(in Mathematics and Music - Springer, 2002)


François Nicolas

En français



What is logic in music ? Is there any practice in music that can be called logic ? Is the nature of this possible logic in music musical or mathematical ?



These questions entail today a considerable degree of subjectivity.

Today we are witnessing a proliferation of the power of calculation applied to music, witness Ircam. What might be called musical reasoning risks of lose some of its rights : the right to direct and to canalise this surplus power in calculation that is extraneous to music and invading the terrain of well-established operations, traditionally conducted with pen and paper. The increased capacity of calculation can no longer be ignored by composers. Their task is to identify the lines of force capable of aliment their own thought.

Reasoning in music, what use could and should it make of this power of calculation which arrives unasked ? How can it be made to serve music, How could this power of calculation serve music, without itself serving as a foil for new techniques that invade and assail it ?

Musical thought cannot, in this situation, avoid reflecting on the relations between reason and calculation, reflecting, that is, on logic, for logic that is necessarily involved in such an examination.

Twentieth century mathematical logic shows that there is a twofold excess in the heart of these relations :

1) Rationality exceeds calculation, particularly in cases of the undecidable where rationality is not calculable

2) Conversely, calculation exceeds reason : for mathematical model theory (1) shows that a coherent theory generates ipso facto the existence of a countable model whose nature is pathological (completely foreign with respect to the " natural " model of the theory).


A twofold excess therefore, capable of reorganising in the musical universe a double excess in relations between music's written and the audible dimensions :


1) On one hand musical reasoning goes beyond what is written, beyond literal formalisation. Not everything in music can be written, that is literalised (usually the impossibility of writing everything is affected in music to the dimension of timbre). It follows that the impossibility in question is not a technical impossibility in abstracto (timbre can always be digitised by projection onto the calculable) but an impossibility in the conditions themselves of the musical act, within the frame of musical writing itself.

2) On the other hand, the coherence of writing, the consistency of a literal order has often been taken (particularly by serialism) as a guarantee of a sensitive musicality, of a musical meaning as it is heard. If the eye is coherent, the ear will be obliged, and able, to follow it (2) .

In other words, with the end of thematisme, of tonality and of metrical structure, an gulf has been created between two systems of music ­ I mean " serious " or so-called " contemporary music " here discussed whose sensitive power is established on the basis of writing -, it is a gulf which separates its sensitive and its literal order, writing and perception and, more generally, score and hearing.

The point of my initial questions is this : how are we conceive music if its universe is irremediably divided into two orders that it is impossible to conceive will ever be fully complementary ? How are we to write music intended to be heard if it is impossible either to write something listened to or to listen to something written ?



To answer these questions, do we have to start with a definition of what musical logic is ? I do not think so. It may be argued, instead, that for a musician there is no more meaning in defining musical logic as there is indefining music itself. Indeed, for a musician there is no good definition of music. Furthermore, such a definition cannot exist for him, any more than a satisfactory definition of mathematics can exist for a mathematician. To be more explicit : I do not claim that in general no satisfactory definition of music or mathematics can be arrived at. The encyclopaedias, for example, tell us that music is the art of sound ; and so it is ? They also state that mathematics is the science of figures and numbers ­ which is less convincing !-. It is not the to the point that such definitions should so often prove unsatisfactory. It is possible, I believe, to frame definition of mathematics that would be completely satisfactory (satisfactory for me, perhaps because I am not a mathematician in the strict sense of the word, but only a friend of mathematics). The definition I take from Alain Badiou's philosophy : " mathematics are ontology " (3) ; therefore : mathematics are everything that can be said about being, about being as such. This is for me an excellent definition, but I am convinced that for a mathematician, especially for what the Anglo-Saxons call a " working mathematician ", this definition will be inefficient. He might feel gratified to find himself raised to the dignity of onthologist, but this will not make him a subject in his own enterprise. In the same way art might be defined philosophically as " the truth of the sensitive " and music as " the truth of the audible ", or, even better, " the truth of listening ". But, as definitions, these statements (whose philosophical origin is the same as the above definition of mathematics) will be without efficacy within musical thought.

Thus, for a musician, one does not define music, even though word itself remains capital for him, a word he cannot do without.

A musician spends his time asking himself in anguish if, by interpreting such or such work, by composing such or such other, the happy outcome will really be music. And one musician is all too ready to disagree with another : " This is truly music ! This is certainly not ! ".

Eventually, for a musician, the word music is a name, precisely a proper name that acts without any need of a definition, just like the woman you love, to your eyes, has a name that is enough to enliven you without any need to be defined.

If music cannot be defined, could musical logic be ? Surely not, again, because of the too strong proximity between the names.

Just a very simple demonstration scheme ad absurdum : if one could define musically what musical logic is, one would be so able to define musically what music is. From the impossibility to do this (see the previous axiom, or axiom of proper name) follows directly the impossibility to define a musical logic. QED


If one cannot define musical logic, what is there left to be done ? We have to identify where the logical dimension operates in music. We have to distinguish the logical proceedings of music.

In a way, we not so much have to speak about " the " logical music or " a " logical music, but more about " logics in music ", that is what could be named logical in music by setting out to locate its proper job : its specific field, its singular operations, its particular effects.

