Musical and Mathematical Structures. Discussion with Michał Heller

Abstract

The article is a discussion of the statement of Michał Heller, who in a few paragraphs of the book titled Filozofia i wszechświat (Kraków 2006) made a comparison between mathematical and musical structures and stated clear parallelism between them (pp. 113-118).  A number of inaccuracies have resulted from a too cursory treatment of that compelling problem. The level of the statement made by Heller, i.e. "mathematical structures in physical models or theories are to the structure of the world like a musical score to a piece of music" is inaccurate; if the score is to correspond with physical models, and it is composed of sound structures itself, the structures in that statement can correspond to structures only and the structure of the music has a mathematical structure already in its basis. In the further course of his argument Heller imprecisely uses the term of "notes", one time as the score, the other time as characters written in it or as sounding elements of a piece of music. Another contentious issue concerns the organization of the material of the mathematical and musical structures. In classical music an orderly collection of sounds of a certain piece of music is always a purely arbitrary and limited selection of sounds, ordered according to the pitch of sounds; it is not driven by any other rules. In comparison to the mathematical structure it presents itself rather as a system which is similar to numbers that are scattered according to rules which are not mathematical at all. The score itself does not constitute any type of musical structure and a specific musical structure appears only in an interpretation (during a performance and reception). A significant difference occurs also at the level of characters. The content of a numeric character is stable even when it is isolated from the whole mathematical system. It always means one and the same thing, i.e. a specific quantity.  A note, by contrast, becomes completely meaningless when detached from the stave.  The numerical character of the sound structures is not self-contained. It exists only when it serves music. Meanwhile, mathematical structures, although they support the world according to Heller, are fully self-sufficient in their own field, which is mathematics. They can exist for themselves there. Thus, mathematics would have a double existence: one for itself as art and the other one for the world as a basis.  By contrast, music has a single existence only; it exists for the aims of art. With its entire splendor it does not have enough power to exist for itself. While creating sound structures, a composer considers neither the numerical basis of sound nor its acoustic sense, or any other sense connected with music. Finally, mathematical systems probably often relate to the mysteries of existence in a way which shakes all our knowledge and vision of reality. How dissimilar are musical structures here, though? There is only one certain thing there: both mathematical and musical structures can deeply move a human spirit. That is where, perhaps, their greatest similarity lies.