DEUS EX MACHINA, UNCOVERING ALDO CLEMENTI\'S SYSTEM 1
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DEUS EX MACHINA, UNCOVERING ALDO CLEMENTI’S SYSTEM1
MICHELE ZACCAGNINI 1. INTRODUCTION
I
TALIAN COMPOSER ALDO CLEMENTI,
who passed away in 2011, has left us with a substantial oeuvre. Though not a household name in the contemporary music scene, he maintains the solid reputation of a composer who over the decades produced a highly idiosyncratic yet consistently refined oeuvre.2 Clementi’s late period, usually identified as the “diatonic period,” consists of a collection of sound panels built on simple, diatonic melodic fragments: Clementi . . . moved away from the structuralism of his early works towards a compositional method based on the selection and subsequent polyphonic elaboration of modal and diatonic materials drawn from past European music, particularly the music of Bach and Brahms, but also Stravinsky, Tchaikovsky, Liadov, Chopin, Schumann, Mozart, Purcell, Dufay, troubadour melodies and Gregorian chant.3
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While some aspects of Clementi’s compositional procedures during his diatonic period have been brought to light and discussed in previous scholarship,4 others have only been objects of speculation. My research is concerned with unfolding the procedures Clementi used to choose his themes and their relative canonic transpositions, and with assessing the aesthetic criteria that led the composer to adopt and develop such procedures. I have had the unprecedented privilege of being able to study the composer’s autograph sketches for this purpose. My wholehearted gratitude goes to Anna Clementi for granting me the access to these sketches.5 In addition to unfolding the secrets of the composer’s process, this research is also concerned with providing an assessment of the composer’s legacy in today’s musical landscape. I will ultimately attempt to reinterpret the composer’s oeuvre under a new light, bringing to the surface those qualities that make it aesthetically relevant, perhaps even seminal. In one of the most celebrated romantic Italian poems, L’Infinito, a hedgerow stands in front of the poet, impeding his view; eventually the simple hedgerow is all it takes for the poet to leap into an indiscernible obscurity, an infinite unknown. Clementi’s themes are related to Leopardi’s hedgerow, for despite their simplicity, they eventually destroy any recognizable trace of a musical narrative and bring the listener into an indiscernible, obscure mass of sound. The only way to truly appreciate Clementi’s late music is to abandon the moorings of the traditional dialectical musical narrative, viz. a dynamic unfolding of musical ideas, and accept its static and unfathomable complexity to eventually fall into what Leopardi described as a sweet shipwreck—un dolce naufragare. But poetic metaphors only apply to the description of the perceptual experience that Clementi’s music might induce. The description of Clementi’s compositional procedures, on the other hand, does not tolerate the intrusion of any such “poetic licenses.” In the second section of this article, a detailed description of the composer’s procedures will give the reader the opportunity to look over the composer’s shoulder while he is at his desk, planning his sound masses. And what will become apparent then is the uncompromising determinism that guided the composer’s hand, his precise and almost mechanistic ways of planning and putting sounds together. This article will focus on one important work of Clementi’s late period: Aus Tiefer.6 To expedite my assumptions about the composer’s procedures with the actual results, viz. the score of Aus Tiefer, I have relied on one of the modern algorithmic composers’ favorite tools: OpenMusic.7 Software like OpenMusic8 is effective in recreating
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Clementi’s process, and is also significant in showing the algorithmic nature of Clementi’s procedures and its resemblances to recent computerized processes commonly categorized as algorithmic composition or computer aided composition (CAC). 2. ERASURE OF CONTRASTS AND THE (IR)RELEVANCE OF THE COMPOSER’S PHILOSOPHICAL VIEWS Clementi’s late production focuses on a consistent exploration of the idea of stasis. Each of the pieces of the diatonic period possesses an unmistakable duple nature: a smooth, uneventful surface and a deep, impenetrable complexity. Deprived of any clear contrast or sense of development, the complex textures are devoid of any forward-moving narrative. Clementi achieved just what he intended to achieve by eradicating the dialectical elements that had traditionally established the narrative qualities of music: To think of music as a discourse, even subconsciously, is fallacious: it inadvertently creates an ambiguity which turns [music] into a caricature of an arc that describes a useless orgasm. [Narratives such as] exaltation and depression are no longer valid; they are modest symbols of an extinct dialectic, no matter how disguised they might be. A forte followed by a piano, a high note followed by a low one, a “sweet” timber followed by a “raw” one are in themselves dialectical, a sonatistic cell of the greater Sonata Form.9 But why did the composer feel the moral imperative to deprive music of its pseudo-linguistic vestiges? Clementi had been deeply influenced by the notorious discussion between Adorno, Benjamin, and Brecht about the future of art in the age of consumerism. He often quoted Adorno as one of his main influences.10 He absorbed the philosopher’s skepticism about the future of art and the devastating consequences of the establishment of the “culture industry:” The musician [today] differs from the one of the great Romantic Era, he is no longer a beacon that guides humanity as a whole. Mankind does not seek art anymore but only comfort, usefulness, pleasure and entertainment. We need to face the reality that there is no more need for art.11
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Furthermore, Clementi incorporated these very sentiments of loss and disruption into his music: It’s true: art and music are dying of saturation. Therefore either we have the courage to be silent, or we can continue to do things . . . that simply convey the agony and the dissolution of the represented object.12 Clementi’s tortured realization of the defunct state of music did not stop him from producing new works, though. Craftsmanship, he argued, was the one element that survived the shocks of the twentieth century, the only refuge for the modern artist: There exists no contradiction between a pessimistic and apocalyptic view on the future of art and the need to put notes together well, to continue a precious musical craftsmanship. For many the crisis of art becomes an alibi.13 Although the modern composer could still find comfort in a meticulous work of artisanship, he would never be able to reach the peaks of his past peers and would be better off by altogether dismantling music’s most “noble” elements, viz. its narrative qualities, and limiting his act to a humble homage to the past. The meta-narrative of a dying art-form described by the art-form itself is an idiosyncratic feature of Clementi aesthetic apparatus, one that links his thought to other artists of his time: from Pirandello’s meta-theater to Cage’s questioning of the role of the artist. An important caveat that will come out of this article is that we should consider the above statements just for what they are: the artist’s poetic interpretation of his own oeuvre. If we had to fully buy into the composer’s narrative, his entire production could be seen as a pointless stylistic exercise or a mere homage to past music. But, as will come out of this research, the composer’s negative thoughts—such as the erasure of contrasts, the lack of virtuosic elements or sonic effects—actually generate musical innovations. The music strangely acquires new subtle features such as a compelling static complexity and unpredictable repetitiveness of sonic constructions à la Escher. It is possible that Clementi consciously or unconsciously used his own pessimism to carve a space of creative freedom. In this sense, the cultural crisis of the twentieth century became an occasion for Clementi to altogether abandon the elements that had made music great in the past, and to explore the non pseudo-linguistic qualities of
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music. The composer’s eschatology, as tortured as it was, eventually granted him the freedom to engage in the perceptual exploration of static textures, repetitiveness, and the idea of spatial organization of rhythms—topics that will concern this very research and that are essential components of Clementi’s music. We cannot positively assess whether the composer’s philosophical views were actually the cause of his radical abandonment of the dialectic of contrast and of mechanisms of tension and release, or whether, conversely, it was his fascination for stasis and textural constructions that generated his convoluted Weltanschauung. The question is ultimately irrelevant. A question that remains relevant is: once music is deprived of all of its dialectical vestiges, what is the listener left with? Clementi’s music provides one possible answer to this very question. 3. AUS TIEFER: A CASE STUDY Aus Tiefer calls for a female chorus (4 soprani, 4 mezzosoprani, 4 contralti) and twelve instruments (3 flutes, 3 oboes, 3 B b clarinets, and 3 French horns in F). The piece is a collection of nineteen panels, each of which presents a particular combination of melodic fragments taken from the Lutheran chorale Aus Tiefer Not. Clementi builds the entire piece, almost thirty minutes long, using just two melodic fragments and their inverted forms, layering them through canonic constructions. These fragments, once presented, do not change throughout the entirety of the piece. In fact the melodic fragments, their transposition levels, instrumentation, orchestration, and dynamic levels are all immutable from beginning to end. The melodic fragments are repeated within each of the nineteen panels, and panels repeat at different points in the piece. The way fragments and entire sections are repeated in the piece does not let the listener infer that any transformation of the material is actually taking place: its basic elements are never really varied or re-contextualized. One can safely say that the dynamic-transformative quality of these elements seems to be missing. The idea of development or even the more basic idea of variation is clearly not part of the piece’s narrative.
