**What can we hope for from
the musical logics established in the twentieth century?**

Sunday 19 march 2000 (Ars Musica - Brussell)

(Translated by Dave Meredith)

We shall depart from the following two premises:

1) It is astonishing not only that there is something rather than nothing (Leibniz) but also that what we have (which is, indeed, not nothing) is presented as a collection of things that are stable and that have form, rather than as a pulverulent chaos. This surprising fact (of a philosophical nature: Kant) concerns not so much reality itself but the appearance of reality---not so much ontology but logic. 2) In music, what is important is not so much the nature of the objects themselves that are incorporated into the musical structure (rhythms, harmonies, timbres...) but rather the type of consequence drawn from this incorporation. Consequently, in music the logical aspects of the structure appear to be more important than the ontological aspects (that is, the nature of the musical "beings"). In other words, in music the relationships between entities (and therefore the context) prevails over the nature of these entities themselves, the extrinsic properties (that is, the situation) are more significant than the intrinsic properties (that is, the structure). By bringing together these two premises it is possible to shed light on musical logic at work.

But why is it only in the twentieth century that we have become explicitly concerned with musical logic? On the one hand, to what extent was this evolution in musical thought contemporaneous with the mathematisation of logic that was being carried out at roughly the same period? Specifically, how does musical logic differ from mathematical logic? And on the other hand, what is the significance of this new musical responsibility placed on logic in contemporary works? What is the musical significance of this importance given to logic that modern-day computerisation only serves to emphasise? Does this concern with logic have its origins within music itself or does it originate outside of music? In particular, does it derive from the growing importance of computing in the art of composition that has arisen by virtue of the computer's technical power for calculation? Could there be some sense in hoping---as it has sometimes been proposed that we should---for musical advancements resulting from the use of computers? More generally, to what extent can we hope for purely logical advancements in music?

Some first steps.

In the twentieth century, we have begun to pose explicit questions about musical logic in the context of seeking the source of music's 'coherence.' This has happened contemporaneously with the demise of tonality and of thematicism---that is, those very principles that up to that time had provided coherence in musical works. Tonality and thematicism imply their own formalisable systems of coherence (tonal functions, motivic implications...) but these 'logics' were founded upon natural principles and not upon decisions: tonality was founded upon physical principles and thematicism upon psychological ones. The 'logical' dimensions were therefore built to a large extent upon ontological foundations or upon basic principles expressed in terms of musical entities (tones and themes). In sum, let us say that musical ontology was over-determinating musical logic. In the twentieth century, composers have found themselves facing a void: the need to make decisions that cannot obviously be made on the basis of physical or psychological evidence. The physics of tonality drew its strengths from many sources, its claim to 'naturalness' being 'wrung out' (in all senses of the term) by chromaticism. As for the psychological consciousness of the theme, the discovery of the musical unconscious (through Romanticism and Schoenberg) has done considerable damage to the belief in the omnipotence of self-awareness. Which begs the question: what role should be given to unconscious mental computations (to consider the issues of inspiration and intuition...)? Which, in turn, leads to the importance assigned to the question of musical logic as such: if the composer takes this or that (a priori arbitrary) decision, what will be the consequences? And if these initial decisions are arbitrary, at least their consequences won't be. In any case, within which logic can these decisions be expressed? Would there not be several such logics that would be satisfactory? What coherence can we hope for beyond the arbitrariness of singular decisions? To take this as a starting point, musicians have turned to mathematical logic to see if it could inspire them and help them to clarify the precision of their mode of reasoning. This raises certain questions such as "Is musical logic intuitionist and modal rather than classical?" and "What is a temporal logic?" Does musical logic resemble predicate logic and propositional calculus (and if so, which order or predicate calculus)? Thus the mathematisation of logic and the logicisation of music are empirically synchronous movements which leads one to question whether or not they are truly contemporaneous, that is, whether or not they share the same period of thought.

Let us make two distinctions:

A) Computation and reason

- Computation is a chain of deductive inference. Computation is part of reason. But reason goes beyond computation because it includes acts that are not deductible, that is, decisions (decisions about existence: these are the existential axioms; but also strategic decisions: to aim for a particular computational goal, for example, to wish to disprove such-and-such a proposition...). We know (from Gödel) that computation cannot subsume reason because one can always demonstrate that certain propositions are undecidable, one can thus compute very precisely the incomputability of certain propositions that for all that are not rejected as being irrational because the existence of these uncomputable (undecidable) propositions has been computed.

