Translated-university Paris 8 - Vincennes - Saint-Denis Ufr 1 - Arts, Philosophy, Aesthetics Up 8 Discipline: m

Page 1UNIVERSITY PARIS 8 - VINCENNES - SAINT-DENIS UFR 1 - ARTS, PHILOSOPHY, AESTHETICS THESIS to get the grade DOCTOR OF U NIVERSITY P ARIS 8 in BEAUTY , SCIENCE AND TECHNOLOGY OF ARTS Discipline: Music Presented and supported publicly by Guilherme Carvalho Title: MUSICAL PERFORMANCES OF IDEAS MATHEMATICS Supervisor: Horacio V Jury: Martin L ALIBERTE Mr. Makis S OLOMOS Ms. Antonia S OULEZ February 2007 Page 2 AGGIONE 2 TOVANT -P ROPOS This work is the result and witnessed a musical journey (composer and interpreter) influenced by mathematics. It addresses the issues of formalization music and, more generally, that of reconciliation between the two disciplines. In this context, it aims to expose a musical thought strategy with which to deal with issues specific to the work of the composer. F OREWORD This work is the result and the traces of a musical trajectory (as composing and interpreter) Influenced by mathematics. It deals with the stakes of formalization in music and, more Generally, with Those of Bringing together thesis subjects. In this frame Particular, it AIMS to the bring forth a strategy for musical thought with qui approach to issues proper to a composer's work. Page 3 3 T TABLE OF CONTENTS TOVANT -P ROPOS .................................................. ............................ 2 I NTRODUCTION .................................................. ............................. 6 . Thought formelle.................................................................................................... 7 . Mathematics presence ............................................... ............................... 7 . Languages ............................................................................................................ 10 . Théorisation....................................................................................................... 11 . Pratiques............................................................................................................ 14 I - E AND Xistence MRULTIPLICITÉ ................................................ 17 1. From the rapprochement between mathematics and music ..................... 18 . Hardware abstraction and constraints .............................................. .............. 19 1.1 Reasons for a rapprochement between the two disciplines .................. 21 . Musical thinking, musical discourse ............................................. ................... 21 to. discourse on music .............................................. .................................. 22 b. musical discourse ................................................ ............................................ 23 . Take mathematics as such ............................................. ..... 24 . Mathematics as "Truth"? .................................................. ....... 27 1.2 Ways to establish a rapprochement .......................................... ........ 29 . Formalization, model, modeling ............................................. .................. 30 . Using mathematical ............................................. ......................... 32 to. ................................................. Application .................................................. . 33 b. parallèle........................................................................................................ 35 c. intermediate situations ................................................ ............................... 37 1.3 From what is formalized ............................................ ............................. 38 . Notation.............................................................................................................. 38 . With the notation: writing and proofreading ........................................... ............... 42 to. writings and logical time .............................................. ............................... 43 b. analysis and proofreading ............................................... ...................................... 45 . Running time (time) ........................................... .... 48 to. distance of a formal language ............................................ ............................ 49 b. validity of this juxtaposition with formal languages ​............................. 53 . Discourse on music .............................................. ...................................... 54 to. theories of music .............................................. .................................... 54 b. analyzes and styles ............................................... ............................................ 56 c. understanding and communication ............................................... .................. 58 2. the possibility of musical performances ................................. 59 . The effectiveness and efficiency of links ........................................ ...................... 61 2.1 First definition ............................................... .................................... 65 . Illustration: Any differentiable function is continuous ....................................... 66 Page 4 4 . Illustration: Princípio Cavalieri ............................................. ................... 71 . Traces of formalization .............................................. .................................. 75 2.2 Second definition ............................................... ..................................... 77 . Illustration: a convex set ............................................. ..................... 80 to. convexity ....................................................................................................... 81 b. homotopies.................................................................................................... 84 . Illustration: Lema 1 .............................................. ............................................ 87 to. partitions....................................................................................................... 88 b. primitivation.................................................................................................. 90 c. déroulement................................................................................................... 91 2.3 Types of representations .............................................. ............................. 92 . Constructed representations ................................................ ............................. 95 . Found representations ................................................ .................................. 97 . Representations métamusicales ................................................ ..................... 100 3. From the senses in a formalization ........................................... ........... 102 3.1 Two sense to "interpretation" ........................................... ............... 103 . Movements of "meaning" ............................................. .................................. 104 3.2 intelligibility and understanding .............................................. ................ 105 . Transmission of content ............................................. ............................... 107 . Signification..................................................................................................... 112 4. arbitrary decisions ............................................. ..................... 114 4.1 The possibility to model .......................................... music 115 .... 4.2 the to distance object model .......................................... ............. 121 4.3 From decisions model between ........................................... ..... 126 4.4 Overcoming the distance between object model .......................................... ..130 4.5 An irreducible distance .............................................. .......................... 134 4.6 The need for arbitrary decisions ........................................... ..... 137 II - G ÉOMÉTRISATION .................................................. .............. 140 1. Figure and forms ............................................. ................................... 141 1.1 Non-Euclidean Geometry ............................................. .................... 141 Figures 1.2 and musical musical contexts ............................................ ..145 . Figure notée................................................................................................. 146 to. illustration: as a melodic line .......................................... ....... 151 . Link to the perception .............................................. .......................................... 152 . morphological eidetic × ............................................... .............................. 155 . Objects ............................................................................................................... 158 2. Musical Spaces .............................................. .............................. 161 2.1 Dimensions and parameters .............................................. ........................ 162 . Illustration: a first musical space ............................................ .......... 165 2.2 Directions and fragments of space .......................................... ............. 167 3. musical objects as functions ........................................... .170 3.1 Variable Sets .............................................. ............................. 171 to. instruments.................................................................................................. 173 Page 5 5 b. game modes ............................................... ................................................. 174 c. hauteurs....................................................................................................... 176 d. positions...................................................................................................... 178 e. dynamiques.................................................................................................. 180 f. durées........................................................................................................... 181 3.2 Fonctions.................................................................................................184 . Indivisible gestures ................................................ ........................................... 186 . Insertion time .............................................. ..................................... 189 3.3 Continuities ................................................ .............................................. 191 4. Areas modular, composed space ........................................... 193 . Several temporal directions ............................................... ..................... 194 . An "extended" size ............................................. ................................ 195 4.2 Locality and globality .............................................. ................................... 197 4.3 The form of a work ........................................... ................................. 199 III - P .................................................. ....... 202 OÉTIQUE MRUsical 1. Définition(s)......................................................................................203 . Language courant.............................................................................................. 203 to. music as a "text"? .................................................. .............. 204 . Stravinsky........................................................................................................ 206 . Schoenberg, Dahlhaus ............................................... ..................................... 207 . Nono, Antunes ............................................... .................................................. 209 . Backes, Ruwet ............................................... .................................................. 213 . A position .............................................. ......................................... 217 2. Poetry and poetics of weak topology ......................................... 218 2.1 clotting time and time felt ............................................ ... 219 . An inspiration in literature ............................................. ................... 220 . Phrasing and integrations ............................................... ..................................... 221 to. in the partition ............................................... ........................................... 222 b. "Lightness" ............................................... .................................................. . 224 2.2 Internal and External Memory ............................................. ...................... 225 C CONCLUSION .................................................. ............................. 228 . Mathematics, a metaphor for the composition? ........................ 229 B IBLIOGRAPHY .................................................. ........................ 231 TONNEXES .................................................. .................................... 241 1. a convex set ............................................. ........................... 242 2. Lema 1 - e partições primitive .......................................... ............ 251 3. Any differentiable function is continuous ........................................... ..254 4. weak topology .............................................. ................................... 269 Page 6 6 I NTRODUCTION Human reason has this particular destiny (...) it is overwhelmed by the issues it can not be excluded (...), but which it can not provide an answer (...). - I. Kant, Critique of Pure Reason A form of abstraction accompanies the so-called western music, and probably that of other regions as well, since its inception. Once a gap or a range is detached from the instrument or voice that the product to be an idea itself, we have a passage of the material to the abstract, a chronometric time to logical time, which carries both wealth and flexibility of what Xenakis call the off-time 1And conceptual and semantic difficulties inherent in any abstraction 2And this detachment in particular. A special thought to the music thus made ​possible, a way to observe and talk about that can be addressed, precisely because of the distance taken with respect to the physics of sound, its time course and understanding. 1 XENAKIS I. [1963], formal music. 2 See in particular GOODMAN N. [1966] The structure of appearance. Page 7 7 . Formal thought In fact, it is very difficult for us to design a speech without music This way of abstracting, it appears as "natural" in the musical thought: it is historically often issue notes, harmonies, rhythms, taken as starting points essential to the definition of music, so that it can be avoid the surgery. Even when these concepts are poorly adapted to a practice (such as for concrete music 3) Or are deliberately avoided (as in scores graphic or verbal Fluxus 4), We still meet frequently a preoccupation with form, how to register events in time and space, and extract (or Less) intelligibility. This formal thought is thus practically unavoidable in our music, and can become an object of study in itself, we believe it is useful to study the ways to talk about these abstract and formal aspects of musical thought. . Of maths We want to address the issues in the text of a critical and constructive oriented mathematics, focused on music in general and composition particular. Specifically, we want to concern ourselves with input from a approximation of musical and mathematical thoughts in the field of understanding of music, understood as organization of sound objects and 3 See redefining vocabulary must construct S P. [1966] Treaty of musical objects. CHAEFFER 4 See eg N M. [1974] Experimental music (ch.6). YMAN Page 8 8 music (listening and writing), but also in those of aesthetic and poetic musical. We will seek to clarify how formal thought in music can address these issues, how they can be made ​and processed. The engine of this work is actually a set of reflections aroused by the idea a reconciliation between the two disciplines, and the fact that we have the personally always thought very similar fashion. Our musical reasoning as for composition and for interpretation, is deeply influenced by a reasoning as a mathematician: we wanted to search and clear the terms of this influence in its many aspects. If sometimes the direct link mathematics seems to fade, as in a discussion on poetry music, the route that leads to these points is no less indebted to this look mathematician on the subject. Mathematics deal indeed abstract forms and formalization, but do not forget that there is mostly accurate question in speech, in the definitions in the methods - and this is what directs this sought after. If one of our goals is then contribute to the accuracy of the discourse on music and ways of thinking about composition and performance, it may be relevant take the preoccupation with a certain musical epistemology as one of lines that run this text. To paraphrase a definition given by Granger Page 9 9 epistemology 5We could say that the musical epistemology should be both philosophical analysis of some musical practices, taken in their procedures and their actual evolution, and also more general interpretation of the meaning of musical knowledge. These two actions, the interpretation of the knowledge and analysis practice, are the source and center of our approach. Of course, here we must distinguish the study of science and knowledge Scientific study of practices and musical knowledge. In particular, the This language is distinct ways in both areas: speech Science is not directly done on or with sensitive data while it is possible to consider, for joints inserted in time, a musical discourse that is. At least some forms of systems used music and musical thought stood outside a language itself, so directly on the sensitive or presentable. 6 Moreover, the very idea of precision Science (in practice and discourse) is not trivially compatible with a musical accuracy, the same definition can vary depending on the context, and sets involved mostly aesthetic considerations that lexical. 5 See GRANGER G.-G. [1994] Forms, operations, objects: "(...) epistemology must be both analysis some philosophical sciences, taken in their effective procedures and their evolution, and secondly more general interpretation of the meaning of scientific knowledge. "(Introduction, p.8) 6 As opposed to the speakable, as we find them in W L. [1922], Tractatus Logico ITTGENSTEIN Philosophicus. This brings us to a first difficulty inherent in any theoretical work on practical music: how to talk about what exactly presentable and sensory experiences that it generates? We obviously can only get around these "holes" in the language, taking into account the meaning of salience, often rich, these detours generate. Page 10 10 . Languages We see that the various relationships of music in language can appear quite naturally into a discussion on the interpretation and knowledge musical: it will also issue throughout our text, in parallel with the formal languages, meaning construction, and reports to the poetic (literary). Us want to integrate these issues to the broader reflection on the insertion of a "way Mathematical act "in music: the construction of musical performances of ideas mathematics. On the one hand, the study of these representations is similar to what Granger appoints métadisciplines 7. They are moving in two directions: first, "to elucidating the possibility of internal operation of a system condition symbolic as a work of thought; secondly, to an explanation of the methods in action in a theory. "Indeed, if we look at the composition of view composer, we are facing an organized set of actions, networked by operations of various types and on them: Operations that combine modify, select, start or discontinue these actions, consider the results, and From then generate other actions and other operations just recharge it network. On the other hand, the musical experience that can not be reduced to a system symbolic (especially when it comes to perception and interpretation) Page 11 11 musically represent a mathematical idea also means taking arbitrary decisions (external to a given formal framework) to act on and from a practice. We focus so as to process changes compositional or interpretive even (in his "gestures", so to speak) when oriented, at least in part, by such a representation. . Theorization It must be stressed here that mathematics will never be used in our work prove anything: they are a tool or a particular contextualization for a thought that bears it, on other subjects as mathematics. More specifically, is for us to develop a discourse on music whose form is directly and centrally influenced by the mathematician thought 8. In particular, when we formalize this is not primarily to extract an application (software, by for example), but because we want to think in that particular way, without did not involve any stiffness. 9 7 GRANGER G.-G. [1994] What is métadiscipline? In forms of Operations Items (p.112). 8 We would like to do in mathematics equivalent to the distinction between "Logical" musical (a work) and musicians (a musician in a compositional process or interpretive). This is made ​difficult by the fact that metamathematics may be strictly part even math (so explicitly with category theory): thinking about Mathematics can have the form of a proper mathematical thinking. We will occasionally the difference between interest rather the results of reasoning and its consistency, or rather how to develop this argument and give it a consistency (Respectively, the mathematical case and the mathematician cases). 9 "It is well known that the accuracy of mathematical results, combined with a poor knowledge of delicate ontology of music, can cause dogmatism in which mathematics is unfairly empowered. "(It is Well Known que la precision of mathematical results, together with a poor knowledge about the delicate ontology of music May Provoke a dogmatism for qui est mathematics unjustly made ​responsible.) - A MZZOLA G. [2002] The Topos of Music (p.9). Page 12 12 Theorizing we want to develop is based primarily on the works and on our experience as a composer and performer. 10 From one point of view it leads to a "naive" theory or axiomatic nor fully formalized. We do not necessarily about looking completeness, subjects, or the theory itself, but rather consistency. 11 The problems are still those of a composer (which is incidentally interpreter and mathematician), and this is reflected in the questions that guide research in the vocabulary we choose to throughout the text, and therefore in the relative importance of different topics constituting the discussion. If elements of music theory emerge from our text, we can a little better understanding of the nature of it by observing its consistency, its objects and its strategy. 12 Naturally, we are aiming primarily a musical consistency: in the last analysis, the conclusions we draw must be musically relevant. In second level, our thinking is trying to keep a mathematical consistency as much as possible, as it heuristically. Similarly, our objects of study and Work is mostly music: we allow even, if necessary, to modify the interpretation of a mathematical idea for musical purposes. Finally, the strategy along the entire research is mainly mathematician: it relies on 10Conversely, the works that we have made ​would not have existed, at least as they are, without theoretical reflection. 11We are not suggesting that completeness and consistency are mutually exclusive here, as is the case in mathematics, but we do not deny a priori the possibility. Page 13 13 precise terminology, on the direct impact of a definition on the reasoning that employs; the choice of elements of a building or deduction is always oriented. But we do not seek to establish a true musical theory but rather to offer directions and theorizing modes. At most, this work could be that the theory of our own music and our own processes compositional: we can see it as a "translation" into more terms precise and sharable, thought and actions at work in these processes. Therein lies Our main difficulty with which can proliferate approaches topics that We are interested in how to realize the network of associations involved in the composition, interpretation or listening (even if that properly ours)? This network is not always based strictly, much less still logically consistent, and so difficult to formalize the strict sense. 13 In particular, more close to our main theme, it is in him that is realized while passing mathematics to music, when the passage is embodied in a formalization or no: formalization is an integral part of the network. If any originality appeared in the approach to issues "Classics" of the composition (such as time, material, shape, intelligibility ...), it would be that the whole approach is causing very 12We take the view that the special nature of a musical theory gives (at least partly) consistency in the triplet-object-strategy N ICOLAS F. [2005b] How musically evaluate mathematical theories of music? Page 14 14 Personal: the effort is provided in the sense of making it communicable and criticism. We chose to run the risk of idiosyncrasy: this approach may not be understood "by one who has ever thought of yourself ideas that are exhibited - or at least similar ideas. » 14 . Practices Again, it is an attitude inspired by mathematics that will guide this discussion put our thinking music (our musical epistemology): we will focus systematically on its shape - its internal joints and its links with other thoughts, properties that allow the report to practice (compositional or analytical) and a way to understand the subjects it surrounds but does not contain. The idea of serving as a point of view mathematician to address a closely linked to the perception discipline may seem contradict a precaution taken Granger, "the transcendental attitude analysis leads us to recognize that mathematics are always away more perceived. » 15 This is only an appearance, we just do not try to do math, but to develop a thought from experience music and from that experience. There is no a priori whether removal is our strategy is a mathematician: we take mathematics as a tool, not mathematical tools. This is Granger itself indicating the possibility of this 13We might even claim that it can not be entered in full (at least by us). 14WITTGENSTEIN L. [1922], Tractatus Logico-Philosophicus (Preface). Page 15 15 approach: "The problem of epistemology is therefore interpretation forms " 16. We discuss the works, formalizing and musical poetics of ways like, constantly seeking to develop dialogue analysis and perception. Us can say with Nattiez that "the analytic act is frozen in time, perception, it is dynamic " 17: Analysis may consider a complete form, perception has access to the shaped construction or emerging. The reference to the perception, impact that the dynamic of the latter has the same definition of an analysis and, particularly, on the possibilities and modalities of understanding of music is fundamental to us. Us can not limit ourselves to observe the progress process without our gaze is modified by the movement of the process itself, in its deployment. This is how we understand the words Molino: The time of the analysis was - is still probably some - the moment where we thought we could bring together the disciplines around a general theory applicable to each individual work: the dream is over. Therefore there can be no epistemology of analysis because Now that analyzes not mean anything specific, because by that we mean any method whose object of study is the music. 18 Although we did not want to keep completely the idea that "analysis" refers to any method of studying music, we will retain the impossibility of 15GRANGER G.-G. [1967] Formal Thinking and Human Sciences (p.11). 16Idem, p.12 17NATTIEZ e part, the musical discourse). , JJ [1973] Foundations of a semiotics of music (2 18MR Olino, J. [1995] Experience and knowledge. Page 16 16 musical epistemology if we consider that a general analysis, instead of several analytical processes that depend on the context in which they are made, and alter. Reflection should well be following the shape of what we study. This text is therefore a critical and reflective study of our musical activity from what we appear, what is somehow immanent: works on composition work and the interpretation (where we include listening and analysis). As we observe the instrumental gestures to take as a material compositional, and that we isolate from their original contexts of the elements of works above us to take them as generators of shapes, we will look the direction of relations between the components of our musical activities (choice, formalizations, associations, defined by a more or less controlled process) regardless of their origin and their logical hierarchy. We chose, as we said, to run the risk of idiosyncrasy, it is because we believe it is actually less: if we can 20 19 to make it more transparent at least part of our thinking, we have opened the possibility of dialogue with other composers, first, and with reflection on music in general. 19Nevertheless, we will use often, economics, the word "analysis" to describe this process. 20We paraphrase N ATTIEZ , JJ [1973] Foundations of a semiology of music: analysis immanent level (which the author has had the misfortune to also appoint "neutral level") is "the study of direction of relations between the component elements work (notes or other sound objects, defined by controlled process), regardless of their origin and their compositional perceptual hierarchy. » Page 17 17 I - E AND Xistence MRULTIPLICITE I am Interested in mathematics only as a creative art. - GHHardy, A Mathematician's Apology In this first chapter, we will look at the possibility of closer mathematics and music in a relevant way to the composer's work and the analyst. This is a discussion that intentionally remains theoretical in its most much: we want to care here the reasons and conditions of this reconciliation, issues that may arise as the formal side than on the side aesthetics. Rather than proposing a particular mathematical theory of music, Here we establish the kind of questions that must be asked a musician, and particularly a composer, when he wants to use mathematical or simply formalize some of his work. We will look in more detail on the types of representation a mathematical idea in music, the sense that such a representation can take, and how it can delineate the relative positions of two disciplines in relation (local or global) it establishes. We aim to show there are relevant ways to musically represent a mathematical idea (By showing that it is possible to represent such an idea, which can obtain a Page 18 18 musical relevance, and having such constructions), and that these representations are manifold. In the following chapters, we will see how these ideas can be confronted with issues raised by others, particularly with concerns the poetic and musical formalization. 1. Rapprochement between mathematics and music We believe it important to start working with a study opportunities represent a mathematical idea in music, and the possible relevance of such representation. The question may arise whether the same reconciliation between these disciplines, before any transition between one and the other, is realizable, and price of each side. In Indeed, if we think of the musical representation of a mathematical idea, we are obliged to take music and mathematics within the same framework of thought, therefore, in particular, bring us an epistemology as it "Versatile", which can be used to talk about tripling musical, mathematical and the interaction between them precisely. It is this "vocabulary" that we want to show, just by its use 21Through this chapter. 21We do not claim to exhaustive definition of this vocabulary, which precede and justify its employment. Instead, it is talking simultaneously of these three fields that caution should be taken and details will be like so many refinements of a purely epistemology musical. Page 19 19 . Hardware abstraction and constraints Before bringing mathematics of music specifically, we can think about the strangeness that there may be wanting to reconcile some activity whatsoever which has any connection with physical concerns. Indeed, mathematics is the abstract discipline par excellence, which has no connection necessary with the sensible world (they can use it, but get rid of them quickly 22). In a sense, we can not do otherwise than mathematical (Or for) the mathematics itself. Nevertheless, all our science, and very especially physics, using mathematics - it's hard for us to designing the formalization of a scientific thought without using abstraction mathematics and pure logic. Naturally, this abstraction itself has never enough to do science: scientific act an experimental component has its root. But to address scientific thought which seeks general from contingencies particular, the use of the abstract makes sense. A tool for thinking this exceeding, and groups conceptually in itself, various content should well be a the most accurate tool and purely formal as possible; Historically, the choice fell on Mathematics 23. 22Striking examples occur in geometry. See in this regard, for example, G G.-G. [1999] RANGER Thought of space . 23Although we seem natural now, this choice was not the only one available, and was made for philosophical reasons as much as practical. V. it, B EA [1952] The Metaphysical URTT Foundations of Modern Science ; particularly Chapter VI. Page 20 20 This view of mathematics as a simple tool of thought reminds us although there can be no justification or recovery of any theory his only mathematical formalization. These judgments can only be issued about content of a theory ( applicable to his farm, somehow "practice"), or the only content that mathematics can directly address are contained formal 24. Of course, the way in which it exposes something involved in that one exposed but discuss the form of a speech amounts to only discuss the internal correction (which still retains paramount). Reasoning without any logical error may well lead to the absurd if he leaves premises equally absurd or simply erroneous (the ex falso sequitur quod libet of logicians: false, we deduce that you like). What a formalization or a glance Mathematics can actually bring to an already established discourse is mainly clarity and consistency in delivering the internal contradictions; refinement, greater accuracy vocabulary may also result. When a speech is built around a formalization, we can hope that it will make single and efficient handling of speech itself - and thus indirectly the manipulation of what this discourse refers. In no case, therefore, relevance Additional is to be expected from the use of mathematics as a tool or framework formalization. Our attention and focus on what can bring to this job work and reflection of the musician on a few different ways to communicate 24V. G RANGER G.-G. [1994] Forms of Operations Objects . We shall return to the subject of Page 21 21 mathematics and music, and finally some elements of the musical discourse or the concretely formalized music. 1.1 Reasons for a rapprochement between the two disciplines What then come to mathematics in music, specifically? If we maintain that appeal to mathematics really makes sense when is to articulate an abstract thought, the question becomes what can be a musical thought, from which it takes place and how. . Musical thinking, musical discourse We will assume, for simplicity, that there musical thought in any act of composition or interpretation of one or more works (musical analysis and listening being forms of interpretation) and also any analysis of these acts (this is the case musicology and theory of the composition). There may not only a musical thought involved in these activities, and there may be musical thought also elsewhere, but we limit ourselves to the work of these three situations. Us still consider that this musical thought is manifested through a speech 25 : A musical work, a partition, an execution, a text. We will call speech musical one that is in some way to an internal part (fixtures and formal contents in music. 25We understand that speech is still registered in time - always in a logical time sometimes also in a timing set time. This registration is essential and we return throughout this work. Page 22 22 joints of its material, its formal content and possibly aesthetics), and which is given to understand by reading or listening to this particular room. In contrast, discourse on music will be the one whose object is the activity of the musician, the speech musical or the same music; it takes place outside the musical. Thus, the shape, harmony, articulation stamps, the course of a play, participate in a musical discourse; the analysis of a work, a treaty of harmony, this text it, are discourse on music. So we want to study what can do mathematics or these discourses. to. discourse on music Of first responses are already appearing throughout the history of discourse on Music: Rameau spoke of music as "psychomathématique science" 26; Simha Arom algebraically analyzes the rhythm of some central music 27. In fact, several formalizations roughly mathematics music speech (Speech and music) accompany the European music history. Can be used particularly mathematics to formalize a theory of music to studying music whose sources and methods are entirely unknown to us or partially, but to articulate the relationship between music and other forms expression, or between music and science. In fact, formally expressing Note that we do not want to lend to the term "discourse" only the content that has rhetoric. In particular, a poem, a story we are also speeches. 26RAMEAU J.-Ph. [1737] Harmonic Generation Page 23 23 relationships keeps the music (or work) with other activities or other speech, it is possible to discover new aspects of these relationships. This is one of the most fertile areas of formalization: the ability to manipulate symbolic elements, establish them abstract relationships that can suggest concrete; this case is even greater when it is a modeling - which can be seen as a formalization equipped with association rules or additional internal operation. b. musical discourse Meanwhile, the musical discourse can also use this reconciliation mathematics: stochastic music and, more generally, the composition Computer Assisted show well. For clarity and richness in the articulation of musical ideas together, to bring out these reports new, or to control more precisely known reports can be called mathematics to. There may therefore be an aesthetic contribution thereof to the track: formalization may suggest new musical relationships and new musical objects (whose relevance remains to be established by use in works consistent). In reality, this could also be said of a rapprochement with any other discipline: new ideas can arise, influenced by a frame particular thought. In any case, whatever one approaches the music may be subjected to all sorts of purely musical order restrictions (outside, so at this 27TO , S. [1985], polyphony and polyrhythms of Central Africa. Structure and methodology . ROM Page 24 24 that closer). This is not necessarily a loss of interest: limitations formalization may just contain a good portion of his wealth. This is both through clarity that is obtained and the limitations that we impose, when adopting a system of rules, such a system is "useful", it allows or demand for new twists to known situations. . Take mathematics as such In addition to formalizing the strict sense, mathematics can offer more just a thought pattern or development of reasoning, a set ideas that revolve and are built in a special way. The different ways to conduct a mathematical discourse can be the basis of ways to conduct a Music speech or music. It is no more then use a tool mathematical thinking in a musical, but to form this thought in the image of the mathematical thinking. Note that, although the result of the production mathematician, the corpus of mathematics itself, can be seen as a set of deterministic processes and purely analytic propositions, the activity mathematician, production of a formal language and its analytical results, is not not deterministic. Pure mathematics research is often guided by choices and tastes rather foreign to its results: "elegance" demonstrations, "Beauty" results ... Without these influences, development would be impossible: yet it was a question of getting as a result of a mathematician activity set of propositions derived from a set of premises, it would always Page 25 25 set this starter set for the deduction. Find starting points, interesting and fertile sets of assumptions and formal ways that exploit at best this fertility, these are typical activities that mathematicians can not be reduced to a deterministic process. However, in the composition, analysis and musicology, we find quite similar situations: an argument that may be more stringent, is followed; "conclusions" are drawn (which can be in the case of the composition, unless of course the "consequences" of their premises); a whole set of rules and patterns are involved. But the choice of these patterns, how to monitor or modify their starting points and support, none of this is given unequivocally, whatever the level of formalization used. Which allows a mathematician to define these joints of his speech can help the set to music; how to create, monitor, and modify the rules can be suggested from one discipline to another. We conduct in this case a reconciliation rather synthetic operations on both sides, before the "Start" a process or series of shares - and this process may, eg in the composition does not itself be directly formalized. This is the contrary to what happens for a formalization, where reconciliation takes place exactly and in the process (where all kinds of analytical operations take place), without the how to make choices that lead to the formalization comes in. A simple example will better differentiate between operations within a formal process and those that precede them. In first-order logic, Page 26 26 one can not deduce anything of the proposal (p ƒ q) nor the proposal (q ƒ r). If I want to infer something, I choose to articulate the (formally) with conjunction: ((p ƒ q) † (Qƒ r)), and I can only assume ƒ (p r). The form of the combination was provided in the formal language, but not the choice of placing this conjunction between these specific proposals. In other words, if I want to get (p formal deduction, I selected as premises (p ƒ r) through a ƒ q) and (q ƒ r), and their combination. There deduction itself is internal to the formal process, it follows written rules before her, and in fact define it. But the choice of premises is outside this formalization 28He moved from the entire set of rules that constitute it: it is this overall vision (thus "outside") that makes it possible. Ultimately, it is a desire to give shape to my speech (here within a formal language). This statement may seem trivial, but we take it in a specific context, in including the form of discourse as manifested in a "middle": a language (Formal or informal), a system, a set of rules in general. This medium defines opportunities form, but it does not create themselves 29: Just knowing that there a set of rules to follow or use not serving the act to follow or use, or give the "details" of the act. Thus, a sense of not formalization, and 28This did not prevent him being formalized in another frame logically further. We believe not only to formal logic which includes the quantifiers and operators need / opportunity, but also to the theory of demonstrations and geometric logic for studying symbolically these decisions and their "form" logic (see eg R G. [2005] ESTALL Proof & Counterexample andOLDBLATT G R. [1984], Topoi, The categorial analysis of logic ). 