Which are therefore in music the properly logical proceedings and how are they working ?




To define these logical proceedings, musical thought must face the following question : strictly speaking, is the logic of music a musical logic ?

The most common answer is that ait is not. Received opinion (the contemporary doxa) has it that the logic of music is not a musical logic, i.e. it is over-determined by different logic from other domains of thought.

There are essentially three ways of placing the logic of music under the tutelage of another domain of thought :


Mathematical tutelage (arithmetic)

The first places the logic of music under the tutelage of mathematics, mere arithmetic for the most part and, commonly, of a combinatory of the first integer numbers. An exemplary formulation of this approach is that statement which underlies St. Thomas d'Aquin's Summa theologica : " Musica credit principia sibi tradita ab arithmetico " (music submits itself to principles which it derives from arithmetic).


Physical tutelage (acoustics)

Others propose tutelage of a physical, more precisely of an acoustical, nature. One might mention here Aristoxenes of Tarento, or Rameau, but it is an orientation which even today seems quite natural.

In this case, music would be structured by a purely physical logic.

A word of warning : this conception brings into play not only specific physical laws (a truism) but it also establishes a tutelage of a specifically logical nature.

Music would not only borrow its material from physics, physics would not only endow music of a shaping framework or furnish the material conditions of its existence (this goes without saying : obviously, there is no music without sounds), but musical thought must also base its logical principles of inference on acoustics, which is quite a different matter. The idea is that musical logic must derive its norms from the physical-acoustical logic of sound itself.


Psychological (and physiological) tutelage

There is a third figure of tutelage, that of psychology. Music would derive its norms externally from a psyche of the feelings and a physiology of sensations. This psychological approach does not go back very far in time, although from the time of the Greeks it is constantly present, and is still present in the modern age. As in the two previous approaches, this tutelage posits a musical coherence based on an exogenous logic : to paraphrase St. Thomas, music should submit itself to psycho-physiological principles.


Musical autonomy

I will defend a quite different thesis : that music as thought is capable of self-normalisation (which by no means signifies self-entrenchment or self-definition, and even less self-demonstration of its own coherence : musical autonomy is neither autarky nor, strictly speaking, a nomology ) (4). What is claimed is that the logic of music is a musical logic. It is not a mathematical-arithmetical, physical-acoustical or psycho-physiological logic.




In terms of this (hypo-) thesis, how are we to recognize logical operations in music ?

I will proceed in two stages. To begin with, I will analyse these questions by means of variations. I will then, more synthetically, distinguish different logical statements in music and try to articulate them.




I shall begin by offering a series of variations involving the logical in music. By so doing, I shall provide my discourse with musical, rather than mathematical, form, since, if mathematics means demonstration, to " make music " means variation.

There are two great modes of variation in music :

The first one, the most common, consists in developing an object in such a way that it changes throughout the discourse. This is the case with Beethoven's variations.

The second, less usual, consists in varying the context of presentation of an object which itself remains unchanged. The simplest way of doing this is to change the lighting, swivelling a projector around the still object, so that each new disposition will reveal a renewed profile. Henry Pousseur has drawn our attention to this kind of variation in the case of Schubert. These two methods have a point in common : they move from the same to the other, more precisely, from the same to various others. Both set out from the statement of an identity (for example our theme would be in this paper the logical in music) to generate otherness : in the first case, generating other objects ; in the second case, showing other facets of a same object. I prefer to work on a completely different kind of variation that, unlike the previous ones, moves from the others to the same. The qualifying issue of this third kind of variation is not the alteration of an initial identity but the extrication of a common trait within a dispersed diversity. The point, in some sense, is to bring close together separate elements which stand apart, without obvious relations, in order to recognise the underlying, we might say incognito operation of one and the same figure within the initial diversity. Certainly, this third kind of variation assumes a particular interest when it is not the pure and simple reversal of the first two types, i.e. when it does not lead up to the presentation in conclusion of the theme which has the same nature as the one presented from the outset by the other types of variations. It is not a question of proceeding as did Liszt in his Fantaisia " Ad nos " or as does Franck in his First Choral for organ, where at the very end we are given the initial theme that establishes the retrospective key of the work.

If I call my first two types alteration and my third recognition, I am highlighting the fact that recognition produces an object which does not readily lend itself to the generation of alterations. Recognition is not a retrograded alteration. While the latter is related to a deduction, the former cannot be linked to an induction.


Here my method will come within these variations-recognition and I will submit the term of logic to different instances in such a way as to delimit what at first appears as a black hole, soaking up all light rather than enlightening the mind.

These variations, some negatively delimiting, others positively comprehending and interrelating, will aim at recognising, under a simple word, a name.




Let us briefly examine the historical moment when the category of musical logic first appeared.


