### Music and mathematics

I'm sure you've encountered a mathematical analysis of some basic musical properties, perhaps rhythm, dynamics or some other quality. What intrigues me, however, is not the use of mathematics on music but a comparison of the cognitive process of mathematical proof and discovery with the similar process of musical composition and interpretation.

I'm not a composer or a mathematician but I love both worlds and it appears to me that they do share important common characters. Certainly everyone has a grasp of basic mathematical knowledge but maybe a few fundamental facts about musical morphology will prove useful.

Classical music is quite simple in that it follows relatively strict rules (=axioms!) that are meant to distinguish pleasant sounds from unpleasant sounds. Several of these rules are immediately obvious to anyone that has tried hitting keys on a piano, while others are perhaps quite cryptic. Obviously the selection of these rules is empirical and is meant to produce an aesthetically acceptable result. Breaking these rules or changing them leads to rather surprising and potentially interesting consequences just like in mathematics where changing a single axiom can generate a host of new applications. Think, for example, atonal music and non-euclidean geometries. Note that a musically "correct" piece is like a grammatically "correct" mathematical proposition: it is well formed but not necessarily interesting.

Now, if we assume that we accept these musical "axioms" we enter a process of musical exposition which is, in a way, similar to mathematical proof. First a musical theme (=idea) is presented in a simple format, just like a mathematical proposition. Then it is attacked from several fronts. A musical theme, which can be quite short and simple, can be manipulated with several fundamental operations like modulation (change of tonality), rythmic variations or even mirroring. Bach's the Art of the Fugue is a very clear example of very thorough processing but it does make your head hurt when you try to play it (quite interesting listening, though, you may give it a try).

The sequence and choice of these operations can augment the musical properties of the initial idea and provide a convincing development of the initial theme into a full piece. Even though the individual ideas are simple, the end result can be quite complex. As an example, Beethoven once responded to a challenge by Diabelli who provided a relatively ... boring musical motif to several musicians to see what they could build from it. Beethoven wrote the famous "Diabelli variations" as a response, to Diabelli's astonishment, composing a multi-page set of 33 exquisite variations.

Musical interpretation can pose similar challenges. Only a solid perception of music's inherent structures can allow a coherent, articulate approach to sophisticated works. Note that this perception can be empirical, sentimental and "right-brained"; "talent" if you'd like, so it's not exactly the same as a rigorous mathematical undertaking.

Absolute nonsense, some people will say at this point. Music is elegant and beautiful, math is (usually) not. This is correct at first sight, but after thorough training one learns to appreciate the delicate implications of the mathematical language. The mathematical means of expression do not offer immediate sensory satisfaction but they do provide the intellectual equivalent of an adventurous and occasionally elegant musical undertaking.

Certainly all the above does not mean to imply that musicians are good in math or that mathematically inclined people would become good musicians. After all, Einstein's violin playing was rumoured to be horrible. It takes different qualities to

succeed in either field, but the basic processes of doing music and working with abstract mathematical entities do share several common characteristics.

PKT

I'm not a composer or a mathematician but I love both worlds and it appears to me that they do share important common characters. Certainly everyone has a grasp of basic mathematical knowledge but maybe a few fundamental facts about musical morphology will prove useful.

Classical music is quite simple in that it follows relatively strict rules (=axioms!) that are meant to distinguish pleasant sounds from unpleasant sounds. Several of these rules are immediately obvious to anyone that has tried hitting keys on a piano, while others are perhaps quite cryptic. Obviously the selection of these rules is empirical and is meant to produce an aesthetically acceptable result. Breaking these rules or changing them leads to rather surprising and potentially interesting consequences just like in mathematics where changing a single axiom can generate a host of new applications. Think, for example, atonal music and non-euclidean geometries. Note that a musically "correct" piece is like a grammatically "correct" mathematical proposition: it is well formed but not necessarily interesting.

Now, if we assume that we accept these musical "axioms" we enter a process of musical exposition which is, in a way, similar to mathematical proof. First a musical theme (=idea) is presented in a simple format, just like a mathematical proposition. Then it is attacked from several fronts. A musical theme, which can be quite short and simple, can be manipulated with several fundamental operations like modulation (change of tonality), rythmic variations or even mirroring. Bach's the Art of the Fugue is a very clear example of very thorough processing but it does make your head hurt when you try to play it (quite interesting listening, though, you may give it a try).

The sequence and choice of these operations can augment the musical properties of the initial idea and provide a convincing development of the initial theme into a full piece. Even though the individual ideas are simple, the end result can be quite complex. As an example, Beethoven once responded to a challenge by Diabelli who provided a relatively ... boring musical motif to several musicians to see what they could build from it. Beethoven wrote the famous "Diabelli variations" as a response, to Diabelli's astonishment, composing a multi-page set of 33 exquisite variations.

Musical interpretation can pose similar challenges. Only a solid perception of music's inherent structures can allow a coherent, articulate approach to sophisticated works. Note that this perception can be empirical, sentimental and "right-brained"; "talent" if you'd like, so it's not exactly the same as a rigorous mathematical undertaking.

Absolute nonsense, some people will say at this point. Music is elegant and beautiful, math is (usually) not. This is correct at first sight, but after thorough training one learns to appreciate the delicate implications of the mathematical language. The mathematical means of expression do not offer immediate sensory satisfaction but they do provide the intellectual equivalent of an adventurous and occasionally elegant musical undertaking.

Certainly all the above does not mean to imply that musicians are good in math or that mathematically inclined people would become good musicians. After all, Einstein's violin playing was rumoured to be horrible. It takes different qualities to

succeed in either field, but the basic processes of doing music and working with abstract mathematical entities do share several common characteristics.

PKT

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