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MATERIAL
The theme of the original chorale is subdivided in two fragments, as shown in Example 1. Throughout the nineteen canons, the piece employs these two fragments either in their original form or in inversion, either alternatively or simultaneously.14 Other than the two thematic fragments and their inversions, two more thematic elements are found exclusively in canons IX and XI: the varied second theme (Example 2) and the organ’s theme (Example 3). While the varied theme can also be seen as a hybridization of the first and second themes, the organ’s chorale stands as a foreign object in the tight thematic organization of the piece. It is taken from a later setting of the Aus Tiefer, psalm 130, and is played by the organ “with its own tempo,” as the score reads.
EXAMPLE
1: CHORALE’S THEME, DIVIDED INTO FRAGMENTS
EXAMPLE
2: SECOND THEME VARIED
EXAMPLE
3: ORGAN’S CHORALE
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Example 4 represents which theme (1st, 1st inversion [1st'], 2nd, 2nd inversion [2nd'], or the organ’s theme) appears in the nineteen canons. By looking at the chart the reader can appreciate the specular quality of the thematic organization of the piece. In considering the appearance of the themes and their juxtapositions we can observe two main structures: • the outer I to VIII and XII to XIX, • the core IX-X-XI. The two identical outer structures repeat in a palindromic fashion after the core structure, which could itself be seen as a palindrome. The axes of symmetry consist of a substructure, which encapsulate canon X between IX and XI. These last two panels are uniquely constructed with thematic variations. Around the central structure we find two symmetrical outer structures.
organ 2 +2 nd
1st' 1st
◊
nd
◊
◊ ◊ ◊
◊ ◊ ◊ ◊
◊
◊
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◊
◊ ◊ ◊ ◊
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I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI XVII XVIII XIX
EXAMPLE
4: PALINDROMIC ORGANIZATION OF THEMES
LAYERING
Clementi exclusively uses canonical procedures in Aus Tiefer. While the technique calls for simply stacking same lines by offsetting their entrances in time, Clementi achieves a remarkable level of nuance in his own canonic construction. In his pieces there are different levels of canon that work simultaneously. They are organized with a Russiandoll-like procedure: smaller canonic structures are incorporated in increasingly bigger ones. Depending on which of the nineteen canons we look at, we can find different levels of canonical organization. One way to hierarchically organize these levels starting from a simpler canonical construction to more complex ones would be as follows:
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• First level. The same theme is played at the same pitch level and same rhythmic values. • Second level. The same theme is played on a different pitch level and with same rhythmic values. • Third level. The same theme is played on different pitch levels and with different rhythmic values, either augmented or diminished proportionally. While first and second level canons do not appear by themselves in the piece but nested within other canonic structures, third level canons do appear by themselves, thereby constituting the thinnest of the canonic settings of Aus Tiefer. • Fourth level. Different themes are played on different pitch levels and with different rhythmic values. This level consists in juxtaposing two third level canons as shown in Example 5. The added variable of complexity at a fourth level canonic construction is the introduction of theme transformations, which in Aus Tiefer are limited to the theme’s inversion. One can already start appreciating the coherence and consistency of Clementi’s methodology. Each level of the canonical construction has an added variable, which adds complexity. The canons are far from being simply a stacking of different voices; rather they are a collection of separate entities consistently built to form larger structures.
SHAPES
Aside from the fragmental content, another element that varies between each of the panels is their “shape.” In each panel, the voice entrances are organized according to shapes such as: • A triangle pointing upwards, • A triangle pointing downwards, • An “hourglass” shape, • A rhombus.
Deus ex Machina, Uncovering Aldo Clementi’s System
EXAMPLE
5: CANON LEVELS (III)
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The last two shapes are compound structures that can be obtained from different juxtapositions of the first two. To clarify how these shapes work in organizing the rhythm of the panels, let’s consider for instance the upward-pointing triangle. In this case, the bottom voices will enter first, middle voices second, and top voices last. Alternatively, in the hourglass-shape, top and bottom voices will come in (almost) simultaneously, while the middle voice will come in last. All these shapes share one common characteristic: they produce symmetrical sound masses with the point of maximal density at their center, producing a gradual increase/decrease of sound. While their rhythm might be different, their ultimate perceptual result is very similar (yet distinct). At the mid-structural level, the listener is presented with gradual changes of textures. The deceptive character of this level consists in repeating some of the panels in all of their most recognizable features, such as fragmental material, transposition levels, and number of voices in the case of panel 1 and 6, 2 and 7, 3 and 5, while changing their “sonic shape”—a feature apparent when looking at the score but hardly recognizable, at least consciously, when heard. 4. SATURATING INTERVALS THROUGH THE “MAGIC SQUARE” In studying the composer’s sketches, it becomes clear that two main kinds of sketching work occupied his compositional process. The first took place on graph paper workbooks.15 As I later found out and will explain shortly, this sort of preparatory work dealt with the choice of the transposition levels that each of the canons was to contain both in number of voices and pitch levels. The second kind employed large sheets of graph paper onto which the composer assembled cut-offs of manuscript paper. This latter kind of work accounted for the rhythmic organization of the canons in their shapes: i.e., the chronology of the voices’ entrances. The quaderni di lavoro (i.e., workbooks) only occasionally show signs of traditional music notation. If Clementi needed to have a notated version of a theme, he quickly and freehandedly drew a stave without switching to a manuscript paper. What the quaderni mainly contain instead is a remarkable number of pseudo-Cartesian graphs (see Example 6). In his interviews, he often referred to such graphs as “magic squares,” without ever revealing the details of their workings, preventing the squares’ secrets—its “magic”—from being revealed. Leaving the folkloristic appellation aside, what we know is that in the sixties Bruno Maderna introduced the technique to Clementi. Clementi later confessed how this explanation had changed his way of writing and thinking about music.
