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UNIVERSITY PARIS 8 - VINCENNES - SAINT-DENIS

UFR 1 - ARTS, PHILOSOPHY, AESTHETICS

THESISto get the grade

DOCTOR OF U NIVERSITY P ARIS 8

in

BEAUTY , SCIENCE AND TECHNOLOGY OF ARTSDiscipline: Music

Presented and supported publicly by

Guilherme Carvalho

Title:

MUSICAL PERFORMANCES OF IDEAS MATHEMATICS

Supervisor: Horacio V AGGIONE

Jury:Martin L ALIBERTEMr. Makis S OLOMOSMs. Antonia S OULEZ

February 2007

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2TOVANT -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.

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T TABLE OF CONTENTS

TOVANT -P ROPOS .................................................. ............................ 2I NTRODUCTION .................................................. ............................. 6

. Thought formelle.................................................................................................... 7

. Mathematics presence ............................................... ............................... 7

. Languages ............................................................................................................ 10

. Thorisation....................................................................................................... 11

. Pratiques............................................................................................................ 14

I - E AND Xistence MRULTIPLICIT ................................................ 171. From the rapprochement between mathematics and music ..................... 18

. Hardware abstraction and constraints .............................................. .............. 191.1 Reasons for a rapprochement between the two disciplines .................. 21

. Musical thinking, musical discourse ............................................. ................... 21to. discourse on music .............................................. .................................. 22b. musical discourse ................................................ ............................................ 23

. Take mathematics as such ............................................. ..... 24

. Mathematics as "Truth"? .................................................. ....... 271.2 Ways to establish a rapprochement .......................................... ........ 29

. Formalization, model, modeling ............................................. .................. 30

. Using mathematical ............................................. ......................... 32to. ................................................. Application .................................................. . 33b. parallle........................................................................................................ 35c. intermediate situations ................................................ ............................... 37

1.3 From what is formalized ............................................ ............................. 38. Notation.............................................................................................................. 38. With the notation: writing and proofreading ........................................... ............... 42

to. writings and logical time .............................................. ............................... 43b. analysis and proofreading ............................................... ...................................... 45

. Running time (time) ........................................... .... 48to. distance of a formal language ............................................ ............................ 49b. validity of this juxtaposition with formal languages ............................. 53

. Discourse on music .............................................. ...................................... 54to. theories of music .............................................. .................................... 54b. analyzes and styles ............................................... ............................................ 56c. 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

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. Illustration: Princpio Cavalieri ............................................. ................... 71

. Traces of formalization .............................................. .................................. 752.2 Second definition ............................................... ..................................... 77

. Illustration: a convex set ............................................. ..................... 80to. convexity ....................................................................................................... 81b. homotopies.................................................................................................... 84

. Illustration: Lema 1 .............................................. ............................................ 87to. partitions....................................................................................................... 88b. primitivation.................................................................................................. 90c. droulement................................................................................................... 91

2.3 Types of representations .............................................. ............................. 92. Constructed representations ................................................ ............................. 95. Found representations ................................................ .................................. 97. Representations mtamusicales ................................................ ..................... 100

3. From the senses in a formalization ........................................... ........... 1023.1 Two sense to "interpretation" ........................................... ............... 103

. Movements of "meaning" ............................................. .................................. 1043.2 intelligibility and understanding .............................................. ................ 105

. Transmission of content ............................................. ............................... 107

. Signification..................................................................................................... 1124. arbitrary decisions ............................................. ..................... 114

4.1 The possibility to model .......................................... music 115 ....

4.2 From the distance between object model .......................................... ............. 1214.3 decisions to model ........................................... ..... 1264.4 Overcoming the distance between object model .......................................... ..1304.5 An irreducible distance .............................................. .......................... 1344.6 The need for arbitrary decisions ........................................... ..... 137

II - G OMTRISATION .................................................. .............. 1401. Figure and forms ............................................. ................................... 141

1.1 Non-Euclidean Geometry ............................................. .................... 141Figures 1.2 and musical musical contexts ............................................ ..145

. Figure note................................................................................................. 146to. illustration: as a melodic line .......................................... ....... 151

. Link to the perception .............................................. .......................................... 152