29We return to this relationship between form and environment in the chapter dedicated to the geometric look that we can focus on the music, and the compositional processes in particular. Page 27 27 alone utility, if it is thought in a complex network of knowledge (and thus opportunities for choice) and if the determinism inherent in analytical operations that make up interacts with the network. When we talk about formalization, then a whole series of decisions and processes external to strict formalization in itself is necessarily implied, as to the definition of a formal language for his job. . Mathematics as "Truth"? In terms of aesthetic contributions that this rapprochement with math can offer, there are sometimes some research of "absolute" music doubts that mathematics could be seen as a paradigm abstraction, independence of any contingency. An abstract formalization it would also create an abstract music, freed from the constraints of 30, Came without physical world (and so the influence of the time)? For some music, the organization according to a formal scheme is at the center of the composition, so that "resistance" to physical contingencies is reached: we believe including Baroque counterpoint, which leaves almost always include both whatever instruments (defined heights) that act. But this example is misleading: it can lead us to believe that all music where the stamp is simply carrier information (not structural information itself) also resist well in the transcripts. Of course, when the stamp carries, transcripts are Page 28 28 possible where the original song is recognizable, but it would be too reducer to believe that in any transcript all the music is preserved. Indeed, historically transcripts were always interpretations specific, so the action on starting work. Even those where appearance purely formal is central (the most famous example is probably the Art of Fugue , Bach), the choice of stamps eventually awarded this abstract information, which the will put in a condition to listen, is never trivial. In addition, beyond this structure logic remains all the aesthetic content 31, The interpretation and implementation must be at the heart of the work of the interpreter. This prolongs the Art of Fugue is therefore its relative independence of the stamp, but must be the same as that which prolongs the orchestration of the Symphonie Fantastique by Berlioz, or use of cello pressure , Lachenmann. Although a formalization can overcome a room of some physical constraints, it allowed at least under the influence of interpretation (especially that of listening). Another way to use this is the absolute abstraction mathematics, yet in order to obtain a degree of continuity of the work, would be construct that expresses this work around immutable principles such as those of mathematics. Such a subject would it not better chance of being always significant if he could find, as some feelings, through the centuries and cultures? It has been hoped, but this condition is largely insufficient to ensure the quality 30V. FICHET L. [1996] Scientific Theories music, XIX e and XX e centuries , J. Vrin, Paris. Page 29 29 of a room, the same way that a piece that represents Nature or Love is not justified by its subject alone 32. Indeed, all these "demonstrations" to music or of a work does take into account precisely to remain completely formalized, none of the external aspects formalization in question the decisions taken by the composer, external to the subject he treats, are nevertheless a crucial point in the development in music from it. Similarly, in the theoretical field, decisions prior to formalization are at the center of the system or model that follow: can be modeled from set theory or from polynomials, regardless of what will be modeled, but the theoretical results will be different according to this choice. The boundaries of a theoretical corpus chosen as part of a formalization already indicate that the author wants to promote or eliminate its considerations. Again, mathematics can not justify anything by their mere presence in a musical thought. Our attention must therefore be as much about what closer, and what, on how it is done. 1.2 Ways to bridge the gap Here we want to discuss some ways to bring mathematics and music, in order to further define better what shall we have the representation music of a mathematical idea. These ways sometimes overlap, and we aim 31Including at least as the set of assessments that we can do the work. 32And yet that was the case, it remains to prove that "Nature" and "Love" remains through centuries and cultures each one and the same. Page 30 30 not to be an exhaustive list of reports that can be established between the two disciplines. . Formalization, model, modeling It should be clarified before any of the terms we've used those formalization, model, and modeling. We say that there formalization of discourse music or music when there is the possibility of manipulation (sometimes indirect) objects of this discourse through a formal language or simply a set abstract symbols (that is to say, originally external to the speech itself) with syntactic rules. Thus, a partition is a formalization of the musical discourse; there pitch-class set theory 33 is a formalization of discourse about music (and can be used to formalize the musical discourse). A model is a formalization special: it is a set of proposals (usually abstract), and relationships between the proposals, which symbolically express certain aspects of an object or a phenomenon, so that if there is a relationship between two proposals then a relationship between the aspects of the object expressed by these proposals, and that relationship expresses thereof 34. A modeling is to build a model from an object (Or set of objects) particular and precise theoretical framework. 33V. FORTE , A. [1977] The Structure of Atonal Music , Yale University Press, New Haven. 34We take the word as well as its most common usage, and not as the model theory takes in mathematics, for which a model is an object (or set of objects) in which we find the properties of a theory . Page 31 31 The main difference between a model and any formalization lies in the fact that the model has an internal organization that goes beyond mere syntax: it has a "functioning" Autonomous (logically necessary links between some of its parts, sequences to the image of a causal ...), which expresses the operation of the modeled object. In modeling, the model is 'like' the modeled object (or vice versa), while a formalization in general does not seek to establish such a link. The organizations that we can find in the manipulation of symbols of a simple formalization (such as those we do with the elements of a partition) are superimposed to the syntax of these symbols, but are somehow independent: a formalization can be used in several logic, while in the very definition of a model is already a way specific chaining proposals to make "deductions". So our rating is a formalization that can be used both to manipulate elements of a speech Baroque music that stochastic elements of music; but only the latter models part of his musical discourse by the mechanical gas 35 - Certain behaviors musical are "like" gas behavior. Similarly, we find in G. Mazzola a model in music theory from the geometric logic and algebraic topology 36: Passing a composition process to a partition is 35V. XENAKIS I. [1963] Formalized Music , Stock Music, Paris. 36MR G. [2002] The Topos of Music , Birkhäuser, Basel. AZZOLA Page 32 32 "Like" the passage of a logic diagram for a Heyting (or proposals in such logic) 37. Note that a formalization and modeling are not necessarily mathematics and that, in any case, the idea of ​expressing certain symbolically aspects of an object involves not express many other any structure thus allowing formal abstract manipulation is necessarily a simplification of what is manipulated; such a construction is thus defined by the choice of what will or not expressed. . The use of mathematics We speak here of use or without use of mathematics propose to do again, not necessarily serve as a direct way of tools formal or do calculations 38. Recall that our focus is on mathematics not only for the formalization, but mainly by the central place in this discipline the idea of precision . If we turn now to what in mathematics, will be used in that reconciliation with music, both situations define the choice of "material formal ". On the one hand, the musician may be interested in a particular outcome of mathematics, such a formula or theorem, and make it applied to all 37 Also M G. [2005] The possible role of musical logic to some AZZOLA mathematical intellectuality (Conference of 16 April 2005 at the Music Seminar in Mathematics ENS). 38They will appear later in the illustrations by musical examples. Page 33 33 or part of his work, his interpretation or his theory. On the other side, somehow the opposite of this interest in a single result in mathematics, it may consider comparing what he wants to formalize a music theory mathematics, an articulated set of formulas and expressions, all carriers their demonstration, and make a parallel between these sets. to. application If our focus is on an isolated result in mathematics, we we can use it as a tool in the construction of a work or a interpretation. We will call this one-time use of a mathematical idea a implementation of this idea. The idea is to go directly to the selected result to the music, by a kind of "translation" to-one with the result. This approach assumes that mathematical consistency will be forwarded to the musical discourse or music by this method, and this principle implies a commutative or equivalence relations expressed in mathematical result and those can exist so musically relevant between musical or theoretical objects in question. An example of mathematical propositions application to speech 39, Where two Music is provided by Xenakis for the construction of his play Herma equivalent expressions of set theory are used to organize between these sets of heights. Within these assemblies, another application 39XENAKIS I. [1963] Formal Music Page 34 34 organizes the heights, according to the theory of probability this time 40. In the field music theory, applications are found in many treaties, particularly towards the end XIX e century41. Among them include the classification of the agreements and the "construction" of tonal harmony made by C. Durutte from polynomials 42. It is obviously not talking about equating the quality and the results of these two applications: only how to pass from mathematics to music is similar, the decisions and actions around this passage differ significantly. A well made application establishes a correspondence between objects of discourse music or music and "objects" of the mathematical theory used as a semantics (at least partial) of this theory, but semantics is not limited to simple objects, as it would be in a formal language of the first logic order imposes between objects of discourse coming from the theory relations. The speech to which the mathematical result is applied "work" as this, regardless of the characteristics he might have had without the application; otherwise said in a strict application any judgment about the musical discourse or the music is really about the theory used. This is practically a modeling but built in reverse, mathematics to music. It is quickly account the issues raised by this approach: an application poses 40The fact that Xenakis organizes the heights stochastically "by hand" in these sets does not take the fact whether it is a good implementation of probability theory: the crucial point remains the impact of this theory about writing and composer already knows enough applications formally strict order to imitate them. 41For an overview of this topic, see the book F music ... ICHET L. [1996] The Scientific Theories Page 35 35 musician constantly question whether these "inherited" relationships of the theory are musically valid or relevant, and what should be there for them to change the become. One who makes an application must always (active) saute its thought of the musical or musicological side. Deviations Xenakis over its formal schemes, although often cited as inconsistency argument music, are instead clearly the construction of a musical validity parabove any purely mathematical or logical validity 43. When Durutte emphasizes rigor and musical relevance of the results obtained only by his polynomials, it moves away from a theory compatible with musical aesthetics while it claimed to formalize. b. parallel The player can also focus on a whole set of articulated mathematical propositions, on an entire theory. In this case, it is no more translate word for word phrase, but rather to implement a structure or general form of a speech to another, and we call this use of Mathematical a parallel between the selected theory and music. Attention is both on the sequences and relationships of concepts and objects on these objects themselves. Again, we imply the possibility of a transfer of coherence 42DURUTTE C. [1876] Summary of elementary Technie Harmonic , quoted byICHET F L. [1996]. 43 Especially when Xenakis informally recreates a stochastic arrangement of notes within sets of heights Herma , it overcomes the problems of a strict application all keeping an accurate overall sound result, genre originally obtained by formalizing well Page 36 36 between two forms of thought, but here it goes through more or less recoveries of one of the major forms of discourse by the other, and not by a set direct equivalencies. F. Nicolas establish, in its theoretical discourse, two parallel: 44And between integration and hearing 45. between number and work The main difference between this approach and a parallel application a result is that, for the application, it is closer to the music fragment isolated mathematical thinking, and it must be done from time to time in some aspect musical object at a time; for parallel, we can not realize this occasionally reconciliation, all chosen mathematical theory is related with all Music speech or music that you want to formalize, and nothing can be isolated to one side or the other. The application is independent of the relationships that can have its terms (that is applied and that which is applied) with their "Environments" of origin; in a parallel, the context is kept intact both sides. Without defining a direct interpretation in music of mathematical objects, parallel wins the certainty of not imposing relationships that could be non-musically valid; formalization is music to abstraction Mathematics: we formalize relationships and properties that already exist. This However abstraction leads to losing precision has application in the established link between mathematics and musical entities: they are more manageable accurate (v. E XNAKIS [1963]). This is precisely the musical representation of an idea, as we define it further. e ICMPC, 44NICOLAS F. [1994] Many notes and musical work , Proceedings of the 3 Liege. Page 37 37 "Directly" (with symbolic paraphernalia itself), and the parallel often fails as an operating tool for composing 46Do actually settling between theories. From a more pragmatic point of view directly, we could say that an application involves metonymic processes (substitution) while a parallel rather a metaphorical process (comparison). c. intermediate situations Application and are parallel, we said, extreme choice of "material formal "if they are strict: a single element or the whole of a theory. The mathematical formalization will be found most often between these two poles, employing multiple results simultaneously, more or less directly, without necessarily jeopardize all theoretical relationships they could maintain (With each other or with the rest of the mathematics). Several different applications articulated (ie controlled) based on musical criteria as much as mathematics may be, for example, a formalization of broader scope than the simple time application of a formula, but still keeps an operational nature practice. Again, the choice of the musician before the formalization itself are crucial: decisions on the intended use, the context in which such use may 45NICOLAS F. [1997] The third hearing is the right (from the musical audition designed as an integral gration) , Musicæ Scientiæ, No. 2. 46This does not detract from its usefulness in general, among others to think directly operating tools the composition themselves, or to operate within a scan. Symmetrically, we seems that the application, as we have defined it, is more immediately usable in practice Compositional in theorizing. Page 38 38 or will be, the generality of the tool, and so on, will place this rapprochement between mathematics and music instead of the side of a parallel or an application. 1.3 What is formalized Now observe what in or around a musical discourse or the music can be approached using patterns or formalizations more or less strict, not necessarily mathematics. We subsequently seek the inside these formalizations can be linked to mathematics. . Notation For a musical discourse, the notation (when it exists) is perhaps the appearance most obviously formalized. Indeed, our traditional notation arises (or at least stabilizes) as a reduction of its instrumental few parameters, including control is relatively simple and that bring the sound instrumental in concerning its manipulation by whoever note of his vocal. This instrumental notation inherits the vocal scoring many of its features; in particular, it favors certain aspects of the instruments sound that easily found in the voice sung as a defined height or duration "singable" (neither too long nor too short) at the expense of those who may be more difficult to identify them, such as attack transients or gestures of the instrumentalist. This kind of inclination Us already shows what can be more easily controlled by any set of rules from this rating or moving "around" her; it may also indicate the Page 39 39 need to expand and supplement, and in which direction 47. These limitations, which are in fact the concentration of this notation around the syntactic object that is the "note" however, are not only harmful to the effectiveness of this tool: simplifying and discretization of sound, in this procedure goal (to be able to handle this symbolically a) allow precisely to handle various structures more "big" as the note itself same (reasons, melodies, variations, counterpoint ...). More recently, the discretization Digital sound Mane ratings for managing more structures "Small" as the note. The notation whatsoever, performs a cut on the continuous sound to express in its syntax, and cutting, which is never neutral, allows logical articulations of the sound continuously. In this sense, it is almost a formal language 48: There is a finite amount of primitive symbols (range, keys, notes, bars ...) and syntax, sufficiently precise to be implemented in music notation software, accompanied by a kind of semantics: the symbols written signify instrumental gestures with enough precision, but simultaneously enough flexibility for several instrumentalists can "read" the same partition understandably, and assimilated as the same piece of music 49. Of course, the similarity ends there: partition and formal language have 47This need has accompanied the rating since its inception: appear according to the musical needs dynamic controllers, joint or attack symbols; the ornamental details evolve or disappear. MR ONTEVERDI In Il combattimento di Tancredi e Clorinda [1624] indicated with "that if lascia arco e strappano if the con rope due ditta" which later would note is simply pizz. 48As, for example, those of F REGE G. [1879] Begriffsschrift , or R USSEL , B. & WHITEHEAD , AN [1910-13], Principia Mathematica . 49We identify here, and it is a voluntary simplification, the "result" of the partition of gestures instrumentalists (more on this). Note that this view also allows us to consider a CD as a kind of partition must also be "interpreted" by a player. Page 40 40 very distant targets, which appears precisely in the fact that score indicates something much larger and more blur (instrumental gesture) than can indicate the proposals of the first order logic (a truth value). In addition, while the equivalence of sequences of different symbols are at the heart of Use "Fertile" of a formal language (they allow deductions), they are rarely useful around do musical notation, precisely because they do not generate Content (logical equivalences here being the same species as a substitute white by two black linked, for example, or by a programmed object patch which reproduces its behavior indentique). There is thus, in the notation Similarly, no substitution or deduction rule that would extend the syntax without preserve the 'semantic' powers (the modus ponens the first logic preserves order): you can not write the same thing in two ways truly different. 50 What is lacking in musical notation is precisely what we do would not find in a formal language (or at least in the first logic order). To overcome this situation, it is always possible to add to the rules strictly syntactic (as it internal) of musical notation other rules operating on its contents, which allow to pass from a series of symbols to a new sequel. It therefore goes beyond the notation as simple representation of a sound "Simplified" to make generating new materials, and it becomes possible to 50This indicates a special bond between syntagmatic and paradigmatic aspects of Page 41 41 building parts only fragments from the notation: increases decreases reversals, transpositions and demotion of a melodic line or a waveform; resumption of a rhythmic pattern on a new set of notes, or a melodic motif on a new rate; simplification and enrichment of a fragment by removal or addition of passing notes, or granulation is ... also possible to "refine" the syntax notation by imposing restrictions Additional to the sequence of symbols of the rules of counterpoint and harmony can thus be seen (and documented) as constraints that remain primarily syntactic (which may explain why he did not just follow the to produce consistent music). It is clear that these rules are an order logic and higher than the rating complexity as simple transcription; it is in fact better for the composition decorated when supplemented with both ways to create the new material and restrictions on this creation. There is Interestingly, these additions to the basic syntax notation are not necessary for interpreting (and hence to the analysis): it does not act on the partition, but from her. An interpretation that is built outside the notation, in its interior it can only find relationships between suites symbols and their use 51. musical discourse. We return to this subject. 51We shall return to this opposition between constructed and found in a work. Page 42 42 . With the notation: writing and proofreading Note before going further, that any rating already implies a system of rules, and can therefore be considered as a (first) formalization. If we invert this finding, we can think that the essence of a formalization, whatever it is, is to be a kind of notation , how to implement a a special thought to what is less understandable or otherwise manipulated (or the least is another way). Indeed, we believe that the main motivation behind formalizing the idea is to manipulate indirectly symbolically, which is formalized. The syntax and rules for handling these symbols do not establish not of themselves (they are not given by the symbols in itself), but depend some precise formal thought. They are in fact a kind of definition or of realization of this thought: we could say that they are allowing that thought to occur when we use 52. When we manipulate objects indirectly through a formalization, so we follow a logic that is not directly theirs, we encadrons according to another thought, which is external to them 53. In particular we believe the changes undergone by the idea that sound can have a composer where it should be noted; this difference in thinking between different 52W cf. ITTGENSTEIN L. [1935] The Blue and Brown Books "thinking is essentially the activity is to operate with signs. (...) Moreover, if we speak of the place where thought takes place, we have reason to say that this place is the paper we write "( Blue Book , [6] - [7]). The idea a thought present in his own writing also reflected in the Remarks on the Foundations Mathematics (W L. [1939]). Note, however, that syntax and rules ITTGENSTEIN Handling is not strictly speaking a write , even if they are indispensable. 53This does not mean that necessarily driven objects have their own logic, intrinsic or they have one that is inaccessible directly; but a formalization is not neutral, and implies Page 43 43 notations involved probably also noted the difficulty of electroacoustic music in a traditional partition. 54 Each formalization leads to manipulation Symbolic different, thus different operations on objects . to. writings and logical time This view leads to a central place in the research we carry the idea of writing , understood precisely as the creative use and criticism of any symbolic system. The deployment of a thought by notation defines a directional sequence of symbols: it establishes an order, or more interdependent orders. We can say that writing is thus part of a logical time , in the sense that we can follow it in its development. This is already written, sometimes we can follow the sequence linearly, but the usually the present order in the writing itself is rather paradigmatic that phrase 55 He does not only depend on the sequences (so to speak "Physical") symbols, but also, to a large extent, the relationships can have between them distantly rated symbols of one another 56. This time designed by a rating is then not rigid, it allows jumps, shortcuts, returns. In how to handle one (and its consequences) that other approaches (and even other formalizations) does not necessarily imply. 54Naturally, the most trivial limitations of the score, as the relative imprecision and stamp temporal articulation, are the source of this impasse, but we would say that the partition offers type of joint (not just scale) of often incompatible with sound joints, same macro-temporal, electroacoustic music. 55 These terms are often present in the vocabulary of the language, cf. for example SAUSSURE F. [1916], Course in General Linguistics . But we want to keep them here without any implications of this discipline, in the sense suggested by their etymology. Thus, we understand phrase as it relates to the sequence and combination possibilities and paradigmatic as it relates to the comparison and substitution possibilities. Page 44 44 Indeed, we can follow a logical sequence between the first theme and development of a classical sonata without dwelling on the second theme or transitions naturally unavoidable when listening; In the same way, when the composer writes a passage he can use items that have been used, or which will become a wholly material later in the work, and that look in future does not take place (in the same way, anyway) when listening. This double organization, both on purely logical or abstract relations (paradigmatic) that according to the same note (phrase), gives a special topology logical time. It is thus necessarily separable (and usually separated) by a time chronometer , and this is crucial: the logical time is reversible and foldable to will 57, Writing can be superimposed on itself, be influenced by its own results realized at multiple scales (time, complexity, abstraction ...) very different simultaneously. The idea of writing makes possible thereby that of proofreading , which is also essential, in our point of view, for the possibility of composition or depth analysis. Indeed, we want to understand a compositional process as a series of actions on and within a network knowledge, choice and operations, led by different states and reactions of this same network 58. In other words, to write consistently requires constant able to read what is already written; the active memory of previous states of the network 56This is even more strongly the case when it comes to what is being written. 57We return to the folds of time and logical spaces in the chapter on geometric. Page 45 45 involved in the determination of the following states. Note that this memory is here the crucial element: we can actually consider a purely mental work organization and generation of the musical material, such as that which occurs in a improvisation as a kind of writing - then the logical time may have its reduced topological complexity (for limitations on the amount of information manageable simultaneously), it is nevertheless fundamentally different from chronometric time by its malleability. 59 However, we will keep throughout this Work the idea of ​writing associated with that of a rating, as the Deputy hear above. b. analysis and proofreading Similarly, we can include analysis as a network (Internally organized) readings and re-readings of a work at various scales; or more just as the organization of this network. If an analysis must be more than simple monitoring events of a work, it is indeed that it allows the setting relationship possibly distant objects, it offers new content from these relationships and relationships between internal and external factors at work. In some so, faced with a series of objects, the analysis is to build and give meaning 58Just likelevels that proposed "complex involving aRemarks pluralityabout of Music Composition operating ", in V Vaggione H. [2001]systems Some Ontological AGGIONE Processes . 59We leave here the principle that improviser has a musical vocabulary internal (not necessarily shareable writing or speech) that allows it to refer indirectly to musical elements, that is to say not only by recalling a sentence per se, for example, but by associating it to other content. Limitations on the complexity we can handle mentally tell us again, if Page 46 46 a logical network of links 60 between these objects, and between them and a context likely to exceed the work; it proposes to raise or build 61. Naturally, the paradigmatic syntagmatic joints from joints more often a part of this network before the analysis, exactly as a context (historical, stylistic, sociological, aesthetic ...) who oversees and directs its constructions. Several types of relationships between objects (musical or symbolic) are already defined before the analysis begins, so we know what style we are dealing, sometimes simply by the choice of a method (as is the case with the analysis Schenker, which is an analytical method provided with a syntax 62). But it is also possible to seek the highest possible abstraction of these contexts prerequisites, as in contrastive analysis 63. In all cases, this network knowledge, choice and operations that the analyst defines or redefines and through which he offers his interpretation depends on the possibility of re-reading of the work: no her, he does not have the ability to recall things that are elsewhere in time logical that where there bears his immediate attention, and thus act on the linear order and were needed, that a rating is not just a way to preserve ideas, but both the work (both in music and elsewhere). 60Here we describe these links logical just to differentiate them possible links materials (for proximity or congruity) there is not necessarily the possibility of formal sequence consisting therebetween. 61This is the case even for the paradigmatic analysis proposed by R N. [1972], language, music, UWET , JJ [1987] General Musicology and semiotics . Indeed, Nattiez itself poetry and N ATTIEZ stresses that the analysis of the problem (paradigmatic or similar) is primarily that of segmentation a statement on its syntagmatic axis (lectures at ENS Paris, 1992, cited by V OisinF. [2003] The contrastive musical analysis ). 62See SCHENKER H. [1935] Der freie Satz ( The free writing , trans. N.Meeùs, ed. O.Jonas) Mardaga, Liège, 1993. 63See VOisinF. [2003] The contrastive musical analysis . Page 47 47 unique juxtapositions of objects to extract a sequence of ideas comparisons and functions. The idea of ​re-reading, set out in this way may seem trivial but actually immediate impact on formal tools of composition and analysis: Adaptive search algorithms, constraints by generating material (on the existing material), analysis of melodic or harmonic structures (for a room), depend to resume as data "input" results. Except computer music, similar situations arise: work out the details a room and their participation in global form, to connect or highlight roles of the same material at two different locations of a workpiece, are actions that require more than a single reading of the musical content. Thus, our focus throughout this work will focus on the music when written (either time of its composition, or during a scan, as transcription), and the issues of formalization of this writing (and around) 64. Of course, writing is in a different logical time time course of the music not only because the logical time can fold, but also because of the time reflection caused by writing (and rereading) of anything 65 - It is usually in real time. 64 In these cases, writing takes for the music a comparable way to that it takes for the mathematics: it is in her that "exercised" thinking it is the logical trace (and the logic) of musical or mathematical world. It is here that we truly the realization of a thought, we were talking about before. 65This time of reflection is somehow chronometric time when the shares in and is part the logical time of the composition and analysis. Page 48 48 . Running time (time) Besides the musical notation, the course of the music in time may also be subject to rules formalized. Thus, a sonata, a rondo or fugue (especially as particular sequences) have fairly precise definitions. Inside even a "note" we can establish a system of standards for building microas sound (depending on attack, changing its spectral density, its granulation ...). In most meso- or macro-time event (such as exposure themes of a sonata, or sonata itself), these rules have a syntactic scope restricted, as the notation: towards "outside", they do not allow build new sequences from a given course; to the "inside", they do not reach on time scales or smaller retail a certain order. Thus, only from the construction rules of a roundel, it can proliferate other "forms"; similarly, in the recapitulation a sonata, the main topics to be included in the main tone, but no constraint is given as to the changes to the transitions themes or changes that will suffer the second theme to go to this tone main. When micro-temporal sequences are taken into account, we more frequently find examples of flow rules which act as a "note" to its interior, and define opportunities for others "Notes" from the first (we think including synthesis algorithms Page 49 49 granular 66 ; certain steps in stochastic music 67Although treating time scales that exceed the micro-time, as share these properties). to. distance of a formal language The formalization of the deployment of the music in the time away yet more than the rating of a formal language. Indeed, the syntax of these sequences is of a higher order than that which organizes a transcription symbols are joints are between sequences but also in their interior, but exhaust this "syntactic inside." We often know, within certain scales time (on the order of a phrase or pattern, for example), how to define different sequences of sequences without having completely defined the sequences themselves, or their opportunities for development: for example, it is possible formalize modulations (or modulating sequences) Terms that apply in different materials. But beyond this syntax is the problem of semantic these deployments. We know how to interpret a note on a staff with a gesture instrumental or vocal, the indication is clear enough; but denotes the form of a roundel, for example? The simple answer to this question is that such a form would denotes herself, she is content. However, this would place that Granger called the zero degree of formal content, where there is "an interpenetration 66As those of R OADS C. [2002] Microsound , MIT Press, Cambridge. 67Especially when Xenakis uses the physics of gas as a model. V. X Formal music . I. [1963] ENAKIS Page 50 50 Full content and forms "," complete adequacy of operating and the objectal "68; or ways to deploy the music (as a roundel, for example) provide us with precisely the information on the possibilities of evolution and articulation (temporal or understanding, perception) musical objects conceivable in the inner world of the room where the manifest ways - if I know I listen to a roundel, I know the chorus comes, even if it comes back changed. The Objects of this internal universe are, at least in part, "previously specified by diverse and multiple operations of thought (...) that what seems capital are longer forced to be exercised only on the sensitive area bounded by our perceptions " 69. So we did a nontrivial formal content of a form music: it acts on an object that is not defined within oneself, which exists beyond its mere manifestation (local) time 70. But these possibilities evolution of musical objects are precisely what defines the shape in question we here want to understand the shape as what is possible worlds, special opportunities to correlate certain operations. 71 Listening to it 68GRANGER G.-G., The concept of formal content and formal content and dualityRANGER , in GG.-G. [1994, p.46., p.62], forms, operations, objects . 69Idem, p.47. 70Here we take the risk to assign a formal content to the music, while Granger argues that it Only logic, natural language and mathematics that can arise a real formal content. We believe that there is a similarity between music and natural language in the constitution of their consistency (unfounded organicity and almost in formal, like the language of games Wittgenstein), firstly, and secondly a resemblance ideas form in logic, mathematics and music, which allow the emergence of formal content. Granger placing formal content somehow upstream synthetic judgments a priori of Kant, we would go up to say their existence in music is quite compatible with (or suggested by) the fact that we always know what music has no authority to define it. 71We will see in the chapter on the geometrization how it is equivalent to consider the form of a work like geometry or as a formal space. Page 51 51 are actual developments of the musical objects (those that actually take place) suggest this form in time. In other words, that I listen to a Rondeau informs me that the chorus comes, but it is only listening to this return that I know how he came back, so I know the shape of this particular rondo (Among others, it is only listening as I can see that it is indeed a rondo). The formal content of a musical form can not not exceed this content is inside the shape 72. You could say it is actually intensional , in the logical sense the term: it allows to define and understand this form. Listening by against, is shows that the expansion of the form, or a part thereof: a particular case of this which could have other achievements 73. This finding may lead to a discussion knowledge and recognition of musical forms: are only extensional defined with each listen; There he has as unitary extensions (this that would prevent the appointment of shapes); can we know a musical form intensionally / extensionally, and how; shape heard for the first time it can be a special case of a "form" more general? Us here will avoid these arguments, it seems, more directly to questions cognition that formalism or music. Note simply that or analysis 72But not to the objects themselves, manifest in the form of a special way. 73To simplify, we can say that the intension of a concept is all the necessary properties and sufficient for something to be part of this concept, or be designated by him; and its extension is the set of all that is indeed part of the concept or designated by him. Page 52 52 interpreting a form is based on its extension, while the composition of a form is working on its intension 74. Without this precise semantics, one does not strictly speaking a language forms - no vocabulary, syntax or rule of well-defined operation, even form of the generating elements (or not a discrete set of these). Maybe be paradoxically, it is precisely this that makes this area particularly conducive to formalization: each of these elements must be fully composed for a form is divided into several time scales, logical, and it is during this formalizing a composition, or simply a more or less flexible set of rules may occur. Note that this applies equally to the production of a work, which becomes a special case of application of these rules and where they exist and change, as for the analysis, when it comes to producing a formal model that the work analyzed contains (as a sub-structure among other possible simultaneously). The composer built the set of rules that will follow and also the possibilities modification of this set; the analyst sets its standards of investigation and determines and the possible sets of rules from which, according to him, is the one the work analyzed below. It is in both cases the composition of a process, consisting of Rules interactions with their users, which is the formalization of conduct music in time. 74Or with the intension, if it is a form already established. Page 53 53 b. validity of this juxtaposition with formal languages Here one can question the relevance of comparing and rating work on the form of a side and the other, formal languages; we have just seen in Indeed how they differ. If we insist on this parallel, it is because we consider any formalization as a kind of notation, so as writing, and the question of a relationship between writing (especially if it is formalized) and Language formal arises quite naturally. Moreover, we believe that through this approximation it is possible to begin a discussion about musical semantics. More Specifically, we can start looking for what may designate objects music 75And especially how. We subsequently develop this subject related with a discussion of "musical poetry"; for now, we can say more just one of the central issues in an investigation of semantics music is to define and differentiate what music shows and what it says 76; a Another equally important issue is the limits (as it internal ) a "Musical language". The merger with formal languages ​allow us to make more precise the notion of formal content 77 to address references of a possible musical meaning, internment and externally to a work. 75Broadly: sound objects in a local context, phrases, sections, up to a whole work. 76These two actions are mutually exclusive for Wittgenstein's Tractatus [1922]. 77GRANGER G.-G. [1994] Forms, operations, objects . Page 54 54 . Discourse on music Alongside these formalizations of musical discourse, several aspects of discourse on music can also be formalized. It is perhaps inevitable talk about music occasionally using terms that are external to him the reference music theory to other disciplines, and vice versa, is a constant Throughout the history of mankind. Thus, the concept of musical form in the period 78; links are forged between Classic feeds as many grammar Biology architecture, painting and music by Bauhaus 79; the sounds of instruments and voices have sizes and strengths comparable to objects of everyday life in Kamayura Xingu 80. But these extra-musical references to not always a formalization, because it is not always about indirectly manipulate objects or musical concepts. to. theories of music Direct association and operating between music and other disciplines that remains our most famous because founder of a particular vision of music (and the world) is certainly that of the Pythagorean school. We have here 78 PREDA -SCHIMEK H. [2003] A look at the genesis of the form of theories, between classicism and Romanticism (1790-1845) , Musurgia, X / 2. For the author, the formal hierarchy of classical form, replaces the Baroque vision of music as rhetoric , is supported by at least three models: the periodic , from dance music and poetic verse; the hierarchical model, from grammar and biology; and evolutionary model, borrowed probably also to biology. 79KANDINSKY W. [1926] Point and Line on Plan . Of course, the discussion focuses on Kandinsky painting, but the vocabulary is so divided, for him, the two disciplines that could almost be the opposite movement. A different link between music and architecture through the thought that one found in X I. [1971] Music: architecture . ENAKIS Page 55 55 an application of mathematical principles not only a musical practice (Training ranges, for example) but also an understanding of what can and be music. This is probably the first mathematical theory of music 81Where musical questions raise math problems (such as the division of the octave into equal parts leading to an irrational number) and conversely (as the ratio of number, measurement, and pitch of a note). This theory was consistent (free internal contradictions), but also complete, because it taking into account all the music they could find; that is, to our knowledge, the only one that was well, probably thanks to the "limits" (geographical and others) the music of the time and especially mathematics and then reports qu'entretenaient philosophy 82. e century alone, another theory making a new use of In the eighteenth mathematics (but also physics) becomes a reference for thought 83. One of the differences music: Sound vibrations Corps studied by Rameau 80M Cf.ENEZES BASTOS RJ [1978] A Musicologica Kamayura. Para uma da antropologia comunicação No. Alto Xingu , Funai, Brasília. 81But this is not the first musical application of mathematical principles: theory, purely arithmetic twelve liu dates from 2637 BC. AD, China. See K Elkel M. [1988], the Music worlds (cited by CHOUVEL J.-M. [1995], The Physics and Aesthetics. An epistemological analysis terms of knowledge of the harmonic phenomenon ). 82 Several mathematical theories of music have existed, but they are most often inconsistent or incomplete. It is possible that the one proposed by MAZZOLA G. [2002] The Topos of Music , comprehensive and consistent, although its complexity and orientation (as much as a mathematician musician) can make it difficult to dialogue with other theories (not mathematical) widely adopted. It seems, however, have the merit of not excluding any musical practice, formalize several existing musical theories and propose new math problems. 83RAMEAU J.-Ph. [1722] Treaty of Harmony reduced to its natural principles , and R J.-Ph. [1737] AMEAU Harmonic Generation (loc. cit.). Before him, Zarlino in the Istitutioni harmoniche [1558] employed the Page 56 56 has this approach from those that precede it is the use, albeit in a prism Mathematics, physics. But it is indeed a formalized approach: the relationships between physical entities dictate the relationship between musical entities (As the movement of the basic low). Virtually all formalization of music theory until XX e century, will refer to one of these two visions to symbolically manipulate music: the numerical ratios more or less simple, or one of the harmonics of a vibrating body. b. analyzes and styles Other formalizations instead focus on the analysis and understanding a style or a particular system. Thus, Schenker offers a new rating for the analysis of tonal works atonal music 84 ; Forte uses abstract algebra to study 85. Of course, any tool for the analysis can be taken up by the composers, but some have independently developed their own formalizations of the same composition. Thus we find in algebraic manipulations the serial school 86 ; an arithmetic proportions Stockhausen 87; theory fractions to build a natural range that "correcting" the Pythagorean; but Rameau the first exception to this research based exclusively on numerical proportions. 84SCHENKER H. [1935] The free writing . Incidentally, Schenker also seeks to justify its approach by natural harmonics. 85FORTE , A. [1977] The Structure of Atonal Music . 