It is a category which seems to emerge at the end of the XVIIIth Century, in a very specific context. I shall quote at length the musicologist Carl Dahlhaus :

" [...] The forces in compositional technique that made possible an autonomisation of instrumental music may be brought together under the concept of musical logic ­ a concept closely allied to the idea of music as speech. [...] In his " Fourth Grove of Criticism " of 1769, Jahann Gottfried Herder could still speak of logic in music with unconcealed disdain. [...] Herder dismisses as a merely secondary force the logic that in music lies in the context of the chords [...] Herder, it seems, was the first to use the term, but only with Johann Nikolaus Forkel, two decades later, did the concept of musical logic gain aesthetic respect. "Language is the garment of thought, just as melody is the garment of harmony. In this respect, one may call harmony a logic of music, because it stands in approximately the same relation to melody as logic in language stands to expression." [...] Where Herder contrasted, Forkel mediated, calling the harmonic regulation of tonal relations musical logic " (5) .


All in all, according to Dahlhaus, the emergence of the category of musical logic should be characterised by the following elements :

1) The theme of a musical logic appeared when music tried to conceive its own autonomy with respect to natural language, when music had to apprehend its en-soi and its pour-soi.

2) The theme of musical logic was at once correlated with the model of language, music tending to reflect on its autonomy through the modality of a specific musical language.

3) Musical logic had as its immediate test field its ability to link harmony and melody, in an historical period in which counterpoint and polyphony (that had until then regulated both the horizontal and the vertical dimensions of musical discourse) had given way to the accompanied melody.

4) The question of logic, defined as the possibility of regulating new relations between sounds, was placed at the heart of the musical discourse, not in its periphery : musical logic was the harmonic centre animating the melodic surface.


These are the four initial features :

1) The question of musical logic is posed when music attempts to think of itself as an autonomous universe and to reflect on what it is that guarantees its inner consistence.

2) Musical logic is originally conceived in correlation with the category of language.

3) Musical logic works on the unity of a divided field (divided between melody and harmony).

4) Musical logic is a centre that animates a peripheral appearance.



The begin of Johann Sebastian Bach's fugues introduces a very simple matrix that can be represented as follows : the theme, classically called the subject, continues in a counter-subject at the same times as it is repeated in an altered form, in what is called a reponse. We thus have the association of an altering repetition (the reponse repeats the theme by transposing it and changing it ­ with possible transformations) and an extension into another melodico-rhythmic figure (the counter-subject). Hence : the one of the subject splits into two (into a counter-subject and a reponse) in the movement of its self assertion.




I believe that this " logic " can be compared to the principle of non-contradiction that finds its emblematic expression in Aristotle's Metaphysics, where the principle is asserted as the first and irreducible foundation of all coherence of logos (6) .

To this principle of non-contradiction can be oppose a musical principle that I shall call the principle of forced negation : any musical object, once asserted, must come to terms with its opposite, compose itself in becoming. In our short example, the theme exists by becoming other through a split [scission] according to a twofold alteration : that of the counter-subject and that of the reponse. This is the first feature that contradicts the idea of any parallelism between classical logic and musical logic, posing indeed the idea of an anti-symmetry (or of an orthogonality) between these two logics.



We now have a case of a negative variation (variation of delimitation). Let us give an example from the composer who has made himself the bard of parallelism between mathematics and music : Iannis Xenakis.

The first page of Herma, a work for piano dating from 1961, presents a material whose pitch structuring process is stated to be stochastic. This seems plausible considering the erratic character of the material, except that ­ and it is no small matter - we find from first bars a pure and simple twelve-tone series, which is not the outcome of any probabilistic draw. A mistake in calculation, we might ask ? Or liberating gesture on the part of the composer, who defies the laws of his own calculation in order to effect a musical ratio emancipated from the mechanical enchainment ? An examination of the score as a whole tends to invalidate this hypothesis, since this twelve-tone gesture is not followed up : we find no altered reiteration nor any influence upon the dominant stochastic gesture that will go on and on producing tornadoes of notes. Here the word illogical imposes itself thanks to the musical principle asserted above (the principle of forced negation) which a minima demands that an assertion (in this case, that of a twelve-tone series) should entail some consequences and not remain without a continuation, an dusty axiom lying unused in the corner of a theory. Worse than an unnoticed axiom : a useless axiom ! (7)



Let us return to composers of a different dimension, say Mozart, and examine this short extract of the development in the Piano Concerto number 25 (in C major K. 503)




Musical example


We are faced with a chemically pure example of a thematic development in which the theme asserts itself as self-consciousness, that is, as a capacity to norm its inner alteration. In fact, we have here a sequence in which the theme is repeated three times, first transposed from initial C major to F major, then to G major and, finally, to A minor. It can easily be linked it to the pitches forming the head of the theme, and deduce that the theme has shifted in accord with a macroscopic path that is isomorphic to the microscopic structure of its outset.

We are dealing with a fragment of development that exemplifies what could be called thematic logic. (8)



In his symphonies and string quartets, Haydn plays a game of surprise that he loves to repeat. He likes to surprise us at once and then to repeat this surprise a second time with a knowing wink, creating an expectation to which we shall become dependent : does not this repetition simply mean that a third occurrence is on the way ?

Haydn is addicted to this " good luck comes in three " game, in the course of which he sometimes deludes our expectations, and sometimes satisfies them, surprising us by insisting on continuing three times. It is an example which seems to me to show how music contravenes a fundamental logical principle, that of identity (something doubly asserted is always the same, independently from its different instances). The principle of musical logic, anti-symmetrical from this identity principle, might be called a principle of differentiation and defined as follows : any musical term which is doubly asserted undergoes an alteration ; i.e. : no term which is repeated is identical to itself. Furthermore : in music to repeat means ipso facto to alter.



