Mathématic and Music

Mathematical logic and musical logic in XX siecle.

Paris, Ircam, 3-4 december 1999.


Mathematical logic, musical logic.

One notices a double evolution in the principle of the 20th century: logic has become more mathematical, thereby loosing its ancient status of philosophical discipline; the question of a musical logic has become explicit as a search for a "coherence" specific to music. This musical evolution is contemporary with the end of tonality and of thematism which were the principles that until then, ensured the coherence of musical works. Tonality and thematism certainly implicated specific logic, which could possibly be formalised, but which were often based on natural rather than axiomatic foundations: tonality was based on physics, and thematism was based on psychology. The "logical" dimensions therefore stayed, for a large part, dependant on ontological foundations, or foundations in terms of musical being (key and theme). In the 20th century, composers found themselves before an ontological void. They had to make decisions that didn't come from physics and psychology in an obvious way anymore, but which remained all the same the starting or articulation points of symbolical calculations, expressing an internal logic of the musical form and material. As for logic, it looks at progressions that are universally valid due to not being attached to such or such position of existence. By becoming more mathematical it has also acquired added calculation power, trying with success formal reasoning on itself, one emblematic result of which is the non-fulfilment theorem by Gödel. Mathematics (or its theories) itself could start where axioms of existence intervene, such as those of the theory of ensembles. We may then ask ourselves the following questions: do formalisms built on musical "reasoning" (the mixed ensemble of its rationality, decisions, aims, and of its calculations, deductions and progressions) have anything to do with such or such formalism of logic? What formal coherence, described in a logical way, can exist in a work, beyond the arbitrary of singular esthetical decisions? Can logic, expressed mathematically help musicians to clarify the specificity of their way of reasoning? Taking the problem the other way round, can musical logic be a source of inspiration for mathematicians?

From a text by François Nicolas.