Deus ex Machina, Uncovering Aldo Clementi’s System
EXAMPLE
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6: AN EXAMPLE OF A SKETCH OF THE MAGIC SQUARE
[Maderna] had a system that he called “reading and rereading of the material.” He explained the scheme of it on a small piece of graph paper. It was an extension of the twelve-tone technique, from the perspective of Webern. All it took was half an hour, but it had been so upsetting that I understood that I needed to start from scratch.16 In my research, I was not able to learn what Maderna actually taught his pupil. Nonetheless, through a careful study of his quaderni, I deciphered how Clementi implemented such technique. In order to work within this graphic environment, the composer had to first translate the theme into a series of numbers from 1 to 12, each number representing a note of the chromatic scale. The numbering is always consistent and independent of the octave placement (see Examples 7 and 8).17 Once the composer had translated a theme into a number series, he proceeded to build matrixes of the twelve transpositions for each of the four mirror transformations: original, inversion, retrograde, and retrograde inversion (Example 9).18
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C C# D D# E 1 2 3 4 5
EXAMPLE
EXAMPLE
F 6
F# G G# A A# B 7 8 9 10 11 12
7: NUMERIC SYSTEM
8: FIRST THEME’S NUMBERING (PITCH LEVEL 7)
EXAMPLE
9: TRANSFORMATION MATRIXES
Deus ex Machina, Uncovering Aldo Clementi’s System
EXAMPLE
EXAMPLE
10: EXAMPLE 8’S THEME REPRESENTED IN THE SQUARE
11:
#
LAYERING OF THE FIRST THEME ON LEVELS F AND G
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He then chose which level of transpositions and which of the transformations could work together in a canon. This is when the magic square technique plays its crucial role: without exception, one can find an implementation of the square right next to the four matrixes. Let me illustrate the workings of the technique through an example. Given the theme of Example 8 and its numeric translation {7, 12, 7, 8, 7, 3, 5, 7}, I can “map” the theme’s intervals as dots (or plus signs or x’s etc., as in Example 6) in the 12 by 12 square. In order to do so, the theme has to be divided into a set of intervals: {(7 12), (12 7), (7 8), (8 7), (7 3), (3 5), (5 7)} In Example 10, each point represents an interval. For example, the coordinates (7 12) are an ordered pair representing F#-B, a descending perfect fifth, or an ascending perfect fourth.19 The graph can thus be used to chart the theme’s intervallic content.20 The Example 8 theme will then have the graphic rendering shown in Example 10. This way of visualizing a melody through dots in a square allowed the composer to quickly grasp the intervallic content of a theme. A question now arises: why would the composer go through the trouble of a graphic mapping of the intervals? As I have already mentioned, these graphs were far from being just a trifle the composer adopted to occupy his spare time. On the contrary, Clementi heavily relied on them to make his compositional choices. Only after the composer had mapped the themes into these squares would he be able to settle on certain levels of transpositions and transformations. 21 To clarify the importance that the choice of a level of transposition plays in Clementi’s writing let me stress how Aus Tiefer’s thematic material, as shown in Example 4, is limited to two themes and their inversion (excluding the exceptions of IX and XI). Furthermore, these themes usually appear separately. This extreme, self-imposed economy of materials, forced the composer to work exclusively with elements such as transposition levels of the same theme and their rhythmic organization. Clementi used two “rules” to achieve the “perfect” combination of levels, one positive and one negative: • Saturate the intervallic space with multiple versions of the same theme, • Avoid repetition of the same interval at the same level by different lines.22
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Let me explain what I mean by intervallic space by looking at Example 10. The dots in the graph represent the dyad of one level of transposition of the first theme (F#). These dots partially fill the graph. If I add another transposition of the same theme seven more dots will be filled into the 12 by 12 space (144 slots in total). To the theme on F #, graphically represented on Example 10, I can then add another version of it, say on level G, obtaining a graph such as the one in Example 11. Such operation will accomplish a higher degree of saturation. But this sort of layering cannot happen freely: different levels of transposition cannot replicate the same dyad. As an example, let’s take the above mentioned first themes on levels F# and G. Their dyad content would be numerically represented as: F #: G:
{(7 12), (12 7), (7 8), (8 7), (7 3), (3 5), (5 7)} {(8 1), (1 8), (8 9), (9 8), (8 4), (4 6), (6 8)}
The two sets of dyads don’t have any element in common. But if I add level G# instead of G, a dyadic intersection will be created: F #: G #:
{(7 12), (12 7), (7 8), (8 7), (7 3), (3 5), (5 7)} {(9, 2), (2, 9), (9, 10), (10, 9), (9, 5), (5, 7), (7, 9)}
The dyad 5-7 (E-F#) is in fact present on the two different levels of transposition of the theme: F# and G#. Avoiding dyadic overlaps of different levels of transposition is the second rule that the composer strictly followed in building his canons. Aside from the technique itself, another original feature of Clementi’s process comes to light: the composer’s concern lies not in the single notes of a musical line but in its dyads. While repetition of notes across the lines is common in any of his canons, fragments cannot replicate the dyadic motion of one another (i.e., from that common note to another common note). In other words, the “absolute value” of a dyad (e.g., the interval of a perfect fifth) can be replicated in different lines but not at the same pitch level. A particular theme, and its determined set of dyads, will then contain the “constraint” by which the number of voices of the canon will be derived. These operations give different results depending on the themes used, some being more prone to multiple-voice constructions than others. The experience that the composer acquired in this technique made him more aware of the qualities of a melody in relation to its dyads. Many of the sketches in the quaderni are studies of themes to explore their polyphonic potentials according to Clementian rules.
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The first fragment of the Aus Tiefer chorale proved to be one of the composer’s most successful, because it allowed him to build up to twelve real voices with no dyadic intersections between the original and the inversion. This sort of operation—namely, maximizing the number of voices while avoiding dyadic intersections—determined each of the canons’ numbers of voices and levels of transpositions. Clementi’s reasons for adopting this particular graphic representation should now be clearer: the magic square provided him a quick way to check different levels of transposition indicating whether they caused dyadic intersections with others; in traditional notation this operation would prove more time consuming. 5. THE VISUAL APPEAL
OF THE
MAGIC SQUARE
At a higher level of generality, the square represents an abstract space that allows the composer to work almost as a visual artist, creating dense textures such as those he admired so much in abstract expressionist painters.23 Clementi used the square as a pivotal procedure that let him into a spatial rather than temporal dimension. While the technique clearly constitutes an efficient tool in mapping dyads, it also gives a synchronic representation of a melodic fragment: the theme appears as a shape, disjointed from its temporal constraint. The postmodernist exploration of physical space operated by Cage and others is reinterpreted by Clementi as the exploration of a conceptual space created through compositional artifice and otherwise hidden to the listener. In this kind of space, intervals constituted the bricks with which the composer proceeded to fill all possible slots, as many as the dyadicintersection constraint and the limited thematic material allowed. But filling the slots of the square was an operation of calculation, a puzzle, comparable to “compositional Tetris,” rather than a free-hand, stream of consciousness act. In Example 12, various graphs are illustrated in which a computerized rendering of the magic square is realized for each of the canons contained in the piece. Each of the dots represents a dyad and each dyad is played by one level of transposition only. The reader can thus appreciate visually the textural difference between each canon. In III, V, and X we can appreciate how the composer achieved an almost complete saturation of the space, remembering that no dot is repeated and that only one theme is employed. In particular, the constructions of the three canons that start the piece are built as a
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gradual stratification of one canon over the other. In Canon I, the map of dyads is a mirror image—across the square’s diagonal—of the one of Canon II. Canon III is a super-imposition of Canon I to Canon II (shown in Example 13). This is possible since Canons I and II do not share any dyad.