In 1952, Boulez pointed out, with justified vigour, what he called a misinterpretation [contresens] (9) in the understanding of the twelve-tone system by its inventor. He refers to those cases in Schoenberg in which the twelve-tone series logically structures the melody (following the laws of its own order) whereas the harmonic accompaniment of the melody is governed by a principle of distribution of the remainder : to construct the chords, use is made the pitches of which the melody has not made use and they are grouped in small packages and somehow or other associated with the proper horizontal order.

Boulez rightly saw in this a failure of serialism to structure a material that remains subject to the outdated rule of the accompanied melody.

If one could index the success of tonal harmony (as it appears in Forkel) to the existence of a musical logic, then Schoenberg's failure effect this in terms of the twelve-tone principles has to be considered as an instance of illogicality. (10)



The next negative example highlights what I propose to call musical tautology. Take, as an example, Ligeti's Coulée, a score for organ dating from 1969. One need only listen to the opening of the piece to have a look and then look at the score to understand how the relations between ear and eye are purely functional and musically redundant. In other words, the proper order of writing is, in this case, brought back to its bare nucleus : univocal codification and, as a consequence, the dialectic between writing and perception, where the former feeds the latter, has been reduced to a truism.



Last example but one, drawn from Brian Ferneyhough's La Chute d'Icare. At the end of the piece, after the cadence of the clarinet and at the beginning of the coda, something surprising happens : three instruments (the piccolo, the violin and the cello) successively state a little regular pulsation in a context of writing that leaves very little space for this type of regularity, the prerogative of music in previous centuries. What is singular here, what raises an unusual " logical " problem, is the fact that this intervention occurs so late in the work that it annihilates all possibility of consecution and imposes a retrospective examination, that acts as a revaluation of what preceded rather than as an opening for what follows (the work is almost finished). Hence the impression that this short moment is a logical problem in listening that might be said to be of the inductive order (11) .



The last example is taken from Monteverdi's Madrigal Hor ch'el ciel e la terra. The initial accord in the tonic is repeated so intensely that, despite its function of rest (according to the tonal logic), it inevitably acquires an increasing degree of tension. As a result, it will be the arrival of the long retarded dominant chord that will act as a relaxation, precisely where, in tonal logic, the dominant chord ought to create tension and call for further resolution.


Musical example


This example shows how a work assumes a musical logic of its own (in this case a tonal logic) which is not to be submitted to codified and operationalised chords progression (as happens with a logical law of inference, as the modus ponens) but in order to bring into play lines of force and energy fluids that the work will be free to distort and to change.

From this point of view, tonal logic must appear as the construction of a magnetic field that can be traversed in any direction, provided one has enough energy to deviate the trajectories traced out by the field.



To summarise briefly, these variations raise the following points :

1) Mathematical logic and musical logic are not so much parallel as anti-symmetrical. One could systematise this anti-symmetry by opposing to the three main logical principles of Aristotle three elements characteristic of musical dialectic :

- the principle of differentiation versus the identity principle (see variation 5) ;

- the principle of forced negation versus the principle of non-contradiction (see variation 2) ;

- finally, where Aristotelian logic prescribes the principle of excluded middle (there is no middle position between A and not-A, hence I must choose between one or the other), musical composition would suggest a principle of obligatory middle : any musical term must entail another term which is different from the evolving negation of the previous term. It is a kind of neutral (12) term, for it is " neither the one nor the other ".

In conclusion, these three principles would suggest that musical thought should benefit from a confrontation with stoic logic (13) rather than with Aristotle.


2) Let us call musical tautology any correlation between two orders that is merely a univocal and mechanisible functionality.


3) In music, logic would act upon the juncture of two dimensions : e.g. the two dimensions of the horizontal melodic and the vertical harmonic, or again the macroscopic and microscopic dimensions where we find that logic is the link.


4) Ferneyough's example poses logical questions that have less to do with the work's musical structure than with its singular dynamics : how does a given work deal with the logical principles of a musical nature that it inherits from the musical situation in which it is set.





It will perhaps have been noticed that my mode of demonstration, by variations which aim to define an object by the arrows that point it is intimately related to the basic idea of topos theory, for which the whole network of arrow-relations is more important than the object itself. The existing common point between a field of mathematics and a musical necessity is by no means a matter of chance, as we shall see.

