Formalism in Music Composition

There is nothing new in the mathematical approach to music composition which characterized many of the main stream pieces after the 60’s of last century. Recall Leibniz’s words: music is the hidden exercise of arithmetic performed by the soul though unaware of its process of counting. To a Platonic and Pythagorean oriented mind that would give an explanation for the universal power of music, its qualification as a mathematical discipline and thus as a subjacent structure of reality. Music would represent a kind of algebra for the intuition of time just as geometry deals with our intuition of space. Obviously, from a Kantian point of view, where space and time are pure forms of intuition or the conditions for our intuition of the world, such idea is meaningless: space and time are not objects, nor relations among objects, but simply the way that occurs our perception of the world. Other minds (non-human) would perceive it differently. In any case, mathematics prestige and strength, have dominated music theory from its very beginnings back in the Greek world.

What characterized music composition after the Sixties in relation to mathematics, was not its mathematical background but its formalistic approach to mathematics: the believe in the possibility of organizing the art of sounds according to deductive formal principles. In giving account of his pieces (Conclusion Partielle), Pierre Boulez was not merely satisfied with the description of his methods, but wanted to have to a deductive method that would justify every action. With that in mind he would require that his audience would perform a considerable amount of abstract reflections when listening and understanding his work. Such program is meaningless. On the one hand, because Godel’s theorem had already finished with any pretensions of completeness for a musical calculus, on the other, because there are social elements in the determination of any aesthetics whose semantics makes relevant Tarski’s theorem, and music cannot be understood fully just in musical terms.

I heard Iannis Xenakis saying that beauty or ugliness are not valid criteria in relation to music. From his point of view, the criterion should be the quantity of intelligence contained in a piece. He did not offer a method for the determination of intelligence in a piece of music, though his compositional processes implied that the more mathematically formalized a piece, the more its intrinsic intelligence. Here again he was merely speaking in platonic terms of formalization, something that encounters the aforementioned objections of those basic theorems, plus an even wider critique from the perspective of intuitionist mathematics. Mathematics unquestionable validity requires a belief in a final objective order for the universe beyond the ordering given by man in his sciences.

Why should we understand music? Well I can think of a better question than that: when is a piece properly understood? Furthermore: who determines that a piece is properly understood, under what ontological principles occur such determination?