I
II
III
IV
V
VI
EXAMPLE 12: DYADIC DENSITIES (FROM X TO XIX THEY REPEAT AS A PALINDROME)
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VII
VIII
IX
X EXAMPLE
1
+ =
EXAMPLE
12 (CONT.)
II III
13: GRADUAL INCREASE OF DENSITY
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6. THE ALGORITHM In the material I gathered from Clementi’s sketches, I could only find some of the graphs and matrixes of Aus Tiefer. This lack of material prompted me to reconstruct the composer’s multi-step procedure with a computer algorithm that would reproduce the process behind the quaderni. This approach helped me test my assumptions about the composer’s process against the outcome of his work: the score of Aus Tiefer. Such an approach not only proved successful (the algorithm’s results were consistent with the canons of the piece) but offered insights about the algorithmic approach the composer adopted. In this section, I will rely on such an algorithm and show the reader the different steps of my analysis. The charts produced by the program OpenMusic,24 while less visually appealing than the ones of our composer, will be neater and easer to read. MONOTHEMATIC CANONS
In my simulation, I will focus on the process from the moment the composer chose his thematic material and started assembling it into his macro structures. As we have already seen in discussing the magic square, the first step will be to subdivide the theme into dyads. 25 In doing so, each note, except for the first and last, will be repeated twice, since it will be both the arrival for one interval and the departure for the next. For instance, the first theme on level 5 (E) will be {5, 10, 5, 6, 5, 1, 3, 5}, while its dyads will be {(5 10), (10 5), (5 6), (6 5), (5 1), (1 3), (3 5)}. Example 14 illustrates a pretty simple implementation of an OpenMusic patch that reproduces just the dyadic subdivision that I will need. The two boxes “notes->1-12” and “intervals” contain simple arithmetic operations that allow the notes to be translated into couples of numbers. I can apply the same algorithm to the remaining eleven levels of transposition and obtain a total of twelve sets of dyads (different transpositions of the same ordered theme—represented as its ordered dyads). Since the composer’s idea is to fill the dyadic space with the use of a single line, transposed on different levels, avoiding dyadic intersections, I will proceed in building another algorithm to check which levels of transposition dyadically intersect with one another. In particular, the first Aus Tiefer theme will intersect only some of its transposed peers.
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EXAMPLE
Perspectives of New Music
14: TRANSLATING A MELODY INTO A SET OF INTERVALS WITH OM
The goal of the algorithm illustrated in Example 15 will be to look for empty sets of dyadic intersections. The example shows the construction of the algorithm: the theme is transposed onto twelve levels (sub-patch “transpositions”), which are then divided into dyads. The sub-patch named “find-intersection” then matches each transposed version of the theme with each of the others, reporting any overlap in the dyadic content. The patch named “inters?” will “ask” whether there is any intersection between two levels of transposition, reporting the statement “true” (“t”) if the answer is “yes,” “false” (“nil”)26 if it is “no.”
Deus ex Machina, Uncovering Aldo Clementi’s System
EXAMPLE
15: THE SELF-INTERSECTION MATRIX ALGORITHM
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To avoid confusion, we need to be aware that the table shown in Example 16 differs from the magic square in that it reports whether a certain level of transposition of the Aus Tiefer first theme has at least one dyadic intersection with another. For instance, checking row “D,” I learn that the transposition level starting on D will have dyadic intersections with C, D (unison transposition), and E. Such an operation will result in a twelve by twelve matrix of t’s and nils. I will call the resulting matrix the “self-intersection matrix” since it reports dyadic intersection between a theme and itself on different levels of transposition. This matrix will possess specific qualities: rows and columns corresponding to the same pitch level will have the same content. In other words, the matrix has an axis of symmetry in its diagonal; also the diagonal will always contain “true” statements since it represents the unison transpositions.27 The matrix tells us that in the case of the first theme (Example 16), in addition to the unison or octaves, two more transpositions are to be avoided for each level of transposition: the major second above and below.28 These results allow me to “simulate” the process that the composer adopted to choose specific levels of transposition. These choices also determined the number of voices that could be included in seventeen of the nineteen canons.29 Once a starting level has been chosen, there will be only a limited number of transpositions that can be added without replicating intervals. In the case of the first theme the number is exactly six. C#
D
nil
t
nil
t
nil
D#
t
nil
nil
t
E
nil
F
nil
C C
C# D
F# G
G# A
A# B
D#
F#
nil
nil
nil
nil
nil
nil
nil
t
nil
nil
nil
nil
nil
nil
nil
t
t
nil
t
nil
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nil
nil
t
nil
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t
nil
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nil
nil
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t
nil
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t
nil
nil
nil
nil
nil
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nil
t
nil
t
nil
nil
nil
nil
nil
nil
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nil
t
nil
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nil
nil
nil
nil
nil
nil
nil
nil
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nil
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nil
t
nil
nil
nil
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nil
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t
nil
t
nil
t
t
nil
nil
nil
nil
nil
nil
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t
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t
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nil
nil
nil
nil
nil
nil
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t
nil
t
EXAMPLE
A
A#
F
t
G
G#
E
t
16: SELF-INTERSECTION MATRIX (FIRST THEME)
B nil
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Starting from the level of the original Lutheran chorale, B (Example 1), I will proceed adding one voice at a time. Each time I add a new voice I will have a further limitation in choosing the next transposition. The levels that don’t have any intervallic intersection with B, and are therefore available to be utilized in a Clementian canon, will be indicated by the statement “nil” in the matrix. From the highlighted row in Example 16, I can extrapolate the following available levels: C
D
D#
E
D#
E
F
E
F
F
F#
G
F#
G
G#
A#
(B)30
G#
A#
(B)
Proceeding from bottom to top, if I keep adding levels of transposition, A# will be added next, producing a new set of available transpositions: C
C#
D D
D#
F#
G
A
(A#) B
Remembering that I can only use levels of transposition that don’t intersect any of the preceding ones, the next level available will be G, since A would create an intersection with B, while G # with A#. Adding G, I will get: C
C
C# C#
D D D
D#
E
F
D#
E
F
E
D#
F#
G
F#
(G) G#
F#
G
G#
A
A#
(B)
A#
B
(A#) B
And so on. . . . Proceeding until no other level is possible without creating an intersection I will have: C
C C C
C#
D
D#
E
F
D#
E
F
E
D#
C#
D
C#
D
C#
D
(D) D#
D
D
D#
(D#) E D#
F
F
F#
G
F#
(G) G#
F#
G
(F#) G F#
G
F#
G
F#
G
G#
G# G#
A
A A A
A#
(B)
A#
B
A#
B
(A#) B A# A# A#
B
B B
24
Perspectives of New Music
I will be able to pick only those levels that contain no intersection with all of the preceding ones which are indicated by the string of notes. Reading the columns of the preceding table those levels are, starting from level B, D, D#, F#, G and A#. Once the sixth transposition is added, no further addition will be available without intersecting one or more of the preexisting ones. The layering has to stop at six real voices. 31 In Example 17, canon II is shown with levels of transpositions consistent with my analysis. These levels are kept throughout the entire piece when the first theme is played.