86BOULEZ P. [1963], Thinking music today , Gonthier, Geneva. Note that several of these manipulations were present at S , A. [1911] Harmonielehre ( Treaty of harmony ) and CHOENBERG some even, of course, in the Baroque counterpoint. However, they become in the Boulez constitution of the musical "The world of music today (...) was born from the expansion of the concept of the series ". The case is similar to the following examples: formalizations there are more a simple composition tool. Page 57 57 probabilities Xenakis 88. It is no longer here to build a formal framework for particular piece, but to formalize the possibility of construction of these frameworks, so that the interaction between the rules selected for the composition, or the analysis of the room, and user is itself subject to other rules, an "order" superior. It is possible with this approach a formal definition of which may be a style or school 89And the act of this definition is not completely separated, in the work of composer, the definition of its own syntax, its "language" individual, his "Accent" in a style or defining or selecting an agreement or its especially inside of a room. The structure of a composition process includes this additional layer of rules that apply to other rules: the process remains consisting of several simultaneous levels of decisions that interact 90. The titles of the "schools" of composition, which could learn from these formal frameworks the largest are given generally well after producing several works more or less independent, do not always think about their inclusion in a "Genealogy" of this kind; style definitions and systems are thus the most often extensional (they summarize post of properties). Of course, precise distinction of logical differences in rule sets, including 87See COTT , J. [1979], Conversations with Stockhausen , Lattes, Paris (especially pp. 211 ff, pp. 251 ff). The subject also appears in some of the Letters to Pierre Boulez ,TOCKHAUSEN S , K. [1953-1960] LivretAutumn Festival program in Paris in 1988, Contrechamps. 88XENAKIS I. [1963], formal music . 89A pretty good approach to these notions is at B , J. [2005] The musica iperAboni-SCHILLINGI sistémica . The author builds the concepts of form , style and system from a double standard necessity and causality applied to sound phenomena, generating sonèmes , morpheme and stylemes . 90Thereupon see VAGGIONE H. [2001] Some Ontological Remarks ... Page 58 58 composer does not necessarily have to care, may become more important when is to establish a hierarchy between the rules inside of a room and a set parts, as can the analysis of a style or construction of a theory. c. understanding and communication Further away from the actual composer or analytical activity, thought formal can be conducted to study the elements that allow understanding of a work, or ensure its intelligibility. Thus, Lerdahl and Jackendoff studying structure of our perception of tonal music through a system of rules comparable to that which can be made to formalize the understanding of a 91. Antunes also addresses the least meaningful elements language Music with tools from linguistics 92. The possibilities for communication through music or communicate a musical message can also be addressed in a formalized way, for example, in the Information Theory 93. Although these subjects often remain outside the scope of our study, the research are conducted focusing more on the structure and functioning of apprehension and understanding of music on the music itself, the results can have a significant impact on research musical performances of mathematical ideas. Indeed, when we seek any effective these representations, understood as an addition to listening opportunities or 91LErdahl F. & J ACKENDOFF , A. [1996] A Generative Theory of Tonal Music , MIT Press, Cambridge. 92TO NTunesJ. [2001b] The Semanteme , in Emotions & music, EDK, Paris. 93V. MOLES , A. [1972], Information Theory and Esthetic Perception , Denoël, Paris. Page 59 59 understanding, we can not overlook what is known of the interpretation process themselves. In particular, it will interest us when we discuss the way in formalization. 2. The possibility of musical performances The discussion above has two dichotomies (musical discourse and discourse on music; application and parallel), which we will use for approach and define the representation of a mathematical idea. We will look to us positioned at the intersection of these two oppositions to study the possibility of bringing locally 94 mathematical ideas, more or less isolated from their contexts purely scientific and musical ideas, articulating the internal and external aspects of work. One can search in the areas of scoring, shapes and theory music which can lead to performances, not just formal elements or formalized; we add to this the possibility of obtaining some "Musical poetics" 95. Of course, we will consider an application or parallel as we define those above, are a step in the direction of a representation - indeed, formalized using mathematics is already included 94That is to say, not only using the punctual, but not necessarily not so overall. 95We are in the third chapter on the subject of a musical poetics, its definition and sense that can make that expression in the context of our work. Here we can accommodate the notion vague enough of "meaning, expression, message" from the music. Page 60 60 in his work or its interpretation. Here we want to go further than these first approaches allowing us to consider multiple simultaneous views on mathematics (or the idea to represent) and especially on the composition process or of interpretation. Naturally, this approach forces us to constantly make decisions about what, in mathematics, will be used and what, in music, this will Content: we are always seeking correction and musical relevance, and can thus be led to move away from a strict mathematical consistency. Actions game in the côtoiement of these two disciplines here are primarily those of a musician 96 ; considering that there is a mathematical content that goes into the music, the result final content can only be a purely musical , in which mathematics may have been help. If we still use the term "representation" to talk about this use of mathematics thus can the "deformed", it is because there is here an indirect manipulation of ideas and mathematical objects as well as the music, even with these. In the same way we take position in both musical discourse and the discourse on music, we will use a speech on the mathematics as much as mathematical discourse itself. 96We consider the actions of a musician closely with their perception and understanding of the music; without these actions, the very idea of ​a musical relevance does not make sense. To resume Vaggione words, "things like thesis , constraints, choices, and so on, would not not musically relevant if they were devoid of implications relating directly to issues of action and perception, that is to say, revealing a commitment to action that depends on the perception as a supervisory body "(in V H. [2001] Some Ontological Remarks ... ). AGGIONE Page 61 61 . The effectiveness and efficiency of links When taking mathematics as more than a tool or a model reasoning, the question arises whether it is possible to link workforce between these disciplines, leading to an action musically noticeable (and relevant) both on internal and external components to a work - which is not necessarily in concerns an application or parallel. We will consider the effectiveness of a link (or even its effectiveness ) will be built on its intelligibility, and necessary here to distinguish between intelligibility musical side and another side mathematics: of it, a link is effective if it allows us to grasp more clearly mathematical concepts (or simply formal) that are present; of one, if an effect on the perception or understanding of musical content, it enriches listening. Of course, we seek rather the latter result. The question of the effectiveness could only relate to the ability of the Music to approach a topic like any other, but here it is not any theme. In Indeed, given all the formal aspects that may be present in thought 97. He music, the establishment of such effective and efficient link can not be trivial however, already existed, at least in the past, sometimes become almost "mythical" and 97We are thinking here of the theoretical establishment prior to this link, sort of living conditions history. The fertile use of mathematics is obviously more to prove, but must be always rebuilt, hence the importance of theoretical inquiry. This is discussed by several authors; include, among others, the texts ofICOLAS N F. [2000] [2005a], M AZZOLA G. [2005] and ANDREATTA , MR. [2005]. Page 62 62 Ancient Greece 98. For the Pythagorean school, the relationship between mathematics and Music is immediate: the being is many, the relationships between sounds can only be mathematical relationships. Later, with Aristoxenes, music theory is more related to sensation and physical, understood as empirical discipline. These currents seminal and contemporary relatives were taken in the Middle Ages (especially between XI e and XIIIe centuries, bringing Pyhtagore Aristoxenes of Plato and Aristotle), no only as philosophical foundations of music, but also all science. It was at this time that music and formed his theory as were known in Europe for four or five centuries to come 99. Our culture Musical therefore undeniably a mathematician heritage. However, other elements more or less formal (or formalized), and among them those we examine above, gradually are superimposed on this legacy initially simple. The musical discourse becomes more complex and rich, but obviously it This differs from a mathematical discourse. This did not stop the music of the twentieth e century whose speech would be if we believes the previous argument, as far removed from a really speech mathematics, to often call the mathematics declared way up 98The that follows certainly very thin, we did notofwant to Mathematics do more than, just flyover.historical We referreview in particular to S isZABO , A. [1977], Thebut Beginnings Greek Vrin Paris, and BURTT EA [1952], The Metaphysical Foundations ... also remember that C HOUVEL J.-M. [1995] The Physics and esthetic ... quote links in a past even more distant. 99It is interesting to note that this "opposition" between Pythagoras and may in some Aristoxenes so end up several times throughout the history of Western music. Thus, it can be seen in classical and pre-classical theorists who tend to follow Zarlino or Rameau, and to the choice of a physical model of sound by the spectral school as compositional paradigm away from the kind of Page 63 63 take up precisely as a direct source of enrichment and complexity his speech, nor bring closer the results of a musical work results mathematics, the opposite. A contribution of this kind is possible to precisely through a link at least perceived (but we believe that here there was actually built pragmatically) between these two forms of discourse, despite the distance. If we Now consider that this call to another discipline is actually done for internal developments in music and aesthetics, and that such development can be done by adding external components completely with the terms apprehension of them, it must be concluded that the ability to perceive (and higher reason to build) a connection between music and mathematics has never ceased to exist, in the Western tradition since medieval times. By accepting this possibility, we accept simultaneously mathematics and music can not be mistaken: there would be no meaning to perceive a link that crosses a zero distance. The essential element for this perception is precisely this separation, and how it evolves 100. This is again to Within this space between the two disciplines, or more precisely over him, that we can build a relationship. To be effective, it must be possible to find one side and the other of at least homologous, in which it serial algebra. Xenakis remains for us one of the only ones to have served without preference of both visions. 100The deployment of this distance in time is critical to the ability to link us seek. Indeed, it would require little than have anything to just be different mathematics to be able to connect to it. We leave here a situation of relative proximity Page 64 64 will build. Any complexity in the musical discourse to which it is possible to assign a certain "logic" should be used as a starting a connection with the mathematics because in our culture, in a logical sense is always "Mathématisante" we can approach any process where there are series of reasoning with tools or logical-mathematical comparisons. However, we readily concede the musical discourse logic (or even more), which articulates together the various levels of temporal unfolding of a work and sets out to virtually in real time at the hearing, the syntax of sound material used. Thus, points to the establishment of a link between mathematics and a work music exist, and at least these (because there may be others) do not depend the period or the school to which belongs or will belong work. Such As all musical works is at least part of the music, it is possible to establish a link between mathematics and music. Note that the effect of this relationship on music listening may be accompanied by a effect on the understanding or grasp of mathematical entities and structures involved, but this is not necessary, unless you want to make music program or strictly mathematical theory. The contribution that can give the musical representation of a mathematical idea (or more) is musical order first. This is not to perceive in a room, building real numbers or a Boolean algebra, but simply to give more to understand which, by various influences on both sides, leading to a distance and an independence increasingly Page 65 65 in a room in the component or by interpreting it: thus establishing a effective link (as we définissions above) between music and mathematics possible. 2.1 First definition We say that there are musical representation of a mathematical idea when it is taken as a subject for the composition or for the analysis of a work (or a fragment). The mathematical idea or object in question is not taken only as a tool or reasoning model, but also the basis for metaphorical or metonymic operations or the musical objects 101. This approach, although very simple, is already bearing fruit. It relies, like the second definition we then give, on or near symmetry that we want to collect between mathematical and musical thoughts, rather than on mere supposition of a possible transfer of coherence from one to the other, or a musical enrichment provided by a formalization. We believe in effect musical thought is essentially formal: it sets various relationship kinds of elements on many different levels of order and complexity, and constructed from these relations, and relations between relations, as thought mathematician; in both disciplines, questions of aesthetic guide decisions on many levels. From another angle, the musical thought differs radically mathematical thinking: it can not be complete disregard great - but not absolute. Page 66 66 any physical element, or be represented exactly by a discrete set of symbols. The music is inextricably linked to time by the perception, mathematics are less necessary. By following these intuitions we have made ​any differentiable function is continuous (2001) and Princípio Cavalieri (2002). These parts are directly inspired theorems (hence their securities), which are represented in this first definition, at several levels of musical material and its progress 102. . Illustration: Any differentiable function is continuous At the base of any differentiable function is continuous 103is a demonstration of this theorem, fragments of which are written in French and other notation mathematics. Among the various ways to achieve this result, we have chosen one that serves many inequalities, especially gender inequality | xx that indicates a neighborhood around the point x 0| <Δ, 0. We then wanted to represent musically these neighborhoods, primarily by oscillations around a "point" fixed: Fig.1: oscillation around a note 101In painting and sculpture, some works by Bernar V ENETfollow this direction. 102The following comments do not naturally want to complete analyzes of these works. We simply serve to illustrate the discussion. 103For soprano, baritone, flute / piccolo, clarinet / bass clarinet, horn, cello, percussion. View partition attached. Page 67 67 Fig.2: oscillations of heights, speed and intensity The idea of ​changes in a relatively narrow vocal range is also extended to sounds: Fig.3: random variations in a harmonic field Page 68 68 Fig.4: variations "close" to some sounds The demonstration is also based, of course, on the definition of the derivative a function. We took it literally, as it appears in the demonstration, spoken or sung by the singers. But we also used the most often associated with the branch image, the image of the tangent to a curve, shown at the start of the workpiece by a glissando (linearly) cello has the middle one note in common with an ascending-descending figure of bass clarinet. Fig.5: curve and right "tangent" All these representations are in turn the other starting point musical manipulations inside of the room. Thus, one finds the "curves" which Page 69 69 share in common, and variation tone associated more directly the text. Fig.6: curves and their "common" Fig.7: word "colored" by the instruments Page 70 70 All these mathematical ideas are represented locally throughout the room, none of them takes only a few steps at a time (although there has reversals: the same inequality can be presented in many different ways, for example). The sequence of these performances is free, but the following when even the general form of the demonstration: each section of the piece exploits inequality and the text of a demonstration section - follow the singers most often the parts written in French, or comment on the meaning of technical terms or geometric ideas in it, and interpret written those instruments in mathematical notation. Thus, the insertion of this part in the clotting time is the image of logical time given by the mathematical text: it becomes the representation of a demonstration, only more fragments thereof. To articulate this level of formalization, we inserted a second demonstration the same theorem before the conclusion of the work 104. This serves aesthetic reasons both musical and mathematical: it is a more elegant proof than ours, and allows a drastic change materials while keeping the representations previous. These few examples already show how this passage mathematics to music remains subjective understandable when a text is absent. Indeed, there is no systematic link glissandi around a Note to inequality, much less articulating a mode change to a 104My. 68 to 80. This is a demonstration ELLima, which are used as limits (see L , EL IMA Page 71 71 demonstration change, if these are the words of the singers. We are of the opinion that any transition of mathematics (or another discipline) to music, which did not support a text that reveals the will as "imprecise" that one, in that it probably will not come back to this music which has been represented. But this does not diminish the value or utility of representation in itself, in our case, this is what guided the selection and arrangement of many of the musical material, to the overall shape of the workpiece. The compositional process need not be discernible as such in the finished work: it is not less determinant of outcome. . Illustration: Princípio Cavalieri In Princípio Cavalieri 105The representation is not based on the statement or the proof of the theorem, but on its use. This is a result that allows calculating the volume of a body from that of a body already known calculate volume: both bodies have the same volume if their cross-sections in the same height have the same surface. 106 Thus, the application of the principle of Cavalieri returns to calculate areas on the cross sections of two bodies and to prove that they are always equal. It is this similarity between operations on one and the other body causes the joints of this piece: "reasons" rhythmic, harmonic or [1992]). 105Snare, harp and electronics in real time. In the partition extracts follow, the symbol* indicates a change in the electronic treatment for one of the instruments or both. Page 72 72 stamps as well as electronic transformations, and pass from one instrument to the other, or come back in different contexts. Fig.8: adaptation of the same material in each of the instruments Similarly to the interest we have focused less on the theorem itself on its use, the representation is made here with employment and changes musical material, not with (or in) the material itself 107. Many times it is thus the assembly of the two instruments with their transformations represents a calculation surface and reappears modified in another context - or, more precisely, creating this other context. (Figures below) We are dealing with a mathematical idea of a different order than in any differentiable function is continuous , further from the results that the Principle of Cavalieri provides concrete (one volume, and along the way surfaces) and somehow external to the theorem itself. This similarity that we want to represent music between operations in each body is not specified in the statement of principle, beyond an elementary algebraic definition (there will be questions multiplications, additions ...) which remains also implied. 106We find the statement of this principle, taken numerous examples of its application and overview History in L EL [1991] Dimension e em Forma Geometria , SBM, Rio de Janeiro. IMA Page 73 73 Fig.9: rhythmic material and exposed stamp at the beginning of the play Fig.10: The same "calculation" again later; real-time treatments are also changed We meet nevertheless face extremely elements similar 108In each sample application of this principle. This is because the choice of its use is not trivial: we use the Principle of Cavalieri "Build" knowledge volumes, building them from stronger simple and known. Calculate the volume of a solid "new" with this tool therefore for 107The rhythmic and harmonic content are actually indirect references to Catastrophe UltraViolet [1974], Jorge A NTunes, Which celebrated this year its 60 years. 108As the actual use of the distance between the cross section and the base of the body, the volume of a cone, the surface of a disk or ring ... Page 74 74 often to repeat such a course, and that is the idea of ​the reconstruction that has been the basis our representation, as we do not have a volume or a method 109It only remains that the repetition of certain operations, slightly modified departure each time, which should lead to an acquaintance, or at least new information, on the relevant body. This situation has led to the inclusion of metaphors and metonymies in our first definition of a representation: we think in fact that the influence of Mathematics is always present here, although no formal reporting tool. Us make the changes and musical similarities "like" those that take place in mathematics. This allowed us also to that room to add being composition other representation, independent of the Principle of Cavalieri, and was not in the original motivations of the piece: the intervals reappearances some irregular gestures and objects allows us to say that this play "pass" by some places several times, with different orientations, and that its overall shape could then be comparable to that of a node . This is still a time of a metaphoric comparison, we are here to serve any results of the extensive theory nodes; However, this merger has influenced the arrangement of repetitions in the room: it is a mathematical idea who participated in the form of the work. 109This is a choice, we could define a sound object as reference body. By eliminating as the starting point, we are interested to benefit the operation of the building as its result. Page 75 75 . Traces of formalization In a workforce such as Cavalieri Princípio , the issue of transfer a material between the instruments can become a central element of the process compositional. Indeed, the differences were not such in the harp and snare they inevitably come into play in the definition of these transfers, and in the electronic composition 110 - Representation is highly constrained by the instruments, and this influence is spreading into retroactively Construction of the starting materials. This finding, though musically trivial, once again underlines the difference between the mathematician and musical thoughts, despite their similarities: physical stress, such as the characteristics of an instrument, modifies the flow of musical thought, the consistency of any "language musical "may be only in its nominal consideration. The language formal mathematics employed by a well-defined grammar, and admits at least semantics; for it to be formalized, the music did not clearly a single grammar, much less a semantics. Despite this, the activities of producing music and mathematics to produce at least this in common: their results still exceed the physical evidence of them, which is their track hardware. Thus, the musical work is not the partition or the support where it is 110For example, filters applied to the snare can add some harmonic content; delay lines tight on a galvanized sound of the harp can the closer to a percussive bearing. Page 76 76 recorded, and a mathematical theorem is not the sequence of symbols used for write and demonstrate 111. Even if we consider as a concrete result of the activity the composer of "Western tradition" 112 that the partition or later magnetic or digital information on the carrier of an electroacoustic part, which would already be a gross reduction, this encoded information, recorded in any so out of time, carry within them the opportunity to be decoded to be put in the time, several "readers" different in different ways. All results of these readings is obviously not encoded on the sheet or tape magnetic, but the definition of the possibilities of such a set is part of the job composer. In addition, the significance of a musical work is not fixed by the one who consists of: in the words of Boulez, "the author, as insightful as it is, does can imagine the consequences - near and far - what he wrote " 113. Similarly, in mathematics, the same theorem can be proven ways different in different contexts, which aim to give different applications. Each of these demonstrations and applications of these lights theorem in some angles: they enrich its meaning in theory that the product or who employs, But one family treatment applied to both instruments simultaneously may also participate a morphological unit in the set. 111We will assume here that statement almost as a postulate, as it is beyond the scope of this text deeply discuss what or where there is a musical work or a theorem. This is a discussion could start with S B. [1947] Introduction to Bach , and N ATTIEZ , J.-J. CHLOEZER [1987] general musicology and semiotics , for music, for mathematics and around WITTGENSTEIN L. [1939] Remarks on the Foundations of Mathematics . 112We will call "Western tradition" theoretical corpus and practice of music as it developed in Western Europe and its colonies (when there is direct influence of this music European) from the XII e century. 113BOULEZ P. [1963], Thinking music today . Page 77 77 and deepen the knowledge of formal tools used in the demonstration. The purely formal end result remains unchanged, and only enough to demonstrate a theorem, but this complexity in ways of accessing this result can not be given by a single application or a single demonstration. The wealth of a theory also depends on the choices made for the building and, once built, the choices made from it. However, all of these choices can not be completely formalized114, And therefore can not be given within the theory. A music formalization should participate in this multiplicity of points Possible view of a room. More specifically, the representation of an idea math, according to the first definition, presents the composer as a additional way to understand the relationships it builds; She then leaves a mark on these relationships, which will be perceived as such perhaps by the composer himself, yet to act on the shape of the work. Other ways to represent an idea Mathematics may leave marks more directly on the objects themselves, but they will always tend, as here, to be steganographic: their presence becomes part of the object, and you must know in advance to be able to spot them. 2.2 Second definition We say that there are musical representation of a mathematical idea when fragment (at least) of the musical discourse or the music has an internal organization 114V. W EGNER P. [1997] Why Interaction is More Powerful than Algorithms , Communications of the ACM, 40 (5), pp.80-91 Page 78 78 similar to the mathematical concept in question. More specifically, we'll talk representation as soon as we can operate on the speech fragment like a mathematical object: by taking care of its shape and its formal content. 115 Specific ways to formalize we presented above, applying and parallel, not in itself, strictly speaking, representations in the sense of our first definition: the application is too limited to be a subject, parallel does not act on the musical objects. Nevertheless, they are approaching the idea of indirectly manipulate both mathematics and music. To refine these methods, we need to look more closely at the two "situations" that are thus set relationship: each is a logic carrier links (possibly dynamic) between its constituent parts; the choice of the tool with which the formal is established Reconciliation is not trivial, it reveals and directs some of the goals and vision from that formalizes (its shape can be more or less "compatible" with the so 116 the wants to associate musically). In addition, we want to take into account the relationships between the operations on one side and the other (that is with them, this time qu'auront place metaphors and metonymy). These are the thoughts on the internal organization of musical and mathematical situations that motivate this second definition. 115This definition is narrower than the previous performances in his senses will also representations according to the first definition. Recall that in speaking of the form of a mathematical object we believe the reports as it may have with other objects in its internal properties - cellesThese determine these, and vice versa. 116Note still just a set of applications fitted with an adequate metaphor can become a representation - we think particularly to the formula speed = space / time applied X ENAKIS ([1963]) to calculate glissandi, decorated in a gas particle image. Page 79 79 The centerpiece here is the idea of ​operation: it is this concept that links working with mathematical objects and work of musical thought when there representation. To clarify this merger, we can try to understand what means to work with a mathematical object: we must know its origins (from What branch of mathematics it comes) understand some of its properties, how it interacts with other mathematical objects and concepts (which applies to him, to which it applies), and so on. We can say that all this conceptualization of a mathematical object (which finally constitutes the knowledge even this object) is operative nature we must know what to do with it 117. In particular, this object is a "sense" that when math on an operation any; it can only refer to other objects or ideas that have the "sense" similar, or can operate with hers. Similarly, this is how we want to work with the musical discourse and its parts, taking into account if of "original" material, creates, or if it has been generated from other pre-existing objects ( "Origins" of the speech fragment in question); knowing its internal properties, intrinsic; knowing what are the opportunities for interaction, processing and action with, or on other objects. We want to know a musical object 117 This view follows the W L. [1939] Remarks on the foundations of ITTGENSTEIN mathematics . Such a view was already present in this author, based on the sign (in the language) in the Blue Book [1935], "If we had to name something that is the life of the sign, we should say that it is his use . "(P.40) Page 80 80 by operations which we can use and which may concern 118. Us then will focus on the organization of such operations, the contradictions and tautologies that may arise among them, with the precision that it is possible to extract to talk about this object or handling. Next to this, and in parallel, a mathematical object is often worked through writing (or more precisely his writing, one that defines it). Our representations thus realized also through operations on the writing, which will be these cases in the heart of the mathematical passage to the musical. Parts a set convex (2004) and Lema 1 - e partições primitive (2004) were written from that outlook on mathematics. . Illustration: a convex set We say that a set is convex if all the segments connecting two points of this set are fully contained therein. This notion of convexity, let the homotopy (continuous deformation of a path between two points) are represented in a convex set 119. We start with a melody inspired Brazilian popular songs, and the idea of ​a texture made rapid cycles notes, more or less long (the shortest being trills or tremolos ). Both starting points before the representation itself, they give us the objects with 118This idea is present in V H. [1995] Objects, representations, operations . It is to AGGIONE us to the base which may be a "composition-based object"; We shall return to few benefits of this approach to the compositional work. 119Clarinet, accordion and suspended cymbal, or clarinet, accordion and electronic sounds; v. partition attached. Page 81 81 which we will represent the convexity and homotopy. On one hand, this is the set these cycles, which are constantly used as background to the development of the melody that we want convex. On the other hand, we find in the melodic line intervallaires and rhythmic content directly from the songs, but not always juxtaposed or in their original order; we want each of the lines written either homotopic, in a sense, an original melody. Part cymbal and 120 electronics are not directly involved in the representation of these ideas. to. convexity For all the notes in the present cycle part is convex, it necessary that the segment joining any pair of these rings is contained in this together: it must appear in full in the music. We still have to define what can be a "segment" uniting two notes cycles a and b: it will be for us a suite ordinate cycles c c have a maximum of 3 0C 1... C n, Where 0c= A or c n= B, such as c i and i-1 difference in scores between them (that is, if we take them simply as assemblies such that | c i -c i-1 |, | C i-1 -c i| Á 3) but also that c c are close "in i and i-1 instrument ", that is to say that the actions to perform differ only by 3 moves up and the keys or keys used in c i are contiguous to 120One could say that they are trivial homotopy: all electronic sounds come by continuous transformations cymbal sounds, and there are really only two sentences to separate the cymbal in the score (the last action is a downgrading of the first). Page 82 82 those used in c 121 To ensure this convexity, we then built i-1 . suites cycles (mostly from a trill) following the close therebetween. Thus, all cycles is already compound with the property of being convex: each new cycle extends a segment that connects the previous; the process of transformation or development of these figures is fully presentation. Fig.11: a "cycle segment" for the accordion Here we have a mathematical idea that guides the work on the idea and the musical material, not only in terms of notes; connectedness and convexity "The instrument" are in fact often the cause of actions. All cycle modifications involve physical movements in the "topology" own of each instrument (in the case of the accordion, we could say that we make continuous deformations of the buttons on course). The imposition of convexity on all these movements (interpreters) generates operations directly on the musical material. It also suggests how to consider the 121Most of the time, the distance between two adjacent rings is less than or equal to 2; us recognize the distance of 3 driven by the passage of a trill to a sequence of a chromatic ambitus more wide, for example. Note also that we identify octaves; This is somehow justified the accordion, the different voices Registrations are overlays at different octaves; we wrongfully extend the principle to the notes of the clarinet. Page 83 83 convexity in the rating - it must indeed be represented in a manner consistent with instrumental playing. The representation of this idea in the instrumental topology allows the compositional work to operate the instrument in the same way it would operate with the morphological characteristics of a sound material. 122 Note that, for precisely here that we wanted to formalize it can not ensure an accurate perception of the convexity in the first place because what the guarantee is not well defined. Indeed, how should be seen a musically property originally on fragments of plane or space? The most picture often associated with a convex set is that of a space "without spikes beyond "(a circle is convex, a star is not), and in this sense actually very little exceeds sets cycles only , as they are written. But listening can integrate with other elements of the room before judging of any convexity: the Separation of these cycles as a standalone set of objects makes sense when writing, it can go to their inclusion in a broader context. We believe including overlays of a cycle with fragments of a melodic line in one instrument: these may give the object resulting "spikes" which break the convexity. 122This idea is central to our compositional approach. It will be part of the formalization General in the chapter on the geometrization. Page 84 84 Fig.12: superimposing a melody to a cycle, you can lose the convexity listening In addition, the segments joining the points of a set, mentioned in the definition convexity are theoretically line segments 123; or the transformations we propose does not necessarily suggest a sense of linearity. In bringing this way a mathematical idea of ​music, we begin to manipulate almost as much as musical objects we transform and adapt the room needs. Again, the advantage of this representation does not lie the possibility of detecting such in an analysis, but in the way work it offers. It leaves a trace here into smaller objects written the instrumental part and allows to consider a mathematical idea like material compositional. b. homotopy The melodic lines and rhythmic patterns of a convex set inspired by several Brazilian popular songs of the early and mid-twentieth e century, but especially Máscara Negra , Zé Keti and Lamento , Pixinguinha. Fig.13: Theme Máscara Negra , Zé Keti 123In a more general vector space, that are combinations linear both ends. Page 85 85 Fig.14: Theme Lamento , Pixinguinha These, Tune appear in pieces into a convex set . The fragments are transposed, stretched or contracted, but are not present in their format original. The melodic lines nevertheless remain the benchmark for representation of homotopy we chose. If we have a path between two points of a set, we say that another way between the two same points is homotopic the first if there is a continuous function that transforms into each other while preserving their ends (this function is called a homotopy ). We will take each of the original melodies, and possible transpositions as a path between its first and last note, by focusing instead on "components" of this path at its ends (Fixed) intervals, melodic figures, distances, rhythms ... This is distorting these gradually as we get homotopic paths first. Unlike repeated cycles to represent the convexity, all intermediate transformations between the original fragment of melody and its last state are not present in the partition; Sometimes the reference is still quite clear when we tend to a suitable homotopy (which is constant in a neighborhood of each end). Page 86 86 Fig.15: start and end of theme Máscara Negra as they appear in the score; between them, the line of the clarinet is obtained by successive homotopy of the original theme 124At If you notice tuned the reference to the popular song in these cases no power probably identify the source, this relationship is much less apparent when several stages of deformation accumulates on the melodic line before it enters the partition. The play is thus not made direct references, but remains evocative of this universe external to it. Naturally, this is not the representation of homotopy that is responsible for this evocation 125But it gives consistency in how to work these references, allowing to integrate them more completely to the other elements that make up the "musical language" of this piece (And our work in general). This representation is combined with the previous and helps create logical links between different levels of instrumental writing. In particular, certain transformations of fragments of these melodies approximate notes cycles, the other piece of base material 126. This can further contribute to blur the perception of a convexity in all cycles or that of a link 124There are only two in the score: my. 59 and 64, with raised. 125It would be possible to object that this representation suggested by operations are themsame fairly similar to those that might already be in development techniques XIXe century, so the homotopy would be directly responsible for anything. While this may be true locally, we could not achieve the set of transformations that we have without actually used this reference to the "image" of a mathematical curve to the fixed ends which is deformed. Page 87 87 continuous melodic line with an earlier, but it dissipates reconciliation borders between the tools used and instead puts forward their OF RESULTS (and thus the opportunity to judge their relevance). Unlike previous illustration, the overall shape of a set convex represents no mathematical idea. In fact, only the clarinet parts and accordion part in performances, and only at a local level; there Cymbal part and the electronic sounds are independent, and are compounds with sound and formal result of the activities of the other two instruments. Throughout connection between music and mathematics (or other discipline, in fact), it found on both sides of the aspects that are less "covered" by this merger, or not at all. In the case of this play, they are not among the details of writing or formalization, but are placed in the foreground as well as those who bear the representation. . Illustration: Lema 1 In short piece Lema 1 - e partições primitive 127 we chose represent some aspects of the integration operation in calculus. The integral (Riemann) is associated with both the calculation of certain areas and the inverse of derivation, primitivation, through the fundamental theorem 126Especially in my. 32-36. 127For solo violin, c. partition attached. Page 88 88 analysis 128. We are interested here in the organization and the articulation of musical material, the interval partitions (used for area calculation) and the idea that primitive of a function are in some sense more "regular" that function. The basic materials are a pace similar to that of a jitter: , and the game rating touched on the violin: "false harmonics" natural harmonics and artificial. They only indirectly involved in the representation that we will build: these special game modes place the instrument in its upper register and shrill, and it is on this that we will work, by partitioning the image of the the interval over which is defined a function to integrate 129; we give also to the rhythmic cell the role of this function. These operations will be a way to handling the instrumental and rhythmic material, do proliferate. The choice of starting cell and game modes are arbitrary here: we take them for their references (various connotations) to virtuoso solo violin repertoire; organizing the part is also largely independent of the representation. to. scores To calculate the Riemann integral of a function, we first calculate its are top and bottom on a partition (subdivision) of the gap where it 128If a function F: [a, b] ƒ½ has an integrable derivative, then F (b) -F U bF ='(t) dt. (a) to 129Note that in the definition of the Riemann integral, this interval is always closed, so that would probably be more accurate to consider "the extreme high register of the violin" as an interval Open heights. We will see that this difference does not profoundly affects what to do Page 89 89 is defined: each is a first approximation of the area between the curve and the axis abscissa 130. Then we refine the partition (for adding points in its interior) and we find that these are not stray (and may be closer). At Last, we define the lower integral (respectively the upper integral) as upper bound (resp. lower) taken on all partitions of the interval, the are lower (resp. upper). A function is Riemann integrable if these two integrals are equal. A first partition of the upper register of the violin is given by the harmonic natural and artificial; it is refined by adding these harmonic trills and by glissandi harmonics, and finally by "false harmonics": notes touched, played with tremolo or sforzato on positions that do not correspond to natural harmonics, and that produce clusters more or less dirty in hyperacute (These are actually several notes quite "far" in the harmonic series of the open string). This construction provides a collection of dots and small intervals roughly relatives, distributed throughout the register we want to address. Note that it not necessarily occur in the order that we followed, sounds more acute, but sounds that cover a range of increasingly large. It is on this collection, taken without particular order, we apply the function to on the partitions of the interval, but requires an adaptation of the representation when it considered in full. 130These amounts are just the surface in a step function , that is to say, sums rectangular surfaces: Am i(Ti-t i-1), Where m i are the values ​of the levels of this step function and t i the points of the partition of the interval. These bearings are the minimum or maximum value Of[t i-1, T i] in the integrand for the lower or higher amount, respectively. Page 90 90 include: the basic rhythm (and some variations of this cell) is presented on these register of share points. 131As these subdivisions and these rhythmic elements are discrete, separable, the fact that the interval is open partitioned poses no problem convenient for representation: we dots inside it without us regardless of its ends (in fact we operate in a sub-interval). From the point of view mathematically, this corresponds to the operations in one of the parts of the integration, geometry; it may order these subdivisions from the simplest to the richest. Musically, by cons, we obtain a material that has not yet directionality 132: Harmonically, everything is given by the instrument itself rhythmically, we have very little variation. Partitioning does not generate alone, for this representation, a singular insertion time. b. primitivation According to a simple analytical insight, we can say that the original one function f is another function that evolves according to f, whose rate of change is given through f. Thus, if f suddenly changes value in a discontinuity, its primitive change at the same point its growth (it will make a shift), but without 133, necessarily make a "leap" discontinuous as f: in this sense the operation 131We could say, from another point of view, this is rather a range of game modes is and partitioned, and on which we apply the rhythmic element as a function; but this is not This interpretation has guided representation. The ability to set game modes intervals reappear in the chapter on geometric. 