. morphological eidetic ............................................... .............................. 155

. Objects ............................................................................................................... 1582. Musical Spaces .............................................. .............................. 161

2.1 Dimensions and parameters .............................................. ........................ 162. Illustration: a first musical space ............................................ .......... 165

2.2 Directions and fragments of space .......................................... ............. 1673. musical objects as functions ........................................... .170

3.1 Variable Sets .............................................. ............................. 171to. instruments.................................................................................................. 173

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b. game modes ............................................... ................................................. 174c. hauteurs....................................................................................................... 176d. positions...................................................................................................... 178e. dynamiques.................................................................................................. 180f. dures........................................................................................................... 181

3.2 Fonctions.................................................................................................184. Indivisible gestures ................................................ ........................................... 186. Insertion time .............................................. ..................................... 189

3.3 Continuities ................................................ .............................................. 1914. Areas modular, composed space ........................................... 193

. Several temporal directions ............................................... ..................... 194

. An "extended" size ............................................. ................................ 1954.2 Locality and globality .............................................. ................................... 1974.3 The form of a work ........................................... ................................. 199

III - P OTIQUE MRUsical .................................................. ....... 2021. Dfinition(s)......................................................................................203

. Language courant.............................................................................................. 203to. music as a "text"? .................................................. .............. 204

. Stravinsky........................................................................................................ 206

. Schoenberg, Dahlhaus ............................................... ..................................... 207

. Nono, Antunes ............................................... .................................................. 209

. Backes, Ruwet ............................................... .................................................. 213

. A position .............................................. ......................................... 2172. 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 ............................................... ........................................... 222b. "Lightness" ............................................... .................................................. . 224

2.2 Internal and External Memory ............................................. ...................... 225

C CONCLUSION .................................................. ............................. 228. Mathematics, a metaphor for the composition? ........................ 229

B IBLIOGRAPHY .................................................. ........................ 231

TONNEXES .................................................. .................................... 2411. a convex set ............................................. ........................... 2422. Lema 1 - e parties primitive .......................................... ............ 2513. Any differentiable function is continuous ........................................... ..2544. weak topology .............................................. ................................... 269

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I NTRODUCTION

Human reason has this particular destiny (...) it isoverwhelmed 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.

1XENAKISI. [1963], formal music.2See in particular GOODMANN. [1966] The structure of appearance.

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. 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

3See redefining vocabulary must construct S CHAEFFERP. [1966] Treaty of musical objects.4See eg N YMANM. [1974] Experimental music (ch.6).

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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

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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.

5See GRANGERG.-G. [1994] Forms, operations, objects: "(...) epistemology must be both analysissome philosophical sciences, taken in their effective procedures and their evolution, and secondlymore general interpretation of the meaning of scientific knowledge. "(Introduction, p.8)6As opposed to the speakable, as we find them in W ITTGENSTEINL. [1922], Tractatus LogicoPhilosophicus. This brings us to a first difficulty inherent in any theoretical work on practicalmusic: 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 themeaning of salience, often rich, these detours generate.

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. 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

mtadisciplines 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)

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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

7GRANGERG.-G. [1994] What is mtadiscipline? 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 orinterpretive). This is made difficult by the fact that metamathematics may be strictly parteven math (so explicitly with category theory): thinking aboutMathematics can have the form of a proper mathematical thinking. We willoccasionally 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 ofdelicate ontology of music, can cause dogmatism in which mathematics isunfairly empowered. "(It is Well Known que la precision of mathematical results, together with apoor knowledge about the delicate ontology of music May Provoke a dogmatism for qui est mathematicsunjustly made responsible.) - MAZZOLAG. [2002] The Topos of Music (p.9).

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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. 12Naturally, 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 theyare, without theoretical reflection.11We are not suggesting that completeness and consistency are mutually exclusive here, as is the case inmathematics, but we do not deny a priori the possibility.

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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 ICOLASF. [2005b] How musically evaluatemathematical theories of music?

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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. 15This 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).14WITTGENSTEINL. [1922], Tractatus Logico-Philosophicus (Preface).

