn(R) and a MacroScore denotator M , a

first operation of f upon M is defined by anchor note action:

f · M = {(f · AN , SN )|N & M} (25.7)

This type of action is very useful if we want to transform just the anchorsand leave the relative positions of the satellite notes invariant. For example,if the satellites encode an embellishment, such as a trill, then this is the rightoperation in order to transform a trill into another trill.

This operation is easily generalized to any set S of nodes in the tree ofMacroScore denotator M , such that no two of them are hierarchically related(one being in the satellite tree of the other). The above situation of formula(25.7) referred to the top-level anchors. Suppose that S consists of nodes N .For non-satellite nodes, we have the above function. Suppose now that such anode N is a satellite pertaining to a well-defined anchor note A(N). Thinkingof that anchor note as a local coordinate origin, we may now apply a trans-formation f & #"

GLn(R) to all selected satellite nodes of A(N) by the aboveformula (25.7), yielding a transformed set of satellites of the same anchor note.We may apply this operation to each set of satellites of given anchors occur-ring in S. Since there are no hierarchical dependencies, no contradiction orambiguity appears, i.e., no note will be transformed together with one of itsdirect or iterated satellite notes. This means that we are simultaneously ap-plying f to all satellite sets of S. In other words, we take the disjoint unionS =

$Sk of satellite sets Sk pertaining to specific anchor notes Ak and then

apply a simultaneous transformation f to each of these Sk. We denote thisoperation by f ' S.

There is another operation that we may apply to a set S with the aboveproperties. This one takes not the relative positions of S-elements, but theirflattened position and then applies the transformation f to these flattenednotes. It is the operation one would apply in a hierarchical context, suchas a Schenker-type grouping, but without further signification of the hierar-chy for the transformational actions. After the transformation, each of thesetransformed flattened notes is taken back to its original anchor note. For ex-ample, if s = 1, and if N = (AN , SN ) is a satellite of level zero anchornote A(N), then we first flatten the note (once), which means that we takeN " = (A(N) + AN , SN ), we then apply f to its new anchor A(N) + AN , yield-ing N "" = (f(A(N) + AN ), SN ), and we finally subtract the original anchor,yielding the new satellite N """ = (f(A(N) + AN ) # A(N), SN ) of A(N). Thisoperation is denoted similar to the above operation, i.e., by f · S.

25.5.1.2 Extensions of Wallpapers

Let us now review the construction of wallpapers in view of a possible creativeextension. Mathematically speaking, a wallpaper is a structure that is producedby repeated application of a sequence of translations T . = (T t1 , T t2 , . . . , T tr )


acting on a given motif M of notes. Each T ti of these translations is repeatedlyperformed in the interval numbers of the sequence I. = (Ii = [ai, bi]), ai ( bi,of integers, what means that the total wallpaper is defined by

W (T ., I.)(M) =%

ai##i#bi

T#1t1T#2t2 . . . T#rtr (M) (25.8)

This formula has nothing particular regarding the special nature of thedi!erent powers of translations. This means that the formula could be gener-alized without restrictions to describe grids of any sequence of transformationsf. = (f1, f2, . . . , fr) for fi & #"

GLn(R), thus yielding the generalized wallpaperformula

W (f., I.)(M) =%

ai##i#bi

f#11 ) f#2

2 ) . . . f#rr (M) (25.9)

which also works for negative powers of the transformations, since these are allinvertible. In our context, the motif M will no longer be a set of common notes,but a denotator of MacroScore form. Therefore, we may replace the naiveapplication of transformations to a set of notes by the action of transformationson such denotators as discussed above. This means that—mutatis mutandis—we have two transformation wallpapers for a set S of nodes of a MacroScoredenotator with the above hierarchical independency property: the relative one

W (f., I.) ' S =%

ai##i#bi

f#11 ) f#2

2 ) . . . f#rr ' S (25.10)

or the absolute one

W (f., I.) · S =%

ai##i#bi

f#11 ) f#2

2 ) . . . f#rr · S (25.11)

This generalizes the transformations and the motives in question. A lastgeneralization is evident when looking at the range of powers of the interveningtransformations. Until now, these powers are taken within the hypercube D =&

i Ii of sequences of exponents. However, nothing changes if we admit moregenerally any finite “domain” set D * Zr and make the union according to thesequences of exponents appearing in D:

W (f.,D) ' S =%

(#1,#2,...#r)$D

f#11 ) f#2

2 ) . . . f#rr ' S (25.12)

or else the absolute one:

W (f.,D) · S =%

(#1,#2,...#r)$D

f#11 ) f#2

2 ) . . . f#rr · S (25.13)


These are the generalizations that we need to describe the transformationsin Structures in a uniform way. The situation in Structures I has been describedabove. The nature of the multiplication relating to Structures II can also becontrolled by the above constructions; we omit this here and refer to [69].