132We'll talk about directionality of a material or object when following a direction and that it is manifest in him. 133We again look constraints of the instrument as a material for composing. Page 91 91 primitivation "smooth" the contours of a function, reducing discontinuities and abrupt changes 134. It is this image that we have chosen to keep to represent the integration a function (again the timing jitter) on the high register of the violin. Us take and characteristic fragments of the cell, reducing their salience by links or increases to thereby obtain a new set rhythmic elements (and therefore times); it is this new "feature" that we will want to browse the extreme treble of the instrument. Here again appears after the need to adapt the representation to external restraint to its formalization in Indeed, how to cover everything registry violin if we can not even define? We chose to take an approximation of that totality, suggested by the representation of partitioning: the limit of scores more refined is the entire interval as to the amounts on these partitions is full. We then apply (almost everywhere) the primitive function that we have build on the finest scores we had: one given by "False harmonics," and we still refine by glissandi with this mode of play. c. development The shape of the work is here again very little influenced by the performance: the integration related transactions were used primarily to provide and organize the confinement 134Technically, the primitive F is more regular as it is differentiable once over f. Page 92 92 musical material not deploy. The syntax was done first by seeking to give some directionality to each of the two families of objects that representation generates, independently: each side of the integration is performed in a separate part. Other formal punctuation appear, also external formalization, that link between the two sides and vary the statement of speech, contributing to the unity and complexity of the work. The role of formalization in this room is the one to allow manipulation of objects and to operate with them as reference points taken from the outside of the music. 2.3 Types of representations We said above that we are interested to consider any formalization as a kind of notation, a particular way of approaching a situation and to manipulate objects. In a very direct sense, then, is defined with a formalization rules 135 governing such opportunities for manipulation. We can therefore consider "rules for composing" and other "rules for interpreting" (or analysis), sets that are necessarily distinct, even if they come from a even formalization (and although they may still have elements in common). The previous sections of this chapter have focused more clearly question of the existence of musical representations of mathematical ideas, rather than their multiplicity , illustrations for a constructive part of this Page 93 93 "Demonstration" of existence. But these works, as well as examples are taken also to review the plurality of ways of representing an idea: at any time there is a question of necessity of such a representation - arbitrary decisions (so substitutable by others) they are based. This also shows that, in a Somehow, the finished work is among these multiple representations possible 136; we could say that "what happens" actually in the works finds a counterpart in the set of rules used to compose (all those that interact during the compositional process and define). Symmetrically, one work can be analyzed from different points of view, without none of them really is comprehensive; formally, the rules we can follow during a scan, or those we can assume, when we analyze, the work "follows" does not fully make this work. Although he was 137We would not exhausted algorithmically possible to find a partition all ways to enrich their listening by analysis or, more generally, interpretation. We can say that when we set rules sets to approach a musical work, we always have the following dips: 135That will or not a formal statement; we saw in Lema 1 indirect manipulation gesture instrumental which were themselves not formalized. 136Under the terms that we introduce below, this is equivalent to saying that several formal spaces (or simply compounds) can be obtained separate from the same modular spaces. 137Which, to our knowledge, is not yet possible (except trivially, of course, by a process direct imitation): we do not know of "resynthesis" a partition that reproduces correctly placement of dynamic, for example. Page 94 94 rules to analyze 2 work 2 rules to compose. 138 It is intentional that we prefer here speak of immersion (or for injection ) rather than inclusion in the strict sense (hence the sign 2 instead of 7) we want to emphasize that the "rules to analyze" are represented or interpreted in place by a set of properties that is their isomorphic, but it does are not themselves; in the same way that follows the rules actually do the work are not found in identical among the "rules for composing" but a sub all of these can be associated with them. 139These are, among others, the interactive aspects different actuations of these rules, the level of analysis and composition, that prevent strict identifications through these levels. The interpretation of such rules is between these groups and the work. To arise representation of a mathematical idea through a formalization, analysis and the composition and operating in opposite directions to achieve a work 140. This 138This pattern is reminiscent of the semiotic tripartition (see N , JJ [1987] Musicology ATTIEZ General and semiotics , for example). Without fully adhere to a semiotic position (Including, above all, in terms of the neutrality of a partition), we feel good to distinguish aspects poietic and esthésiques music (although they may overlap) and to be able to consider a separable object composer and its settings at least for an interpretation (as a partition we can take for itself , for example - we prefer the term "immanent" for this level of observation). 139Of course, talking about rules "actually followed" is problematic, at least because it is difficult to define this term without reference either to the act of composing, or to the analysis (also the problem of the general definition of neutral level in musical semiotics). We want just keep in mind that a work exceeds its construction (even during it), in the sense that even the composer may surprise. We return to these 'surprise opportunities. " 140By cons, if it is to formalize a music theory, both paths must be followed simultaneously as a theory must be used to produce new works and study of works existing. This is what we find for example in M G. [2002] The Topos of Music and AZZOLA BAboni-SCHILINGI , J. [2005] The iper-sistémica musica . Page 95 95 defines two different types of representation 141Which also reflect the separation between musical discourse and discourse on music. . Constructed representations To represent a mathematical idea, a composer must build a work where this idea will be part of the musical discourse, at a certain level of its organization. Although it may be local, such action always affects the work in full, since all speech organization levels are linked and interact: at listening, insertion into the chronometric time prevents complete separation different scales 142; during the composition (inevitably also includes listen) is the same organization that puts all processes related representation is defined in formalizing the above work ready, it is 143. There made at the same time as the set of "rules" that the composer is required (the formalization in question is actually a part of this set) and it unfolds in the work at the same time that it is written. We will speak in this case of a constructed representation , including the apprehension by the listener can only be done during listening to the work. The effectiveness of this representation depends precisely on this Listening: it is not strictly independent of the formal framework through which 141Where we can separate a third subtype. 142We will see further forward than this corresponds to a decline in musical spaces (in a composable logical time) on themselves. 143As said V H. "There is no contradiction between attention to detail and comprehensiveness (...): the AGGIONE detail, thus understood, is everywhere and in every moment "([2000], Musical Composition and Resources IT: Approach Questions ). This important development report and these interactions are a subject Page 96 96 builds the representation (for one can imagine different formalizations more or less effective on the representation of a particular idea), but it should not settle on prior knowledge would have the auditor of the formal, or more precisely the mathematical idea represented. Is a constructed representation can to listen must register in various time scales of conduct of the work not only of his writing. If a formalization in the composition process, and in particular a constructed representation, can affect only one level of musical discourse because their interdependence, which is affected then the whole shape of the work, which is collected at different scales simultaneously. Quote here the exemplary characterization form of the concept given by Granger 144: The notion of form at first seems inseparable from perception space and therefore give first as empirical concept. This it is, indeed, in its most obvious incarnation. However, Gestalt psychologists have done in the same perspective of a empirical determination, two discoveries that guide us towards a more fundamental reinterpretation. The first is the extension of the concept form non necessarily spatial perceptions; the second is the essential role acknowledged the opposition "form-substance". One and the other suggest the possibility of designing the shape as untied entitled to any realization in a spatial perception, or even any perception. On the other hand, the opposition-shaped bottom is reduced ultimately to the purely logical opposition of yes and no, presence and recurrent in this author, cf. eg [1997] Dealing with object networks and [1998a] Son, time object syntax. Towards a multi-scale approach to computer-assisted composition . 144GRANGER G.-G. [1994] Forms, operations, objects Page 97 97 the absence that characterizes a border situation, a necessary condition to the appearance of a form. Of course, there is no denying the role played by predictive text input, spatial or not, a form as continuity, continuity as it internally, but always correlative of a break, a recess. As thinkable as a concept, the form is precisely this correlation grasped in both perceived forms, space or sound, or perhaps even falling autonomously other registers of meaning, in the form of a speech (not only its sensible realization), of reasoning, sense, of a thought. Specifically retain the ability to design the shape of any untied implementation, and as internal continuity and correlation: it is this view that allows to take this concept as operating independent of the figures which it may apply and own operations defined by a context. If constructed representation is the shape of a work or certain objects inside it, then this is also the contents of a room options contributes. All the musical discourse and interaction of the work with a situation special music can be enriched. Such a representation enables new without imposing contents, and its effect is thus first musical and perceptual place. . Representations found On the other hand, the analyst seeking representation should highlight the a pre-existing musical text, fixed and therefore independent of the analysis or representation in question. The result of each representation is then a enrichment of listening possibilities and understanding of the work to a certain Page 98 98 level of his speech. As before, this ends up enriching opportunities listening to the entire room, but here it is given in addition to listen to the east initially at a single level of musical speech, and then, for consistency the work145Extends at other levels of the organization. The representation is defined in this case a formalization which succeeds the work, so it unfolds in time different from the work (and composition) and in a speech on music. We will speak in this case of representation found whose apprehension the auditor should begin to do before listening to the work. Note that the results of such a representation for the perception and understanding are possibly very different from those of a constructed representation: here we can artificially isolate a time scale or fragment to study without taking necessarily consider its participation in the overall shape. Furthermore, attention is drawn to a mathematical character even if the contribution exceeds the listening perception of the stroke, it remains dependent on its understanding. Such representation suggests new content, and its effect takes place primarily outside listening. The choice of the word "found" does not mean that such representation is less built by musician than is a constructed representation as we have defined. Instead, we want to highlight a representation 145It is not a question of the consistency musical of the work in its all, but rather a consistency just logic. Must be pre links noticeable between the various fragments Page 99 99 found mostly depends on the analyst: in all research, all possibilities findings is defined by what is sought or, more precisely, by way of the look (in a certain sense, this way defines the purpose of research). Obviously, it This is not to say that a representation is found completely imposed on a piece music without dependent; but it is possible to interpret a work of infinite different ways, so that all kinds of representations (which are established precisely through an interpretation of the formal framework chosen for analysis) can be "Found". These interpretations are not independent of the work, but they are composer and formal setting (whatever it is, if it exists) that he established for the dial. So find the representation of a mathematical idea in a room Music does not indicate that the intention of this representation was present at the composition, but this does not invalidate the analysis that makes it appear - you have just make the reservation. 146To try to avoid too distant representations it is assumed that the intentions of a composer, or what one deems valid Music speech than the representation suggests passes from one of these fragments to other. 146Include, as an extreme example of what can lead forgetting that reservation, book , V.HOUTEN K & K Asberger M. [1992], Bach and Number . In this book, the authors 'prove' their analysis of the work of Bach that he said in his plays, among other things also surprising, the date of his own death. The "facts" purely formal presentations are clear: the authors did not change the scores they study, and we can not deny the possibility of finding numerical combinations they offer. Their interpretation remains unsustainable. The rule sets immersions, we were talking about earlier, clearly taken by van Houten Kasberger and as inclusions in the strict sense and deduction follows that if a rule applies to analysis, it was used in the same form and with the same intention for composition. Okay Obviously, if an analytic operation "on", it is because the structure of the work permits, and So because the "rules" used for the composition actually generated this structure. But the operation used in the analysis is not necessarily present itself among the "rules" composition . Without it, we could also get in Bach references to John Cage (cf. P , ASCAL e century A. [1997] The number in the musical composition in the twentieth ). Page 100 100 to find a part analyzed (two distances whose evaluation is already asking itself even reflection), limitations may be imposed as criteria almost aesthetic simplicity of the idea represented, for example, or simplicity of the representation (understood as the set of analytical operations that make the idea perceptible shown). . Métamusicales representations If we conceive a theory in which are represented ideas mathematics, these representations must be first out of production specifically musical, serving to support the thinking about music. They manifest while a front portion of the formal sensitive, and act on this passageway rather than formalization or musical material directly; perhaps it would be more accurate in this cases, the name of the representations métamusicales 147 of mathematical ideas. We nevertheless happen that métamusicalité almost immediately at the same musicality, and the effects on the understanding of the music, because such a theory, for consistent, must ultimately address the reality and the sound works, their compositions and their interpretations. Thus, in a formal theory of music has, at métamusical, another immersion rule sets in the opposite direction this time: rules [thinking] the composition 2 rules [thinking] interpretation. 147Here we take the notion of Metaconcept to G RANGER G.-G. [1994] Forms, operations, objects (Conclusion): "A Metaconcept relates, not directly to an experience, actual or potential, but a representation of the experiment. "The métamusical would be all the concepts in a reflection on the musical experience. Page 101 101 In other words, thinking about the experience of the composition is a case particular reflection on the interpretation and understanding of music: the composition is part of the identity of the work, and the composer, appropriating its own musical past, is no exception to the interpretation of other works. 148 In fact, this immersion can take place only at the level métamusical: if affirmed during the approach to a work, it would, for the immersions set more above, an identification of the composition analysis, and work to defined rules for each of these processes, which would be a reduction of at least brutal content of a piece of music 149. The representations of mathematical ideas in a music theory are therefore established at a level different from those of the composition and analysis put to work . These are then necessarily representations constructed, since they leave no specific work (and therefore can not be found ). However, they are only the candidates to be representations found in an analysis in the light of this theory. Indeed, a composer follow this theory, which would represent these ideas in mathematics his work should reconstruct these representations in order to integrate them into the network rules and interactions that it establishes itself to compose. 150 This attribute of the dual 148This scheme can also be read as follows: dial is a particular way of knowing a work. The métamusical level, it actually has more rules, strictly speaking, to analyze that to compose, as it is easy to see in any composition textbooks. Note that this immersion is métamusicale, it excludes the reports to a particular work. 149It is also this error métamusicale identify a relationship to a musical relationship are , K. andAsberger K M, [1992], Bach and number . V.HOUTEN 150This passage from the musician métamusical practice is even more laborious than the theory question is mathematically consistent. Indeed, this consistency tends to set "hard links" between mathematical objects, which may hamper them in isolation, or divert (unless Page 102 102 métamusicales performances well above indicates that both the analysis and composition, thus do not affect, in a first step, the potential apprehension of a particular work. When switching to musical activity, representations of mathematical ideas in music theory are characterized either as representations (re) constructed either as representations found, and can then become actually visible. 3. Meaning in a formalization As we have seen, a musical work is always shifted from the established formal framework to address it: she exceeds when analyzed, and do not exhaust when composed. This indicates that although formalization, whatever it is, is not enough to meet (and more so to determine) a work. It is within this space between the formal framework and the room (or objects same musical) it is possible to truly develop an interpretation of rules play to then be able to contribute to listening. It is also in the relationship established between the "materiality" of the work (Taken as an object at runtime, but we also think of the partition or a digital support) and different thoughts around it (which it should be concurrent or not) that we can speak of a consistency in the logical sense, the musical discourse. to be inconsistent, precisely). The limiting case of course, is a theory mathematics of music, which "thoughtful musician (...) is here strictly cluttered" according N , F. ([2005b] ICOLAS Page 103 103 If there can be no question of meaning in a formalization in music, or with it we must indeed be considered a consistency somewhat "shared" between aspects formal preceding work (and are the basis for the formalization) and those who emerge listening. Naturally, some apprehension of form is unavoidable when we are confronted with musical objects; this is the coincidence (even partial) with prior formal framework is not always given a priori but must be built. 3.1 Forth to "interpretation" The different meanings of the word "interpretation" require some attention to its use: it is a term that can be applied at two different times of understanding of a work. We first first understand the interpretation as transition from an abstract object (a set of rules, for example) to a concrete object corresponds to it (such as a partition, a text, an instrumental performance). There is important to note that the terms of such correspondence are to be defined along the passage itself, and which plays the role of a concrete object in a particular interpretation may be the container of the abstract object into another: for example, Partition is the "concretization" of certain rules when made; it contains a set of rules to follow when played. Specifically, we How musically evaluate mathematical theories of music? ). Note that in this case, the Page 104 104 see in this case this passage as a share to and from an abstract, such as building (creative at least in part, so) of a new particular concrete. On the other hand, we can also hear the interpretation as assigning meaning to an object. It is no longer here in the construction of a new object concrete, but the establishment of logical links 151between parts or aspects of it which is interpreted. These links can rely so heavily on phenomenal elements the object (in the case of a sound stream, silences or intensity peaks, for example, or fragments of repetitions) as a network of associations and interpretations prerequisites. Again, the precise nature of these links can not be fixed in advance: interpretation is in this second sense as a work around an object and from he, who built a new abstract one object with meaning ; segmenting the object interpreted, which defines the elements between which links can be established, depends interpreter (and, a fortiori, an interpretive context). . Movements of "sense" When we bring these two uses music, they imply two contrasting views on a musical work: the performer and that of the listener. Indeed, we could simplify and saying that one follows rules while it creates or deduced; and we assume the existence of a "sense" in both interpreted rules (according to the first sense) and the result of mathematical ideas are present in the theory , not just represented. Page 105 105 interpretation (in the second). But if we consider that in both cases there is the transition from one type of structure to another (from one partition to gestures, sound stream a mental image, for example), this "sense" is not as localized, and we could say that he gives himself entirely with this passage. From this point of view, the action interpretation is always that of complete understanding, precisely through this transposition (which is necessarily creative). We can not, of course eliminate differences between the act of performing a partition and that to understand a piece of music, but the two meanings of "interpretation" that we give to if we move closer to the idea of sense: if we take it as a possibility organize and activate a network of knowledge or gestures 152. 3.2 Intelligibility and understanding When it comes to spoken language, we can differentiate a step that above the same attribution of meaning to a sentence: it must be intelligible , use known and properly spoken words. With this we can already judge whether it is 151Remember that we do not specify as the presence of formal deductions, just relations abstract. 152A gesture is also learned knowledge, as it were "housed" in the body (therefore not necessarily accessible or describable in the same way as knowledge "abstract", often verbal origin); we believe such gestures of a musician, he organizes and involves by a partition. To study the issues around representations of mathematical ideas, Here we turn naturally rather the side of the verbal or intellectual knowledge, which can more direct approach, it seems, syntactic, semantic and aesthetic issues that are arise. Place the body in knowledge remains a vast subject, among other philosophy and anthropology; we refer to the work of A PEL, K.-O. [2005] The "a priori" the body in the problem of knowledge and Q P. [2000] complex messages through communication UETTIER gestural sequences . Note that conversely it may issue "gestures" of thought (see X I. [1963] Music ENAKIS formal (av. Introduction): "materialize movements of thought with sounds"). Page 106 106 syntactically correct, then further questioning on its semantic correctness and meaning 153. This problem is very different in music, where we do not always have "words" known to support a musical sense, and sometimes not the equivalent of a proper pronunciation. One can not "go" from the construction of a musical sense to the basic elements; more precisely, these elements or collected in the segmented sound stream by the auditor are already only liable to wear a "meaning". The idea of ​interpretation, as we see it at the moment, is central to think the intelligibility of a musical work. We will say that the work has a meaning when you can "understand" when you can make meaningful connections between a few at least one part of it, without much internal contradictions (it is interpreted ). 154 Which is well understood, the "meaning" of the work is the same logical network links, which one can remember, we can "read" in memory. It is important to note here comes in our ability (facing each individual work) to establish such network, especially whether to make the entire work. Indeed, we could say that the incomprehensible music is the impossibility or inability (sometimes Temporary or just local) to detect a sense - that is to say, build it from the sound stream. The question may arise about sharing this construction, between 153In particular, when we state that a sentence is nonsense, something provided it sets out feel about it, the sentence is apprehensible that way at least. Nonsense can not be unintelligible. 154We take precautions for all parts-related and the complete absence of contradictions, as we believe that through the establishment of logical links between all the parts of a Page 107 107 listener and composer: it is he responsible for all the consistency of the work, what can it offer to the listener that ensures that the work will be interpreted? On the other hand, the listener is not it, in the end, the only one who can assign meaning to it hear? If a piece of music is always the result of a process of composition, and if interpretation (ie listening) is also constructive action we are faced with a transmission and reception (or at least apprehension), and we can look at the transition between these two poles. . Transmission of content The notion of form is of course related, at least since the beginning of the period classic, organization of musical discourse, and thus understanding. as we said above 155 But 156, Classical forms have only syntactic scope limited in musical writing: they do not consider the details (meso-temporal) and may, as they have defined, emerge from any base material. Although critical to the idea of form as a container , which may receive 157 work without contradiction, have understood he has to understand this work, there therefore haveany nothing to hear it we (andwould not the more reason toeverything listen). 155P cf. REDA -SCHIMEK H. [2003] A look at the genesis of the form of theories, between classicism and Romanticism (1790-1845) . 156Sect. 1.3 Procedure in time (time). 157Indeed, we can detect an "almost sonata" in Threnos , K. Penderecki (1959): a "Exposure" three "themes" contrast, the third is morphologically close enough of first (which could also be seen as an introduction), and a coda that combines elements; a "Development" of kinds of articulation of the second "theme" shaped cannon and with some appearances of elements of the other two; a short transition (with new sounds in the room) leading to the 'recapitulation' morphologies first and third "themes". This interpretation certainly leads the definition of sonata form beyond its limits, not least as the lack of tones in Threnos ; it nevertheless keeps to use dialectic of material, which is central to this definition. This example also points out, more dramatically than in Page 108 108 various content, comes relatively early 158The classic shapes remain points mark for a long time: it takes the free atonality, and then especially Webern, always forced to see and deconstructed. It seems it is at this point that form and content are beginning to be inseparable in practice and thinking of composers in general. But this is probably Xenakis who first, asked the question specifically the understanding of musical discourse, or simply sound events in Under a global organization directly related to local arrangements of elements discreet. 159 The central innovation of his thought, for this discussion, is for us to treat organizational elements to a certain level for a scale or level higher (logical or temporal) and the use of probability distributions or mechanical throttle to define, from global behavior of large masses sound, articulation and writing of the details that make them up. Xenakis was especially busy (in what its formal composition tools directly address) large sets of notes, with which it has been able to operate as on "notes" of a species or a new scale. 160 We want to follow a similar direction to think as a piece of music, locally and Overall: not only we see the details of a piece emerges its classic pieces that knowledge or perception of this form is not essential to appreciation of the work or understanding. 158It is found at H ANSLICK E. [1854] From the beautiful in music . 159In X ENAKIS I. [1963] Formalized Music (cI). 160Of course, Xenakis was not indifferent to the question of the overall shape; the "sketch" of Pythoprakta presented in formal music is clearly shown. Page 109 109 overall shape throughout the listening (listening to the given object is the succession and superposition details), but also that understanding and organizing this overall shape is given by the interaction, listening and writing, several other local forms (logical links networks, not just separate objects). We come back to the possibility and the ability to assign meaning to what is listening, this time considering "local insights" that can change when faced with other forms that follow or complement. This understanding constantly redefined course construction of the listener, but responsibility of the composer to identify opportunities; more precisely, the process establish the compositional, whether or not they part of the ideas or entities consciously manipulated by the composer. The question may then arise of Approaches to this "flow forms" to compose the intelligibility of a room. Several approaches are possible, but we can use some information theory elements to identify some issues that need to be common to all. The attribution of meaning to a sound stream depends, at least, the ability to detect there the "contrast" 161 to be able to segment the one way any, and then establish links between well defined elements. Without segmentation, the object is understood that atom in the etymological sense, inside 161Here we want the term in the broadest possible sense, we consider that the perception of a change, whatever it is, has already caused some contrast. Page 110 110 which can not build relationships. 162The joints in the sound flows therefore the basis of the information it can carry; the links can be established between the they define segments depend on the nature and arrangement of these points of mark. 163The message most difficult to pass , that which contains the According A.Moles more information is also easier to make about because it looks like the background noise: the receiver can not be aware of the intentions the transmitter (no redundancy). This is the message the most fragile : the lack of redundancy does not predict, so check, which will follow From the above, the slightest error destroys the precise content of the message. In Specifically, this message does not spontaneously captures our attention, it lacks that Moles called aesthetic value . One can not be interested in it (and even more so the understand) if it is said we priori it must be interesting, and we need to know to organize (possibly providing effort). 164 The activity of the composer is imbued with these considerations when considers the possibilities of apprehension and understanding of what he writes. Plus one message is structured, it is intelligible; more it is redundant, the less original - 162It could be in relationships with other such atoms, but this of course assumes segmentation prior to a higher level that identifies them. 163MR , A. [1972], Information Theory and Aesthetic Perception . Recall that for this theory, OLES the amount of information objects to the amount of redundancy in a message: a repeated element indefinitely is the message with the least information; the total lack of redundancy the door maximum information. 164We are getting closer and by other words, the criticism of Xenakis serial music: the effort to make sense of this music is too large, this effect depends too much information Prior to listen, and this reduces the aesthetic value of the work. Page 111 111 the balance is to be performed again in each room. As a starting point, we can see that the absence total aesthetic value (according to Moles) can not rarely happen in the case of a written music: like this exists as Sound flows only through an interpretation or reading (an active process, so), it can be properly issue of spontaneous attention. Furthermore, in a situation of concert, we know (or assume) in advance what will be heard is recognized like music, and this judgment in some ontological door so the possibility other judgments, aesthetic. However, some spontaneity of listening is inevitable, and must be taken into account by the composer. Indeed, the landmark classes that are active for recognize "music" and begin understanding, shall not prevent any If the appearance of "salience" which, combined or alone, provide new support for the direction of constructions. Moreover, we can not eliminate all associations that make a listener, and are an inseparable part of its assessment music: there is no limit to what they can cover, and may vary very strongly between tapping a song. 165 The object that is hinted must withstand these actions of listening: if the composer can not impose uncertain associations and links networks logical to the listener, it may nevertheless provide a consistent set of landmarks, 165We could say that the absence of these variations between reduced interest in tapping the play: it do revolves more with what remains to him outside. The size and organization of these associations is discussed in SAIDE. [1991] Musical elaborations (ch.3). Page 112 112 allowing the work to be entered in several possible ways. We believe that in fact the composer is limited to this: to offer an object that is as readable as possible, carrying an intelligible complexity. The idea to musically represent ideas mathematics fits this compositional strategy: we aim the possibility of build highly structured works while controlling comprehensibility. The organization of information in mathematical discourse is our copy in its "economy" in that it does not seek to eliminate all redundancies, but build a structure with them. . Meaning The discussion on the concept of musical sense naturally raises the possibility a musical semantics . Of course, one can completely separate sense and do Semantics in languages, but the use we want to make these ideas Music is simply to analyze, on one side, that is "provided with direction" and the other, "This means something." Both ideas often overlap, but are not confused (they can not be confused in music). Beyond the senses as we have defined it, a work may designate any something that is somehow "outside" to the network, it can return to contents which precede or surround (text or context, for example). Us associate "musical semantics" this possibility of referrals and designations: the work has a meaning when one establishes links between at least some of its parts and a consistent family of content. To build a perceived significance Page 113 113 other than himself, the composer must share with his listeners not only the family of content, but how to organize that supports these links to work. Without this common vocabulary, it can not lead the nominations: the associations of ideas are made "despite himself" in the interaction between morphology the piece and the Auditor's memory (although they may coincide with the intention starting). We are thinking here, speaking of meaning, as simple judgments to consider a work "sad" or "happy" or to a work which is an aspect of nature: the composer can direct these understandings that from when a reference directory exists for these ideas in music, and that all listeners this listening base 166. The example for us the most striking access Direct to the vocabulary of the listener is Richard Strauss: if he could write music that effectively pretends to be sad (like the last part of Till Eulenspiegel ), it is because it could be assumed that his audience would have in memory, at least passive, almost two hundred years of tonal music, with its codes and habits. There is Interestingly, this manipulation of the tonal system virtuosity appears when it is no longer the only one whose composers have when already pushed beyond its limits, and sometimes abandoned. Without this support on a "lifestyle musical "widely shared meanings depend on other pre Associations 166Of course, a "shortcut" to represent an aspect of nature is to put the exact sound directly into the work (through registration, for example). But this only bypass Page 114 114 listening to exist - an explicit relation to a text or context, for example. We return to this point in the chapter on musical poetics. Note before proceeding that in which we accept these terms, is remember the meaning of a work leads to recall at least fragments of this work; this is not necessarily the case for its significance. 4. Arbitrary decisions As we have seen through the comments on Lema 1 and a set convex , the simple representation of a mathematical idea does not exhaust the process compositional: many operations remain outside the chosen formalization. This can of course be attributed in part to a kind of "incompleteness" of representation (Other elements, other compositional operations might have been able to part), but rather we want to support the idea that true wholeness in sense that while the composition process would achieve representation is impossible. We will address the issues raised by formalization of music to study this passage from the idea to its representation musical and decisions that surround it. problem, not eliminate: if there is an act of composition, even this sample is placed in a context which can change the understanding, away from this direct reference or at least complicate the reference. Page 115 115 4.1 The ability to model music When it comes to addressing the practice or music theory through a formal schema, the adaptation of it about her is crucial. This adaptation must be expressed by a certain quality in the interpretation of expressions and results purely formal in musical terms. A perfect formal corpus would be one that addresses all the music in its results, and the results all indicate a musical reality. Of course, this description of a theory full and consistently brings us closer to Model Theory. It just explores the relationship between theory , ranging as a set of propositions (axioms, theorems ...) in a formal language, its achievements , understood as sets in which the proposals of the theory are interpreted. Model theory makes use of the word model which is not the usual: a model in this sense, is an achievement in which all the propositions of a theory is true. We will keep in this chapter (as did in the first one) common usage, which includes a model as a diagram formal (abstract) assisting in the understanding and prediction of concrete situations. It This is not, on the contrary, to deny the contributions that can give an approach model-theoretical points that will be addressed. We will also close to this point of view, times. But the considerations that will occupy us as rather aesthetic (or artistic in the sense of the practice) and will not target necessarily integrate model-specific results (in the usual sense) the corpus of results specific to the modeled object. For its part, model theory establishes a " Page 116 116 double reciprocal connection: that of syntax and semantics that of mathematics with metamathematics. It is their intersection lies the work the theorist of the models. » 167 It is the connection between syntax (in theory) and semantics (in the passage to the object) that we do not always seek to establish. In the study of musical representations of mathematical ideas, we are led to observe, at least at certain levels, operation of a musical work through a "prism" mathematical, either for analysis or for the composition. This inevitably constitute the first step towards a mathematical model of this work, which could be extended to other works, or music. Indeed, addressing a work in this way is already making predictions and draw conclusions from abstract results, external to the work, or at least compare these abstract results to "Facts" music. Note that this is a common situation in many Science: when trying an experiment in a certain resemblance to a pure mathematical situation, in some data values ​at the expense of others whose Leaving aside to constitute a formal framework capable of thought and comment predict the results of the experiment. This formal framework is the origin of a mathematical model for this experiment, and comparisons between structures musical and mathematics may be at the origin of such a model for music. However, the most famous mathematical tools (and almost all, actually) have not been developed in order to model a musical work. Unless we develop a tool 167SINACEUR H. [1991] Body and Models , Vrin, Paris Page 117 117 specific mathematical for a work or group of works, the matching model object must then either be through a judicious choice of the object (often the case for the built representations) or by transforming interpretations usual model chosen. It is in all cases to have a semantic operational, a way to pass objects and ideas to those of the model of the object. This passage is, as we have seen, in the heart of every musical representation. However, a fundamental difference is noted between the models of Natural Sciences and any modeling music. For those, which is observed is Nature, or one of its aspects. The investigation focuses on objects independent, somehow 168, Of the observer. Physics research share principle that "laws" of nature do not change, regardless of the aspects valued by a theory. Thus a physical theory can be rebutted by experience or be restricted to a few special cases. In music, this is only Obviously not. That is if "laws" strictly musical may exist, they are not immutable, and are not discovered or verified through scientific experimentation 169. They are composed , and therefore possibly subject to all kinds of changes, including those suggested by the musical experience. 168We know from quantum mechanics, it is impossible to observe a phenomenon without edit. We just want to say that this is not a creation of the observer or someone that it would be equivalent as to the creative possibilities. 