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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 momentwhere we thought we could bring together the disciplines around a general theoryapplicable to each individual work: the dream is over.Therefore there can be no epistemology of analysis becauseNow that analyzes not mean anything specific, because by that we mean anymethod 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

15GRANGERG.-G. [1967] Formal Thinking and Human Sciences (p.11).16Idem, p.1217NATTIEZ, JJ [1973] Foundations of a semiotics of music (2 epart, the musical discourse).18MROlino, J. [1995] Experience and knowledge.

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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. 19

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. 20

We chose, as we said, to run the risk of

idiosyncrasy, it is because we believe it is actually less: if we can

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: analysisimmanent level (which the author has had the misfortune to also appoint "neutral level") is "the study ofdirection of relations between the component elements work (notes or other sound objects, defined bycontrolled process), regardless of their origin and their compositional perceptual hierarchy.

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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

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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 itsemployment. Instead, it is talking simultaneously of these three fields that caution should betaken and details will be like so many refinements of a purely epistemologymusical.

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. 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 RANGERG.-G. [1999]Thought of space .23Although we seem natural now, this choice was not the only one available, and was made forphilosophical reasons as much as practical. V. it, B URTTEA [1952] The MetaphysicalFoundations of Modern Science ; particularly Chapter VI.

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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. GRANGERG.-G. [1994] Forms of Operations Objects . We shall return to the subject of

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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 timesometimes also in a timing set time. This registration is essential and wereturn throughout this work.

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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 "psychomathmatique 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 hasrhetoric. In particular, a poem, a story we are also speeches.26RAMEAUJ.-Ph. [1737] Harmonic Generation

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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

27TOROM, S. [1985], polyphony and polyrhythms of Central Africa. Structure and methodology .

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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

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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,

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one can not deduce anything of the proposal (p q) nor the proposal (q r). If I want

to infer something, I choose to articulate the (formally) with

conjunction: ((p q) (Q r)), and I can only assume (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 r) through a

formal deduction, I selected as premises (p q) and (q 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 believenot only to formal logic which includes the quantifiers and operators need / opportunity,but also to the theory of demonstrations and geometric logic for studyingsymbolically these decisions and their "form" logic (see eg R ESTALLG. [2005]Proof & Counterexample and GOLDBLATTR. [1984], Topoi, The categorial analysis of logic ).29We return to this relationship between form and environment in the chapter dedicated to the geometric lookthat we can focus on the music, and the compositional processes in particular.

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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 30, Came without

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

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

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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. FICHETL. [1996] Scientific Theories music, XIX e and XXe centuries , J. Vrin, Paris.

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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 throughcenturies and cultures each one and the same.

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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 takesin mathematics, for which a model is an object (or set of objects) in which we find theproperties of a theory .

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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 ofits 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. XENAKISI. [1963] Formalized Music , Stock Music, Paris.36MRAZZOLAG. [2002] The Topos of Music , Birkhuser, Basel.

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"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

37Also M AZZOLAG. [2005] The possible role of musical logic to somemathematical intellectuality (Conference of 16 April 2005 at the Music Seminar in MathematicsENS).38They will appear later in the illustrations by musical examples.

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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 objectsin question. An example of mathematical propositions application to speech

Music is provided by Xenakis for the construction of his play Herma 39, Where two

equivalent expressions of set theory are used to organize between

these sets of heights. Within these assemblies, another application

39XENAKISI. [1963] Formal Music

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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 takethe fact whether it is a good implementation of probability theory: the crucial point remainsthe impact of this theory about writing and composer already knows enough applicationsformally strict order to imitate them.

41For an overview of this topic, see the book F ICHETL. [1996] The Scientific Theoriesmusic ...

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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 par-

above 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

42DURUTTEC. [1876] Summary of elementary Technie Harmonic , quoted by FICHETL. [1996].43Especially when Xenakis informally recreates a stochastic arrangement of noteswithin sets of heights Herma , it overcomes the problems of a strict application allkeeping an accurate overall sound result, genre originally obtained by formalizing well

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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:

between number and work44And between integration and hearing45.

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. XENAKIS[1963]). This is precisely the musical representation of an idea, as wedefine it further.44NICOLASF. [1994] Many notes and musical work , Proceedings of the 3e ICMPC, Liege.