As far as logic ­ as well as other subjects ­ is concerned, I believe that there exists no direct link between mathematics and music and that all attempt to relate them passes (that is has to pass) via philosophy. Any attempt to link mathematics directly to music (14) can only be effected within what I shall call an engineering problematic, that is in the mode of an application of mathematics to music. This relation based on the applicability is completely independent from the content of mathematical thought and uses only results susceptible of formulisation, in other words, the classical situation in which mathematical rationality takes the shape of a pure calculation equivalence between the two sides of a sign equals (" = "). The most dominant approach today is unfortunately that of reducing mathematics to a collection of formulas that can be applied to, or transposed as, music. Xenakis has built his reputation on operations this kind

My thesis is that any assumption of a relation between music and mathematics must proceed by way of philosophy, not through a compendium of calculations. If there is a question of contemporaneous thought between mathematics and music, not of vassalage or of application, then it is philosophy to which we must delegate the setting up of a conceptual space capable of containing it. This is so because musical thought is not scientific but artistic, so that direct links between, for example, mathematics and physics, have no counterpart in the case of mathematics and music : in the former case, such relations are rendered valid if one assumes the ontological character of mathematics (for, everything that makes sense for being as such [l'être en tant qu'être] makes sense ipso facto for any being [étant]). But, more precisely, music is not a science, and musical logic is not an acoustical logic



After having explored some logical issues which relate to music, let us attempt a more synthetic elaboration by offering what mathematicians would call a " compendium of results ".


In our chosen field of music, I propose to understand by logic everything that in formal terms conditions possibilities of existence.

Not all conditions of existence are logical ; logical are those that entail possibilities of existence in formal terms.

To give an elementary example, the so called logical rule of modus ponens (if A ->B and if A, then B) takes into account the validation of B by assuming A and A ->B, independently of whether A really exists nor if the implication A ->B really subsists.

Thus logic does not concern itself with things that really exist. It is concerned only with the prescription of a coherence of the possible, without taking into account effective realization.

As Leibniz has it, logic establishes a configuration of possible worlds but it delegates to God the task of determining the unique world that really exists.


With this delimitation, I suggest we distinguish three proceedings of logic in music :

- the writing of music ;

- the dialectic of musical pieces ;

- the specific strategy of a musical work.



A central point in musical logic concerns the difference between a structural level of music and the concrete and singular level of a work.

A tonal logic may exist, for instance, but no one work would ever display it as such.

Only a treatise of harmony would be capable of accounting for it. From the point of view of the work, which is what most interests us, music is influenced only superficially by this logical prescription (in the case, of course, of tonal music), without being completely subject to its law.

The work, on the one hand, is endowed with a " need-to-say " [devoir-dire] which is in fact a prescription involving its being ­ i.e., the necessity of asserting its unity as a musical being [étant] -, what is usually called a " piece " of music. At the same time, the work takes upon itself a kind of strategic prescription.

Hence, the general process of inference of a work - or the consistency of its " need-to-say " ­ needs to be separated from its strategy - or insistence of a " want-to-say " [vouloir-dire] -, for which a singular process is operating.

In what follows, we will describe as a piece of music this first element of the musical opus (the general process of inference or consistency of its have-to-say), and use the term work for the second element (the strategy and its singular process of inference, or insistence of the want-to-say). The piece of music is the level at which the opus establishes itself as a being, existing in a situation. The musical work is the level at which the opus takes the shape of a project, of a musical subject.


There are three proceedings :


-Writing influences in a formal way the coherence of a possible world of music : it represents the logical proceeding of music as a universe.

-Dialectic influences in a formal way the consistence of a piece and the possibility of its unity : it is the logical proceeding of a piece of music.

-Strategy means the logical proceeding of a musical work seen as a subjective singularity. Its influence formally concerns the insistence of the work, that is the possibility to sustain a musical project throughout the dimension of the piece.


We will now proceed to clarify one by one each of these proceedings by disengaging their specifically logical aspect.



Musical writing can be thematicised as a logical dimension through two reflections :


Writing and sound material : topos theory

The first point concerns how musical writing takes into account the sound dimension. The answer to this question may be logically clarified through topos theory. In mathematical category theory (or topos theory), logic appears as a logic of universes. The collection of logical operations may be characterised as the relationships between the totality of objects in the universe and the so called subobject classifier. The validity of any logical connection in the universe is given by a particular and well-determined point of this universe. Therefore, following Alain Badiou's philosophical interpretation of mathematical category theory (15) , it can be stated that a logical operation corresponds to a centration of the universe (16) .

This new approach to logic leads us, from a philosophical point of view, to differentiate being in situation or being-there [être-là] and appearing.

This could clarify the problem of a musical logic for the following reasons :

1) The relation between the score and listening to it may be conceived as the relation between the being-there of music and its appearing to the senses.

2) The logical centration in music may be characterised as a centration on the writing : musical writing is what takes into account the musical dimension of the sound created. By distinguishing what in music does exist and what does not, writing states the validity of appearing (17) . It also validates in a musical situation the real existence of any appearance of sound ; for it would remain a mirage without the effectiveness provided by the writing itself. Who, after having heard a piece of music, has not been led to compare the consistency of he seemed to have heard with the score ? By so doing, you show you are familiar with the transcendental use of writing, even if it remains an intuitive use that is not analysed as such.

From this point of view, musical writing takes into account the passage from a sound level to a musical level by structuring the musical logic through the sound situation.


Writing and listening : model theory

My second point is : how does musical writing relate not only perception and hearing, but - more specifically - the musical listening ?