25.5.2 The BigBang Rubette for Computational Composition

The BigBang rubette was implemented during a research visit of one of theauthors (Florian Thalmann) at the School of Music of the University of Min-nesota. It allows for graphically interactive gestural actions for transformationsand wallpapers on ScoreForm denotators. We shall not describe all transfor-mations in detail, but show the typical gestural action to be taken for a rotationof a denotator (see Figures 25.5 and 25.6).

Fig. 25.5. Rotation (right) of the first bars of Beethoven’s op. 106, Allegro (left).The rotation circle shows the mouse movement on its periphery; the original is alsoshown.

The user loads (or draws) a composition (a denotator in MacroScoreform) M . This is shown in the left half of Figure 25.5; the example is thefirst bars of Beethoven’s op. 106, Allegro. This composition is shown in theplane of onset (abscissa) and pitch (ordinate), but the user may choose anytwo of the five axes corresponding to the note parameters and perform alltransformations on the corresponding plane. After having selected with themouse (drawn rectangles around the critical note groups) the notes from this


Fig. 25.6. Here, a relative rotation is performed on the two satellite sets, with theirtwo anchor notes at the rays’ centers. The original positions are also shown.

composition to be transformed, the user next chooses a rotation center byclicking anywhere on the window, i.e. the center of the circle on top of Figure25.5. Then pressing and holding the mouse button apart from the selectedcenter, a rotation tool appears, showing the current angle in gray. As long asthe mouse is not released, the rotation simultaneously acts on the selected notegroup. The rotated music is also immediately played when the user holds themouse still. The user may hold on and redo his rotational movement on thecircle. The visual result in our example is shown to the right of Figure 25.5.

As to the relative rotation, Figure 25.6 shows the result of such an action,together with the original composition. To achieve this operation, the userchooses a set of satellites throughout the given composition. We have chosentwo satellite groups derived from the composition in Figure 25.5. Then theuser chooses one anchor note and defines the center of rotation relative to thatanchor. Here, our center was chosen near the anchor of the right satellite group.Then the same gestures are performed as in the previous rotation. The circleis shown in Figure 25.6, and all chosen satellite notes are rotated relative totheir respective centers. Here, we see two rotated groups: the left one stemmingfrom the initial chords of the composition (red), the right one is overlappingwith the original selection. Again, the user may hold on (without releasing themouse) and redo the rotation after having listened to the result.

The selected notes will remain selected, and the user may then add anext transformation, and so forth. This enables a completely spontaneous anddelay-less transformational gesture in musical composition.


A similar procedure realizes wallpapers as defined in equations (25.12)and (25.13). Let us illustrate the wallpaper construction for a motif of top-levelnodes, as shown by the darkened set on Figure 25.7. The user selects this motifand then switches to wallpaper mode. Now, whenever a transformation (andalso a composed transformation, such as a translation followed by a rotation,much like with single transformations) is defined by the previous gestural ac-tion, the union of all iterated transformations of the motif is simultaneouslyshown (and heard). The range of iteration (the powers of that transformation)can be set at will. For a second transformation, the wallpaper mode is clickedagain and allows the user to perform a second transformation, and a third,fourth, etc. The user can also switch to another parameter plane when addingnew transformations, and thereby create wallpaper structures in less evident,but musically precious parameters, such as loudness and voice. The example inFigure 25.7 has two transformations, each of them being a translation followedby a rotation and then a dilation.

The BigBang rubette also allows for multidimensional alterations andmorphing. These are deformation operations, which alter given notes (on spec-ified levels of the Macroscore hierarchy) in the direction of another composition,which might be anything, or just a single point of attraction. We do not discussthis technique further here and refer to [117] for details.

Fig. 25.7. A wallpaper is built from a motif (darkened). Two transformations areused—both are translations followed by rotation and shrinking dilation.

25.5.3 A Composition Using the BigBang Rubette and theBoulettes

Here is a composition, logically named restructures, which Guerino Maz-zola and Schuyler Tsuda co-composed using of the above techniques, start-


ing from the raw material MC shown in Figure 25.4. We also applied thealteration techniques implemented in the BigBang rubette but will not dis-cuss this technique further here. The composition can be downloaded fromhttp://www.encyclospace.org/special/restructures.mp3.

This composition has four movements. Each movement is transformedaccording to a specific geometric BigBang rubette technique, which we describein the following paragraphs. After executing these operations, the twelve voicesof each movement, which are avaliable as twelve separate MIDI files, wereelaborated by adequate orchestrations. This was realized by Tsuda, who isan expert in sound design. He orchestrated and attributed the MIDI files tospecific sounds in order to transform the abstract events into an expressivebody of sound.

The first movement (Expansion/Compression) takes a copy of MC , then“pinches” the satellites (but not the anchors!) of part A in the sense that thefirst (in onset) satellites are alterated 100 percent in pitch direction only toa defined pitch, whereas the last satellites are left as they were (0 percentalteration). The satellites inbetween are pinched by linear interpolation. Thesame procedure is applied to part B; however, this time the pinching is 100percent at the end and 0 percent at the start. This is shown in Figure 25.8.