- Conversely, we know from Lowenheim-Skolem that there is too much computation in reason: every model of a given domain is also a model for another, entirely different domain that can be regarded as being "pathological" with respect to the first. It can thus be shown that the computational approach always operates in a way that is more or less deviated to that in which we would like it to operate. There is therefore a surplus of power in computation that reason cannot control. In logic, as in mathematics, there is a dialectic between reason and computation that is not exactly the same.

Now for the second distinction:

B) Logic and mathematics

- Logic concerns tautologies---that is, chains of reasoning that are universally valid (not specific to a particular existential position). Logic does not contrast sharply with factual existence: if A implies B and A exists, then B exists (and this is independent of the existential significance given to the letters A and B).

- Mathematics begins with existential axioms (for example, those of set theory: empty set, the axiom of membership, the axiom of infinity, possibly the axiom of choice...). Mathematics constructs theories (spaces in which reason and computation can take place) founded upon such a priori existential positions.

Let us now consider the case of music:

A) Rationality concerns itself with that which is explicitly transmittable; computation deals with the mechanistic. B) Musical logic is concerned with the dynamic issues of coherence and consistency. The distinction between logic and mathematics finds its parallel in musical informatic in the distinction between the rules of musical coherence and the diversity with which these rules are instantiated (that is, musical existential decisions that control the musical flow and that depend on a specific chord as opposed to a rhythm, and then on precisely which type of chord it is, and so on...)

We are therefore faced with two notions of the relationship between music and mathematics: 1) The first, which I call the engineering relationship, begins with a proven mathematical formula and then proposes that musical formulae can be derived from it by translating and transcribing the symbolic letters in it. For example, one might take a stochastic formula (an equation that establishes a mathematical equality between two terms) and then apply this formula in the musical domain. In this case there is no musical computation as such. Moreover, there is not really any mathematical calculation either involved in this formula for although the original formula might have been a product of mathematical reasoning and computation, it has been taken by the engineer as a given result, divorced from its original context. The engineer is not interested in how the formula can be mathematically derived. He treats it as a "dogma" and his motives are purely operational. The view of the musician-engineer is therefore that musical logic can be obtained by a transference of mathematical logic by means of this formula, which it suits him to use as 'ready currency.' Musical reasoning is supposed to be guaranteed by its isomorphism with mathematical reasoning. This type of attitude comes under the heading of metonymy since it involves the substitution of a musical entity (or a musical letter: a note) with a mathematical letter. 2) The second relationship seems to me to be more promising. It consists of examining music's own internal logical interplay (that is, its own rational and computational logic) with a view to determining whether there exists some resonance between the internal dynamics of music and those of mathematical logic. Here we have a relationship between musical logic and mathematical logic that falls under the heading of metaphor.

To sum up, do the formalisms on which one can build musical "reasoning"---that is, the sum total of its rationalities (decisions, goals...) and its computations (deduction, chains of reasoning...)---have anything to do with particular formalisms in mathematical logic?

Does this concern with musical logic work to shape musical works themselves (by ensuring coherent development, for example) or does it rather play a part in the inception of a work, that is, in the pre-compositional stages? In other words, is musical logic something that forms part of a work or is it an aspect of the context in which a musical work is conceived? Should we therefore speak of Schoenbergian logic (which would be specific to the works of Arnold Schoenberg) or should we rather speak of the logic of twelve-tone music or the logic of serialism (which would be characteristic of certain types of music rather than the works of certain composers)? Is a logic set to work or is it musically rather put into play? At the end of the day, are we talking about the logic of particular works or musical logic in general? In any case, what can we hope to reap from the musical logics that have been employed over the course of the twentieth century?

Some alternative directions for discussion.