169We do not consider here the results of acoustics or psychoacoustics as laws musical but physical and bio-psychological. Only use music, which requires at least aesthetic considerations (and therefore unscientific) can be musical. Page 118 118 This may, indeed, engage with the scientific experiment: development Technology has always influenced the musical development. This contrast between the two kinds of model is evident in the fact semantics: it is necessarily less accurate for a model of the music. Indeed, it is possible to associate a physical transformation to a formal concept stock, as the derivative of a function at a speed, music the very idea speed becomes vague, if we maintain the relationship between a definition of "distance" traveled and the time to go: these "distances" and this time depend on perception which is irregular, non-linear. A precise semantics for a model mathematical music would address not only the perception of how deterministic (which may, ultimately, be conceived), but also the contents aesthetic. But these, at least in our culture where music is speech individual , are inherently non-deterministic. We can not accept the trivial existence of Universal Music (independent aesthetic content the artist) to be established by a semantically consisting mathematical model. This semantics must therefore be "inaccurate" flexible. A rigid description of the Music by a model which describes the physical world (such as mathematics) would place in the position of an alternative description, but valid, Page 119 119 of this world, which also can not take place. In the words of P. Manoury 170"The role of art is not to propose a precise definition of the world." Work a musical work by the "prism" of mathematics going on by the proposal of a mathematical model for this work. The choice of the branch mathematics (or set of ideas within this branch) to serve based model already shows how shall establish the interpretations of concepts abstract in music data. It effectively defines a network of "preferences" within mathematics: the importance of multiplication, for example, is if we treat different topology or Arithmetic, and interpretation of this operation can not be the same in a topological model in a model arithmetic. It can also not be represented in one of these models. This reinforces the impossibility (at least problematic) strict consistency of these models: the operation is the same in both models (the importance it gets there does not affect its definition), a consistent semantic always interpret the same way. Once selected the model base, which is thus a refinement, accuracy of the mathematical point of view adopted from the beginning, all the interpretations are constructed from a selection of aspects of the work (or music) that will be addressed by the model. It may be a level of musical discourse a time scale, articulation between different levels or scales of these, or yet the relationship of a work with other works or other disciplines that 170MR P. [1991] Borges points of view , in Six Musicians Quest Author , Pro Musica, ANOURY Page 120 120 music (own mathematical included). It will be difficult for many of these aspects simultaneously: a model is constructed indeed for a particular situation, may or may not be generalized or will be integrated with models built for other situations. The difference between a model of music and scientific modeling appears here again: Mathematical models of physical world are articulated and integrated at multiple scales (time and space) and this through experimentation and observation. One can, for example, address the movement of stars in a galaxy with very mathematical tools similar to those of the kinetic theory of gases. In music such integration can be discovered . It establishes that a decision of the musician. The organization of all time scales of a work by the same "formula" is more the exception than the rule171. Of course, the idea persists longer will it be possible one day fully describe the physical world at all scales simultaneously, by a deterministic model. But the natural sciences, the unity of purpose, aim rather the integration of their models to keep them fully independent. Finally, each interpretation of an object or an abstract property is to Through the identification, in the modeled work object or musical properties they are structurally similar. It is able to operate with these objects and properties in work in the same way as in the model, or in a manner Isle-lès-Villenoy Page 121 121 close enough, and thus take the model as a tool for manipulation of these musical entities. We stated above, it is during these interpretations comes crucially, creative musician. Indeed, it is he who sets musical manipulations, their relevance and applicability: the structural similarity and operational between musical and mathematical objects depends on these definitions. Mathematical manipulations in the model induces then a particular set musical manipulations among those that may occur during the composition or analysis. This completes the construction of a mathematical model of a work (or music), which is formed by a set of mathematical ideas to its base, one aspect of the work (or music) it addresses, and a way to associate abstract operations of concrete operations (interpretation). We can say a musical representation of a mathematical idea is a set of objects, relationships or own structures at the musical discourse or temporal scale tackled by a model which is based this idea, this set of operations is provided and manipulations induced by this model. 4.2 The distance between object model In modeling, modeled object and the formal framework as a basis to preexisting model, separate, establishment of it. For the establishment of a 171In particular we believe in the work of K. Stockhausen, whose appearance integrator across all time scales has been repeatedly analyzed. See eg R IgoniM. [1998] Stockhausen ... a spacecraft launched toward the sky , Millennium III Ed., Rouen. Page 122 122 mathematical model of the music, the situation is no different: we also observed in the first chapter that the distinction and distance between mathematics and music are essential to their reconciliation, so are more let alone the definition of a model. If music and mathematics precede model and are independent, it can be seen as a way operational manage the distance between them. Indeed, the multiplicity of choices to make during its construction allows placing this distance, and sometimes minimize, if is wanted. The various comparisons between the structure and the theory of the subject (The work modeled, or music in general), and their respective sub-structures, indicate "anchor points" on each side, as it were, for the most effective interpretations for the passages between abstract and concrete. From the functional point of view, a model is then a kind of formal language describes the modeled object, which bears in itself the characteristics of the object on which will be made abstract and concrete manipulation. This "language" manages the distance between the abstract (himself) and concrete (music) by the content of its statements. A Compared with the point of view of the Tractatus Logico-Philosophicus 172, L. Wittgenstein, on the relationship between language and the world to which he refers us will shed light on the distance between the music and its mathematical model, as we understand it. In the Tractatus , considered the language is a formal language (logical) in the strict sense: it consists of symbols devoid of meaning and denotation Page 123 123 intrinsic, which revolve around a well-defined syntax. Its goal is accuracy, "it that can be said in general, can be said clearly "(Preface). A difference important is established between words (specifically, with the language) and show (without this language), "which can not be shown can not be said "(4.1212). This language contains So the possibilities of precise knowledge of the world, which are identified by the world himself: "The limits of my language mean the limits of my world "(5.6). Wittgenstein asserts, however, the existence of proposals out of his formal language, and an "off-world" on which nothing can be said (which therefore can not know precisely): "It is the peculiar characteristic mark of logical propositions their truth can be recognized in one symbol, and this fact contains in itself all philosophy of logic. Thus, it is also one of the most important facts that the truth or falsity of the non-logical propositions can not be recognized in the one proposal. "(6113); "There is certainly the ineffable. This shows , the Mystique. "(6522). As part of a mathematical modeling of the music, we initially have a formal language of mathematics. The "world" it refers to here is the music, or more precisely those aspects of it that models, with which a comparison (conducted by the musician) established similarities structural. The limitations of this language are so much those of this world will be found no more than formal properties (those that language could express) in objects musical than mathematical objects which they are interpreting, and by the 172Originally published in 1921. We follow the German-Portuguese bilingual edition, Edusp, São Paulo, Page 124 124 very construction of the model. Nevertheless, precisely fitted interpreting abstract concepts in music, we can make proposals non-logical (From the point of view of language considered here), in the sense that it is impossible to recognize their truth or falsity within the formal language alone. Proposals for where such a judgment necessarily involves evaluating properties musical (Thus not mathematics) may be set out in the formal language (in altering the functions of meaning to signs, for example), but they show beyond its borders. This indicates that the world described (specifically) by this language may be related , but it is probably not convex : it does not always contain a "straight line" drawn between two of its points 173. The Tractatus ends with the famous aphorism "about what one can not speak, one must be silent "(7). We believe it is necessary to understand that here on what we do can say anything specific, we should not try to say something specific: the result would be nonsense. This, of course, from the point of view of the book, for which a proposal is interesting (possible even) for knowledge if it is says. In the case of a model of the music, two forms of knowledge is out: that of the model (mathematical) and that of the object (musical). Can therefore do not be silent about what we can not say anything specific for any of this knowledge , the result which can be assimilated by the other. Music is obviously not the language 1994. The following quotations of this text are followed by the number of the aphorism where they are. 173We also believe that this non-convexity is the general case: what is beyond the limits of a language may be shown from the inside of this language, proposals to incomplete meaning or Page 125 125 its model as the world is the formal language of logic, according to Wittgenstein. It is, at most, a part of this world 174And the language of his model, with its interpretations in this small part of the world, is not strictly formal. Topic who thinks and represents music through a mathematical model is a position contrary to that of the subject Wittgenstein who "do not belong to the world, but is a limit of the world "(5632). It is the world of music, at least through the changes his perception will cause, and its limits are much broader than those of musical world. In addition, "the world is beyond my control," says Wittgenstein (6373), but the music is fundamentally dependent on the will of the musician. The distance between the music and the model appears to us as something dynamics of thing, as opposed to the fixed distance between the world and logical language: "In logic, process and outcome are equivalent. (So ​no surprise) "(6.1261). This distance is divided internally by the opposition between propositions "logical" and "Not logical" contained within the model, incompleteness interpretations, the dependence game and independence between modeled musical aspects and those that the model does not address. The will of the musician not only changes the music itself, but also the meaning of mathematical ideas in the model: model the music is much interference on music materials on misinterpretation, for example, which generate successive approximation of a point "surrounded" by the world that describes the language, but do not own. Page 126 126 employed mathematical tools. A compositional and analytical process that arises from a mathematical model, or generates a dynamically operates in the possible structural identifications between music and mathematics. The result, by his static characteristics 175, Can only be remarkably different from the process. 4.3 Decisions to make to model We saw in the first chapter that any rapprochement between music and mathematics through external choice thereto. In the particular case of the establishment of a mathematical model of the music, the independence of these choices is underlined by the abstract possibilities of the theory and structural wealth the object. The founding act of the possibility of such a model is likely to want to use of pure mathematics. By itself, it already marks the arbitrariness of modeling: this decision naturally can not be dictated from within the practice mathematician or musician practice. Without taking into account the will Initial, we would be led to consider an immanent and necessary link between the two practices and thoughts. But even in this configuration, the active judgment musician, and, accordingly, a network of arbitrary decision, is inevitably 174But rather we would defend the idea that only part of the music in the world described by logic. 175A musical work and analysis are not static all perspective - their interpretations and the influences they may exercise demonstrated - but fix , once completed, these reconciliations and structural identification. Page 127 127 present in modeling 176: In the choice of mathematical ideas that will serve as based model, and those that will abstract object manipulation concrete (musical); Also in the selection or the composition of the special features this object to be manipulated; in selecting manipulations effectively carried out finally that rarely deplete all all manipulations possible 177. A difference should highlight here between modeling music by mathematics and another by physics. We have compared a mathematical model of music with that of physics: the opposition is now between two different models (built on different theories) for the organization musical. Some similarity between the two is inevitable, given the context theory in which we treat the physics itself, which is mathematical. Thus a reconciliation between physics and music will certainly make use of tools mathematics, but applied directly to the physical only . Relationships will be established in such a model, between musical objects and physical facts, not "Facts" of mathematical ideas. Freedom in associations is then immediately distinct from that found in a mathematical model: 176Unless considered immanent link the strict inclusion of music in Mathematics: if this one is a proper part of them, the idea of ​modeling is meaningless. This is the perspective implied by J T. [2001] Objects Found Mathematics . The composer OHNSON supports the idea of a music objective , as opposed to music "happens subjectively and ideas personal feelings. " However, we believe that the results still bear traces of decisions arbitrary, non-mathematical, in particular as regards the arrangement of notes in time and in height (as the breaks in the catalog Agreements and heights chosen for Rational Melody No. 14 ). 177The decision to use all theoretically possible manipulations is also arbitrary. Page 128 128 sound phenomena are subject to the restrictions of the physical world. Thus, when struck a chord, and that vibrates its half, we always get what we call an octave. Here arises the problem of interpretation in the model, and the necessary distinction between mathematics and physics in this context we have identify a natural phenomenon, mechanical, musical phenomenon. If this identification may seem natural, and lead to an identification of the octave with the Report mathematical February to January, it's probably because we keep this in mind the shared rope image into two equal parts. But the visual relationship between distances, which can be established through this rope and disappears when the octave is obtained singing, or a wind instrument. The physical relationship between the frequencies of sounds obtained is perhaps still the same, but we can not say that a relationship purely mathematical or to the origin of the understanding of the played musical interval or sung, or at least not necessarily proportional relationship that between them the numbers 2 and 1. We can not accurately measure and compare these frequencies only with instruments for physical experimentation, so extra-musical and extra-mathematics, then to get the ratio of 2 to 1. A comparison between the sung notes and those produced by a stringed instrument for indirect the image of the division by two, as is arbitrary: it presupposes similarity (Physical and / or aesthetic) between the production of sound by voice and by a rope justify the passage of a proportion of one instrument to another. Thus music is is always accompanied by a physical fact, but its interpretation is not given a Page 129 129 priori in the fact itself. In our culture, a mathematical interpretation is required because all our modeling of the physical world is mathematical. Us find again the importance of a prior commitment to use mathematics as a tool for knowledge of the world: the interpretation of an octave as the ratio between two numbers, the combination of a sound spectrum to the Fourier transforms of periodic function is only possible because we seek a priori a mathematical consistency in physical phenomena we observe. In this context, identifying the association (required) is a musical with a physical done association with a mathematical idea would affirm that the organization of the world physical self is necessarily mathematical 178. This separation between these two models of music does not, However, using the mathematical understanding of sound phenomena (physical) as part of a mathematical model, in particular to define the interpretation musical abstract manipulations. Thus, digital manipulations can be interpreted, for example, as manipulations on frequencies expressed by real numbers, the derivative of a function as the stiffness of an attack. The model physics can then offer musical performances for a few relationships mathematics it fits between model and object, as part of the network of associations not mathematics that make this passage from the abstract to the concrete. Note that the 178Such a claim is metaphysical, and it is not within the limits of this work to propose the discussion. Simply specify that this is not our view: we believe that this statement Page 130 130 physics, and the relationship of sound phenomena in physical-mathematical model of world and becomes part of the arbitrariness in the purely mathematical model: the choice to make use of these relationships above operations and manipulations ready model, motivation is therefore outside any formal constraint. The intervention of the musician on the distance between music and mathematics while door Also on weight (possibly zero) that must have for his model and effectiveness its analysis or composition operations, the distance between music and physics. Specifically, this means evaluating the distance and differences between his and music , or even between hearing and listening . It is interesting to note that considerations these differences will often arise in music, prompted by different approaches the sound phenomenon and its potential significance. This simply inserts more mathematical modeling of music in all collisions possible theoretical (and valid) its organization or understanding. 4.4 Cross the distance between object model If a model can serve as a tool for manipulation of a concrete situation through an abstract framework, it is because it acts on the separation between these two poles, as in providing a cover by comparison and structural identification. The arbitrary decisions accompanying the construction of a mathematical model of the music and give shape to the distance between these two disciplines. These is reducing as much as the physical reality of mathematics. For a discussion of the evolution of Page 131 131 remain separate, but interpretations and established links make this distance any more. It becomes "populated", so to speak, ideas and associations non-musical and not mathematics which alter the perception and understanding. We have seen that distance, and deployment time is needed to approximation of mathematics and music. We can refine this statement, adding that its deployment to independent directions it separates disciplines is essential to a reconciliation that seeks to be operational (as one who seeks to form a model). Without this independence, Distance is "empty" and impassable: cross effectively means it assign a content expressive , which just creates the possibility of linking between them structural characteristics and interpret the non-musical (here purely formal) in the musical 179. This content, as arbitrarily assigned by the musician, is it possible to know that distance, to assess the similarities and contrasts that interpretations highlight. However, the knowledge and control of the distance between an object and its model are what can make sense of this: a model all Points identical to its object is not a model itself (it would be a also great card and said that the same ground that it represents). Differences and similarities not only allow to obtain operational simplifications and these ideas in physics, see EA B , op. cit. URTT 179Note that this expression can take many forms, depending on the model which implies: mathematical expression, metaphysical, aesthetic or religious. Page 132 132 significant gains on interpretive aspects covered by the model (and selected from those present during its construction in the subject), but Also, and here again affirms the difference with a model of the physical world, they allow to enrich the musical forms and relationships that do not exist before the modeling. Using a template is always address only part of the object modeled. In the case of music, this portion is changed by the modeling: the reduction of a musical situation (potentially an inexhaustible complexity by formal means) the only items that will be formalized generates new elements, she grow to its interior the modeled part of the music. The distance between mathematics and music, when so invested with creative possibilities music becomes the very relationship between the two disciplines: cross is a process art , either by calling or by analyzing, with what it implies judgments aesthetic, philosophical or scientific. These judgments are also present regardless of the distance to travel between ideas or structures in the relationship model. The musical object on which it operates through abstractions Mathematics may be structurally very similar to that abstraction, or have with it a common trait: the transition is always an operation a musical mathematical operation, and this passage is nontrivial. Note that this view excludes the possibility of discoveries in the field of music, at least through a model. In physics, an experiment suggested by the mathematical model strengthens or dismantled: it acts on the model. Page 133 133 In music, follow the suggestion of the model leads to create new materials and new musical relationships: it also acts on the object. One can find in because physical experimentation objectively reveals the physical properties of world (as the model can address). She can not reveal the properties music of the world because they are subjective and depend on appreciations aesthetic. You can discover the sound properties (they are physical), but they become musical that at their job. In the words of H. Vaggione 180, "The musical sound (...) is always what happens after a composition of act." In the instrumental field (acoustic or electronic), for example, new stamps arise from experimentation (with new game modes or new instruments, for example): they are purely aural discoveries, which may subsequently be used musically 181. A simple "catalog" of stamps or sounds can not be a musical work. Also for analysis and representations found, we will not talk about discovery is changing the model, establishing in theory and in its transition to new concrete relationships or interpretations, it is possible to emerge something new in a room already over. So here the model is modified or enriched, it is not 180VAGGIONE H. [2000] Musical Composition and IT Resources . 181A note must be made ​about improvisation, understood as composition in real time . When the experiment is possible in an improvisation, and to discover new sound properties simultaneously with the composition, it is only by means of a decision compositional beforehand that controls and limits. Composer in real time requires knowledge deep sound and musical possibilities of the instruments used, so that a "discovery" Sound is predictable in many ways, and is always preceded by a judgment on its relevance time work where it should fit. If improvisation makes use of experimentation, she said associates Page 134 134 only objective properties of the analyzed work, but also, and above all, by what the subjective perception of the analyst adds, that in no way is there to the origin. This lays as equivalent, at least from the point of view of efficiency or a model the possible correction the analysis (and a fortiori interpretations) seeking to approach the intentions of the composer and those who seek no. All are subject to the arbitrary aesthetic decisions that their under predicate, subject to change and revaluations 182. 4.5 An irreducible distance Many times, the musical structures that we approach by modeling mathematics appear to us as extremely wealthy formally, and very close to the mathematics used in the model. It is tempting to say, in these cases, that there is between music and mathematics more than a metaphorical relationship, an independent association of any model between the two. While recognizing the possibility of existence of such a link 183, We believe it is important to consider its potential consequences. Could it indicate the closest works mathematics than others? Would it be the measure of a certain "correction" mathematics of the work? In many ways, mathematics is a immediate aesthetic content, and a musical object thus obtained is not a discovery, but the result of an act of composition. 182We believe such different interpretations "correct" Baroque music who have seen e century. day throughout the twentieth 183We will not be placing, however, the only formal level, as we stated in the first chapter. Page 135 135 theoretical corpus consist "ideal" for the universality and immutability of its statements. And bring a piece of this formal structure could lead us to judge by purely mathematical properties rather than musical. We must then also ask if such properties when they are perceived in a room, not not just the strength of the model used to address it. Note first that Mathematical modeling is needed to perceive: the comparison between structures of musical origin (audio or written) and mathematical structures the fact arise. But we have seen that the arbitrary is always present in the passage a mathematical operation on a musical operation, regardless of the structural differences or similarities between ideas and objects put in contact. There Compared to other mathematical entities may arise other properties formal or even inconsistencies. Moreover, other comparisons (which bring into relationship of the different elements) are often possible with the same mathematical idea. Any property of a work has a musical component that can not be completely ruled out and is interference on the "purity" of the relationship of the work to another discipline as music. So we can not speak of "correction" in mathematics music without a significant reduction in the scope of this concept. The distance between music and mathematics, though it may seem to decrease with A particularly consisting model is irreducible and beyond a certain Limit: one of the choices for modeling, arbitrary decisions (who can not deduct formal operations). Ultimately, all music is also Page 136 136 remote mathematical ones than the others, it may be no more music "Scientific" or more "natural" than another. More contact between a workpiece and a mathematical formalism, in the form of suggestions or formal influences, does not indicate more proximity between this piece and mathematics. At most can we talk about the "success" of a particular model, if it is reasonably free mathematical contradictions, or if it is rich in interpretations that propose new musical relationships, for example. Obviously, this success can not be measured in turn, as compared to the intentions of the composer or the analyst models: the Quality is always sought musical first. There or a model is formally correct in its interpretations, another may be more musically fertile, accepting larger differences between musical objects and mathematics. Some "mistakes" Mathematics is then justified in a modeling of music as "mistakes" counterpoint can find place in a room. Again, the modeling process for the composition differs from that for the analysis. The composer dynamically sets as well as their rules ruptures, the analyst must accept the exception to a rule he proposes for a work, or change the rule. On the other hand, an analysis can be quite simply from general rules and goings of a musical use common, like, learned to conservatories, tonal harmony or serial contrapuntal manipulation, while a musical work, as simple as it can not be born of those rules alone "teaching". But analyst and composer Page 137 137 interpret purely formal rules musically relevant rules when use a mathematical model, and this firmly maintains the separation between mathematics and music, which can be reached only at the cost of choice came from Outside these two disciplines. 4.6 The need for arbitrary decisions We have established throughout this chapter some conditions for the construction of a mathematical model of a musical work, or an aspect of music. Against arbitrariness of several decisions to make in this sense, the question could arise if such a construction is valid if the formal approach of musical phenomena should not be done differently (logic or philosophy, for example). Indeed, if the connection between mathematics and music is limited, and there is no purely mathematical operation in music, what is the Depending on the choices of the analyst or composer trying to establish a connection between these disciplines? This question leads us to consider what distinguishes the work of these musicians a fully technical and formal procedure. To answer this, we must then turn to the non-linear and interactive aspects of performance production musical or musician. The internal organization of the work allows the listener and analyst explore it as "relief" they perceive it. A starting point and a method to monitor its progress (time or discursive) are sufficient to develop a interpretive trail of this work. The multiplicity of possible interpretations indicated Page 138 138 not only the richness of this structure, but also a sort of resistance the work to a definitive interpretation, which would exhaust the expressive resources. This is this introjection of the work that the composer develops: the "reality" is internal the more stable and capable of explorations it is self-consistent, what is 184. The composition is thus envisaged the creation or Free intrinsic contradictions organizing a musical world in some way separated from the "real" world (which, fact, contain), in that the constraints are different for the exploration and understanding of each of them (and not necessarily higher for the music world contained in the "real" world). This creation (the process and its result) is responsible possibilities of apprehension of the work by an auditor or an interpreter (also listener, as the composer himself). Freud compares this creative act for the literature, the play of a child 185: [E] ach child playing (...) creates a cleaner world, or, to speak more Rather, he arranges things in his world after a new order, his convenience. It would be wrong to think as he does not take this world seriously; on the contrary, he takes his game very seriously, it commits to large quantities of affect. The opposite of play is not serious, but ... reality. (...) The creative writer does the same so that the child plays; he created a fantasy world that he takes very seriously (...) while clearly separating it from reality. 184For literature, this issue is addressed by T TolkienJRR [1964] On Fairy-Stories . See also H. VAGGIONE [2001] Some Ontological Remarks ... , for the relationship between substance and reality within of a musical work. 185FREUD S. [1908 e ] Das Dichter und das phantasieren ( The literary creator and fantasy ). Page 139 139 The comparison extends to music: the composer creates a world to be explored by the hearing. If the consistency of the world depends on its internal coherence, Mathematical modeling can strengthen it, and can then allow a wide variety exploration (interpretations, analyzes ...). Arbitrary decisions accompanying this modeling are of the same nature as those that guide the music listening: they reveal the expressiveness of the composer or performer, they are the expression notions of coherence and musical depth of these and the listener. Judge and make independent choices of a deterministic formal system is essential for communication through music. The transition from a discipline other than mathematics (Such as painting, sculpture, literature ...) to the field of music would require also independent decisions of his theoretical framework. The challenge of modeling is always similar: a distance between "model" and "purpose" is to cross, one way which is given in one or the other, but built in the collection (and apperception) of the relationship between them has one who crosses. Page 140 140 II - G EOMETRISATION Take care of the senses and the sounds Will take care of Themselves. - Lewis Carroll, Alice in Wonderland In this chapter, we will use a geometric vision of the concept of form as it is configured from the appearance, thinking mathematician of non-Euclidean geometries. This is to consider the form rather than as a something intrinsic to an object, but rather dependent insertion into a context. Specifically, we will, like mathematicians, geometry as a work not on objects but operations on objects; accordingly, the form (geometric) be loosed objects in themselves and depend on these operations. Point of view of music, so let us also be interested in operations, and properties defined through operations; and we will examine to what extent it can untie the form (music) of the objects that suggest to take a musical space as a formal object. We want with this we look more directly on manipulations (Ideas and material) that may occur during a compositional process, which may aim to representations of mathematical ideas. Again, it is the of maths as a conceptual benchmark, alongside reflections Page 141 141 musical, which will be the focus of our theoretical elaborations. Our objective remains As always thought to be a framework in which to evolve with precision, rather a comprehensive and rigorous formalism; the accuracy of formalization, when it is present, is not provided a less important part of the consistency of this frame. 1. Figure and forms The distinction between figure and form is crucial in geometry: it is what allows separation between the operation and that on which we operate. This complete separation the independence of geometric reasoning in relation to the senses, and thus moves center of the geometry of the study of objects ("inherited" the concrete world) to the study of areas where objects are located. This is precisely what we want to transpose musical vocabulary we so we will look at some aspects of these notions of mathematical side then look for their musical relevance. 1.1 Non-Euclidean geometries The origins of geometry 186We find as the science of measurement land (hence its name), completely linked to material objects. But very 186The following historical overview is intentionally brief. We refer once again to the work of SZABO , A. [1977], The Beginnings of Greek mathematics , but also to S , M. (org.) [1989] ERRES Historical background of science , and G G.-G. [1999] Thought of space . RANGER Page 142 142 quickly the concept of geometry will move to a more abstract discipline, which give birth later to the Elements of Euclid, which deals with certain figures "Ideal". These figures are certainly always directly inspired by the experience sensory, measurement and proportions, but they are increasingly taken " itself "gradually loosened material constraints to be studied only for what to call their shape . This set of properties associated with an object but in a way, transcend matter, and constitute a kind of abstract generalization, will be placed increasingly at the center of attention geometry. But the link with the perception remained quite strong during a period very long: the geometry objects were much abstract and studied for themselves, but always supported by intuition, by physical objects that were (Platonic) imperfect representations. We can say that it stands Euclid the first of a supremacy of intuition on reasoning, striving to and axiomatically formally demonstrate results that are obvious. But it is there, from our perspective, a project that confirms this "geometric intuition" rather it renews: validation no longer dependent on direction, but it confirms the widely 187; Euclid does not care to show (or even study) which, in itself is wrong. In fact, when it comes to Euclidean geometry , we are not talking 187In terms of the geometry and the objects at that time. This investigation rational has also always used to warn against the inaccuracies of the senses in general - we do not see necessarily things as they are (in several senses of that word, throughout the history of philosophy). To Page 143 143 only theorems actually demonstrated in the elements , or five axioms employed there, but we also sub-along very often adequacy of this geometry to our sensory experience, our inclusion in space. We can say that this relationship is manifested through the history of geometry in the attachment of the notion of form to a face , or more precisely in the idea that the geometry objects are in direct correspondence with figures (even they are these figures) 188. It was not until the "discovery" of non geometries Euclidean, and in particular of projective geometry, to be able to dissociate and form figure. Indeed, if we can build a consistent geometry on transformations do not keep figures (as the projection), and if you put in highlight properties that remain unchanged by these transformations, we must rethink the subject of same study, and the idea of ​shape. This can not be the figure, which was until this visual support to geometric reasoning, which loses much of its force for the invariance operation . This is what real change paradigm that is reflected in the words of F. Klein: "projective geometry took birth than when one is accustomed to consider as completely identical to primitive figure and all those who can deduct in projection, and state the classical geometry, we can say that it is usually similar to the assertion that there was actually a distance (platonic) between object and idea. 188Even if they are never exactly representable: it is the uniqueness of this correspondence which is crucial here - a triangle always has the air of a triangle, there can be no doubt that this is not no hyperbole. This geometry remains the science of measurement on a "field", admittedly abstract but whose properties are those of space as we know it with our senses. Page 144 144 projective properties in order to highlight their position vis-à-vis independence changes made by the projection. » 189 Fig.16: an ellipse and a parabola have, in projective geometry, the same shape. When this is defined as an ellipse or as a pair of straight parallel has more the appearance that we are used, without this leading to contradiction, we must broaden our concept of form and separate from the appearance of objects: that "is" a pair of parallel or ellipse will well be the following certain rules relational and functional, which is maintained through some operations, rather than what can be represented in a figure an example of respect these rules. These operations actually define a context in which the figures are placed, and in turn defines how to approach these figures: this context is the space where they are listed, and each well defined space can cause a different geometry. As discipline, the geometry ceases gradually being the study of objects and their shape (intrinsic) to become studying this surrounding space and different forms of spatiality as support for the form 190. This implies a change in the role of intuition 191If the two figures above seem to indicate a difference in form is because we look a Euclidean eye (they have the 189Quoted byRANGER G G.-G. [1999, p.71] Thought of space (our emphasis). 190On this concept, cf. yet G RANGER G.-G. [1999]. Page 145 145 same form in projective geometry, but are in fact distinct forms in the Euclidean geometry). It is this habit that must get rid of, and the task is not least the notion of figures distinct or similar makes sense to us according to a Euclidean model. The form, as we must consider in order to speak of different geometries, must be independent of any figure: these are the figures that somehow "will" shape of the geometric context of space in which they are taken. In other words: a figure has no form in itself, it has the given shape by the geometry in which we see, for the use that fact192. 1.2 Musical figures and musical contexts This is of course in a musical context that we want to project these ideas: we aim to achieve as a way of speaking of form in music (several scales) highlighting its operational aspects, which tend to some independence objects. The idea is again that such an approach can lead to more precision in the treatment of this subject; In particular, we believe that over the concept of shape will be removed from material constraints (objects 193 sound or music in 191This is also what is apparent from the form definition given by GRANGER G.-G. [1994] Forms, Operations, Objects , which we quoted in the first chapter. 192We are getting closer and the relationship between words and their meaning proposed by W L. ITTGENSTEIN [1935] The Blue Book and brown notebook . Again, we will take this idea for music and a "sense" musical. 193We want in this section, as in almost all this work, keep an approach somehow naive of "object" (especially if it is equipped with the "musical" adjectives "sound") without making the text definitions certainly philosophically more accurate and complete, but less nimbly manipulated. We just take it as a 'grammatical' class: what we Page 146 146 this case), the more it will be discussed as a part of which is a direction music (we still see that the limits appear to independence). This Role of the form (local and global) in intelligibility and expressiveness of a work we particularly interested. 194 . Figure noted We can, in a certain light, compare the musical work done with our traditional notation to that made with the figures in geometry, as we quoted above. Indeed, the rating is supposed to represent the musical objects and allow them indirect manipulation. But it also allows for a better understanding these objects, even though it still shortcomings and inaccuracies: it can be, for example, easier to recognize a melodic demotion or reversal of a complex agreement in writing that listening. Thus, we find the concept of operations on a object that are inspired by the possible operations on its representation (and sometimes limited thereto). In addition, the rating is being established as a simplification of its instrumental to a few selected characteristics, it excludes certain manipulations: refers, that is manipulated, which is what we work; we will clarify further the distinction between sound and music that we will use. This position is also consistent with the importance we want to give to the study of operational aspects in the process, although they may vary their objects. Further discussion of this definition is proposed in the work of V , A. [2002] ILLA [2003], and SOATUU DIASA. [2005]. 194We will return when we discuss the modular and composed spaces. Page 147 147 those that remain confirm the "intuitive" Input music 195, As the Euclidean geometry confirms the apprehension of space. We see thus that there is risk of identifying the subject of study and work this is on the partition 196: It could be limited to the knowledge that gives writing our object. It is precisely the incompleteness of the notation that reminds us the existence of other functions and properties, excluded from the partition, which is attached to noted.197 Not only they are the basis of the change in the rating, they also used to capture the relationships between them can maintain different writings (eg at different scales). If we can call a musical figure is obviously not only what is on a partition, the two things could at one time be least conceptually separated today (a musical figure could be "Sufficiently good" shown in the score). But we must consider what is as shown , and which could be other ways, as the real object on which the job is done. In fact, we already have a grasp of what can become a musical material before writing, and this apprehension is not fully 195 it was at the at least;this butreport we could argue athat, after scoring wellAsestablished andtime, expanded, continues longeven time: it will probably be the twentieth e century concrete music for failing to note a musical idea yet clear, understandable. 196That is if it is always a risk, properly speaking: the music has been a material Theoretical enough to accept this type of approach - we think including motets over forty voices of the early Renaissance and musica mundana the Quadrivium (although it predated our rating as we know it). One could say, a radical perspective pragmatic, that the object works a composer who made ​a partition is that this partition even (and therefore the figures around) - but that would be to deny any reference to an instrumental gesture. Page 148 148 covered with writing 198. As we said in the first chapter, the idea writing is related to proofreading, and thus a symbolic device that allows operative approach (particularly articulate phrasal manipulation and paradigmatic); but any symbolization imposes a distance between the symbol and it achieves indirectly symbolic device relies, by definition, on a level sub-symbolic that it can not exhaust. 199 It is not a question here of a any "resistance" of the object symbolization, but simply because a formalization (and any notation) focuses on some aspects of this it addresses at the expense of others. We can establish different symbolic approaches a set of objects, adapted to the needs or point different view, and it is the presence of a common sub-symbolic level that binds all. 200 It is precisely this sub-symbolic level that we would like to place musical figures, then the distinction between figure and form that we 197Incidentally, this underlines that, although a full notation was possible, it would not necessarily useful. 