Instrumentation 1: Voice 1 = grand piano, voice 2 = scraped, bowed,rolled, and struck suspended cymbals, voice 3 = electronic mallets, voice 4 =solo cello, voice 5 = pizzicato strings, voice 6 = electronic space strings, voice 7= plucked e-bass, voice 8 = grand piano, voice 9 = electronic percussion, voice10 = timpany, voice 11 = electronic toms, voice 12 = electronic bells.

Fig. 25.8. First movement: variable pinching the satellite onsets—100 percent pinch-ing at start and end onsets, no pinching in the meeting of end of part A and start ofpart B.


Fig. 25.9. Fourth movement: sucking down the anchors, expanding their durations,lifting the satellites in part A, then progressive pinching of notes in part B.

For the second movement (Space-Time), we took another copy of MC andexpanded the onsets and the durations of the anchors of the second appearanceto the double, which yielded the situation shown in Figure 25.10.

Instrumentation 2: Voice 1 = strings, voice 2 = flute and horn, voice 3 =grand piano, voice 4 = sine waves, voice 5 = electronic voice, voice 6 = grandpiano, voice 7 = trombone and tuba, voice 8 = electronic strings, voice 9 =triangle and finger cymbals, voice 10 = bowed piano, voice 11 = clarinetes,voice 12 = electronic bells.

Fig. 25.10. Second movement: expanding onsets and durations.

For the third movement (Rotations), taking again a copy of MC , andfocusing first on part A, we apply a retrograde inversion to the anchors, andthen in a second operation also to all satellites relative to their anchors. Wethen take part B and apply a rotation of all satellites, relative to their anchors,by 45 degrees in the counterclockwise direction. The result shown in Figure25.11.


Instrumentation 3: Voice 1 = sine waves, voice 2 = oboe and bassoon,voice 3 = pizzicato strings, voice 4 = marimba, voice 5 = horns, voice 6 =electronic mallets, voice 7 = temple blocks and tam-tam, voice 8 = grand piano,voice 9 = electronic percussion, voice 10 = sine waves, voice 11 = trombones,voice 12 = electronic bells.

Fig. 25.11. Third movement: retrograde inversion of anchors and satellites in partA, rotation of satellites in part B.

Finally, for the fourth movement (Coherence/Opposition), taking again acopy of MC , we take part A and pinch to low pitches the anchors and dilatetheir durations, whereas the satellites are pinched to high pitches. In part B,we also operate such separation of pitch of satellites from anchors, but we alsoexecute a progressive pinching of the pitches toward a fixed pitch toward theend of the composition. The result is shown in Figure 25.9.

Instrumentation 4: Voice 1 = glockenspiel and electronic noise, voice 2 =glockenspiel and electronic noise, voice 3 = grand piano and electronic noise,voice 4 = harp, electronic noise, and pizzicato strings, voice 5 = sine waves,voice 6 = finger cymbals and timpani rolls, voice 7 = electronic bells, voice 8= grand piano, voice 9 = Chinese opera gong and low and high gongs, voice 10= bowed cymbals, voice 11 = triangle and bass drum rolls, voice 12 = triangleand bass drum rolls.

25.5.4 Was This “Creative Analysis” a Creative Success?

Let us summarize the e!orts we have described above in a computer-aidedrecomposition of Boulez’s Structures. In a first step, we have opened thewalls of Boulez’s serial approach by a mathematical restatement of his criticalstructures. Here the technique to open those walls was the usage of modernmathematics—more precisely, Yoneda’s lemma.

This conceptual extension is set up starting from the formally very naiveapproach of music theorists or composers and then using universal theoriesset forth by advanced insights that are driven by the most abstract and pre-cise style of (mathematical) thought: topos theory. This style of extension canbe criticized as being “overdressed” with respect to the technical level of our


conceptual reconstruction. But it is clear that such extensions would never bepossible when sticking to the naive style of musical set theory. We definitelyneeded the topos-theoretical perspective to find those extensions that enabledour recomposition.

We then applied the conceptual extensions to an implementation in thecomposite software RUBATO!. This was not a creative action per se, but itwas the embodiment of those theoretical concept extensions, which is necessaryto eventually make music. This implementation is mandatory when we speakabout the instrumental reality of music. And it also shows the role of computerprograms in music. They are not creative per se but mediate between com-plex thoughts and instrumental realization. In our case, the complexity of therecomposition was far beyond human score writing, and we could achieve thecontrol of a variety of system parameters in Boulez’s compositional experiment.

In this sense, the program was not only an embodiment of abstract ideasbut also a creative environment for inventing new variants of the thoughtfulcreative analysis. We hope that the concrete results of our e!orts have givensome suggestions of how to work e!ectively in Boulez’s spirit for future creativeanalyses.

Structures Recomposed · 2014-07-05 · 25 Boulez: Structures Recomposed In [19], Pierre Boulez...

Documents

Transcript of Structures Recomposed · 2014-07-05 · 25 Boulez: Structures Recomposed In [19], Pierre Boulez...