POLYTHEMATIC CANONS
The canon of Example 17 which appears in different “shapes” on II, VII, XIII, and XVIII is what I referred to previously as a third-level canon. Moving up the ladder of canonic levels, the composer started adding new thematic material to thicken the canonic strata. In fact a whole new set of possible transpositions can be added if I introduce new themes. Clementi starts doing so by adding the theme’s inversion (see Example 18). Interestingly enough, we can notice that, unlike what has been observed in the case of the self-intersection matrix, a theme and its inversion do not cause any intervallic intersection if taken at the same pitch level. In other words, the set of intervals: {(7 12), (12 7), (7 8), (8 7), (7 3), (3 5), (5 7)} has no element in common with the set: {(7 2), (2 7), (7 6), (6 7), (7 11), (11 9), (9 7)}. Let me remind the reader here how the composer’s concern resided in a theme’s intervals rather than its individual notes. It was the intervals and not the notes that had to be unique to a line. If we compare the two lines, while the number 7 will appear multiple times in both lines it will never be approached or left the same way between the original and the inversion: the elements of the sets are couples rather than single numbers. But if the same level is allowed for the first theme and its inversion, other levels of the inverted theme will create intersection with each level of the original.
Deus ex Machina, Uncovering Aldo Clementi’s System
EXAMPLE
17: FIRST THEME SETTING (II)
ORIGINAL F
#
INVERSION F
EXAMPLE
#
18: FIRST THEME AND ITS INVERSION ON F#
25
26
Perspectives of New Music
We can proceed to look for those intersections in a way similar to the self-intersection matrix. The algorithm used before can be tweaked to allow for more than one theme to be analyzed. This slightly different algorithm will compare two different themes and find intersections at different levels of transposition. As Example 19 shows, the algorithm is able to start from one input, the theme, invert it through the patch “inversion,” and then, similarly to what has been showed before, translate both the original and the inversion in numbers (patch “notes 1->12”), produce all possible transpositions of both the originals and the inversions (patch “transposition”), break them into sets of intervals (patch “intervals”), and eventually match each of the originals with each of the inversions to look for intersections. I will call the resulting matrix the “poly-intersection” matrix, and I will differentiate an inversion’s level from the original with an apostrophe (see Example 20). The statements of the “poly-intersection” matrix should be read as follows: the theme on row C will have no intersection with inversions C', D', D#', E', F#', G', G#', A', A#', B', while it will have intersections with C#' and F'. We can now proceed with the simulation of the composer’s process and start adding new voices using the theme’s inversion. In doing so, I will have to make sure that those new voices do not intersect any of the preceding voices, which, as a reminder, are D#, D, F#, G, A#, and B. The matrix will help us in this process. For each of those original levels, I will report all the inversions corresponding to “nil” values (Example 21). The theme’s inversion levels capable of being stacked with the preceding third level canon will then be: A, A#, C#, D, F, and F#. Is this enough to put the theme and its inversion together? “Not so fast,” Clementi might say. There is a further test that these inverted themes will need to satisfy in order to comply with the no intersection rule. What we learn from the Example 21 results is that those levels of inversions don’t intersect with their original counterparts on D, D #, F#, G, A#, and B, but we would also need to check whether they produce intersection among themselves. Sparing the reader another tedious demonstration, I will just mention how the self-intersection matrix built on the inversion of the theme will be identical to that of the original theme (the intervals are the same, only inverted, and as we have noticed earlier, the square technique does not account for the direction of the interval). On the self-intersection matrix reported earlier, one will discover that no intersections are to be found among levels C#, D, F, F#, A, and A#. Now we can put the six plus six voices together being certain that no interval is repeated among different levels of transposition.
Deus ex Machina, Uncovering Aldo Clementi’s System
EXAMPLE
19: POLY-INTERSECTION MATRIX’S ALGORITHM
27
28
Perspectives of New Music
C' C
C#
nil
C#'
D'
t
nil
D#'
E'
F'
nil
nil
t
F#'
G'
nil
nil
G#'
A'
nil
nil
A#'
B'
nil
nil
nil
nil
t
nil
nil
nil
t
nil
nil
nil
nil
nil
D#
nil
nil
nil
t
nil
nil
nil
t
nil
nil
nil
nil
nil
nil
nil
nil
t
nil
nil
nil
t
nil
nil
nil
E
nil
nil
nil
nil
nil
t
nil
nil
nil
t
nil
nil
F
nil
nil
nil
nil
nil
nil
t
nil
nil
nil
t
nil
nil
nil
nil
nil
nil
nil
nil
t
nil
nil
nil
t
D
F# G
G# A
A# B
t
nil
nil
nil
nil
nil
nil
nil
t
nil
nil
nil
nil
t
nil
nil
nil
nil
nil
nil
nil
t
nil
nil
nil
nil
t
nil
nil
nil
nil
nil
nil
nil
t
nil
nil
nil
nil
t
nil
nil
nil
nil
nil
nil
nil
t
t
nil
nil
nil
t
nil
nil
nil
nil
nil
nil
nil
20: POLY-INTERSECTION MATRIX (FIRST THEME AND ITS INVERSIONS)
EXAMPLE
D:
C'
D#: C' F #:
C'
A #:
C'
G:
B:
C#' D'
E'
F'
D#' E'
F'
E'
F'
C#' D'
D#'
C#' D'
D#' E'
C#' D' C#' D' C#' D' C#' D'
D#'
F' F'
F'
F #' F #'
G'
F #'
G'
F #'
G'
F #' F #'
F'
F #'
EXAMPLE
21
G'
G#' A'
A#'
G#' A'
A#'
A'
A'
G#' A' G#' A' A'
A#'
B' B'
A#'
B'
A#'
B'
A#' A#'
We can now go back to Example 5 and appreciate the construction of the twelve real voices canons, which, with the addition of the twelve instrumental “shadows,” form a sonic mass that hides such a refined constructing principle.
Deus ex Machina, Uncovering Aldo Clementi’s System
THE
29
“SECOND THEME”
The second theme is taken from the second part of the chorale theme in its original form (Example 1). This fragment proved much more resistant to Clementi’s method of construction, since it easily creates intersections with its own transpositions or its inversions. This will become immediately clear once we extract the now familiar selfintersection matrix of the second theme (Example 22). We can observe the far higher number of “true” statements in respect with the selfintersection matrix of the first theme. Starting from level B and making the same set of operations already done on the first theme, I will select A# as my second level and then G. After adding the third level there will be no more possible transpositions. Similar results will be reached starting from the other levels. C
D#
D
C#
D
C#
D#
F E
F#
F
G
F#
E
G#
G
(B)
A#
B
A#
B
(A#) B
A
(G) G#
A#
G
Given the limited amount of levels available, the composer decided to increase the thematic variance. To compensate for the lack of a thick polyphonic layering, the composer chose to use two levels of the original second theme against one of its inversion in setting the second theme. The only instance of the second theme inversion also appears on B: similarly to what we have seen in regard with the first theme, the second theme also allows for setting one level of transposition with its inverted form at the same level. This is shown in the poly-intersection matrix of Example 23. If we assume then B and B' (transposed) as our starting levels, then we can extrapolate the set of available levels of the original theme from the B' column: B: B':
C
C#
D
D# D#
F E
The next original level will be D#.