To be optimistic hope requires that we have have already won some victories, that we have already taken some forward steps and that we are not concerned only with holding our ground (Rimbaud) but also with universalising our position. Indeed, to hold our ground amounts to universalising our position and not simply to keeping things in their current state (local and constrained). What, therefore, have we achieved in the twentieth century in terms of musical logic that would be worth holding onto? The history of the category of musical logic has appeared (as Dahlhaus reminds us) in the nineteenth century: "The notion of "musical logic" was given pride of place not only by Herder (who, it seems, was the first to use the term) but also, twenty years later, by Johann Nicolaus Forkel: 'Language is the clothing of thought, just as melody is the clothing of harmony. From this point of view, one can define harmony as being the logic of music since the relationship that it has with melody is almost the same as that which obtains between logic and linguistic expression.' "(The Idea of Absolute Music, p.94--95.) The twentieth century cannot therefore claim to have invented the term 'musical logic.' On the contrary, it has shared its usage and its significance. Thus it was basically the serialists that made use of the term, in opposition to what might be called the 'debussy-ist' point of view which condemned the explication of musical logic. On what level, therefore, does what we call musical logic operate? Is it part of a musical work or part of its inception? At this point we must make the following distinction:

1) With respect to the inception of a musical work, musical logic can
be conceived as being the relationship between a perception and its written
representation (from the serial point of view); or, less strictly, the relationship
between what is heard and the score. This subscribes to philosophical view
of logic as the correspondence between what is and how it appears to be,
between "being" and "being there" (Alain Badiou): logic
concerns the coherence/cohesion between "here" and "being
here," between localisation and being. 2) With respect to the musical
artwork, musical logic can be understood as what we nowadays call the musical
dialectic of a work. The fact that there are different types of dialectic
(disjunctive dialectic: Kierkegaard / conjunctive dialectic: Hegel - synthesis
by negation of negation - but also Plato's problem of mixtures; negative
dialectic: Adorno; structural dialectic: Mallarmé...) suggests the
possibility of different types of musical dialectic. Can these two sets
(philosophical dialectics/musical dialectics) be correlated term-for-term?
If Charles Rosen has clearly associated the musical dialectic of the classical
style with hegelian dialectic and Adorno has (less clearly) associated his
negative dialectic with the dialectic of Mahler and Berg, and Célestin
Deliège has brought together Mallarmé's structural dialectic
and that of Boulez, is it possible to systematise this type of philosophico-musical
correspondence? In particular, how would one characterise a post-serial
musical dialectic? If anti-serialism has often denied the need for explicating
musical logic, does this mean that to hope for a twentieth-century musical
logic means to strive for a neo-serialism? Would it be possible for there
to exist a logical goal in music that is truly post-serial? In this case,
the aim would be to hold onto the ground gained by serialism in order to
make progress towards the explication of musical logic in a post-serial
style of thought. This would require the invention of a musical dialectic
of cross-relationships rather than processes, a musical dialectic of mixtures
(where one connects mathematics via the philosophy of Albert Lautman) rather
than syntheses, a musical dialectic that paradoxically relativises the power
of the negative...

**RÉFÉRENCES**

**Mathématiques**

J. Barwise : Handbook of Mathematical Logic, Amsterdam New York Oxford, North-Holland Publishing Company, 1977.

R. Goldblatt : Topoi - The Categorial Analysis of Logic, Amsterdam London New York Tokyo, North-Holland, 1984.

**Philosophie**

T. Adorno : Dialectique négative, Paris, Payot, 1992.

A. Badiou : Topos ou Logiques de l'onto-logique, Paris, Documentation de Séminaire, 1996.

Mathématiques du transcendantal, Paris, Documentation de Séminaire, 1998.

J. Cavaillès : uvres complètes de Philosophie des sciences, Paris, Hermann, 1994.

I. Lakatos : Preuves et réfutations - Essai sur la logique de la découverte des mathématiques, Paris, Hermann, 1984.

A. Lautman : Essai sur l'unité des mathématiques, Paris, UGE, coll. 10 / 18, n° 1100, 1977.

**Musique**

P. Boulez : Penser la musique aujourd'hui, Gallimard, Paris, 1963.

C. Dahlhaus : L'idée de la musique absolue, Genève, Contrechamps, 1997.

C. Deliège : Invention musicale et idéologies, Christian Bourgois, 1986

G. Mazzola : The Topos of Music, À paraître, Birkhaüser Verlag, Basel, 1999.

F. Nicolas : Quelle unité pour l'oeuvre musicale ? Une lecture
d'Albert Lautman, Lyon, Horlieu, coll. Séminaire de travail sur la
philosophie, 1996.