Abstract
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?
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References
See Löwenheim-Skolem theorem, to be discussed later.
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.
Cf. L’Être et l’événement (Seuil).
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. Prom this point of view, musical logic is no more defined than the concept of set or the concept of membership in set theory.
The idea of absolute music (translated by Roger Lustig), Chicago and London: The University of Chicago Press, 1989 (pp. 104–105).
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).
See my article against Xenakis: Le monde de l’art n’est pas le monde du pardon (Entretemps, n° 5, February 1988): http://www.entretemps.asso.fr/Nicolas/textesNic/Xenakis.html
See my essay in the concept of theme: Cela s’appelle un thème (Cf. Analyse musicale n° 13, October 1988): http://www.entretemps.asso.fr/Nicolas/TextesNic/Theme.html
Relevés d’apprenti, p. 268.
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).
For a more detailed discussion on this point, see Une écoute à l’œuvre: D’un moment favori dans La chute d’Icare (de Brian Ferneyhough) Compositeurs d’aujourd’hui: Brian Ferneyhough (éd. Ircam-L’Harmattan, 1999).
In an etymological sense: ne-utrum.
One could refer to Claude Imbert’s philosophical works. For example: Pour une histoire de la logique (PUF, 1999).
Unfortunately, I am not aware of any attempts to work the other way round, from music to mathematics. [Note of the translator: an interesting counterexample is given by the problem of construction of musical rhythmic canons, as formalised 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 (http://www.ircam.fr/equipes/repmus/)].
Cf. Court Traité d’ontologie transitoire (Seuil, 1998).
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.
In philosophical terms, the writing measures the there of sonic being-there, the da of the Dasein.
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.
Cf. Les moments favoris: une problématique de l’écoute musicale, Cahiers Noria n° 12 (Reims, 1997) : http://www.entretemps.asso.fr/Nicolas/TextesNic/momentsfavoris.html
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 œuvre au XXe siècle? (forthcoming).
See Gödel’s well known theorem.
See later, Löwenheim-Skolem theorem.
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.
Cf. Charles Rosen’s works, e.g. The Classical Style (The Viking Press, 1971).
Cf. Célestin Deliège’s works, in particular Invention musicale et idéologies (éd. Christian Bourgois).
Of an incognito, in Kierkegaard’s terminology.
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).
Cf. La singularité Schoenberg, éditions Ircam-L’Harmattan (Paris, 1997).
Cf. La singularité Schoenberg, op. cit.
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).
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Nicolas, F. (2002). Questions of Logic: Writing, Dialectics and Musical Strategies. In: Assayag, G., Feichtinger, H.G., Rodrigues, J.F. (eds) Mathematics and Music . Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04927-3_6
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