F#
G
G# G#
A
A# A#
(B) B
30
Perspectives of New Music
Once we add the D# level no other levels can be added since from the self-intersection matrix we get the following available levels: C
(D#) E
D
F#
G
A
B
which exclude the G# and A# levels available from B and B'. Even in the case of the second theme on canons IV (shown in Example 24), VIII, XII, and XVI, the choice of number of real voices and levels of transposition is strictly dictated by the composer’s system. C#
D
nil
t
nil
t
t
nil
nil
t
nil
nil
C C
C# D
D# E F
F# G
G# A
A# B
t
D
G
nil
T
nil
t
nil
nil
nil
t
nil
t
nil
t
F
nil
nil
t
nil
t
nil
t
nil
t
nil
t
t
nil
G#
A
A# t
B
nil
nil
nil
nil
t
nil
nil
t
t
nil
t
nil
nil
nil
t
nil
t
nil
nil
nil
t
nil
t nil
t
nil
nil
t
nil
t
nil
t
nil
nil
t
t
nil
nil
t
nil
t
nil
t
nil
nil
t
t
nil
t
nil
nil
t
nil
t
nil
t
nil
nil
nil
t
nil
t
nil
nil
t
nil
t
nil
t
nil
nil
nil
t
nil
t
nil
nil
t
nil
t
nil
t
t
nil
nil
t
nil
t
nil
nil
t
nil
t
nil
nil
t
nil
nil
t
nil
t
nil
nil
t
nil
t
C' C#
F#
E
nil
EXAMPLE
C
D#
nil t
22: SELF-INTERSECTION MATRIX SECOND THEME
C#'
D'
D#'
E'
F'
nil
nil
nil
t
nil
nil
nil
nil
nil
t
F#'
G'
t
nil
nil
t
G#'
A'
A#'
nil
t
B'
nil
nil
t
nil
t
nil
D#
nil
t
nil
nil
nil
nil
t
nil
t
nil
nil
t
t
nil
t
nil
nil
nil
nil
t
nil
t
nil
nil
E
nil
t
nil
t
nil
nil
nil
nil
t
nil
t
nil
F
nil
nil
t
nil
t
nil
nil
nil
nil
t
nil
t
t
nil
nil
t
nil
t
nil
nil
nil
nil
t
nil
G
nil
t
nil
nil
t
nil
t
nil
nil
nil
nil
t
t
nil
t
nil
nil
t
nil
t
nil
nil
nil
nil
nil
t
nil
t
nil
nil
t
nil
t
nil
nil
nil
nil
nil
t
nil
t
nil
nil
t
nil
t
nil
nil
nil
nil
nil
t
nil
t
nil
nil
t
nil
t
nil
F#
G# A
A# B
EXAMPLE
23: POLY-INTERSECTION MATRIX SECOND THEME
Deus ex Machina, Uncovering Aldo Clementi’s System
EXAMPLE
24: SECOND THEME’S SETTING (IV)
31
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Perspectives of New Music
7. REPETITIVENESS AND DECEPTION Clementi took the task of avoiding any traditional musical narrative to the bitter end. According to the composer, any sort of juxtaposition of contrasting elements in music could be interpreted as the beginning of a musical discourse, a discourse that he was determined to eradicate from his music.32 The slightest variations in pitch material, dynamics, or even timbre could be enough to trigger music’s pseudo-linguistic features. Clementi’s lucid realization of a “narrative bias” in westerners’ perception of music—a bias that he was determined to “rectify” in his music—is a key moment that defines Clementi’s aesthetic. The erasure of contrasts is, therefore, far from being a choice based on trite ideology (i.e., a solipsistic intellectual fixation of sorts); on the contrary, it is instrumental to achieve a specific perceptual outcome for his compositions, one that would prevent the listener from locking in a narrative perception of the music.33 Clementi achieved his goal of eliminating contrast altogether from his music by means of a simple tool: repetition. For Clementi, repetition is a complex matter that deals with different levels of perception. The case of Aus Tiefer exemplifies how Clementi used repetitions at different levels of perception. To simplify the discussion, we can observe how repetition takes place at three different levels: micro-structural, macro-structural, and meta-structural. At the micro-structural level, we find simple melodic fragments repeating ad libitum. The simplicity of repetition of identical, easily identifiable, melodic fragments is, though, bound to carry the listener into indiscernible sonic masses. The simple, easily recognizable melodies gradually drown into a maze of sound where no solid grounding in terms of harmony, rhythm, or dynamic punctuation is provided. Mattietti rightly notices how Clementi creates a perceptual experience similar to that of Escher’s absurd constructions.34 Escher’s drawings feature familiar elements such as staircases, absurdly rearranged, creating a perceptual game of deception. Douglas R. Hofstadter in Gödel, Escher, Bach: an Eternal Golden Braid accurately describes the game: Those staircases are “islands of certainty” upon which we base our interpretation of the overall picture. Having once identified them, we try to extend our understanding, by seeking to establish the relationship which they bear to one another. At that stage we encounter trouble.35 In Aus Tiefer, the saturated repetitive textures of the different panels are immutable. Their material does not change, lacking any structural
Deus ex Machina, Uncovering Aldo Clementi’s System
33
arc; the piece does not unfold but rather “folds” onto itself in a palindromic fashion. While at a local level counterpoint is deprived of its transformative qualities, at a structural level formal construction lacks its traditional dialectic of contrast. At yet another level of perception—the meta-structural level—many elements in Clementi’s pieces are “repeating” familiar elements of musical tradition. There are many elements that in general link Clementi’s music with the tradition. Procaccioli rightfully points out about Tre Canoni that there is an inherent classical character in Clementi’s music: In Tre Canoni, one finds at least two facts that in spite of the original intentions, . . . tend toward determining a reference to . . . tradition: the use of contrapuntal relationships and the control of form.36 Similarly in Aus Tiefer, the listener’s expectations are teased by the familiar tunes of a Lutheran chorale, and by the familiar instrumentation and canonic construction traditionally associated with it. But even though there are many elements that are remindful of the tradition of music, there are some important deficiencies that prevent a straightforward labeling of Aus Tiefer and other pieces from Clementi’s diatonic period as a “celebratory revival” of traditional music. In fact, if one lets the meta-historical fogginess created by the familiar fragments and canonic construction evaporate, something else will become clear: there is something terribly amiss in Aus Tiefer, the horror vacui of a complete lack of a traditional musical narrative that any piece of traditional music, even a simple chorale, possesses. Therefore, Clementi recuperates some elements of musical tradition, but only as objets trouvés : like a survivor of a lost civilization rummaging through remnants of the past, playing with relics whose original purpose has long been forgotten, he employs themes from the past and uses canonical constructions depriving them of their original nature. As Procaccioli notices, [Clementi] . . . deeply changes the nature and meaning of counterpoint and turns it into a symbol, in an evocative, abstract manner: counterpoint as symbolism.37 In some sense, one might say that the piece is telling us not what it is but rather what it is not. If at the micro-structural level the parallel with Escher comes to mind, at the meta-structural level, Magritte’s nonpipe seems to be the most accurate visual representation (Example 25).
