Mathematical symmetry

The relationship between mathematics and music is hotly contested, but remains a frequent topic of discussion. What is undeniable is that some correlation, however vague, must exist. If not, the notion would be heard far less often. Indeed, there are various pieces of suggestive information, ranging from brain wave studies which indicate activity in similar locations when listening to music and doing mathematics, to the basic inclusion of music & mathematics (with astronomy) alongside each other in the quadrivium of the medieval university curriculum. More concretely, mathematical ratios (whether rational or logarithmic) form the basis of intervallic relationships and indeed geometric properties inform various musical facts. However, at some level, this is much like saying that mathematics is closely connected with e.g. cooking, because our temperature scale includes numbers and the very fundamentals of radiative heat transfer involve polynomial relationships. In short, we can inject math into anything, because that is what it was made to do, to aid in our ability to describe complex phenomena.

Seen from this angle, any relationship between mathematics and music should not be surprising, especially given the abstraction inherent to classical music itself. We can give with far more certainty the mathematical relationships governing notes than we can the emotional content they might contain. If we admit a relative closeness to mathematics & music as disciplines, where does it lead us? For many, this evokes grand mathematical theories of musical affect, usually prescriptive. Yet, as remarked earlier, it is almost absurd to give music theory some preeminence in creation. There must be an artistic act, a musical conception which is indifferent to the vagaries of numbers on a page. Perhaps it is reassuring to have an equation describe something, but it often tells us little about ourselves. In short, we might just as well say that music inspires mathematics. After all, without phenomena by which to motivate it, where might mathematics be?


Inspiration is a difficult topic, and one about which I would like to write more. Within the context of the present article, we do know that mathematics has inspired music, especially in this century. From Xenakis with his complex formulaic schemes, to the algebraic geometry of serial manipulations, to innumerable other mathematical ideas grafted into sound, advanced mathematics has made a decided impact on contemporary music. If we grant mathematics a special relationship to music, does it prove the excellence of this music, provided the maths are valid? Of course not, but that is the sort of implication this topic can sometimes have. One also sees the reverse implication, that music derived mathematically is necessarily sterile. Perhaps true, if that is its real derivation, but too often such aspects are secondary to the composition, even if they are easier to write about in liner notes. The latter can have a definite impact on our perceptions. Indeed, science itself can often proceed from direct intuition which is only subsequently notated. It is entirely believable, especially in this age of scientific glorification, that some mathematical idea could be the source of a profound inspiration, but it cannot be a musical end in itself.

There are some prominent earlier examples of composers & works which, while not generally taken as a priori mathematical, are often perceived in a mathematical way. For instance, Bartok's exploration of Hungarian folk music led him to use additive rhythms, a technique which naturally works out in "golden section"-type forms by way of the mathematical relationship between the two ideas. Likewise, many people describe Bach's music in terms of mathematical perfection. I find more the mentality of the engineer than of the mathematician in Bach, but perhaps that is a personal idiosyncrasy. Nonetheless, the idea that he was inspired by mathematics is prevalent, if vague. Going farther back, John Dunstaple was frequently praised for his knowledge of mathematical proportion, and of course he was one of the groundbreaking composers to use the intervals of a third as the basis for vertical consonance. Yet, one cannot go too far with these ideas. After all, there are plenty of other prominent composers for whom a specific connection with mathematics would be completely illusory. Alternately, one might wish to make something of the fact that Dunstaple was also renowned for his knowledge of astrology.


For me, there are few lessons to be gained from this discussion, and I write it as much for the fact that I happen to have been both a composer and a mathematician as that I find the connection compelling. Perhaps there is something to be learned from that. By way of other remarks, one could perhaps look equivalently at a composer's knowledge of literature for an investigation of his phrase structure. Certainly literature is the forefront of communication, and lexical structure can translate to some extent. Looking back at symmetry, it is clear that any music will have some symmetry, whether horizontal or vertical or both, except for works which are specifically designed to avoid it (probably requiring an even greater knowledge of symmetry to achieve). Mathematics is perfectly designed to describe symmetries, down to the level of mere number (whose basis is repetition). Yet, I view it as a mistake to assign a causal connection here, as most of these relationships were intuited before they were described and standardized. The underlying question perhaps breaks down to asking how much of musical composition involves mechanical construction of forms and how much involves basic material coming to mind. It is easy to say that the great works transcend such an artificial distinction, but it is perhaps more difficult to examine in any great detail.

Administrivia: I have another vacation coming, pushing the next column from two weeks to three. After that, things should settle into the new schedule of every two weeks. Summer seems to be lacking in writing inspiration, for some reason. Perhaps that is a worthy topic on its own, but some long-delayed ideas will be articulated in this space soon.

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Todd M. McComb