Relating writing to the singularity which is the musical listening implies a specifically logical dimension that can be expressed as follows : in music writing is what calculates and demonstrates, whereas listening starts precisely at the point from which things cannot be expressed and ordered according to some strict rules of perception (18) . The privileged moment [moment faveur] (19) of a work, when listening takes wing, is founded on a logical condition : that some demonstrated things cannot be shown, that speaking about something really existing remains meaningful, even if it cannot be presented according to the traditional schemes of showing. Mathematics provides us with many examples of this.

Musical listening, which entails a thought linking the sensitive aspect to the intelligible, is effected when sensitivity splits off from sheer perception and gives way to a new principle of intelligibility which no longer depends on showing, but embraces a musical rationality of infinities for which no representation is possible.

How can writing take into account this new schema of sensitivity ? More precisely, how can musical writing influence the possibility of such listening taking place and continuing to operate in a work ?

This question implies a logical dimension which can be related to what the 20th century mathematical logic called model theory, a theory that analyses the articulation of reason and calculation.


The mathematisation of logic and, thus, its literalisation has, since the end of 19th century, led to a split between, on the one hand, the scheme of the letter and, on the other, that of its interpretation ; in other words, an horizontal barrier separates a purely syntactical scheme from a semantic one. This barrier does not disappear, in model theory because its semantic interpretation of objects is not concerned with logical connectors, which are confined to the syntactic field. It is clear that logic develops, on the one hand, in an horizontal dimension (that is the propositional calculus) and, on the other, in a vertical relation between syntactical inferences and semantic interpretations (this is model theory).

This fact enables us to characterise what musical logic is concerned with : the relations between writing and musical perception are similar to those linking syntax and semantics. From this point of view, musical logic is the science of the way in which writing progressions dialectise themselves into sound consecutions.


By interpreting the mathematical duality theory / model by projection on the musical duality score / audition, it is possible to " interpret " different logical / mathematical theorems of our century by proposing the following theses (20) :


1) Musical listening proceeds according to some determinations that cannot be written as such. (21)

2) Any score is compatible with at least two radically heterogeneous listenings. (22)

3) Any consistent musical writing guarantees ipso facto the existence of a possible listening. (23)



My second question concerns musical dialectics. As already pointed out, music can be organised according to dialectical principles that are anti-symmetric to those of Aristotelian logic. These principles are organised by schemas of inference that are equally logical, of the type " If then ". The principle has already been established in the two axioms of forced negation (see variation 3 : " If A, then not-A ") and obliged middle.


Four dialectical issues

In music all these inferences assume a more systemic character when we consider the strictly logical base of different compositional styles. The fact that in music logic means dialectic becomes obvious once we have observed that all musical historical situations have set up a specific dialectic issue with respect to the works they embraced.

1) In the case of the baroque fugue, the dialectic issue was that of a split [scission] of its single subject (into a counter-subject and a response : see variation 2)

2) In the case of the classical sonata form, the dialectic issue was that of a resolution of two deployed opposite forces (24) .

3) In the case of the romantic opera of Wagner, the dialectic issue was that of a transition between the multiple entities that make it up.

4) As to Boulez'serial work, the dialectic issue was that of an inversion [renversement] of order (25) .


Dialectic of the same

We return here to our previous statement : any dialectic is characterised by the fact that a musical variation (in its broad meaning) is considered as an alteration of a principal unity. We might be tempted to say that it concerns classical musical dialectic and we might remark that there is no reason at all to restrict musical dialectic to this classical dialectic (as if one were to confine mathematical investigations to bivalent classical logic, that of the excluded middle).

I have already remarked that there exists a possible alternative which involves my work as a composer, an alternative that I called the recognition-variation [variation-reconnaissance]. Let us attempt a brief description of the conceptual space of its realisation.

The idea is to delineate a musical dialectic which, contrary to the classical dialectic, goes from the others to the same, in a kind of conquest of the generic, of anybody, of the anonymous. The alterity would be a starting point, the first evidence so that what is astonishing and precious will be attached to the universality of the same, rather than to the differentiation of particularities.

Of course, as has already been pointed out, this dialectic cannot be a retrograded alteration, transforming deductions into inferences. It is a dialectic that must make up operations of its own that definitely cannot be a mere inversion of classical operations.

I suggest that we should adopt Kierkegaard's approach, in particular, the three following operations : reply [reprise], reconnaissance and reduplication.

The reply (that is coming back forwards) is a second occurrence which turns out to be the first one, whereas the reconnaissance (of an unknown ) (26) is a first occurrence which turns out to be the second one.

The reduplication is a reflection that seals the one of a single gesture thanks to the how [comment] that reduplicates the what [ce que], the enunciation that validates the statement, the making [faire] that seals the saying [dire] (27) .

These three formal operations concern two different aspects of the two : reply and reconnaissance concern an ordinal two (for they fix an order and determine what is primer and what is not) whereas reduplication - and its Hegelian counterpart redoublement - is concerned with a cardinal two (which means that it is linked to the quantity and determines how and whether 2 may be identified with 1+1 or not).

These operations, I believe, could provide a logical alternative to the development of classical music. In fact, they already exist in contemporary music (see, for example, the works of Eliot Carter or Helmut Lachenman). Our task is thus to acquire consciousness of what has already been done, of casting new light on the principles that already exist, something like what happened with the axiom of choice at the very beginning of 20th century, an axiom that many mathematicians of the previous century had already used implicitly.