34
Perspectives of New Music
EXAMPLE
25: RENÉ MAGRITTE, “THE TREACHERY OF IMAGES” (1929)
8. CLEMENTI’S LEGACY: SONIC AUTOMATA The ultimate goal of this paper has been to notice how Clementi’s late production, while starting from cosmically pessimistic premises, is far from constituting a moot point in the development of musical aesthetics.38 On the contrary, Clementi’s music revisits the very premises of musical narrative and carries the listener into a particular and original kind of musical experience. If, generally speaking, a piece’s narrative is found in the dialectic of contrasts and/or the more sophisticated idea of development, Clementi’s complex sound artifacts appear to have no such a thing as a narrative. But Clementi’s late production deals with a different sort of contrast: that between the apparent calm of the surface and its indiscernible core. In particular, Aus Tiefer perfectly exemplifies both the un-narrative qualities and simple/complex dichotomy of Clementi’s late production. The apparent simplicity of its traditional themes, as they are voiced in canons, the textural, contrast-free layering of material gently carries the listener into familiar territories of traditional music. Subtly but inevitably, the simple elements that start the piece undergo a gradual and hypertrophic layering that transforms what seemed a discernible or even predictable surface into an incomprehensible maze of sound. By abandoning the use of traditional musical dialectics, and by systematically erasing any trace of them from his music, Clementi radically deprived his act of any attempt to express anything other than the sounds being played. The best way to understand the composer’s oeuvre is to consider it as an exploration of complexity, and not just of
Deus ex Machina, Uncovering Aldo Clementi’s System
35
any complexity, but of that which surprisingly springs out of the most simple initial conditions and through the most basic rules of iteration. Seen this way, the composer approached his work with experimental rather than expressive goals. The algorithm Clementi created over the years aimed at generating the most complex sonic textures starting from the perceptual clarity of a fragment, or, in other words, to undercut and ultimately erase the initial clarity of the fragment by the sole use of the fragment itself.39 In my own compositional research I have come across the work of a mathematician whose work bears a striking resemblance to the one of Clementi. Some of Stephen Wolfram’s research is based on the observation of the behavior of simple programs. In particular, Wolfram’s A New Kind of Science, published in 2002, is an in-depth study of so-called “cellular automata.”40 Oversimplifying, Wolfram’s magnum opus describes how simple programs—i.e, a simple set of rules —can produce highly complex and unpredictable results. Wolfram like Clementi, is fascinated by the observation of the outcomes such programs produce when run ad libitum. In the following quote, Wolfram describes his “eureka” moment that led him to embark in a long research of cellular automata: our everyday experience in building things tends to give us the intuition that creating complexity is somehow difficult, and requires rules or plans that are themselves complex. But the pivotal discovery that I made some eighteen years ago is that . . . such intuition is not even close to correct. . . . what I found—to my great surprise—was that despite the simplicity of their rules, the behavior of the programs that I looked at had behavior that was as complex as anything I had ever seen.41 Wolfram’s fascination and experimental fixation with the world of simple programs and their occasionally unpredictable outcomes is easily relatable to Clementi’s own fixation for elaborate sonic masses as they are generated from simple elements. Instead of attempting to direct their experiments in a specific direction, both Clementi and Wolfram control their calculations to guide them towards specific musical or scientific goals, and let the “rules” run by themselves only to observe the result later. Wolfram’s image-rich book is ultimately a work of scientific research unconcerned with issues of aesthetics and artistic endeavors, but Wolfram’s approach to science can be easily related to Clementi’s, for the scientist, as the artist, becomes the wondering observer of simple experiments and their unpredictable results.
36
Perspectives of New Music
Visually, there is a striking similarity between the images in Wolfram’s book and the specific way Clementi represented his music. The triangular shapes of both the Automata (Example 26) and Aus Tiefer’s panels (Example 27) are an intriguing aspect of the odd couple Clementi-Wolfram. To my knowledge, the composer never acknowledged Wolfram’s work among his influences, though I would not be surprised to learn otherwise, given the multiple similarities.42 In conclusion, the modern field of Cellular Automata provides a fecund interpretation of Clementi’s legacy. Through the words of Wolfram, we can appreciate how Clementi’s work tapped into a “New Kind of Music,” aimed at exploring rather than narrating. The composer is an observer rather than a story-teller, observing and wondering about the origins of complexity. the Cellular Automata we set up are by any measure simple to describe. Yet when we ran them we ended up with patterns so complex that they defy any simple description at all. And one might hope that it would be possible to call on some existing kind of intuition to understand such a fundamental phenomenon. But in fact there seems to be no branch of everyday experience that provides what is needed. And so we have no choice but to try to develop a new kind of intuition. And the only reasonable way to do this is to expose ourselves to a large number of examples.43 “Exposing ourselves to a large number of examples” is what Clementi did for his last forty years, reinventing his role and opening new doors for aesthetic exploration.
Deus ex Machina, Uncovering Aldo Clementi’s System
EXAMPLE
26: CELLULAR AUTOMATA
EXAMPLE
27: AUS TIEFER
37
38
Perspectives of New Music
NO T E S 1. I would like to express my deepest gratitude to Allan Keiler, YuHui Chang, Joshua Fineberg, Guido Zaccagnini, Anna Clementi, Garbiele Bonomo at Suvini Zerboni, and Manuele Morbidini. 2. “Rare is the composer who claims such a consistent niveau of quality, most of all in the twentieth century. . . .” Dan Albertson, Aldo Clementi: Mirror of Time (Abingdon: Routledge, 2009). 3. Stephen Snook, Liner Notes, Aldo Clementi: Works with Guitar, Geoffrey Morris (guitar) (Mode Records 182, 2007), compact disc. 4. E.g., Gianluigi Mattietti, Geometrie di musica: il periodo diatonico di Aldo Clementi (Lucca: Libreria musicale italiana, 2001). 5. The Sacher Foundation in Basel recently acquired the collection of sketches I accessed in Clementi’s studio on the Via Cassia, thereby inscribing Clementi in a very select circle of modern composers. 6. The reasons for choosing this piece over others include its considerable size, both in instrumentation and duration, the relatively recent year of composition, and the lack of scholarship concerning it. 7. “OpenMusic (OM) is a visual programming language based on Lisp. Visual programs are created by assembling and connecting icons representing functions and data structures. Most of the programming and the operations are performed by dragging an icon from a particular place and dropping it to an other place. Built-in visual control structures (e.g., loops) are provided, that interface with Lisp ones” (http://repmus.ircam.fr/openmusic /home). 8. The procedures can be easily replicated in other software environments or programming languages. While the composer positively proceeded with the sole help of pencil and paper, the use of the computer greatly and effectively helped expedite filling in the gaps of my research material. In today’s computerized world, anybody can perform highly complex operations in real time. In music composition, programs have been developed to take advantage the capabilities of computer processors to present composers with musical result that would have taken weeks or months to develop on paper. This is one reason why some overlook such deployment of computer software to compose as a product of “laziness” on