198We believe, for example, the classic commentary on Wittgenstein sound of a clarinet, we know it without being able to describe or describe this knowledge. (Cf. W ITTGENSTEIN L. [1949] Philosophical Investigations , aph.78) 199The term "sub-symbolic" gets different meanings depending on the discipline that employs; we take it here simply indicate which is not directly manipulated and which exceeds or does not exhaust, in sense, a symbolization. A very thorough discussion is in V H. [2006], Symbols AGGIONE signals, music operations . 200See the comments of H. Vaggione during maintenance "Schoenberg-Wittgenstein ', OLOMOS in S , MR., SOULEZ , A. & V AGGIONE H. [2003], Formel-Informal: music-philosophy . We could question the uniqueness of this level: do we always speak of the same sub-symbolic when it "guesses" symbolisations behind different? Can we accurately communicate this substrate? Reply adequately to these questions beyond the scope of this work; we adopt the view that it is possible, at least enough to adapt his vocabulary to communicate on this substrate as if it were unique. Page 149 149 did for geometry: equipped with a spatial intuition, we built a geometrical figures (and, originally, shapes) that attaches to this intuition, and may later deviate without loss of consistency between the very figure of its invariance under specific operations; similarly, equipped with an "intuition Musical "(which we could then take as the apprehension of this level subsymbolic) we would build a writing that was suited him, and who could depart apprehension of the original, for example by suggesting new materials, separating and the object manipulation it undergoes and its integration in a context particular. However, such a project can succeed completely: even if a write adapted to this "musical intuition" is possible (that is also the reason for a writing), the separation between the object and its manipulations or context is only ever partial music. Indeed, if one can distinguish easily figure and fitness mathematics is also because studied and manipulated object is always abstract physically absent: it can be represented by one or more figures, it has properties that give it its shape and can be written without using the figure. In music, objects have a concrete existence, phenomenal: one can not help therefore assign internal properties, intrinsic (specific characteristics which apply to perception). Additionally, their definition (and their manipulation) often depends on aesthetic judgments, it can vary; it is then sometimes impossible to write or describe an object without presenting himself 201. 201To changes in the definition of the object are added here the case where the object is defined so Page 150 150 We can say that a sound object is always in a context from when it is perceived: our categories of apprehension are never neutral, they submean analysis of this object operations, the interpretation of a phenomenon as a particular vocabulary; the same identity of an object, the separation of which can surround, is subject to change, since it can "lose" a sound object in a particular environment 202. When we invest in our listening a set of sounds, we insert thereby in a context we already impose on it a family operations (and we therefore exclude others). Thus, it is inevitable to consider the properties of a sound object that are external to musical contexts where it can meet, it is nonetheless essential to take into account the operations used to enter these properties. We nevertheless maintain a distinction between what is inserting in a musical context and what is somewhat independent of the insertion. This is possible (and useful) if we take the idea of ​a figure associated with notation or in writing. As we noted earlier, geometry proposes to separate the form of one of his presentations in a figure to consider it rather as a family of invariants according to some operations, and this leads to study the space that define these operations, which can find objects. Taking musical figure as something that is noticed, we can see how our overwhelming sub-symbolic, as when taking its concrete or direct work on waveform. Page 151 151 form of idea is related to this notation, and start to separate; Moreover, we can and begin to define what can be a musical space from doing operations only on written material (and then consider more situations abstract). This approach allows us to consider the properties of the figure itself, internal relationships that composes such as rating or write: Figure thus truly taken as a "drawing", like arcs in Poincaré half-plane, which are or are not "straight" in this space, though still identified visually (without relation to their function or properties in the space in question) as arcs. By combining the figure to a level only symbolic, it is easier to separate both its form (use that in fact) and the object it shows (once established, the figure is fixed and has its own attributes). to. illustration: as a melodic line Identifying we often make the shape concept to figure denoted appears in a simple example built with operations on intervals: we usually think of as having the same shape melodic lines are graphically superimposed by translation or by symmetry with respect to a straight (Vertical or horizontal), which corresponds to the transposition operations, reversal or intervals demotion. If we add to these operations "Retain shape" the multiplication of intervals 5 modulo 12, we get 202We think of a glissando in glissandi frame, or to change that undergoes its Page 152 152 new figures have the same shape without being graphically congruent, and we have a new geometry and melody lines. Fig.17 If the increase in intervals of 5 modulo 12 retains the shape, these two figures are melodically isomorphic. By wearing the kind attention to the manipulations that we subject objects, so we are seeing the space where they are. The multiplication operation 5 by modulo 12 gives a characteristic space intervals where are these melodic lines: it informs us about the behavior and organization of intervals, not on the intervals themselves. Similarly, if we rather interested to symmetries with respect to a line, we inform us on graphic behavior (in terms of the score) figures, not on figures itself: thus we observe another space than the intervals. The same figure has properties (and thus the possibility of manipulation) different depending on the area where we consider: his form is on this context of operations, it is extrinsic to it (and stronger extrinsic to the object it says). . Link to perception Naturally, this new geometry of the intervals is not supported by the perception of directionality, our "intuition" indicates that these forms when the attack is hidden by that of another sound. Page 153 153 different: it is a reaction comparable to that aroused by the elimination of the fifth Euclidean postulate geometry (we have parallel that approximate or no parallel lines at all). Note that other transformations that maintain geometric shape of the figure on the partition can also break the geometry intervals usually perceived: symmetry (graph) from a straight oblique, for example, may transform a monophonic line in an agreement. This will indicates that the geometry of the melodic lines to which we are accustomed is not in fact given by the graphic partition space, though many operations are historically emerged thanks to him 203But by the operations we choose to be classified as those that retain shape (or the change of a musically acceptable way). The question would then arise if this removal of "intuition" caused by the addition (or removal) of certain rules purely formal poses a problem to the composition. Indeed, if we present isomorphic as very distant objects to material properties, we impose may be a direct transfer of the consistency of written manipulations on the material sound - if the formal manipulations are equivalent, then the understanding of objects should be too. This confusion between composition and listening (between poetics and aesthesic), which is, in our reading, in the heart of the criticism of Xenakis music serial 204Is naturally avoided. But we must not necessarily rejecting these operations 203This is the case of certain types of counterpoint, such as guns and demotion inversion. 204In X ENAKIS I. [1963], formal music . Page 154 154 which may upset the usual associations and identifications, neither the research consistency: it is not a formal tool that is defeated, but its use for as it were improper - that identifies two different families of isomorphisms. On this point, remember that the rules are added to a "usual geometry" of our apprehension can be precisely choose for this particular impact they have on the perceptual result of relationships of "equivalence" written participate in building listening for connections that do not have to be of the same type. Again, the formalization should enrich the possibilities of listening: it gives more to hear, without having to specify what is thus given. This approach to break with a sense strictly related to geometry, for get new listening relations, is often that of Xenakis himself. In Indeed, as indicated Hoffman, using geometric structures in "spaces not the basis of our daily space experience, "Xenakis could "Perform geometric manipulations that led to his musical imagination new solutions and sound unheard, or by defining a formal calculation, either supporting its intuitive ideas . » 205 The geometric formalization (whether or not serve a calculation formalized) thus serves well the material result, although formally avoids some of its constraints. The idea to deviate from what is directly sensitive to generate a concrete musical richness is also not new: the counterpoint Double is not really noticeable as such before the voices are reversed 205HOFFMAN P. [1997] The abstract space in the music of Iannis Xenakis . This emphasis. Page 155 155 actually - morphological properties of this counterpoint naturally present from the beginning, but have a sound and musical impact after an operation further to reveal them. In a way, the true form of counterpoint Double appears only when this inversion when exposed the function of features that were already there yet as a phenomenon (in the sound object). . morphological eidetic × This observation raises an important question of terminology: If the form appears only when it fulfills its function (or more precisely, if the fact able to fulfill this function), can we talk properly properties morphological which do not appear, which have no functional impact? In Indeed, the common usage seems to associate the term "morphological" to features intrinsic to an object, which would be a constituent part of and therefore define the interior; the "morphology" of an object is somehow which resists multiplicity of observations that can make and remains as a common core, as it predates these observations - which remains almost finally subsymbolic . But the etymology of the word refers to "form" ( μορφή) we want here extrinsic and operative, strongly dependent on the context of apprehension and particularly in the case of music at least, a symbolization. In addition, "Isomorphism" in mathematics refers both to that which maintains understood form operably two algebraic structures are isomorphic if one can pass to one another while maintaining the operating characteristics of each side. In Page 156 156 Somehow, the current use of "morphological" refers to what we call here the figures : it is in these that we identify separate properties contexts where we can place them (two different designs on the page can have the same shape, the same pattern may have different shapes 206). We believe it is useful to maintain the association of "form" in operations and "Shape" on ability to abstract from one context (at least one), so the adoption an end to the different etymology seems necessary. To stay in a field Semantic close μορφή, we propose the term "eidetic" (from ε ίδος: form , Figure but idea or kind ) for what is relative to the figures. Naturally, certain precautions are necessary regarding the musical use of this term already present in the vocabulary of philosophy. A primary motivation for this choice, prior philosophical, is the origin in εί δος suffix -oïde , "Resembling" the figure is what remains similar despite the change context or function (so that is independent); geometry we identify Figures visually, so by similarity criteria. For against, in the Husserl's terminology, "eidetic" means the phenomenological understanding of species in general, regardless of the existence (the eidetic reduction ); Plato Rather, it refers to something that is independent of sensory experience, "Immutable form"; Aristotle, the "eidetic" is related to the phenomena by 206A music sample appears in the shape node Cavalieri Princípio ; v. illustration Ch. I, particular fig. 9 and 10. Page 157 157 need to observe in order to know the essence, the essential truth . jobs are closer to the meaning of "typical" or "idea", but we can still see the use we offer as a special (and very short) thereof: if a musical context is a particular sensory experience, or a framework for such an experience, a figure well in the sense that we take the term independent of that experience (again, it remains beyond the context); similarly, the existence of a musical object occurs only in a specific context (at least that of our apprehension methods), and we can consider what may remain apart from that context. The same figure does not exist in these "types" 207 These special. Naturally, perhaps we force in this manner using philosophical eidetic , but we want to rely mainly on its etymology musicological for this use. Indeed, we are not here in the presence of "ideas" and separable representation or objects that carry them; the figures that we consider, although separable from certain contexts, remain largely symbolic. We can not actually have attributes absolutely eidetic, even in the sense that we take this word to suppose that it was possible to grasp such features of an object, we could not do anything with it before you wear previously directed gaze, so inserting into a particular space. More Specifically, we must keep in mind that in a compositional process, subject 207These considerations are naturally not want a summary of the positions of these philosophers, but just a general benchmark that allows to direct our terminology. We refer to a little more discussion, to B works , A. [2004] Introduction to ARBARAS Page 158 158 and background are built simultaneously and interact: there is no real Figure formless, of "context-free" complete. The interest of our differentiation Morphological and eidetic precisely is the ability to take these words relatively, and possibly nested at multiple levels into each other: one clearly differentiates what holds the musical space where we work and what the beyond, keeping the option to change repository and reassign those adjectives. 208 . Objects The foregoing discussion should not prevent us from speaking further objects sound or music. On the contrary, we can see an object as something that "Exceeds" always a figure and a form: in each context, it will have characteristic eidetic not morphological (which have no role in this context); on the other hand, a figure can be a carrier of properties that the object can not not have in itself (such as a graphical appearance), and it can let represented by more distinct figures. 209In addition we can now make a difference Simple terminology between object and sound musical object: we will talk about sound object when it is no question of eidetic properties, therefore unrelated to other objects or a particular context and musical object when these reports are taken into account. , L. (ed.) [1989], Plato himself , and R ODRIGO P. [1995], Aristotle, philosophy of Husserl , G UILLERMIT eidetic and phenomenology . 208This sequence across different scales between object and context, content and container, and all party, was already treatedCHAEFFER by S P. [1966] Treaty musical objects . 209This is reminiscent of the report of a work for a formalization approach: it is always "Shifted" compared to this formalization - she exceeds in the analysis and exhausts by when composition (see cI). Page 159 159 Again, if we somehow ignore the context for the first apprehension of the object, with the assumption that it is shared as a "Maximum context" that would be as general as possible and contain other as special cases (for adding rules or constraints). 210 Of course, thus defining the sound by eidetic properties, we must be careful not to identify its figurative or symbolic representation - we do not do here that oppose the musical and morphologically. Indeed, on a partition or a waveform editor we can sometimes regarded as the eidetic graphic properties (they may not have musical role). However, for they were the properties characterizing the same sound, it should be considered that this partition or the editor are the most general to seize and understand objects; we prefer to take the course as a way to more handle. 211We maintain the distinction between the object and a figure representing the precisely because of the uniqueness of it and its modes of apprehension. Thus adding properties (locally) eidetic objects, a symbolization, and a fortiori a rating, suggesting them as properties Morphological; more precisely, the space where they feature a symbolization of the sound is 210We can think of the eidetic characteristics of the sound object as what gives it its shape in this "maximum" space . The question may naturally arise if this space is truly unique, and even if there at all. We will simplify the discussion on the assumption that we can as reasonably share the ways of understanding the objects defining the contexts in which they can be inserted, and therefore we can consider at least a position of apprehension local maximum (containing the contexts of interest). 211Without this identification, we can nevertheless take into account the reversibility of some symbolic systems, where it is possible to switch symbols to sounds; this is the case Page 160 160 Musical immediately because it is functional. We can say that eidetic, at all levels, is the "salience" of the object: These are properties that emerge in certain contexts and which exceed by offering larger and this as much in the composition process of analysis and listening 212; we can initially refer to an object, and from that the handle, by its salience. 213 Recall that, in all cases, which is given to listening is a set of objects (or, even more surface level, a single object - the piece of music). It is up to the listener to grasp the patterns on its own, to assign meaning (Interpret). Again, we assume that there are ways of understanding common to the composer and the listener, which allow advance construction some salience; but listening, ultimately, that will define locally, different scales and morphological eidetic. In this discussion remains the problem of defining what makes two objects different: what characteristics (morphological or eidetic) that distinguish them, are they separable properties of figures and symbolization themselves, and to what extent? To clarify these differences, or often publishers waveforms and spectra. This reversibility is also discussed by VAGGIONE H. [2006], symbols, signals, music operations . 212The musical use of "emergence" relatively recent drift (and differentiated) from that in cognitive science. The fallout from this idea of spontaneous overflow of a context are manifold; we are interested here in its operational aspects and "morphophoriques" (generators shapes). A depth discussion is proposed and initiated by S M. [2005] Notes on the concept OLOMOS of "emergence" and Agostino Di Scipio . 213We practically find a methodology of work and from the salience among Horacio Vaggione. See eg V H. [1995] Objects, representations, operations . AGGIONE Page 161 161 to measure, we will use musical variables : the elements, aspects or properties in which we operate, and that truly allow us to enter a object. As we observed above, consider the operations that can undergo an object amounts to considering the space where it fits: knowing the variables and their associated operations that define a space, it is possible to say what is proper to it (which will be the basis of morphological features) and what belongs as feature, a figure. This process of isolation , somehow, the distinctive features 214We objects is not unrelated to the construction of qualia Goodman precisely aim the possibility of ordering or at least organize, qualities that do not always bend to definitions, let alone action, strict. 215 2. Musical spaces We are led by the distinction we want to do between form and figure to take care of the spaces where the figures found that we take to handle sound and musical objects; we are defined by operations: they will characterized by what can be done with the figures (and therefore indirectly with 214V. GOODMAN N. [1966] The structure of appearance . Although we were operating, both for composing as to analyze, actually that through these variables, we do not want here to identify an object and its qualities we keep still open, as part of the same musical thought, ability to change contextualization, therefore set of variables, as often as desired. In Specifically, we believe that within a compositional process coexist several "game language "among which it is possible to oscillate. Page 162 162 objects) within it. We are not far from the concept of space as Wyschnegradsky 216, The course of a continuum pansonore (he kisses all sound possibilities), but mostly seen as a space pattern of thought . Motivated by the above discussion, we can think of that would be an elementary operation just to get from one figure to another that it is similar in that space this would compare figures who have close forms, for example, or observe the changes of shape undergoes a figure by "moving" in space. But to do this, we need to define and estimate these similarities and variations - in an example like melodic lines, we take do account that the sequence of intervals, but in a more interesting musical context (and more realistic), the focus is on several aspects of both. The musical action (Compositional or analytical) is also guided by the perception, and it is not generally focused on a single point. 2.1 Dimensions and parameters To think these qualities, 217we start from the idea of ​handling, operation on something, and we all used to compose operations (An object, a section, a room, according to the scale at which we place ourselves). Us 215Unlike qualia , musical variables are directly (and only) operative, and acquire meaning in the process of composition and interpretation: we do not seek to studying the conditions of existence but the working conditions. 216Presented by CRITON P. [1997] sensitive areas . 217The following considerations are made from the viewpoint of the composition, but the constructions adapt to the interpretive process. Page 163 163 Also consider the set of all that can be manipulated according to these operations: it will be the musical variables . Note that these two sets are compounds, and they influence each other. The variables therefore constitute what is manipulable in an object (continuously or not): it appears that, 'where' and 'how' one can take in the compositional process. 218In a way, they are all that has covered in the object, and can be identified (operatively in processes) it to the state of his musical variables. It is important to emphasize that, although by definition an operational category 219A musical variable can consider also give physical (harmonicity a spectrum) provided that they are subject to compositional operations. Where, in a process, a musical variable can vary independently all other musical variables considered, we say that it varies according to a musical dimension of space where we situate (eg if to compose, we believe that it is possible to change pitch without changing anything else, we can talk about the size of heights). Note that the independence of a variable in relation to others is not an intrinsic quality: it depends on the choice (overall) of all of the variables, and a 218The vague definition is based on the fact that all operations to compose a piece is necessarily limited (and usually discrete). She says the organization (also compound) in this set of similar transactions families, and in particular by the choice of musical dimensions among the variables (below). The similarities between operations could also be a qualification (as opposed to a quantification), but there is enough here to consider that one can differentiate operations without to look on how this is done (for the reasoning that follows, we simply need operations, and the variables, distinct). 219On this concept, cf.AGGIONE V H. [1995] Objects, representations, operations . Page 164 164 organization of this set (ie operations that are done). What and how many are these dimensions and also depends on these choices. In particular, there is no necessarily only one possible set of dimensions among the selected variables to work (several sets can be taken separately as independent their complement). We chose here the terminology of variables and dimensions not only for association with the vocabulary of mathematics, but mostly for power differentiate with parameters in music. Indeed, we often see References to "sound settings", indicating as high-intensity-all stamp-term measures such as the spectral density, or simply numeric attributes of a MIDI note. We want to distinguish purely controls digital, that could somehow associate with a single potentiometer, those which depend on a judgment (musical) to be defined. Thus, in a modulation frequency, modulation index and modulating frequency are parameters whose change primarily affects the timbre of the resulting sound. Similarly, we intensity as a variable (and possibly one dimension) music, but the amplitude as a parameter. Indeed, it does not always act primarily on the perceived intensity: in additive synthesis (rich enough), the change amplitude of a component once again affects the stamp before a variation intensity arises. Of course, some situations can identify a parameter a musical variable (such as indexing midicents to height), but we believe Page 165 165 This distinction is relevant: it takes into account the role of our perception and our understanding in defining the elements on which to build operations of a compositional or interpretive process. We could also say a musical dimension, even in isolation, already evokes a certain existence music, which is not the case for a parameter: it seems that saying " C sharp " or " high , "or even" at this point , "provides more information on something that could these attributes that saying " the modulating frequency is 220Hz "or" amplitude is 0.7 "; this are in any case, for us, the best starting points for the construction of an idea musical. . Illustration: a first musical space We begin by considering what is manipulated (musical variables) and operations that we can use. Note that these two categories are composed mostly simultaneously - you choose what you do and with what. There variable list is not meant as a simplified example and is not exhaustive for real musical situation (for both variables for operations): heights (nominal); transpositions fundamental frequencies collected; additions, multiplications intervals; inversion, retrograde, additions, multiplications harmonicity; distance / approximation of a theoretical harmonic spectrum register; transposition Register on an instrument; change travel intensity; increase / decrease (continuous or stepwise) Page 166 166 dynamic 220; increase / decrease (continuous or stepwise) spatial localization; displacement, permutations temporal location; displacement permutation duration; multiplications, additions, subdivisions speed; increase / decrease (continuous or stepwise) rhythmic organization; combinations, subdivisions "Stamp" (as the identifier of the original sound) Reconciliations / removals, fusion / fission "stamps" order of appearance; permutations, demotion As part of the composition of an instrumental piece, we could isolate among these variables thereof, which allow to indirectly treat the other without (much) influence each other: Register relating to an instrument dynamic spatial location duration order of appearance "Stamp" We have built a musical six-dimensional space (or more precisely eight if we decompose the spatial location into three independent dimensions). The operations on each of these dimensions can help us to define distances or a way to "measure" in this dimension. Other musical variables we 220Here we distinguish between intensity and dynamics: it rather refers to the gesture of the player that sound result. Thus, the move to a score high on a violin with a mute lead Page 167 167 consider may come into play here as well: for example, one can choose to subdivide Record each instrument according to a tempered chromatic scale, measure and subdivided into multiple durations of a fixed unit, or to organize the stamps available according to their harmonicity. Soon it will be seen that it is very difficult (if not impossible) to obtain a set of musical variables from which it is possible to extract truly independent musical dimensions and still take into account all variables (in the above case, the "patch" is not entirely independent relative register, for example). But this does not prevent us to design independent manipulation of these variables - and it is this independence operations of interest for the definition of these dimensions. 2.2 Directions and fragments of space A first feature of the musical space in which we work is so operatively given: its dimensionality (quantity and quality of its dimensions). Other characteristics emerge within each dimension (sometimes their definition: how to measure it or "move", for example), or involve several dimensions in an operation (as defined distances in space in full, or transformations of space itself - rotations, translations, reflections ...). This, again, is made with and by the set of operations unused. Musical variables that have not been considered independent evolve probably give an intensity similar to that of a note mezzo-piano without muted, but Page 168 168 according to those that indicate the size and form of directions particular in the space in question; they are of course also manipulated directly, and a change depending on their direction implies a change according to several dimensions. The dimensions are directions that define and enable somehow to reconstruct the musical space any action in or on this space affects (Directly or not), and they are sufficient in order to specify an object. But this construction an object from only some of the musical events can be also 221In the example above, from another set of independent directions we could get an object by choosing an instrument (a "stamp") and pitch, intensity, rhythmic organization and spatial location, instead of the dimensions we had initially chosen; the result could also be written by them. We thought the musical variables associated with the musical objects and sound, yet they were used to characterize the spaces. This is how a space proving to be a particular possibility of objects 222: These are sort of "Pieces" of space, state (or family of states) special variable musical in. In a compositional process, it is the space that allows us to see objects: the operational categories that allow us to capture or call the object must precede the already defined (at least implicitly) a space. Us with a timbre, articulation and significantly different energy 221This is the equivalent of a base change in linear algebra. Page 169 169 can therefore have many different spaces that coexist in the composition of a work: each is derived from a family of operations, a set of intentions compositional . To listen, we have the opposite situation: objects somehow there preceding space; more precisely, it is perceptible only by objects presented to listen. The musical space which is thus made clear is that the objects completely fill: there listening choice but what is indeed the case (it is "might" not listen to anything other than what it was listened). All geometry is given by the inserted objects in time, operations that define it are those effectively levied on and each other. Of course, with the progress of the work, these operations (or at least what is included) is change and multiply: the space in which the entire work is part, the one she completely filled, is redefined continuously. Consequently, for listening and interpretation 223The organization of the musical discourse, its logic (whatever that is), do can only be immanent : it's all joints (and their sequences) that presents all the "rules" actually followed; This is where one can emerge musical sense, as we définissions above. In a way, if there is a any "musical grammar", it should always be done or redone in 222We could say that in this sense a musical space is a logical place , as Wittgenstein takes: "The logical location and the locus agree in the fact that both are the possibility of existence. "( Tractatus Logico-Philosophicus , 3411) 223Our argument is based on listening and sound phenomena, but the reasoning extends quite easily to the interpretation of a score (taking itself): space is also given by the Page 170 170 setting is . 224We come back to the need for the composer to make clear and perceptible a certain organization, make interpreted his work. Us could say that it is the trace of these local redefinitions of musical space (Resulting in the space of the entire room) to be able to organize in the memory the listener. 3. Musical objects as functions We now return to the view of the composition to address the direct manipulation of the material or musical ideas in and with a space. Us want to highlight in our approach the passage of independence musical dimensions when composing an even empty space to their inextricable bond in a compound object, especially when the latter is inserted into time, but already when we consider just relationships and transactions the inside of his own time. If we think of music as objects inserted into a space with musical dimensions, it can be useful to represent and build these objects from of them. We propose a formalization for the construction of a musical object, relying primarily on instrumental music and for periods rather objects, there are no inherently substitutions. It is possible to change a partition if you can borrow, at least in part, the views (and often also on the context) of the composer. 224Even, perhaps especially in the case of tonal music: although the "grammar" above several works, it was rebuilt and exposed again in each of them, and could be Page 171 171 meso-temporal 225But so easy to generalize. Our goal is to arrive at a description of a musical object O according to its 'internal' duration, O = O (d), or O: DƒEspMus, where D is the set that defines and articulates internment the total duration of the object and 226). Us EspMus is a musical space (such as those we have defined above then want to insert the object in the course of the work, and at a time specific: O = O (t0, T) after this insertion, where t is the time course of the work (continuous and irreversible), and t we place the object. 0 when 227 3.1 Sets of variables Each musical variable we choose to handle, we can constitute an assembly with its "values" possible: these are the elements of this which together define the musical object according to each of its dimensions. We want naturally our formalization permits the insertion of a musical object in the time, but also that manipulations are possible off-time , so slender in constraints of continuity and irreversibility. This is why it is interesting to deducted. Historically, it is from works (and not before them) that such "grammars" settle. 225In general, a note to a short sentence. 226More precisely, it is a space composable . We will introduce precisely the distinction between spaces composable and compounds in the following section. Page 172 172 taking any subset (sometimes continuous) "values" as a single element, especially when we work on quantified durations discrete: a By example, is a change in dynamics, but we want to apply this variation in a "note", without being obliged to consider shorter durations this note (or another selected unit). To do this, we will consider sets of parts , rather than isolated elements collections; but we will write often, by abuse of notation and for simplicity, only instead of ENS 2 ENS or Ð (ENS) (the usual notations for all parts), for a set ENS variables and x instead of {x} for an element x "ENS giving {x}" Ð (ENS). This detour also has to introduce as the empty set element benefit |, which we may associate in each set, different meanings without undermining the consistency of formalization. Some of the variables that we take in this section are likely slightly different from those we have introduced above; we think particular game modes and the position on the instrument . We can of course take these aspects of instrumental gesture as variables, since we manipulate effectively in a compositional process. But these are variables that seem more attached to a particular instrument at a general musical space, one might well hesitate to take them as candidates for a musical dimension. Us 227Note that this implies immersion D 2 T, where T is the time of the work, which corresponds to Operations passage (temporal and others) off-time to on-time operations. We return to this immersion. Page 173 173 see that despite this particularity they keep global interest, and it is possible easily circumvent attachment to a single instrument. We do not claim, of course not an exhaustive list of variables used in instrumental music, the sets we construct below are at best sufficient for a compositional process. Nevertheless, they illustrate peculiarities of certain variables, and can be the basis for a more formalized encompassing. to. instruments To clarify our reasoning, we limit initially to specific variables only instrumental music. The first set to define is probably one of the instruments used: TRG = {violin, cello, flute, accordion, piano, voice 228, ...}. Here too, though this set is discrete, it is interesting to take all parts: this will allow us, for example, to treat as a single instrument several violins or more different percussion instruments played by a single instrumentalist, or all brass of an orchestra. We will work rather along with TRG = {violin, cello, string {}, flute, clarinet {}, piano, accordion, voice, Percussion {}, ...}. 228In what follows, we will consider the voice as an instrument; must be well understood instrumental almost always as instrumental and / or vocal . Page 174 174 Here we take the party to consider a partition as a set of instructions given to performers: what the composer wrote is always instrumental gesture , such is the partition semantics as we consider here. We will build then a musical object from the behavior of instruments; assemblies that follow are designed with respect to the actions of the performers in the manner of compose these actions. 229 This vision is directly applicable to certain environments composition computer aided We believe in particular the separation between "partition" and "Orchestra" in CSound language. The broader idea of ​a "virtual" instrument, defined by a set of behaviors in operations or actions, or simply an organized and navigable sound materials, inserts Also in our formalization. b. game modes Each of these instruments defines a set of game modes that are individuals. In all parties, we can identify the empty set to "normal" mode (the one that is understood by the notation "ord." in a partition). Okay 229Here we find the position already in the work Lema 1 and a convex set (and who will be present also in weak topology ) instrumental gesture is taken as a matter elemental compositional. The concern to take the score overwhelmingly like writing an action of the player also has experience as a composer and performer: the traditional considerations of vocal range and possibilities / General technical impossibilities are now largely insufficient to employ the full potential of an instrument - address aspects such as the comfort of a gesture, physical and intellectual effort to provide the preparation time and transition between gestures, habits came from a classical training, etc. can extend so consistent instrumental expressiveness (the opportunity to convey musical ideas, even those call for expanded gambling opportunities). We will see how a general notion of Continuity can participate in this. Page 175 175 Sure, we can also take a combination of several of these sets and consider the Common modes of play to a whole family of instruments: MDJ flute= {Ord. (|), Wind, flattz, {with more and more air}, slap, tongue ram ...} MDJ voice = {Sung, spoken, whispered, shouted ...} MDJ snare = {Ord. (|), Rubbed on the edge, with different wands {} ...} MDJ strings = DJ violin> MDJ alto> MDJ cello > MDJ bass Ord = {pos., Arco ord., Sul tasto, sul bridge { spostando arco } legno tratto behind the easel on the easel, legno + crini ...} We still consider, for reasons of continuity as we explore later, the set of all game modes, all instruments: MDJ = C Instr "INSTR MDJ instr, where we take care to differentiate (by adding new elements if necessary) a fashion "ord. "For each instrument (we can note them by | example). This union also possible to work with a single set of modes game for all instruments. Naturally, this assembly is not directly relevant for the modeling instrBy the behavior of a "virtual" instrument, for which there is no gesture 230. instrumental strictly speaking, let alone different game modes However, if the virtual instrument "reacts" to families organized settings, 230We do not place the sampler in this family of instruments: the gesture performed by the musician who plays it has no impact on the resulting sound (except at best in its envelope amplitude) and is therefore not an instrumental gesture as we understand it here. Page 176 176 we can consider them as so many "gestures" virtual, adjustable in different "modes". c. heights Some instruments can produce defined heights, and we can therefore consider all these heights. But when we turn to parts of this together, to take into account, for example, glissandi or agreements, we include hence continuous sets of heights, which can be taken simultaneously (unlike the glissando , which orders), which is very close to the definition of a white noise band, and two problems arise. First, when we consider a continuous set of heights, the tools we have for manipulating closer frequencies: continuity prevents all index by names, which also affects the reference intervals, and elementary operation with heights as transposition, is thereby very Limited231; the problem arises to define the extremes of this set Furthermore, when we make sets of heights as singular elements the order of these heights throughout a period is crucial, as the comparison between glissando and white noise shows - yet we can not trivially include durations in a plurality of heights. 231Transposition always corresponds to a translation, as in the case of staple heights, but we can not have all the transpositions defined as before. 232. In 232 In most a discreet it could have set the "most note" the serious Sith contrabassoon and " Note acutearea, "in an artificial harmonic violin,serious for example. Page 177 177 The first problem is solved in part when we can identify the hills of fundamental frequencies perceived and work directly on celles233The second problem really arises when we work with thereof. discrete periods: a glissando is reduced to a single height at each moment when we can take these moments continuously. Over periods that a unit minimum, we can assign a "height" { gliss. } to a unit that is already deployed in an ordered time. But this does not take into account the glissando Conversely, neither resolves the difference between a glissando and a cluster on the piano. Us return to this problem from another point of view, which takes into account the gesture the musician and also a sense of musical gesture first or indivisible. For now, keep all the fundamental frequencies perceived and denote by HTR all its parts, admitting that in our list of elements there may have redundancies if we have more of a { gliss. }. HTR = {... if 1 do 2 do Y2, D 2, ... if 2, ..., { gliss. }, {chords}, {} spectra, white noise, ...} In this set, we can, if necessary, combine the empty set (|) to silence or an inaudible frequency; we can also consider that "white noise" is other than the set of frequencies made stationary at the time. Note that when we taking into account a continuous set of heights, it is possible to include spectra 233However, we are taking a serious risk to identify frequency and height, it will of course be avoided. Page 178 178 (Even inharmonic) as elements of HTR, and we can thus associate elements of the package even for indefinite height instruments. 234 d. positions On several instruments, a same height can be obtained in more than one way, and this is still more complex functional description of a gesture instrumental if we limit ourselves to a nominal value or frequency define a note, we do not always have a clear gesture. In addition, these achievements produce separate patches and require different efforts. Us take into account while, in parallel with different game modes, position "on the instrument " POS = {} positions on the instrument, in which we can identify | the "at rest" position. This will allow us not only to distinguish one of the first open string of a cello in the same Height played on the top rope, but also formally address the continuity of instrumental gesture235. It is also by looking at this instrumental position we can define the action that produces an unstable MULTIPHONIC, a sound indefinite height, or even a glissando any. 236 234This is certainly necessary if we isolate the work on heights recognize or break it specter of a snare drum or cymbal does not give a defined height. However, it is important for the rest of our formalization to define generally the behavior of a instrument and for this it is necessary that there is a sense (formal) to such an association - the limiting case would without associating any instrument height defined | in HTR. 235This continuity was the basis of the representation of the convexity in a convex set (cf. cI). 