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"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

45NICOLASF. [1997] The third hearing is the right (from the musical audition designed as an integralgration) , Music Scienti, No. 2.46This does not detract from its usefulness in general, among others to think directly operating toolsthe composition themselves, or to operate within a scan. Symmetrically, weseems that the application, as we have defined it, is more immediately usable in practiceCompositional in theorizing.

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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

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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 needsdynamic controllers, joint or attack symbols; the ornamental detailsevolve or disappear. MR ONTEVERDIIn Il combattimento di Tancredi e Clorinda [1624] indicatedwith "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 REGEG. [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 gesturesinstrumentalists (more on this). Note that this view also allows us to considera CD as a kind of partition must also be "interpreted" by a player.

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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. 50What 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

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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.

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. 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. ITTGENSTEINL. [1935] The Blue and Brown Books "thinking is essentially the activityis to operate with signs. (...) Moreover, if we speak of the place where thought takes place, wehave reason to say that this place is the paper we write "( Blue Book , [6] - [7]). The ideaa thought present in his own writing also reflected in the Remarks on the FoundationsMathematics (W ITTGENSTEINL. [1939]). Note, however, that syntax and rulesHandling is not strictly speaking a write , even if they are indispensable.53This does not mean that necessarily driven objects have their own logic, intrinsic orthey have one that is inaccessible directly; but a formalization is not neutral, and implies

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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 55He 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 stamptemporal articulation, are the source of this impasse, but we would say that the partition offerstype of joint (not just scale) of often incompatible with sound joints,same macro-temporal, electroacoustic music.55These terms are often present in the vocabulary of the language, cf. for exampleSAUSSUREF. [1916], Course in General Linguistics . But we want to keep them here without anyimplications of this discipline, in the sense suggested by their etymology. Thus, we understandphrase as it relates to the sequence and combination possibilities and paradigmaticas it relates to the comparison and substitution possibilities.

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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.

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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 like that proposed Vaggione "complex systems involving a plurality ofoperating levels ", in V AGGIONEH. [2001] Some Ontological Remarks about Music CompositionProcesses .59We leave here the principle that improviser has a musical vocabulary internal (not necessarilyshareable 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 othercontent. Limitations on the complexity we can handle mentally tell us again, if

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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

paradigmatic syntagmatic joints from joints 61. Naturally, the

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 thework (both in music and elsewhere).60Here we describe these links logical just to differentiate them possible links materials (forproximity or congruity) there is not necessarily the possibility of formal sequence consistingtherebetween.61This is the case even for the paradigmatic analysis proposed by RUWETN. [1972], language, music,poetry and N ATTIEZ, JJ [1987] General Musicology and semiotics . Indeed, Nattiez itselfstresses that the analysis of the problem (paradigmatic or similar) is primarily that of segmentationa statement on its syntagmatic axis (lectures at ENS Paris, 1992, cited by V OisinF. [2003]The contrastive musical analysis ).62See SCHENKERH. [1935] Der freie Satz ( The free writing , trans. N.Mees, ed. O.Jonas) Mardaga, Lige,1993.63See VOisinF. [2003] The contrastive musical analysis .

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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.

64In these cases, writing takes for the music a comparable way to that it takes for themathematics: it is in her that "exercised" thinking it is the logical trace (and the logic) ofmusical 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 partthe logical time of the composition and analysis.

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. 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 micro-

as 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

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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 OADSC. [2002] Microsound , MIT Press, Cambridge.67Especially when Xenakis uses the physics of gas as a model. V. X ENAKISI. [1963]Formal music .

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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

68GRANGERG.-G., The concept of formal content and formal content and duality , in GRANGERG.-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 itOnly 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 theirconsistency (unfounded organicity and almost in formal, like the language of gamesWittgenstein), firstly, and secondly a resemblance ideas form in logic, mathematicsand music, which allow the emergence of formal content. Granger placing formal contentsomehow upstream synthetic judgments a priori of Kant, we would go up to say theirexistence in music is quite compatible with (or suggested by) the fact that we always knowwhat music has no authority to define it.71We will see in the chapter on the geometrization how it is equivalent to consider the formof a work like geometry or as a formal space.

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are actual developments of the musical