Deus ex Machina, Uncovering Aldo Clementi’s System
39
behalf of the composer in the best cases or even abdication of the role of creator in favor of mechanized procedures. What these criticism fail to address is the choices that the algorithmic composer has to make in order to create the process. Choosing one mathematical or logical operation over another will produce different results that will represent the composer’s idea or fail to do so. 9. “Gli equivoci nascono solo da quanti, anche inconsciamente, pensano la Musica come discorso e quindi, non accorgendosene, come caricature di un arco che descrive un inutile orgasmo. Esaltazione e depressione sono capitoli chiusi: comunque camuffati, sono modesti simboli di una dialettica già estinta. Un forte seguito da un piano, una nota acuta seguita da una grave, un timbro dolce da uno aspro: ciò è di per sé dialettica, cellula sonatistica di una più grande Forma-Sonata.” Michela Mollia, Autobiografia della musica contemporanea (Cosenza: Lerici, 1979, 48–55), 48 (my translation). 10. “If my music lacks theatrical elements, [musical] effects, neoclassical optimism or any useless positivity, it is to a large degree because of the explicit or tacit influence of Adorno’s thought” (my translation). “Se nella mia musica mancano il teatralismo, l’effettismo, l’ottimismo neoclassico e ogni inutile positività, ciò è dovuto in gran parte all’influenza esplicita o sottintesa del pensiero di Adorno” (cit. in Graziella Seminara and Maria Rosa De Luca [eds.], Canoni, figure, carillons: itinerari della musica di Aldo Clementi: atti delll’incontro di studi, Facoltà di lettere e filosofia, Catania, 30–31 maggio 2005 [Milan: Suvini Zerboni, 2008], 69). 11. “Il musicista non è più una specie di faro che illumina tutta l'umanità, come ai bei tempi dei romantici. L'uomo non cerca più l'arte, ma il comfort, la praticità, il piacere, il divertimento. Bisogna avere il coraggio di ammettere che di arte non c’è più alcuna necessità.” Enrico Cavallotti, “Aldo Clementi: verso la morte della musica,” Il Tempo, Roma, 22 December 1977 (my translation). 12. “E’ vero: l’arte e la musica stanno morendo per saturazione. Allora, o si ha il coraggio di stare zitti, oppure si continua a far cose . . . che esprimono proprio l’agonia e il dissolvimento dell’oggetto espresso.” Mollia, Autobiografia della musica contemporanea, 48 (my translation). 13. “Non c'è contraddizione tra una visione pessimistica, o apocalittica, sul futuro dell’arte e l’esigenza di mettere bene insieme le
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Perspectives of New Music
note, il desiderio di continuare un prezioso artigianato musicale. Per molti la crisi dell’arte diventa invece un alibi” Enrico Cavallotti, “Aldo Clementi,” il Tempo, Roma, 22 December 1977. 14. Other forms of transformations—retrograde and retrograde inversion—while present in Clementi’s sketches are not employed in the piece’s final version. 15. Those workbooks were stacked in his studio and were differentiated by cover inscriptions indicating the dates and titles contained. 16. “[Maderna] aveva un sistema che chiamava ‘lettura e rilettura del materiale.’ Mi spiegò su un foglietto di carta a quadretti gli schemi di questo sistema che era una estensione della tecnica dodecafonica però partendo da Webern. Il tutto in mezz’ora di tempo. Ma mi sconvolse perché da quel memento capii che dovevo ripartire da zero.” Manuele Morbidini, “Musiques informelles: il concetto adorniano e l’opera di Aldo Clementi” (Tesi di laurea in Filosofia, Università di Siena, 2010), 28 (my translation). 17. E.g., F1 and F3 would both correspond to the number 7. 18. In Aus Tiefer the composer ended up using only the originals and the inversions. 19. One of the “flaws” of the magic square is its inability to represent the up or down direction of the interval. 20. By intervallic content I refer to its horizontal intervals—namely, the intervals between each pair of successive notes. 21. In particular one can see in Example 9 a marker inscription quoting “Buono!” (Good!) that comes after pages filled with squares and matrixes reworking the same theme. 22. The F#-B interval that opens the theme in Example 11 cannot be replicated in Clementi’s system by any other line. Another perfect fifth like C-G is though allowed.
23. “It was because I was acquainted with the Informel and Materism [Arte Materica] in paintings (of which I could not see a correspondent in music) with their abolishment of contrasts . . . that I considered a start from scratch.” “Furono l’Informale e il Materismo in pittura (di cui non vedevo un reale equivalente nella musica) e i relativi postulati circa l’abolizione dei contrasti . . . a farmi seriamente considerare la necessità di un ri-inizio da zero.” Mario Bortolotto, “Intervista con Aldo Clementi,” in Lo Spettatore
Deus ex Machina, Uncovering Aldo Clementi’s System
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Musicale 2/5 (1967), 16–17 (my translation). As explained in the Treccani Online Encyclopedia (http://www.treccani.it/enciclo pedia/arte-materica/), “Materic Art is a stylistic tendency that became popular in the 1950s. In a Materic artwork, the matter of which the work is made holds a fundamental role indistinguishable from the image itself, as carrier of a message and meaning. . . . Included in Materic Art are artists of the Informal and Neodadaism, but it is possible to identify precedents in Surrealism, Dadaism, and Futurism.” (My translation.) 24. See note 7. 25. I will use Clementi’s way of numbering notes from 1 (C) to 12 (B). 26. “t” and “nil” stand for “true” and “false” in the Lisp programming language. 27. A theme will not only have some but all of the intervals in common with its own self. 28. A caveat to the matrix reported is that not all “true” statements are equal: the unison transposition will contain all of the intervals while those on the major second will only have two intervals in common, but that does not matter to the composer, who wanted to avoid any overlap. 29. Canons IX and XI represent exceptions, since they follow the rules only partially. 30. The note in parentheses indicates the transposition level to which the t-nil string belongs; the matrix will give a “t” statement, since a unison transposition will create intersections. 31. There are twelve voices if we include the same level’s “shadows.” 32. See note 12. 33. Against the objection that avoidance of dialectical narrative can itself be considered an ideologically driven goal, this article has presented arguments to show Clementi’s work as a collection of perceptual experiments that deal with the fine line that divides the understandable and the unintelligible. 34. Mattietti, Geometrie di musica. 35. Douglas R Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (New York: Basic Books, 1979), 97.
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36. Stefano Procaccioli, Aldo Clementi, a Barometer of the Changes of Our Time? Some Observations after Listening to the Tre Canoni (but not only that), in Dan Albertson, Aldo Clementi: mirror of time II (Abingdon, Oxfordshire: Routledge, 2011), 237. 37. Ibid. 38. Regardless, that is to say, of what the composer wants to say about it. 39. Many questions arise about the choices the composer made in terms of privileging the intervallic content of each fragment while apparently disregarding the verticality of the sonic result. More research on the subject might show more insights on the matter. 40. A cellular automaton is a collection of “colored” cells on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states of neighboring cells. The rules are then applied iteratively for as many time steps as desired. Von Neumann was one of the first people to consider such a model, and incorporated a cellular model into his “universal constructor.” http://mathworld.wolfram.com/Cellular Automaton.html. 41. Stephen Wolfram, A New Kind of Science. (Champaign, IL: Wolfram Media, 2002), 40. 42. Clementi did acknowledge the work of another mathematician, Douglas Hofstadter, as one of his influences. 43. Ibid., 41.
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