236This is probably where the voice is distinguished as the instruments themselves: all "Positions" of the voice are defined within the body of the performer, and it is difficult for a Page 179 179 In some cases, as for bowed instruments, the DJ sets and POS may be sufficient to define or heights that are produced; for the same correspondence in brass, for example, should be taken into account in POS lip pressure, which may be more or less convenient Redundancy POS compared to HTR has a special meaning if we consider 237. This possible as an instrument (formal) separate each voice on a polyphonic instrument, or each instrument one performer who plays several at once (as in a set percussion), or each performer playing the same instrument (such as a piano four hands): The position functions (item) for each of these we allow to keep within the limits of the possibilities for implementing a single instrumentalist or a single instrument. But in the general case, it is irrelevant that there or without redundancy: the position variable serves more considerations of continuity, composer to design their topology (unless it is also a singer). To our knowledge, there is no matter of these positions when learning a singing technique; Once learned, they are rather designated by the resulting effect, or as a special game mode (as boca chiusa ). We keep the idea of ​a set POS = {| ("Resting"), "active"} especially completeness. voice 237We indirectly wrote this aspect of gesture in trombone parts of open path-connected sets (2002, for soprano, mezzo, baritone and two trombones), indicating a fixed position of the slide for more notes: or by imposing changes barrel for a glissando (which thus keeps pressure lips): Page 180 180 fluidity or comfort as a possible musical dimension. We can not not consider POS to the definition of a musical space and keep this set for the definition of the topology of other variables (including MDJ and HTR). As for game modes, the idea of ​position does not apply directly to a "virtual" instrument; we can still closer once by families organized parameters with which the instrument responds. In recital Such "virtual positions", it becomes possible to think of distances (and consequently a topology) in all these "gestures". e. dynamic We still have to consider the dynamics with which the instruments can play. In this first purely instrumental approach, we limit the indications that can be given to instrumentalists: DYN = {... , , ,, , ,, , ... { cresc. }, { Sun }, , ...} This assembly is once again composed of the subsets of the set of all possible dynamic that is continuous, but we'll use as a quantity over these parts (although some of them are continuous) by indexing Page 181 181 often by the common symbols (, ...). Here, too, we can use de | identifying it with a silent gesture 238or the beginning of a sound in the silence. Note that the definition of this set is directly related to the construction an instrumental behavior : dynamic symbolic indications may not be understood in the same way in music without instruments or voices. In addition, this information can have very different results depending on the instrument that the runs 239And it may be useful to index by instrument symbols: Of course, part of DYN lets Order naturally, it is considered with or without the indexing. f. times Finally, we will take a set that allows us to process times and to articulate them. We can first imagine it as all segments of a temporal flow 240Which would take us even after all parts: Ryth = {duration, rhythmic cells {}, {durations accell .}, ...}. Strictly speaking, the only rhythmic cells that we can have in such a all are those formed from all different periods: there is no repeat . flute, gong, piano element in a set. However, we must be able to operate on rhythmic cells 238J. Antunes noted that replacing the note heads by " ? ". See A NTunes, J. [2005] Sounds para os novos Sopros e as cordas , Sistrum, Brasília. 239This is a classic point of divergence between composers: should write the resulting sound or intend to play? The answer often depends on the context in which it arises. Page 182 182 as simple as . Moreover, the evidence we have here are sets unordered, and it is impossible to differentiate from . To work around this problem, we will introduce two operations on Ryth: the first is a simple addition, and gives us the elements of the same set. For S, T "Ryth we define the S + T element "by Ryth S + T = {s + t | s "S and t" T}, where s + t denotes the usual addition times. The neutral element of this operation is the empty set, and (Ryth, +) is a monoid. The second operation is that of concatenation , which is no longer an internal composition law: we get chains or ordered sequences of elements. The set of such chains are finished is denoted Ryth *: This is the free monoid on Ryth whose operation (internal, this time) is the concatenation and the neutral element is the empty set (the chain with no element). This is in Ryth * we find all possible rhythmic cells 241, Along with or without repetition. For purely instrumental music, Ryth (and Ryth *) Is the often with a symbolic identification: traditional rhythmic values. But must be emphasized that this is precisely where a marker in part (organized) these ensembles without insertion into the clotting time (by the choice of a tempo, for example), a white or a quaver do not indicate a duration. Us 240This set is identifiable ½ +Or ½ +> {0} if we consider the zero duration as element in each term is a number. Algebraically, it is a monoid with the addition of durations as operation and zero duration as neutral element. Page 183 183 can in fact when the time scales involved permit, set Ryth as the set of rhythmic values ​(replacing the operation + by bonding notes, for example) and build on it to get a free monoid all rhythmic cells that can be noted. Of course, this approach is not used if the durations (Individual) you want to use does not fall within that proportionalities can approach the traditional notation. In all cases, note that the operation in Ryth induces the opportunity to act within a sequence in Ryth *We can replace the ST fragment by a single element, S + T, without altering the "duration total "of the sequence. Specifically, this operation will establish a relationship equivalence in Ryth length . * from which we can speak of the same elements 242This not only allows the rating simplifications, but 243 This is get new sequences "close" a starting sequence. Also with these operations we can consider several "layers" organization: from the superposition of two equivalent parts of Ryth *, we can get a new, more complex chain. We will as well as "duration" of a musical object string D "Ryth *D = d d D ... , With each of 1 2 n i"Ryth. These elements are still 241Even if limited to a sequence of durations all equal to that of a sample, which can be considered a "rhythmic subdivision" underlying any digital music. 242Two elements will be equivalent if it is possible to pass from one to the other by such substitutions successive. We can note D 1§D 2 when D 1 and D 2 thus have the same length. Note that this term is usually used (in the context of a free monoid) for designating the quantity elements in a chain; we turn away here because this particular little seems useful to musical reflection we offer, and to keep "duration" associated with the entire chain (v. infra ). Page 184 184 algebraically manipulated, even if they are ordered or directed (that is, for example, a series of rhythmic cells, or families of such cells). The term is not so not for us merely a "segment" neutral time, but rather a set where 244. Of course, it is exist relationships (or operations) between the elements inevitable to think a rhythmic cell already equipped with an order, but we want keep the possibility to switch or reverse this order, for example; each immersion D in particular the time of the work set a state of the assembly (and thus fix a form). 3.2 Features For each instrument we consider each of these sets of variables allows us to define a function, that of the "value" of this variable along the length of the object we build, for this instrument. More specifically, for a set of variables VAR, we can consider the function var: INSTRƒÆ (D; VAR) and noted var (instr) = var instr: DƒVAR, where Æ (D; VAR) is the set of D functions VAR, VAR also noted D. By example, mdj violin(D2) Indicates violin playing mode on the second element of the duration D. 243This equivalence relation may, ultimately, be used to define a distance in Ryth *, To from, for example, the amount of substitutions required to move from one item to another. He should nevertheless generalize this measurement at different lengths of elements. 244The "neutral" segment remains a possible case limit, as for a time measured in samples. Page 185 185 We can define a function of "behavior" for each instrument (we keep here as three sets of variables to simplify notation): comp: INSTRƒÆ (D; POS_MDJ_DYN) and note comp (instr) = comp instr: DƒPOS_MDJ_DYN, comp instr(D) = (item (d) mdj (d), dyn (d)), where in fact we mean pos instr, Mdj instr, Dyn instr. We will say that when comp (D) instr is defined for all d "D, it's already a musical object (duration D), we can Note comp instr(D). To create more complex objects by combining behaviors several instruments, we can think firstly a simple union of these elements: O (d) = {comp instr1(D), comp instr2(D) ... comp instrN(D)}, or simply O (S) = C i "INSTRcomp i(D). This idea is, instrumentally, the overlay sound results different gestures. From one point of view, this is actually what happens if several instruments play together, but it may be interesting to note in this superposition relationships that may emerge (for also build). If we write O (d) as a matrix, the organization of each of the sets of variables induces an organization to the "inside" of the object: Page 186 186 (d ) ⎡ pos instr 1 ⎢ pos (d ) instr 2 O ( d ) = ⎢ ⎢ ⋮ ⎢ (d ) ⎣ pos instrN mdj (d ) instr 1 mdj (d ) instr 2 ⋮ mdj (d) instrN dyn ( d ) ⎤ instr 1 ⎥ dyn (d) instr 2 ⎥ . ⎥ ⋮ ⎥ dyn ( d ) ⎦ instrN This arrangement allows to highlight not only the behavior an instrument according to the musical variables (that the function comp was already), but also the relationships between the different instruments in a variable, throughout the columns. We can also think of the progress of O (d) throughout D as a succession of such matrices, which further leads to consider the relationship between instruments and variable depending on the length (and its components). . Indivisible actions With this configuration of juxtaposed matrices we can better consider the possibility of holding an object in several sequences of the same length at a time. Indeed, our rating may suggest that we do account where the same chain D "Ryth * underlies all behavior: a homorhythmic or the existence of a common subdivision (a sequence more "fine" for the greatest common divisor of others). Yet that the latter is in general a possible approach, it eliminates one of the advantages choosing parts sets for our musical variables: mdj functions, dyn and pos are certainly discreet because they indicate the elements isolated and are defined on a discrete set, but these elements may be themselves continuous - that is the case for any change recorded in a single symbol, as a or Page 187 187 glissando . Always take a finer time sequence as for behavior of all instruments in an object can lead to subdivide a glissando if superimposed on sixteenth notes, for example. We want to consider actions that do not admit such a subdivision, because they are essentially continuous : assigned to an element of "D, such 245We must behavior can not be attributed to a period s + t, although s + t§d. * therefore include the possibility of life as either a single chain of Ryth but a set of the same length strings (we always denote D not weigh na notation). The ongoing behavior (indivisible) provide decomposing the network into its smaller "relevant parts" that are not necessarily submultiples of other parties; we will name the gestures first (alluding to prime numbers). We can see these "blocks" constituent of duration in succession O matrices (d) as representing different independent elements variables for each instrument if an indivisible action is superimposed on finer segmentation function behavior "crosses" the matrices of this refinement. We can actually represent and polyrhythms in several directions (even in the behavior of the same instrument). 246 245More generally, we can consider actions that admit only one sequence as duration, to the exclusion of all others that are equivalent to it. 246The following diagram is intended of course not be readable just like a partition: it is only an illustration of a topology of polyrhythm. Page 188 188 Fig.18: A provision of O (D) with multiple segmentations duration D. All together, the "Plans" are the finest sequence equivalent to D, but not all intercept not all behaviors; for each thereof (at a variable and an instrument), a term different of the same length that D is given by the intersections that occur. The idea of ​working with longer indivisible elements that the greatest common divisor periods involved also used in a digital environment where yet everything is necessarily mensurable by the length of a sample. This generalized discretization does not eliminate the multiple integrations of listening, which emerge units and continuities, let alone the possibility of thinking (and therefore composer) of "gestures" indivisible, as dynamic variations, glissandi, or global behaviors of granular synthesis. 247 These are Incidentally, if all the variables of the same instrument follows the same period, a two projection dimensions of the object (eliminating the distinction between the variables) is equivalent to a rating proportional. 247The total discretization granular synthesis is just a tool to work great detail continuity: a large number of spot sounds easily becomes an open set (which not contain its own border), and so lets turn individually by imperceptible steps. A thorough discussion specifically on the articulation techniques based on grains sound is in R C. [2002] Microsound . OADS Page 189 189 associated with much longer times that a sample and possibly articulated them, below which the object is somehow distorted. Note that the matrices thus "traversed" by behaviors incomplete, in isolation, they do not provide information on all aspects of 248 The absence of an element in a matrix (namely the absence the object. articulation, according to at least one variable on an element of D) does not indicate a inactive instrument for this segment of time. Do not act (or not play ) is a this behavior, we can note comp instr(D) = (| |, |) written silent in a partition requires a deliberate and precise gesture, just as a note two gestures are not essentially different. This presence of silence such conduct has the added advantage of including it in the considerations on the continuity of instrumental gesture throughout a musical object. . Integration time We still consider the duration D as an ordered sequence, thus already designed with a certain temporal orientation in mind. It remains, however, handled internally (not least by rearrangement of the first elements). In addition, its inclusion in a clotting time can be done in several ways: we can see O (D) as a figure that can take many forms depending on the velocity (or speed variations) to which it is covered. As part of the 248This reminds us that we can not avoid taking into account the duration of the object; there no "instant" object, no time to speak of. Page 190 190 instrumental music, this is trivially at different tempi (or changes even continuous, tempo); but we can consider how similar figures in electroacoustic music, manipulated beings inserted before in a clotting time and adaptable to different speeds. 249 It is during the insertion of an object over time in the game in the work qu'entrent dependency relationships between musical variables. Indeed, although we have already considered "independent" directions in a musical space, and also ability to accurately define an object in only a subset of the variables used, high mobility remains in their treatment. When all deploy simultaneously, subject to one temporal direction, we lose the ability to handle them separately - because they are not necessarily perceived so individuated. By operating off-time, the variables are separable, and we can (at least locally) ignore their interdependencies to address fragments our musical space (subspace). Crossed by the time this space is somehow folded on itself around the directions, listening, protrude as major or relatively independent. 250 249This is the case, for example, in the micromontage: we can define figures from extracts samples, then to swap, lengthen or shorten (or without changing the pitch of sounds). Algebraically, the situation is very similar to working with notes, although the temporal scales can be much smaller and so here we have access to the interior of the new "notes". A tool copy for such manipulation is the software IRIN , C C. [2004]. Note that we AREAS Here are freed of proportionality between the elements of duration D, present in scoring Traditional. 250Of course, the definition of these directions there, and understanding as a landmark in space musical, is no longer in the hands of the composer; it is a set which can also vary along a listen. Page 191 191 3.3 Continuities The sets of variables are the basis of our definition musical objects as functions, and providing them with certain operations or properties Us can approach the continuity of these functions. Indeed, we can understand a continuous function like bringing in elements close together on close elements in another set. 251 So if we can organize sets of variables to be able to speak of proximity or neighborhood between elements, we have the tools to think the continuity of a musical object. A simple way to define proximities appears if we can measure distances is the case for groups such as HTR or DYN, if we take it isolated elements. But it is difficult to accurately measure the distance between two instruments or two game modes, and an alternative approach to proximity in musical variables is required. What seems most musically relevant is to take this judgment of neighborliness between elements, largely arbitrary and oriented perception within a specific work context, to make neighborhoods formal, of open subsets of the sets of variables. This defines a kind of topology of each variable, a classification of elements according to the criteria, it is possible to choose and to vary, familiarity, manipulation, sound result, Comfort or more. Note that, when the set of variables is provided with a 251This is a "translation" of the mathematical definition: we say that a function f is continuous at a point x if for every open set containing f Y (x), it is possible to find an open set X containing x such that f (X) 7Y. Page 192 192 precise distance, we can define the neighborhood of a simple item like all elements whose distance to it is smaller than a limit. We will say that an object is in a continuous variable if, from a D element to the next, we always stay in the same vicinity of this variable, or in neighborhoods which overlap. We observe and the function that defines the small object to a variable and an instrument. More generally, we can restrict the function of several ways: an instrument, a variable, or a subset (or a single element) duration D. We can then consider a whole musical object topology, given by the topologies of different variables. In particular, it is defining proximities between functions associated with different instruments that we have an approach continuity which takes into account all of the interactions within an object. Note that with a topology or a neighborhood system in POS us insert in our thinking compositional some instrumental usability: the adjacencies or proximity keys and positions on a rope or ropes adjacent fingers more or fewer movements, changes instrument, may well be part of which leads the construction of an object and its continuity. 252 252We take this continuity of instrumental gesture, considered from the point of view of comfort the performer, a true compositional material. Page 193 193 4. Modular spaces, made space By defining our musical variables, we have seen that the ability to enter different objects, and in different ways, leading to the existence of several distinct musical spaces during the composition of a work derived from intentions compositional variety. Each is a context manipulation and evolution objects but remains at its simplest definition, empty of such objects: it conceals forms (potential) it precedes and therefore defines a family morphologies. In compositional process, we will say that such a space is composable : we can operate on its contents without changing itself, it is "open" and manipulated. To dial, we build spaces from variables musical preceding the construction of sound or musical objects. In some so we work on space as a particular type of object itself provides the means to operate on sound objects (these two kinds of operation overlap and interact in a compositional process). But what is itself perceptible , it is the musical objects: it is through their deployment that can manifest musical variables, and accordingly the dimensions of space (and the space itself) where they operate. Variable is a musical, listening, a trail left by a musical object 254: Space to talk about a work already made, it must be considered as " irreversible space , determined as 253Who we are 253On this idea, v. GRANGER G.-G. [1999] Thought of space . Page 194 194 number of dynamic relations, while having both "permanent World" that it is, in all its diverse scales of magnitude determined by the thesis which is manifested in the "compound" of the musical work " 255. This space is indicated by the musical objects (rather than defined before them), on which no longer but that is reconstituted during listening, we will name composed space . . Several temporal directions This distinction between space and composable composed seems important because it allows us to pose the problem better time as a (possible) Musical variable. Indeed, a musical composition is in delayed time and we can be considered by dialing term temporal location, proportion, speed, and many, like so many musical variables, so manipulated in the same way, even in the same way that the heights or intensities, for example. It is thus possible to have a musical space several different directions (or even several independent dimensions) that have to do with time. However, in space made ​time itself as one dimension , to which other is somehow subject: if a variable is seen as traces a path time is necessarily underlying and all these routes are simultaneous in listening. What becomes different temporal variables, they spend composable the compound, and how? 254In our formalization, we can identify this trace in the image of the duration of the function D associated with this variable, var (D) itself inserted into the time. Page 195 195 Perhaps we could say that the space is composable "spread" (because it is empty): nothing irreversibly binds one dimension to another, they are all truly independent. The composition of an object in this space connects of "segments" of these dimensions, but it is only with the passage of time the object space folds on itself as each relationship (at least). Yet Once again, we perceive space composed only through the object, within which there is no possible independence between variables - what comes to listening only can only be analyzed. The only differentiation in the time that remains is effective the time scales: the folds of composable space manifest themselves joints at various scales, that every time a noticeable effect on several different sizes at once (but along a single time). Thus, short heights cycles can bring up by accents longer cycles play modes; very varied rhythmic and quick joints within a all sufficiently large instruments can build a global stamp. Even if writing occurs (because it is symbolic) in a space to the dimensions independent and "whole", which finally is in writing (or, more Specifically, creates) a space dimensions "fractional". . A dimension "extended" The question may then arise if the space made is unique dimensions that characterize (even if they are inextricably tied) are they always 255VAGGIONE H. [1998b] The composable space . Page 196 196 same throughout the same work for several works for several plays, a same work? Several spaces distinct compounds co-exist (listen) as can composable spaces? Any answer to these questions must, in our view, be based on the single dimension necessarily "extent" of the composite space, time. Indeed, the "spatial" properties are made exactly those that emerge by the perceived relationships between (and including) the objects musical. However, the definition of these objects, their identity to listen, time dependent and a particular time scale: the network that brings together multiple constituting a single object, which contains the cutting or may be formed in various relationships inheritance and polymorphism 256 that turn according as they are observed in closer or further. Attributes specific to a dimension, at a certain scale time, can affect another dimension to another level 257; units that appear separate, individualized by their internal morphology may participate the definition of a morphology "large" (in another time scale), and be well together in the closing of a musical object. All this can only arise in a work musical that at as it is heard, and we could say that space compound is unique to each moment - he constantly redefined by what is heard then being put in touch with what remains in the memory of the listener. 256We find this theme at V H. [1995] Objects, representations, operations . AGGIONE 257Examples of transactions made on time but having effects on perceived space are built by VAGGIONE H. [2003] Decorrelation microtemporelle, morphologies and spatial representations of its musical . Page 197 197 But the immediacy of the space compound precisely the result that "spaces-in", those previously collected but especially those levied on other scales than the entire song heard, accumulate in the memory of the listener, forming a complex network of different spatiality and not always consistent (to the image of the various modular spaces several times folded on themselves by inserting objects over time). From a certain overall length, the need arises power "to hold" these inconsistencies in a general organization yet understandable. Recall that we have defined as a musical object as a set of operational attributes, managed within a particular space. We can take this idea on a time scale and wider sense for think what would, from this point of view, as a whole work. 4.2 Locality and globality This accumulation of subspaces long duration, culminating in a single space composed given by the entire work, suggests that we consider characteristics of the entire room (in the global space where it fits) or own only a segment of time which is inside it (local properties). Ideally, we would consider local which can be separated from one context larger, keeping a certain identity and autonomy. Locally, the space compound is nothing other than a composable space crossed by the time, but the whole composed of the work involves simultaneously more spaces. This is actually the simultaneity spaces that characterizes all Page 198 198 whole, be it an entire work, merely a section or even a object. We might be tempted to say that it is local or global scale dependent or temporal logic which we are, but that would only be partially Correct the problem for us. Indeed, although we may redefining multiplicity and unity each time scale, it could have y resort, as we understand that on isolated points in space made ​- yet it no such points 258 because all the modular spaces (therefore all temporal scales) interact throughout the space made: overall, the space compound is nothing other than the articulation of all the modular spaces, it is always connected. To formalize this joint spaces, algebraic geometry vision may again be useful: it is to integrate several areas in one system operations, making each of them a special case of a "general" space or then a part of a formal of a higher logical network. 259Naturally, such network or a space such as "general" should be composable, and his writing is done simultaneously and interacting with that of all the communities (this is what allows integrating all time scales in the same formal process). Us could appoint the global composable space . 258Their possible concrete existence is irrelevant: they would anyway not discernible. 259This "logical nesting" space (and hence geometries) occurs especially with geometries projective and Euclidean. SeeRANGER G G.-G. [1999] Thought of space . Page 199 199 If each composable space (local) is defined by a set of operations, are operations on these operations , or the relationship between them, which define an overall geometry. The problem is that these operations, that we now want to take as objects, can be very diverse nature (to because of the diverse nature of their own objects) algebra that can take them all account does not seem to be trivial. One of the first obstacles to formalization of the space as a network composed of modular spaces or as insertion time of a single global composable space is precisely the transposition of all logical time lines to a single timeline physics. 260 4.3 The form of a work The whole "the greatest" in a compositional process is of course the of the entire work. Thinking the form of a work thus involves thinking about both geometric system that can articulate all spaces that appear throughout a processes (listening or composition) and the geometry in which the work is registered whole thought as object . Indeed, the work itself is a fence, manifest "Salience" and singularities by its internal operations, and must provide "Efficiently" to listening operations. As an object compound, so it builds 260According to MG. [2002] The Topos of Music (ch.47), the theory for this formalization is not AZZOLA yet established. Page 200 200 the space in which it operates, as well as music objects give perceive the internal space of the work. There is therefore a superposition (or confusion) of these two forms of spatiality, that the work sets or reports (compounding area) and that the characteristic of the Interior (the overall composable space). A piece of music listened to would be its own space (which she would be the only object). Specifically, we can consider a family operation or manipulation (with this time on composable spaces) and all that they allow to manipulate (we could appoint formal variables ), to define a new formal space , which is itself a particular possibility of existence of the modular spaces. Thus, these spaces are not arbitrary, but also have a form that arises in formal space. Immediately, a problem arises: the formal space is itself a space composable, and we face the classic paradox of the sets that contain themselves. It is in fact a vocabulary inaccuracy to which he be careful: the formal space is composable because it is handled in the compositional process; but it is not of the same nature as the other spaces (Sound objects) we have named composable because in these the time can be multiple and be all equivalent, while in the formal space Time course of the work stands out as special and important. In addition, formal space is unique (by the uniqueness of the work): he can fall back on itself Page 201 201 even when it is crossed by time; but the operations on the form of a work (At least some of them) must carefully consider a chronology, a organization of modular spaces throughout the work. We are then led to that formal space will not fold with the passage of time, because this time is already enrolled in full - it is well made ​identifiable in formal space, the the entire work indicated at the hearing. In a way, this is equivalent to saying that all that is offered to the hearing at all scales, is part of the form of a work (Or, more precisely, made ​this form); or that the whole of the joints between all the possible sub-spaces of a work creates the logical (or geometry) in which these same joints should take effect. Once the work is completed, the deployment of all musical objects will allow us to apprehend the spaces where they "place". The spaces we actually hear are the ones that define objects, define and give heard. We can think abstractly dimension of heights, develop ranges and tones; if in a part (or fragment) are present only 4 heights, they are good all the space (any size) from the heights of this musical object, and define the topology. Other notes would add to those ones modify this topology - they would redefine the composite space. Page 202 202 III - P OETIQUE MRUsical Adults are allowed to collect and study anything, Even old theater paper bags or programs. - JRR Tolkien, On Fairy Stories It is always with great curiosity and some healthy suspicion that we have heard the term "musical poetics." Indeed, we find in the the most diverse speech, relatives or not our music, or music in general, rarely explained and used in very different ways, incompatible, and sometimes on the edge of contradictory lack of precise definition. We do can be denied that we ourselves around this expression associations that suggest, albeit vaguely, a real center of our interest musical, any content whose study could be significant. In this chapter we take a critical look at this notion of "Musical poetry". We will try first to obtain a definition sufficiently large and clear, and then we will discuss issues that may well arise: the poetic leads in music, questions of meaning, intention and meaning, but also to the ways of doing , strategies to achieve these senses and meanings. Finally, we will project the discussion on weak topology , one of Page 203 203 works created during the development of this research and which includes elements of our answers to these questions. All this study can be done by ongoing dialogue with other writers and composers have always questioned us and influenced visually: this is finally within such a multiple network interactions and interpretations of musical activity that takes place. As elsewhere in our reflection, we want to address these issues with a look and a vocabulary influenced by mathematics and accuracy. 1. Definition (s) To better understand our own relationship to the multiple meanings of "poetic musical, "we will try to illuminate the surroundings of this expression as appears in everyday language and then in the musicological and critical vocabulary. 261 We do not follow a chronological order in the authors whose remarks were discussed, but rather references the critical path itself. . Everyday language A first naive approach the content of "musical poetry" would naturally look for a non-specialist definition, which could indicate the "sense 261And assign importance to non-technical vocabulary may seem inappropriate in a search like this, but we believe that its semantics often interferes with that of a language that wants more precise: the current use of an expression is involved, ultimately, this can be for us the gasoline itself, and thus affects the opportunities we have to redefine or change the point of view on it. By taking into account the influence that we can best get around it. Page 204 204 common "this expression. But this approach gives us a general sense of "Poetic", without considering the possibility of qualifying musical: poetic : nf (...) 1. Compendium of rules, conventions and precepts relating to poetry composition and construction of worms. 2. (for ext.) Theory General nature and destiny of poetry. Theory of creation literary literality. 262 We can adapt this definition to our research in the field of music taking "musical poetic" or as a collection of rules, conventions and precepts relating to the composition of music pieces and the construction of sentences musical , or as a general theory of the nature and destiny of music , a theory of musical creation, musicality . However, we remain far enough an association that can be done easily when speaking of "musical poetry": the reference to the adjective "poetic" and so all figurative meanings (thus more or less waves) of "poetry". The current use of the term then also serve to say "poetic musicality" (or " the poetic music "), a formulation that could make sense but never appears, to our knowledge. to. music as a "text"? This association with poetry, and therefore with a text and possibilities interpretation and meaning it offers, we seem to be at the heart of the difficulties around the phrase "musical poetry". The vocabulary of musical analysis Traditional is indeed often extremely vague, and it seems that this Page 205 205 imprecision also take place when it comes to poetry. A collection of vague definitions in the analysis is documented by JJ Nattiez 263Which states that based in fact on a quick analogy between language and music. Precisely because the "meaning" music can not be compared to linguistic meaning, it is futile to look for musical equivalents the phrase or morpheme. Indeed, the ways to understand a text, to abstract it is, essentially different ways to assign meaning and significance to musical work when it comes to semantics and correspondence between sounds (or words) and ideas; but despite this we speak (and often fairly accurately) as "sentence music "or" form of carriers "elements. The question of the meaning and expressiveness thus remains posed by this parallelism in common usage; we could might even say that it is a set of preconceptions about "musical manner" communicate that leads to the comparison with poetry. This brings us closer second part of our naive definition: a certain expressiveness can be heard as a part, at least, the nature or fate of the music, or as the defines "musicality" and "musical poetics" and to deal with those issues. What Regardless, these links are established within everyday language so sometimes will refer to ways to, sometimes the desired content or music necessary, sometimes a "Function" communicative music. 262REY-DEBOVE J, R EY, A. (org.) [2002] The Oxford Dictionary . 263NATTIEZ , JJ [1973] Foundations of a semiology of music (part 2, Speech Musical, pp.255-279). Page 206 206 . Stravinsky This is the side of the first of these definitions, perhaps, that place Igor Stravinsky in his Poetics of Music in Six Lessons 264. Indeed, there is a question of method execution correctness (and thus for the author, writing), positioning in relation to the history of music ... Though sometimes make considerations on the nature of his activity as a composer (when he mentions the writing context his works, for example), it seems to be in the music, Stravinsky, other poetry that the good work of the craftsman (composer or performer). It is not a matter for him theorize creation, but especially the practice; his thoughts are clearly has posteriori . 265 In addition, Stravinsky voluntarily loosens the musical expression of the report to any text, even as seemingly innocuous or honest as the title 266. There is no for him to musical expression itself (the music can not say anything, nothing express), but stresses the need to respect (almost as prescriptions Medical 267) Imposed by the partition interpreters or the composer is required itself; all content is given by a just execution. In this sense, poetic Musical Stravinsky is also a theory of the nature of music, and musicality: its reduction to how to make converges all these subjects. 264STRAVINSKY I. [1942] Poetics of music (in the form of six lessons) . 265"The theory does not exist. It can be deduced from some compositions. "(In C , R. & RAFT STRAVINSKY I. [1959] Conversations with Igor Stravinsky ). 266"The title of a play becomes an excuse for Free reinterpretations. "( Poetics of Music ) 267"[The Creator] must ceaselessly refine his taste, or run the risk of losing insight. "(Id.) Page 207 207 However, the Requiem Canticles are not immune to generate all kinds significant associations (even when the text is not graspable) and Petrushka actually accompanies an extra-musical activity, emphasizing and supplementing Message, if that narrative ballet. 268 So we do not want to follow Stravinsky strictly in its definition of "musical poetry": it seems more close to a poietic (the same composer's confession). And although it is effectively linked to all poetic, we are on that instead of the notice of Backès: "Recall (...) that the poetic is to study processes is a challenge many the meaning of speech, the expression on the message. » 269Now we just able to tackle these words, knowing how the composer or analyst can work on and refer to it appropriately. . Schoenberg, Dahlhaus We find in an article by Carl Dahlhaus 270clear indications of its how to take the phrase "musical poetics"; we see it as the same occasion what it means, he said, and according to this definition, in Schoenberg. The position is again taken downstream works: it is only with the work ready it is possible to theorize. The history of music is to Schoenberg 268We intentionally include different periods of the works of one where Stravinsky presents its musical poetics conferences: his vision of the subject does not exhaust nor the works he had already composed or those which consist again. 269BACKES J.-L. [1994], music and literature. Poetic comparative test , PUF, Paris. Us return to the words of this author. 270The musical poetics of Schoenberg , in DAhlhaus C. [1997] Schoenberg , Contrechamps, Geneva. The following quotations, unless otherwise indicated, are from this article. Page 208 208 as a process highlighting and making it evident that, in the own music nature is sketched and sketched as a possibility just waiting to be realized as a process whose vector is genius infallible composer. Since this is the genius who decided, the theory is a cura posterior . But note a crucial difference from the point of view of Stravinsky: the theory is no longer just seen as emanating works in general, but as the dependent of a historical situation and a composer. In particular, it is can no longer maintain such musical poetics which held that role until the eighteenth (And what largely adheres Stravinskian neoclassicism), namely systems e century rules. For Dahlhaus, "any attempt to develop musical poetry from e and XXe centuries must be limited to descriptive and works from the nineteenth individualizing rather than prescriptive and generalizing, "and the poetic must be theory of an individual work. The main focus remains on the study of the production of works, but poetic 271In music must now be reconstructed from the interpretation of the works interaction with theoretical about the composer on the meaning (and perhaps meaning) of these musical works. In this sense, Dahlhaus includes poetic music of Schoenberg as an attempted compromise between postulates that the unfortunate quarrel musical aesthetics between formalism and affect theory makes 271For Stravinsky, we should have instead built from the interpretation of the poetic, the way whose work had been done. Page 209 209 contradictory, namely the requirement of a compelling expressiveness at all instant and seamless consistency of the musical event. The expressiveness is thus part of the speech as well as the consistency (for formal speak, according to rules); more specifically, the expression is made possible by the coherence and aims to convey. The musical work becomes a kind of Sound thinking , hardware implementation of a network of relationships between ideas - it is so almost a speech itself (non metaphorical sense). The presence of meaning and expressiveness in the discussion raised by Dahlhaus indicates a particular vision of the "musical poetics" that takes the communication and intelligibility of music as important elements for composition. A workpiece being literally made ​and multiple, that is the consistency of Overall, at all levels, who can guarantee understanding 272 and higher reason, his expressive impact. If it is not a question of what would mean the composer, is not, however, deny the music the opportunity to say something 273. . Nono, Antunes In contrast to this focus on how to make the term "poetic music "is often taken to indicate clearly intentional content the composer or to define a role (aesthetic, social, political) of the music. Thus, we often find in Jorge Antunes direct references to 272"Schoenberg (...) regards as incomprehensible and does not tolerate what is isolated, which is sufficient in yourself. Understanding of detail depends on the logic of the whole. » Page 210 210 274, political climate that receives his works, either in texts and titles ( Inutilemfa reference to the staff of the armed forces or Três Impressões Cancioneirígenas 275Where text phonetically stops on "AI-5" institutional dictatorship ended Brazil), or in his writings on music published in Brazilian newspapers 276Either still in the same sound material (loans to traditional music in Dramatic Polimaniquexixe 277, Horns protesting in Sinfonia das BUZINAS 278). There "Musical poetry" to Antunes is indeed what he is talking with his music, and why he does it: the music is a poetic political music. That said, we do should not avoid taking into account, at least when a text is absent from the inside parts, a whole paratext which contributes to the political dimension to these events. Indeed, the situation in which they occur is often crucial (for political rallies, for example), as well as announcements that precede or follow them, made by the composer, and that direct listening to content or meaning specific; otherwise the effectiveness (or even the existence) of the message could be reached. Concerns of a similar order are at Luigi Nono: political commitment openly marks a large part of its production; there is 273Again, it is said emerges from within the discourse, coherence: "Unlike a very widespread prejudice, expressiveness is also in no small measure, context feature. » 274For trombone and piano (without piano), 1983. Analysis by trombonist R. Feitosa is in TO NTunes, J. (org.) [2002] Uma poética musical brasileira e Revolucionária . 275For flute, viola and cello, 1977. 276Particularly in the Correio Braziliense . See also the preface to A NTunes, J. (org.) [2002] Uma poética musical brasileira e Revolucionária . 277For clarinet, cello and piano, 1985. 278 For choir, instrumental electronic 200 cars, The 'Sinfonia composerdas himself commented on theensemble, composition and the sounds creationand of about this piece in A J.1984. [2001a], NTunes BUZINAS 'o sublime useful eo na fronteira between o medo ea Ousadia . Page 211 211 Also in this composer titles, text and sound materials loaded a specific meaning. 