We now come to my third major logical concern, centred on the specific strategy of the individual work and which leads us to distinguish two fundamental rules from the standpoint of logic.

1) The strategy of each work's must be thought of within a specific inferential framework and not just in a more or less selective deviation in relation with the broader system it has inherited.

2) The work must be brought to an end. It has to finish somehow, without arousing any suspicion of suicide.


Let us briefly analyse both points.


Inferential system

By prescribing a systematic strategy for the work we are suggesting that it must pursue insistently and even relentlessly a musical project all its own, independent of the variety of sound situations it might encounter. In order to be a real musical subject, the work cannot limit itself to simply noting down a punctual clinamen, without consequences. Nor can it content itself with placing some local declination in relation to a musical system that would constitute its global envelope. Such a work would imply a hysterical and unilaterally rebellious subjectivity.

The challenge is a different one : the work should create an expression, a " need-to say " based on its own force. It should create a persisting instress [intension] due to a systemic parti pris and not only to some spontaneous reaction toward a path standardised by a tonal or serial system, or even by some systemic dialectic of the same

I am not asserting here that this systemic character has to be formalised, nothing indicates that it could replace the well-known musical systems. It has more to do with a subjective quality of insistence than with a system that can be codified.

This singular systemic character of the work could be seen as its personal modality of inference, just as a singular mathematical theory adds its own rules of inference.

To give an elementary example, the order relation states that if A < B and if B < A then A = B, which is a new way of inferring the identity of A and B. The assumption that this strategy has to be systematic inverts Boulez'problematic System and Idea, because the principle of the work no longer consists in confronting and deviating the musical system ; on the contrary, it superposes itself on the musical system.

As a radical example of my systematic presentation, I suggest the category of the diagonal, which derives from Cantor's mathematical ideas and owes nothing to Boulez'concept of oblique. As I have given an account of this method elsewhere, I shall not deal with it here (28) .


When the end occurs

If this insistence orientates the desire of the work within the infinite of its situation, it is the end that has the work face the necessity of a conclusion.

The moment of its end, when the work entrusts itself to the subsequent outcome of what it enacts, within the dialogue with other works it establishes, poses a number of significant questions of logic. In order to answer them, let us have recourse to the mathematical idea of forcing. Here too, Schoenberg's manner of working is highly illuminating, particularly for what extends the whiles of his works (29) .


The correlation of two

Hence system and conclusion are connected from a logical point of view : interruption only intervenes because a strategy is involved (in the case of a chaotic collection of events this would be impossible).

The strategy of the work involves the relation between the finite and the infinite within the work. This relation is a product of a logical approach, insofar as it is investigated, as it is here, in terms of a formal examination, i.e. an analysis based on a scheme which takes account of the conditioning of the possible. If the work is really a work, that not only must say but that wants to say (i.e. if the work is a real musical subject which is no reducible to a piece of music activated by the structuring situation), then its " want-to-say " must be part of a singular process of insistence and must reach the point of deciding to come to a conclusion. The fact that a singular " want-to-say " - which is not the same as a general " need-to-say " - thus generates a necessity interlinking all its own, may be seen as a specific example of what has been called, in other fields (philosophy, psychoanalysis,) a logic of the subject. This logic is the formal scheme of inference to which the subject freely submits itself, provided the freedom of the musical work means freedom of determining itself (Kierkegaard) or of considering itself accountable for its own actions (Nietzsche).



If we see logic as a formal scheme of conditioning and inference, then the logical prescription in music will take the shape of a triple injunction, by projection on a threefold level, that of the musical world, of the opus considered as a piece of music or as a musical work :

1) The musical world is a universe of thought, and this means it is not only capable of indent musical being [l'être musical] - by determining what comes about in music [étants] - but of controlling appearing [les apparaîtres], appreciating their existence, insofar as musical writing is capable of defining a central field (the score). It is this that, at one and the same time, is located at the centre of the musical world and capable of providing a centre for music itself.

2) The musical piece will be endowed effectively with unity, being countable-as-one [comptable-pour-une] once a specific dialectic is brought into play, involving the specific musical situation (i.e. the very exceptional status of the music universe) inside which it has been placed. This dialectic governs in formal terms a general scheme of inferences and consequences that has been incorporated by the piece.

3) The musical work will become a subjective process rather than an act of pure subjectivation, provided that it involves a strategy, i.e. an aptitude to make insistent, throughout the piece of music that identifies with it, a singular want-to-say structured [ossaturé] by some singular principles of inference that enable it to exist as a musical project.


To put it more schematically :

-The writing provides the logical coherence of the musical world (30) .

-The logical consistency of musical pieces is related to the types of musical dialectics historically established.

-The logic of insistence of a musical work takes the shape of a specific musical strategy.


In other words : in music logic acts as coherence in the writing of the world, a dialectic of consistency in the pieces and a strategic insistence along each work.


(translated from French by Moreno Andreatta)

(1) See Löwenheim-Skolem theorem, to be discussed later..
(2) It did not imply that the ear should follow the coherence of the eye, just like there is no semantic transcription of the syntactic model in mathematical logic; the only purpose was to show that the ear can find its own way to follow what has been organised by the eye according to a coherence that remains singular to it.
(3) Cf. L'Être et l'événement (Seuil)
(4) Even if the logic of music is a musical one, musical thought cannot demonstrate, infer or even define it, as we have noticed above. This musical logic is a data with which music makes and creates. It is something like an axiom which musical thought decides, without demonstrating it of course, and then puts to the tests of its consequences. From this point of view, musical logic is no more defined than the concept of set or the concept of membership in set theory.
(5) The idea of absolute music (translated by Roger Lustig), Chicago and London: The University of Chicago Press, 1989 (pp. 104-105).
(6) Aristotle asserts it as follows:
"The same cannot simultaneously belong and not belong to the same, according to the same" (3, 1005 b 19-20 ­ translated to French by Barbara Cassin in "La décision du sens. Le livre Gamma de la Métaphysique d'Aristote", Paris, Vrin, 1989, p. 125).
Then "No one can assert that the same is and is not." (3, 1005 b 23-24; ibid. p. 125).
And finally "The most widely accepted opinion is that two contradictory statements cannot be simultaneously true." (6, 1011 b 13-14; ibid. p. 153).
(7) See my article against Xenakis: Le monde de l'art n'est pas le monde du pardon (Entretemps, n° 5, February 1988) :
(8) See my essay in the concept of theme: Cela s'appelle un thème (Cf. Analyse musicale n°13, October 1988):
(9) Relevés d'apprenti, p. 268.
(10) Boulez opposes this treatment to his own, which he describes as follows: "[Complexes] are derived one from the other in a strictly functional way, they obey a logical coherent structure" (Penser la musique aujourd'hui, p. 41).
(11) For a more detailed discussion on this point, see Une écoute à l'uvre: D'un moment favori dans La chute d'Icare (de Brian Ferneyhough) ­ Compositeurs d'aujourd'hui: Brian Ferneyhough (éd. Ircam ­ L'Harmattan, 1999).
(12) In an etymological sense: ne-utrum.
(13) One could refer to Claude Imbert's philosophical works. For example: Pour une histoire de la logique (PUF, 1999).
(14) Unfortunately, I am not aware of any attempts to work the other way round, from music to mathematics. [Note of the translator: an interesting counter-example is given by the problem of construction of musical rhythmic canons, asformalised by the Rumanian mathematician Dan Tudor Vuza. It leads him naturally to non trivial results in the domain of factorisation of cyclic groups. In particular Vuza provides a method of constructing all factorisations of a given cyclic group into non-periodic subsets, by clarifying the properties of the so-called non-Hajós groups. Rhythmic canons associated with such a factorisation have the very fascinating property to "tile" the musical space (that is, no superposition between different voices or holes). Vuza's algorithme has been recently implemented in OpenMusic by IRCAM's Equipe des Représentations Musicales (].
(15) Cf. Court Traité d'ontologie transitoire (Seuil, 1998).
(16) For reasons that I will not develop here, this function can be philosophically named the concept of transcendental, following the very precise meaning that Alain Badiou originally introduced in his philosophical interpretation of mathematical topos theory.
(17) In philosophical terms, the writing measures the there of sonic being-there, the da of the Dasein.
(18) Although one might be tempted to criticise the idea of what is conceived as an adequacy between what is shown [montré] and proved [démontré], in music the category of perception is very close to the philosophical concept of apperception.
(19) Cf. Les moments favoris : une problématique de l'écoute musicale, Cahiers Noria n°12 (Reims, 1997) :
(20) I will not analyse here the chain of propositions in detail. One may refer to my contribution to the Colloquium Ars Musica (Bruxelles - 2000) : Qu'espérer des logiques musicales mises en uvre au XX° siècle? (forthcoming).
(21) See Gödel's well known theorem.
(22) See later, Löwenheim-Skolem theorem.
(23) See Henkin's theorem. This third thesis tends to validate the serial statements that we have mentioned before (as: "perception has to follow the writing"), once one has noticed what follows: if perception has to follow (serialisme), the "real" model does not follow the logical mathematical theories (model theory), for deductions by the latter have no semantic translations in the model; consistency of the model and coherence of the theory are not isomorphic to each other. This means that musical listening does not work by following the writing (listening is not a perception of written structures) but by deploying according to its own rules.
(24) Cf. Charles Rosen's works, e.g. The Classical Style (The Viking Press, 1971).
(25) Cf. Célestin Deliège's works, in particular Invention musicale et idéologies (éd. Christian Bourgois).
(26) Of an incognito, in Kierkegaard's terminology.
(27) Its opposite would be the Hegelian redoublement, when polarisation shows the two divisions of a primary unity (the redoublement concretises the two faces of a same thing).
(28) Cf. La singularité Schoenberg, éditions Ircam-L'Harmattan (Paris, 1997).
(29) Cf. La singularité Schoenberg, op. cit.
(30) More precisely: of the world of music that we are concerned with and which is, as we must keep in mind, just one of the many real or virtual musical worlds (there are such examples as the different worlds of oral music tradition, of popular or of improvised music).