279 But "poetic music" at Nono (Or applied to it) can take a broader sense, especially if we return to the association of music with a text. Indeed, several works of his last period 280employ special way of text, up to let only on a purely instrumental score or referred by Title 281. The textual material is sometimes taken directly as sound material, for direct cutting the sound of words (as in the opera Prometeo ), sometimes as a source of the associations composer only sketches (as in the string quartet). But these approaches are not designed to remove the original meaning of the text for a sense of "purely musical" that may arise as well, but to use it in a new way. In his study of Fragmente - Stille , Doris Döpke 282actually points out that The key is rather to make fruitful, the composition, the reports between the phonetic sound equipment and semantic content. (...) The division of the text into phonemes and the simultaneity of several texts have not 'exorcised meaning, but by cons have developed in a figure phonetic-semantic service of musical expression. " Thus, by deconstructing the phonetic organization or placing to the pages of his quartet, for they are neither read out nor taken as the subject of 279Include among many other Intolleranza (1960), Il canto sospeso (1956), The fabbrica illuminata (1964). 280This division is usually made from a string quartet (1980) and his work studio Südwestfunks in Freiburg; we follow here. 281As the quartet or No hay caminos, hay que caminar ... (1987). 282DOpke, D. [1987], fragmentary reflections on musical poetics of string quartet Luigi Nono , Program Review-Autumn Festival, Contrechamps, Paris Page 212 212 fragment which they entitle or even may be disclosed to the auditor is to Nono infuse poetry he cites his musical discourse, may treat as treat a poem. We could say it done this way the representation a musical poetic idea 283; a similar approach is also in pieces for choir Antunes as Cromorfonética (1969) and Proudhonia (1972, tape). For these two composers, "musical poetics" is what we "Speaks" (or we want to talk) when making music. The choice of the verb here is not trivial, because it is good for them to send a clear message immediately understandable (at least when it comes to political commitment). But we keep in mind the comment Betsy Jolas 284For whom communication through music has perhaps afford to be done in terms as accurate: Will we (...) allow (...) that music has nothing to say since the death of the tone? What other words, it would only make sense that through well-established codes of this system, which we agree to think today it refers ultimately to extra-musical meanings, using a powerful network of agreements known and accepted by all? (...) [There Music communicates with us through an endless stream of pictures sound, the role (...) is to evoke and to relate some lived in the world, the composer, with all of us. Otherwise 283Döpke cites the effect of direct parallel between text and music: mirror games, oppositions agitation and calm ... The fragment associated with worms is "a symbol of operational possibilities and human. " 284JOLAS B. [1993], pictures and sound musicianship , in UCI B -GLUCKSMANN , C. & LEVINAS , M. (org.) [1993]. Note that the author does not speak explicitly of musical poetics in this text. Page 213 213 is, if the music expresses, we manage to say something, it is through an extensive network of representations . This way of constructing a meaning or musicianship through evocations and put in relation seems particularly relevant here because it allows, among other things, to observe the musical poetry we have cited previously from another angle: if it is question of representations (precisely, to re-present an experience of the world of the composer), we can direct attention to the composer's attitude towards his work. The "musical poetics" could then reference to how to wear face the music and the compositional act, which prevents the discussion around the front more or less just a possible message. 285 . Backes, Ruwet So far, the different meanings of "musical poetics" that we were able to articulate emanated parts of one musician at a time, and therefore dependent a particular practice. To our knowledge, very few works address this issue generally or without submitting the study to the presence of a text set to music; this we find most often seems to be focused towards this "poetic musicality" we mentioned above - the musical properties of a text or how such piece is as such a poem (what we might also call a "poetical" musical). 285This is the version of the expression that we think back in works such as those of MR B. [1989], Olivier Messiaen: a poetic of the marvelous , and M G. [1999] A ASSIN AGNIEN poetics of the unexpected . Page 214 214 Independent ways, Nicolas Ruwet and Jean Louis Backès do not hesitate not reporting to the literature, but do not associate their research to speech sung (they avoid Likewise, most of the time). As they do not place as composers, their approach is more general: the term "poetic Musical "takes on a role rather than strictly musicological practitioner. There poetic definition proposed Backès 286 is very inclusive indeed, and quite remote associations that we saw in previous authors: Poetics can be defined in a very modest way, such as presentation and critical analysis of the terms used in the studies literary and even in musical studies. The outlook remains (...): clear, correct knowledge we can have of the reality of literary and musical events. These authors often refer to the language without apply it directly to musical analysis: it is mainly in that it is investigating the construction of a sense 287 (Or his apprehension ) that this discipline is used as a reference plane, which reflect in music can approach or move away. There "Musical poetics" so understood is therefore occupies the ways makes her understanding of music: how to give coherence, unity of a work, internal and external references, especially the relationship between practice and theories 286BACKES J.-L. [1994] Music and Literature. Comparative poetics test . 287Again in the words of Backès (op.cit.), "The study of semantic interpretation is perhaps not to recognize the meaning, but trying to find out how it is built. "This clearly echoes the About J AKOBSON R. [1963], General Linguistics Essays : "The role of the poetic is not as it is believed too often dispel illusions, but to try to describe how some occur representations. » Page 215 215 surrounding it, which would aim precisely to build this understanding. The opposition between way (s) to make music and function (s) of music disappears in favor a point of view which is intended purely analytical (fully posterior therefore) and not normative, which would have no direct impact on a compositional act. Though Stravinsky and Schoenberg can not define their "musical poetic" that result works (their own and other composers), their efforts remain no less oriented composition, therefore upstream of new works. The concept of "Musical poetry" based on language takes the contrary all activity musical understanding as a subject of discussion. It is however important to note that, in the work of Ruwet 288, Research the music (and musical) are clearly separated and interact very little, the author speaking only rarely "musical poetry". For our discussion, his literary comments are important as much or more than music, who turn often rather to a semiotics of musical 289. These are his thoughts on a speech articulated, provided direction (or liable to be), we want to link music and organizations - and that is just "be provided with sense" that can cause problems. Indeed, for the author as much poetry as music are filled with meaning, naturally, but this meaning is not constructed in the same manner, and thus not étudiable with the same tools. If this position is of course not refutable in itself, 288RUWET N. [1972], language, music, poetry , Seuil, Paris. Page 216 216 However Ruwet leads to evacuate some meaning opportunities for music: almost all the musical sense has syntagmatic relationships (discussed especially sense , and no semantics or meaning, in the sense where we ate the above terms). For its part, Backès also the question of the possibility of obtaining Semantic only with instrumental sounds, to "speak" the music: "Is it really possible to make the music speak without using paratext title, comments, programs, various notices, or sung text? "Whatever arises Always separation between literature and music, he is led to the assignment of meaning to music by text, present or implied ("a known melody and provided words proliferated imagination "). But we do not see a musical discourse taken into account; although the unity of a work, for example, be considered by its internal correlations (and shortcomings of these), the very idea of ​a "musical discourse" is usually discarded (it is, for the author, "a metaphor that it is good monitor "). Now we want (or we) speak well of joints, links logical, referrals ... a material that "speaks" not in the sense described above. It is all that we want to place under the term "discourse". This approach of "poetic music" oriented language seems quite fruitful also because it indicates precisely how to present 289The paradigmatic analysis, and much less contrastive analysis that extends (v. V [2003]), is indeed essentially a work on signs more or less complex, "cut" in music by whoever analysis. OisinF. Page 217 217 some problems related to the definition of a musical semantics. 290However, it remains relatively far from our position, which can not ignore the issues and choices own in the composition. . A position Among this multiplicity of meanings of the term "musical poetics", we retain the first link to a poietic : the composer viewpoint, ours, all poetic is primarily a poetics , it can not be sense of expression or construction without a thought on how to make music, action on the act and the compositional process. On the other hand, as do we mean in what is retained below above, we also maintain the link to the expressiveness and meaning for fuzzy matter the definitions: as we see it, a poetic depends on expressive intent , a report layout (or at least its intent) with a network meaning outside music or at work studied. 291 These two considerations are closely related to the role of memory in listening to a work. We indeed consider two types of memory game: an internal memory , associated with links between the elements of the room, their organization and how it contributes to the construction of a musical sense; and 290Other authors also address a "musical semantics", however, without correlation with a (or the ) poetic. See eg B , J. [2005] The iper-sistémica musica and Aboni-SCHILLINGI MR G. [2002] The Topos of Music . AZZOLA Page 218 218 external memory , which is built by references to other objects or ideas as present in the work. A recovery or a variation, recognized as such, are Simple examples of internal memory in action; a literary quote or allude the political context can only take place through an external memory at work. We could summarize our position by associating the term "poetic music "to a poietic oriented towards building an internal memory and a external memory to a work. 2. Poetry and poetics weak topology In 2004, we had the opportunity to work for a month with the New Ensemble Modern, for which we have made ​weak topology . 292We will here an analysis of the room, not in the traditional sense, but to highlight the ways in which we addressed in practice some issues related to the poetic and expressiveness. In most cases, we will follow the order of composition of the work, which coincides generally, but not always, with that of its progress. Here, we also prominent effects on the listener we Wanted (especially in terms of understanding and attribution of meaning) and how to include them in the results of the compositional process (locally, 291We do not exclude and to consider parts that offer no relation outside the music. Instead, we can now speak accurately of a poetic musical "truncated", which removes these expressive intentions. Page 219 219 until the complete works). Naturally, this involves the presentation of views very personal aesthetic (if not entirely idiosyncratic), so unjustifiable in itself; it's not about them we want to build the discussion, but how to put them in. Many times we will refer to American Lessons Italo Calvino 293A text very much in our mind when composing; the room was not written as an application of these principles, but they have directed several decisions in the writing process. 2.1 Clotting time and time felt One of the concerns that have guided the composition weak topology was 294We wanted get a room that seemed to last less than it actually lasts. and of course act on the comprehensibility of the work, but this time contraction that we find in several rooms that have marked us 295Has a value for us aesthetic in itself: it is always a rewarding source of surprise, as we perceive afterwards (discovering the actual elapsed time during playback) or in the course of the work (in a sudden redefinition of the form we believed collect). 292The NEM is a chamber orchestra of 15 musicians in residence at the University of Montreal, led by Lorraine Vaillancourt; weak topology was composed on the occasion of their Forum in 2004 (v. partition attached). 293CALVINO I. [1984] Lezioni americane . Unless otherwise noted, the following references to this author indicate this item. 294The order for the Forum asked a room from 15 to 20 minutes, that have considered a period relatively long. Page 220 220 The room is organized into five sections (introduction, three central areas and a coda), each designed this compact as confinement in connection with parties that precede it. . An inspiration in literature This "implosion" of musical time is not unlike what offers Calvino when talking about speed , large content presented in very few words. Of course, here there is no question of small amount of "words", but rather concepts : If a musical discourse fragment lets represent in our memory concepts with little, if we can reconstruct (or what would be the "essential") from some logical links, we can consider that it "takes up little space," and the time it seems to have to be fully understood is reduced. The example Calvino takes to start his presentation on the speed is Charlemagne legends bewitched by a ring; he detects a verbal link (the words "Love" and "passion") and a narrative link (the magic ring) that provide juxtaposition of the chain of events and founded the economy of this concise narrative. We see compositional strategies we can transpose music thinking of them as a material link and shapes or features link. More Specifically, a certain economy of the "narrative" music can get links 295We believe including the Sonata in B minor by Liszt, the Don Quixote of Richard Strauss at Concubia e R. Schumann symphony or 3 e Nocte J. Baboni-Schilingi but also to certain passages of 4 String Quartet H. Lachenmann ( Grido ). Page 221 221 established between the different forms that are similar figures (or the same figure), and links between the various figures who perform similar functions. . Phrasing and integrations Over a period provided around twenty minutes, the first decision Compositional to reduce the time was felt to have an introduction could be somehow forgotten formally , that would prepare what could be felt like the "real" start of the music (or at least a new beginning in zero.) This is an action on what we call the phrasing of the piece: assigning meaning to a concatenation (ordinate), so that each element plays a role in obtaining a unit (the sequence being understood then as one "Object") in this position and that order . In other words, the phrasing is all functions of each segment, a position and a special order, which emerges a form that includes. 296 Thereby separating the introduction (mm. 1-70) the course "principal" of the room, we chose the first part of a musical space relatively "Narrow" and in great contrast to what follows: it is immediately characterized a predominance of his noisy on the stable sound; heights, they appear the strings and piano, are still in the Mith spectrum of bass. 297 296We will take this definition also for pieces smaller than the entire room. 297Notes written for winds indicate that playing positions that filter wind sounds - a pipe fully open or closed, or greater or shorter length. Page 222 222 With this general uniformity of timbre and harmony, is the dynamic activity and rhythm that articulates the shape of this section. For it to lead to this new beginning without being simply a contrasting part, we wanted to bring the introduction of a proper consistency (As if the room promised change completely in this space) and a directionality (which allows it to go somewhere). These two aspects are of the form here linked and interact 298They allow the integration of contrasts and joints section in a single "object" and provided greater meaning. to. in the partition The first instrumental gestures generate a continuous texture, rather smooth, which occasional salience only appear mostly as traits ascendants (still wind sounds). These climbs punctuate the material; each appearance they allow to close a unit of "meaning." Thus giving an identity common (through their profile) at different times segments, we pay already possible condensation of musical time. Moreover, the pace (formal) these units mark can be used to renew the listener's attention. The first time this object appears (mes.6-7), it is followed by a clear break Texture he comes ( ram tongue dry, interruption of all wind sounds, first entrance of the piano). This cut mark that precedes such an accomplished frase and 298This is not necessarily the case: we can say that gives directionality (some) consistency, but not the reverse. Page 223 223 This gives rise blasts the role of "end-of-frase" 299; two repetitions souffle- the object ram tongue before resuming the starting texture confirm this. There second appearance of the line upward (mes.15-17) is a variation, it is more long but still happens after four texture measurements (more articulate than internment the first time); the recovery is quick, and the input of piano still follows our punctuation. A much longer segment, which explains the different components of the complex wind sounds preceding the third viewing this object, made with the same figure that the second time. The immediate left with a previously exposed material as is, and develops, participates once again to the consistency of the piece: it "knows" in somehow what to expect. The fourth case contradicts the established behavior which the above: the times grew on each occasion the gesture, but this one is shorter than the third, and also follows a shorter segment of the smooth texture; again, we move to a more pointillist texture, accompanied by noise-notes (mes.38-50). The deviation is temporary and establishes a new standard for really a redefinition; return to the previous situation confirms this. It is at this time of recovery a previously established standard that is completed for us the impression that the room whole could develop in this particular space. In addition, the accumulation of these ascending lines, which is added to the accumulation, faster thereof, short crescendos on the ropes and suspended cymbal (my. 63, 67, 70), reinforces the perception 299Incidentally, it also redefines the very first step of the play, which takes clearer sense Page 224 224 a process which passes through the entire section and leads to the second part. The entrance it notes with "real" clear and regular rhythmic articulation, and much stronger dynamics, redefines the process that precedes and in fact, indeed, an introduction. b. "Lightness" Family stamps and joints used in this first section still involved in the making "formally forgettable" by characteristics that closer to the light given by Calvino, especially facing the densities follow the course of the play. We can find it, somehow, the accept that the author gives the concept 300" a literature of language , through which meanings are channeled through a verbal fabric almost imponderable up take that same consistency rarefied "; "Narration of reasoning or psychological process that involves subtle and imperceptible elements , or description which includes a high degree of abstraction "; " a figurative image lightness that takes an iconic value. " We do not, of obviously true meanings channeled directly by the material used, but the economy in stamps leads in fact to a rarefied consistency of the material sound. On the other hand, the directionality of this introduction is based on elements actually subtle (which must overlap to be perceived) and dynamic of a pickup. 300This emphasis. Page 225 225 sweet and the omnipresence of the breath sounds close may refer to a certain "sound lightness". 2.2 Internal and external memories In the composition of this piece, we used a memory External only as poetic resource, to represent what we evoked the Title: objects or vague concepts without clearly defined beginning or end, and whose definition may seem difficult or doubtful. This is the case in these melodies background (2 e part, my. 109, 132, 162, 178; and 3 harmony borrowed from Bach in 4 e part, my. 271 and 284) and e part (my 361-448;. it is the sarabande of the sixth suite for solo cello), but other aspects of the work involved in this representation without using this way to memory: détimbrés sounds the introduction, agreements dal niente and out of phase with the attacks (throughout the room), and even the opposition between a static harmony and rhythmic articulation very varied. On the other hand, it is structuring internal memory to the work we we wanted to build the consistency of speeches and opportunities for integration . There internal memory of a room are in the links that can be established between its parts and its scales: if we hear something that brings us to another party or Another aspect of the same work, this is possible thanks to the internal memory (plus Specifically, this reference is the internal memory). Page 226 226 The first way to organize the internal memory of a work is to establish of norms of behavior of the material which are evident. We have in weak topology of local rules, valid in a section (or subsection), and some global standards which remain for the entire room. Once an established standard, becomes a landmark for listening: it is followed or contradiction, it can be depart to return. The presentation of such benchmarks seem useful to us Since we no longer have a framework that we can consider shared all listeners and would allow these movements 'conceptual' speech (Functional punctuation, proximity or distance of a recognizable state ...). In Somehow, we are talking about musical spaces (compounds): the tonal system was an area known beforehand, in which we could put objects and make a priori assumptions about their behavior (and the impact of these behaviors on the intelligibility of the music); without such a landmark, it is for the same objects present the space in which the workpiece is registered (the space that it is , in fact - that Registration fills completely). The establishment of a standard is a way of make evident the topology of this space: if we can assess distances, we can think about continuity. It is of course possible to redefine a standard throughout a room: it implies a new allocation of meaning to the game elements in listening (redefining these, in fact), or simply a change. But this change does not erase obviously not what precedes: geometrically, it is not simply the Page 227 227 transition from one space to another, but the immersion of a first space in a Second - this is the global topology that redefines a local change. In each section weak topology we can identify several standards that characterize local (even when these standards are momentarily thwarted). Surface of the joints, which punctuate the overall shape, correspond to redefinitions of local standards: if enough is redefined, we a sufficiently significant change to be perceived as moving to another section. Note that "significant enough" depends on integration over time: for a standard to be effectively redefined, not only opposed (by process that moves away to return to it), it is actually a new standard is established, and this takes time. If we see these standards as spaces the room occupied or sets in giving to hear, we can say that we find a neighborhood of this space to be able to judge its topology; the more neighborhood is extended, the better you know - but you may not know it by one dot or a few isolated points. 301 301In a way, this redefinition always takes place continuously (see Chapter II), but here we consider the possibility of a change "locally discontinuous". Page 228 228 C CONCLUSION Não sou nada. Nunca serei nada. Não posso querer ser nada. To leave isso, tenho em mim todos os sonhos do mundo. - F. Pessoa (Álvaro de Campos) Tabacaria We tried to follow throughout this text, faithful to our reasoning compositional approach and thus to study its theoretical peculiarities. The choice of a light filtered by mathematics has allowed us to focus on the study of shape of the elements of this thought, and relations between them and the define. When formal thought is thus placed in the foreground, we run the may move away from physical constraints of the music and perception. In Granger's opinion, "the transcendental attitude analysis we leads to the recognition that mathematics are always further away from perceived. » We sought to support, despite our affinity with this author, an attitude Instead of analyzing immanent , which allows us to see a relevance of mathematics 302GRANGER G.-G. [1967] Formal Thinking and Human Sciences (p.11). Page 229 302 229 taken as a tool of thought, especially when the perception and understanding are at the center of reflection. Indeed, it is the insertion of the musician in his world that shapes its music: it can only be a reflection of the relationship of the individual to its exterior. The study of reasoning (and not just perception) involved in a compositional process is actually at the heart of what we might call a theory of the composition . Moreover, as the role of the visual arts has never was to give to see exactly what the eyes perceive the world, that of Music is not to suggest that the ears perceive: a exceeded "habits" is essential to the perceptive musical expressiveness and an abstraction research can eliminate or question its possibility. Yet the composer chooses to follow strictly mathematical operations compose, there is in the music, in the final analysis as musical rules. . Mathematics, a metaphor for the composition? If it is still possible to establish a relationship between mathematics and music, form of this relationship and its musical consequences are not given a priori in one of these disciplines. It is the action and the perception of a musician (not a mathematician) who carry this form according overwhelming musical interests. He no sense "material" in mathematics without interpretation, and there So no relationship between mathematics and music without arbitrary. However, a Page 230 230 report is still there, we just have to ask the question of its existence to do pop up. Whitehead 303 says that there is "discrimination desires, according to a standard adjustment. The reason the country is doubtful, vague and obscure. But exists. " This imprecision is the same space within which unfolds arbitrary, the choice of the musician on the significance of structures (mathematics and other) game in listening. This "wave territory" is actually also present in mathematics: we can say that the mathematician deals overwhelming elegance and clarity of his speech, the economy and the accuracy of the means employed; he directs his Search by personal taste or trends in a group where it fits, and there puts his "accent" staff; short, there can be no mathematics research without properly aesthetic choices. On the other side, the composer handles abstract forms which, although they can be inspired by elements of nature, do often refer to themselves; it seeks to articulate intelligible speech from a complex symbolic system; he must insert his research in continuity historical, or deliberately choose to break or change the route and points mark. Our work is composed of several of these ideas representations mathematics; we could say that is the musical representation of an idea of the Mathematics. 303WHITEHEAD , AN [1929], The Function of Reason , Princeton University Press. Page 231 231 B IBLIOGRAPHY TO M. [2005] Problems musical and mathematical conjectures. Test of a NDREATTA typology "mathémusicale" Seminar "Music & Mathematics" ENS (Seminar of 19 February 2005) TO NTunes , J. [1969] Cromorfonética , score at Suvini Zerboni, Milan TO NTunes , J. [1972], Proudhonia , score at Suvini Zerboni, Milan TO NTunes , J. [1974] Catastrophe Ultra-Violet , partition Salabert, Paris TO NTunes , J. [1977], Três Impressões Cancioneirígenas , score at Sistrum, Brasília TO NTunes , J. [1983], Inutilemfa , score at Sistrum, Brasília TO NTunes , J. [1985], Dramatic Polimaniquexixe , score at Sistrum, Brasília TO NTunes J. [2001a], 'Sinfonia das BUZINAS' o sublime useful eo na fronteira between o medo ea Ousadia , in Brasiliana no.7, Academia Brasileira de Música, Rio de Janeiro TO NTunes J. [2001b] The Semanteme , in Emotions & Music , EDK, Paris TO NTunes , J. (org.) [2002] Uma poética musical brasileira e Revolucionária , Sistrum Brasília TO NTunes , J. [2005], para os novos Sons Sopros e as cordas , Sistrum, Brasília TO PEL, K.-O. [2005] The "a priori" the body into the problem of knowledge , the Editions du Cerf, Paris Page 232 232 TO ROM, S. [1985], polyphony and polyrhythms of Central Africa. Structure and methodology , SELAF, Paris (also: African Polyphony and Polyrhythm. Structure and Methodology , Cambridge University Press, Cambridge, 1991) B Aboni -S CHILLINGI , J. [2005] The musica iper-sistémica ( The hyper-systemic music , trans. in preparation), ESC, Paris B ACKESJ.-L. [1994] Music and Literature. Poetic comparative test , PUF, Paris B ARBARAS , A. [2004] Introduction to the philosophy of Husserl , Ed. Transparency, Chatou B ARONIM. [1995] The part of the analysis in the composition process , Musurgia, II / 2, Ed. Eska, Paris B OULEZP. [1963], Thinking Music Today , Gonthier, Geneva B UCI-G LUCKSMANN , C. & L EVINAS,M. (org.) [1993], The Musical Idea , PUV, St-Denis B URTTEA [1952], The Metaphysical Foundations of Modern Science , Humanities Press, Atlantic Highlands C AREAS C. [2004] IRIN - Graphical micromontage in Sound Editing and Mixing Tool , in Proceedings of the International Computer Music Conference, Miami: The Interrnational Computer Music Association, 2004, p.219-222 C ALVINOI. [1984] Lezioni americane. Sei proposte per il prossimo Millennio ( Seis Lições para o próximo Milênio , trans. I. Barroso), Cia. das Letras, São Paulo, 1997 C Arroll L. [1865] Alice's Adventures in Wonderland , in Alice in Wonderland , Wordsworth, Hertfordshire, 1995 C HOUVEL J.-M. [1995], The Physics and Aesthetics. An epistemological analysis terms of knowledge of the harmonic phenomenon , Musurgia, II / 4, Ed. Eska, Paris Page 233 233 C HOUVEL , JM & S , M. (org.) [1997] Space: Music / Philosophy , OLOMOS L'Harmattan, Paris C OTT, J. [1979], Conversations with Stockhausen , Lattes, Paris C RAFTR. & S TRAVINSKY I. [1959] Conversations with Igor Stravinsky ( Conversas com Igor Stravinsky , trans. Moutinho SRO), Ed. Perspectiva, Sao Paulo. C RITONP. [1997], sensitive areas , in C , JM & S HOUVEL , M. (org.) [1997] OLOMOS D Ahlhaus C. [1997] Schoenberg , Contrechamps, Geneva D Opke , D. [1987], fragmentary reflections on musical poetics of quartet strings Luigi Nono , Program Review-Autumn Festival, Contrechamps Paris F ICHETL. [1996] The Scientific Theories of Music, XIX e and XX e centuries , J. Vrin, Paris F ORTE, A. [1977] The Structure of Atonal Music , Yale University Press, New Haven F REGEG. [1879] Begriffsschrift ( Idéographie , trans. C. Besson), J. Vrin, Paris, 1999 F REUDS. [1908 e ] Das Dichter und das phantasieren ( The literary creator fantasy , trans. B. Feron) in The Uncanny and Other Essays , Gallimard, Paris 1985 G ENEVOIS H. & O RLAREY,Y. (ed.) [1997] Music and Mathematics , Grame: Hazards, Lyon G OLDBLATT R. [1984], Topoi, The categorial analysis of logic , Elsevier, Amsterdam (Also available online http://historical.library.cornell.edu/math/) G OODMAN N. [1966] The structure of appearance , Bobbs-Merrill Company, Indianapolis G OODMAN N. [1985], Languages ​of Art: an approach to a theory of symbols , Hackett, Indianapolis Page 234 234 G RANGER G.-G. [1967] Formal Thinking and Human Sciences , Aubier-Montaigne, Paris G RANGER G.-G. [1994] Forms, Operations, Objects , J. Vrin, Paris G RANGER G.-G. [1999] Thought of space , Odile Jacob, Paris G UILLERMIT , L. (ed.) [1989], Plato himself (selected texts), The Shard, Sommières H ANSLICK E. [1854] Vom Musikalisch-Schönen ( From the beautiful in music: test reform musical aesthetics , trans. C. Bannelier), C. Bourgois, Paris, 1986 H ARDYGH [1940] A mathematician's apology , Cambridge University Press, Cambridge H OFFMAN P. [1997] The abstract space in the music of Iannis Xenakis , in C JM & S , HOUVEL , M. (org) [1997] OLOMOS V.H OUTEN, K & K Asberger M. [1992], Bach and the number , Mardaga, Liège J OLASB. [1993], pictures and sound musicianship , in B , C. & L EVINAS, UCI-G LUCKSMANN M. (org.) [1993] J AKOBSON R. [1963], General Linguistics Essays (trans. N. Ruwet), Ed. of Midnight in Paris J OHNSON T. [2001] Found Objects Mathematics, Seminar meantime, Mathematics, music and philosophy , IRCAM, 2000-2001 (seminar of 13 January 2001) K ANTI. [1781-1787], Kritik der reinen Vernunft ( Critique of Pure Reason , trans. A. Renaut), Flammarion, Paris, 2001 K ANDINSKY W. [1926] Punkt und Linie zu Fläche ( Point and Line to Plan , trans. J. S.et Lepien) Gallimard, Paris, 2000 K Elkel M. [1988] Music worlds , Vrin, Paris Page 235 235 L Erdahl F. & J ACKENDOFF , A. [1996] A Generative Theory of Tonal Music , MIT Press, Cambridge L IMAEL [1991] Dimension e em Forma Geometria , Sociedade Brasileira de Matemática - SBM, Rio de Janeiro. L IMAEL [1992], Curso de análise, vol.1 , Impa, Rio de Janeiro MR G. [1999] A poetics of the unexpected , master's thesis, University AGNIEN Paris VIII MR P. [1991] The views of Borges , in Six Musicians Quest Author , ANOURY Pro Musica, Isle-lès-Villenoy MR ASSINB. [1989], Olivier Messiaen: a poetic of the marvelous , Paragraph, Aix-enProvence MR G. [2002] The Topos of Music , Birkhäuser, Basel AZZOLA MR G. [2005] The possible role of musical logic to some AZZOLA mathematical intellectuality , Seminar "Music & Mathematics" ENS (Seminar the 16 April 2005 available also on http://www.entretemps.asso.fr/maths) MR ENEZESB ASTOSRJ [1978] A Musicologica Kamayura. Para uma da antropologia comunicação No. Alto Xingu , Funai, Brasília MR C. [1624] Il combattimento di Tancredi e Clorinda , score at ONTEVERDI Universal, Vienna, 1991 MR OLES, A. [1972], Information Theory and Esthetic Perception , Denoël, Paris MR Olino , J. [1995] Experience and know , Musurgia, II / 4, Ed. Eska, Paris N ATTIEZ, JJ [1973] Foundations of a semiology of music , PhD, University of Paris VIII Page 236 236 N ATTIEZ, JJ [1987] General Musicology and semiotics , C. Bourgois, Paris N ICOLASF. [1994] Many notes and musical work , Proceedings of the 3 e ICMPC, Liège N ICOLASF. [1997] The third hearing is the right (From the musical audition designed as inte gration-) , Musicæ Scientiæ n o2 N ICOLASF. [2000] Music, Mathematics and Philosophy: What comes to make it here Philosophy? Seminar Meanwhile, Mathematics, Music and Philosophy , IRCAM, 2000-2001 (seminar on 7 October 2000) N ICOLASF. [2005a] Rai resonance music / mathematics writing shared , Seminar "Music & Mathematics" ENS (seminar of 19 February 2005) N ICOLASF. [2005b] How musically evaluate mathematical theories music? The example of Mazzola's theory , Seminar "Music & Mathematics ", ENS (seminar of 16 April 2005) N YMANM. [1974] Experimental music , Studio Vista, Macmillan, London P ASCAL, A. [1997] The number in the musical composition in the twentiethe century , in GENEVOIS , H. & O RLAREY , Y. (ed.) [1997] P ENDERECKI , K. [1959] Threnos , score from Belwin Mills, Melville P ESSOAF. [1928] Tabacaria , in Obra poética Ed. Nova Aguilar, Rio de Janeiro, 1986 P REDA-S CHIMEK H. [2003] A look at the genesis of the form of theories, among Classicism and Romanticism (1790-1845) , Musurgia, X / 2, Ed. Eska, Paris Q UETTIER P. [2000] complex messages Communication by sequences Gesture: kata in Japanese martial arts school Shintaido , thesis Ph.D., University of Paris VIII R AMEAUJ.-Ph. [1722] Treaty of Harmony reduced to its natural principles , in Complete Theoretical Writings , vol. III, American Institute of Musicology, 1968 Page 237 237 R AMEAUJ.-Ph. [1737] Harmonic Generation , in Complete Theoretical Writings , vol. III, American Institute of Musicology, 1968 R ESTALLG. [2005], Proof & Counterexample , University of Melbourne (available in online http://consequently.org/writing/pc) R EY-D EBOVE, J. & R EY, A. (org.) [2002] The Oxford Dictionary , Dictionaries The Robert, Paris R Igoni M. [1998] Stockhausen ... a vessel launched into the sky , Millennium III Ed., Rouen R OADSC. [2002] Microsound , MIT Press, Cambridge R ODRIGO P. [1995], Aristotle, and eidetic phenomenology , J. Millon, Grenoble R USSEL, B. & W , AN [1910-1913], Principia Mathematica , Cambridge HITEHEAD University Press, Cambridge (republished in 1968) R UWETN. [1972], language, music, poetry , Seuil, Paris S AIDE. [1991] Musical elaborations ( Elaborações Musicais , trans. H. dos Santos) Imago, Rio de Janeiro, 1992 S AUSSURE F. [1916], Course in General Linguistics (publ. Ch. Bally and A. Sechehaye, ed. critic T. De Mauro), Payot, Paris, 1974 S CHAEFFER P. [1966] Treaty musical objects , Seuil, Paris S CHENKER H. [1935] Der freie Satz ( The free writing , trans. Meeùs N., ed. O. Jonas) Mardaga, Liège, 1993 S CHLOEZER B. [1947] Introduction to Bach , Gallimard, Paris S CHOENBERG , A. [1911] Harmonielehre ( Treaty of harmony , trans. G. Gubisch), Lattes, Paris, 1991 Page 238 238 S EDES, A. (org.) [2003] Sound Spaces, research procedures , Ed. Musical Transatlantic, Paris S ERRES, M. (org.) [1989], the history of science items , Bordas, Paris S INACEUR H. [1991] Body and Models , Vrin, Paris S OLOMOS Mr S OULEZ, A. & V H. [2003], Formel-Informal: musicAGGIONE philosophy , L'Harmattan, Paris S OLOMOS M. [2005] Notes on the concept of "emergence" and Agostino Di Scipio , Acts of Computer Music Days (JIM 05), MSH-CICM, University Paris 8, Paris S OATUU DIAS, A. [2005] The sound object: situation assessment and potential , thesis PhD, CICM, University of Paris VIII S TOCKHAUSEN , K. [1953-1960] Letters to Pierre Boulez , Festival of the program booklet Paris Autumn 1988 Contrechamps S TRAVINSKY I. [1942] Poetics of music (in the form of six lessons) , Harvard University Press, Cambridge S ZABO, A. [1977], The Beginnings of Greek Mathematics , Vrin, Paris T Tolkien JRR [1964] It fairy-stories , in Tree and Leaf , George Allen and Unwin, London V AGGIONE H. [1995] Objects, representations, operations , Review Ars Sonora n pp.33-52, CDMC, Paris (available on http://homestudio.thing.net/revue/content/ asr2p30.html) o 2 V AGGIONE H. [1996] Towards a transformational approach in CAD ​, Proceedings of the 3 Days Informatics Musical, IRCAM Paris (Available th on http://recherche.ircam.fr/ teams / Repmus / jim96 / actes.html) Page 239 239 V AGGIONE H. [1997] Dealing with networks of objects , International Academy electroacoustic music in Bourges V AGGIONE H. [1998a], Sound, time, object syntax. Towards a multi-scale approach the computer-assisted composition , in Music, Rationality, Language , Notebooks Language Philosophy n o 3, L'Harmattan, Paris V AGGIONE H. [1998b] The composable space , in C , JM & S HOUVEL , M. (ed.) OLOMOS [1997] V AGGIONE H. [2000] Musical Composition and IT Resources: Questions Approach , Proceedings of the conference Domani Musica e Tecnologia , Milan V AGGIONE H. [2001] Some Ontological Remarks about Music Composition Processes , Computer Music Journal, 25: 1, Spring 2001, MIT V AGGIONE H. [2003] Decorrelation microtemporelle, morphologies and figurations spatial sound musical , in S EDES, A. (org.) [2003] V AGGIONE H. [2006], symbols, signals, music operations , in S OULEZ, TO. & VAGGIONE , H. (eds.), Ways of sounds , L'Harmattan, Paris (forthcoming) V ILLA, A. [2002] Pierre Schaeffer and musical objects: Questions about the possibilities a phenomenology of musical perception , Master's dissertation, University of Paris VIII V ILLA, A. [2003], Pierre Schaeffer: a phenomenological study , DEA, University of Paris VIII V Oisin F. [2003] The contrastive musical analysis , PRISMA 01 Euresis, Milan W EGNERP. [1997] Why Interaction is More Powerful than Algorithms , Communications of the ACM, 40 (5), pp.80-91 W HITEHEAD , AN [1929], The Function of Reason , Princeton University Press Page 240 240 W ITTGENSTEIN L. [1922], Tractatus Logico-Philosophicus (bilingual edition germanPortuguese LH Lopes dos Santos) EDUSP, São Paulo, 1994 W ITTGENSTEIN L. [1935] The Blue and Brown Books ( The Blue Book and brown notebook , trans. Mr. Goldberg and J. Sackur, Gallimard), Paris, 1996 W ITTGENSTEIN L. [1939] Bemerkungen über die Grundlagen der Mathematik ( Remarks on the Foundations of Mathematics , trans. Lescourret MA) Gallimard, Paris, 1983 W ITTGENSTEIN L. [1929-1948], Zettel ( Sheets , trans. J.Fauve), Gallimard, Paris, 1970 W ITTGENSTEIN L. [1949] Philosophischen Untersuchungen ( philosophical Searches cal , trans. F. Dastur), Gallimard, Paris, 2005 (often referred to as Investigations philosophical , in the translation of P. Klossowski, Gallimard, Paris, 1986) W ITTGENSTEIN L. [1951] Über Gewissheit ( On Certainty , trans. J. Tan), Gallimard, Paris, 1987 X ENAKISI. [1963] Formalized Music , Stock Music, Paris X ENAKISI. [1971] Music: architecture , Casterman, Paris Page 241 241 TONNEXES We append here a few scores of pieces composed during our research, and have shown our words in this text. A brief overview of mathematical ideas represented in more centrally each piece before its partition. These introductions can be skipped without loss of Content for reading works. Page 242 242 1. a convex set Convexity Let E be a vector space. For A, B "E, the line segment ends a and b is a whole [A, b] = {(1-t) a + Tb; 0't'1} A subset X of E is said convex if a, b "X © [a, b] 7X Homotopy n, With the same Are æ ç: [a, b] ƒX paths throughout X7½ domain [a, b] and the same ends, that is to say æ (a) = ç (a) and æ (b) = ç (b). A homotopy between æ ç and is a continuous H: [a, b] _ [0,1] ƒX such that H (s, 0) = æ (s) H (s, 1) = ç (s), H (a, t) = æ (a) = ç (a) and H (b, t) = æ ( b) = ç (b) for all s "[a, b] and all t" [0,1]. When there is a homotopy between æ ç and we say that these paths are homotopic . A homotopy H: [a, b] _ [0,1] ƒX between æ ç is said and adapted when there is a (0 <a <1/2) such that for all x "[a, b], H (x, t) is constant in the intervals 0't <to and 1 to <t'1; ie, such that H (x, t) = AE (x) for all t "[0, a) and H (x, t) = c (x) for all t "(1-in, 1] and that for every x "[a, b]. Page 243 Page 244 Page 245 Page 246 Page 247 Page 248 Page 249 Page 250 Page 251 251 2. Lema 1 - e partições primitive Let f: [a, b] ƒ½ a limited function and P = {a = t 0, T1, T ...n= B} a partition [A, b]. For each i = 1, ... n, we will mr = inf f andMR = sup f . i i [ti-1,ti] [ti-1,ti] We define the lower sum s (f; P) and the higher amount S (f; P) of the function f with respect to the sheet P by: n n s ()f ; P = Σ mr ⋅ (t - t ) andS ()f ; P = Σ MR ⋅ (t - t ) . i i i-1 i i i-1 i=1 i=1 We can then define the integral lower and upper integral of f on [A, b] with (respectively): b b ∫ f = sup s ( f ; P ) and ∫ f = inf S ( f ; P ) , P P to to with the upper terminals (u) and lower (inf) taken on all of the sheets P [A, b]. b b ∫ f = ∫ f And we note A function f is integrable on [a, b] when to to ∫bf . the integral of f simply to Fundamental Theorem of Analysis If an integrable function f: [a, b] ƒ½ has a primitive F: [a, b] ƒ½, then ∫ b f = F ()b ()- F to . to Page 252 252 Page 253 253 Page 254 254 3. Any differentiable function is continuous Definition of the derivative in x0Denoted f'(X0) 'To> 0, -p> 0 such that f ()x ()- f x 0 - f ' ()x < ε x - x <δ ⇒ 0 0 x- x 0 Definition of continuity in x 0 'To> 0, -p> 0 such that | Xx 0| <ß © | f (x) -f (x 0) | <A Theorem : -f '(X ) © f is continuous at x . 0 0 Demonstration : Either 0> 0. There ß 0> 0 such that: f ()x ()- f x 0 - f ' ()x < ε x - x <δ ⇒ 0 0 0 0 x- x ⇒ f ()x ()- f x - f '()x ( 0x - x ) < ε x - x < ε . δ 0 0 0 0 0 0 0 But | F (x) -f (x 0) | - | F '(x0) (Xx 0) | '| F (x) -f (x 0) F '(x 0) (Xx 0) | Therefore | Xx 0| <ß 0 © | f (x) -f (x 0) | <A0ß 0+ | F '(x0.) | | (Xx 0) | <ß0(To 0+ | F '(x0) |) ε .δ Thus, ∀ε > , ∃0δ = 0 such as 1 ε + f ' ()x > 0 | Xx 0| <ß 1 © | f (x) -f (x 0) | <A, therefore f is continuous at0x. Quod erat demonstrandum . Another demonstration: - f x0 exists, then there is also the limit If the limit lim f ()x () x→x0 x - x0 ⎡ () () ⎤ lim [ f ()x ()- f x ] = lim ⎢ f x - f x0 ⋅ (x - x )⎥ = 0 x x ⎣ x - x 0 ⎦ x→x0 →0 0 f ()x ()- f x0 lim ( x - x0) = .0 = lim ⋅ x x - 0 x→x0 x→x0 So f is continuous at x 0. QED Page 255 255 Page 256 256 Page 257 257 Page 258 258 Page 259 259 Page 260 260 Page 261 261 Page 262 262 Page 263 263 Page 264 264 Page 265 265 Page 266 266 Page 267 267 Page 268 268 Page 269 269 4. weak topology Let E be a normed vector space and complete (a Banach space). A topology on E is a family of subsets E, among which are the empty set | and E itself, which is stable intersections (œ) and finite unions (>) Any. These subsets are called the open E. A function f: Eƒ½ is continuous if the antecedent by f of every open set ½ is an open E. Let now E 'the space of continuous linear forms E ½ (dual topological E), and f 'E'. We denote by ï f : Eƒ½ the defined application ï f (X) = <f, x> (form linear f applied to the point x E). Considering all f in E ', we obtain a I family of applications in ½, it may be noted that (I . f ) f "E ' The weak topology on E, denoted ì (E, E ') is the coarsest topology on E (With the least open) makes continuous all applications (ï f )f "E.' Page 270 Page 271 Page 272 Page 273 Page 274 Page 275 Page 276 Page 277 Page 278 Page 279 Page 280 Page 281 Page 282 Page 283 Page 284 Page 285 Page 286 Page 287 Page 288 Page 289 Page 290 Page 291 Page 292 Page 293 Page 294 Page 295 Page 296 Page 297 Page 298 Page 299 Page 300 Page 301 Page 302 Page 303 Page 304 Page 305 Page 306 Page 307 Page 308 Page 309 Page 310 Page 311 Page 312 Page 313 Page 314 Page 315 Page 316 Page 317