Many of us think mathematics is addition, subtraction, multiplication, division, percentages, geometry, algebra, etc., and all this mathematics seems to be good, as it has some practical applications in our real lives.
As an undergraduate student of Physics and Mathematics with a keen interest in music, I have often heard of the relations of mathematics and music with some sort of figures and patterns, which I would say is a kind of exploration of metaphors that mixes with the music and mathematics or the process of discovering, comparing (which is the basis of both subjects), and the nature of both.
This article will have a short discussion on two points akin to both fields. In which statements will be supported and denied, and the concluding results will be discussed in the end.
One of the best interpretations of learning the similarities of both subjects is that both can be referred to as languages and functions. We will be trying to connect all the points and also exploring all the metaphors that mix music with mathematics.
Discussion of the first point, a function.
Mathematics and music are like two different functions, and our work is just to try to show a simple way to understand how they connect, unlike a fugue, where different voices meet and mix.
Music is like a function.
In math, a function means you take one thing and get another thing. In music, we take time and get sound. For every time $t$, there is some sound, like a note, loudness, or tone. This can be written like a function $f(t)$, which shows what sound occurs at time $t$. Sound waves in music are actually functions, mostly like sine waves. They come together to make the sounds we hear. In the book "The Math Behind the Music," Leon Harkleroad says that the waveforms are precisely what give music its structure and feel. And we also know that music has patterns, like functions in math.
But can music really be only a function? Functions in math are very exact and always the same. Music is full of feeling and sometimes surprise. Like in jazz, musicians make new sounds and break rules. People feel music differently, even if the sound is the same. So maybe music can be shown by functions, but it is not only functions.
Music is more than just numbers or rules. But even feelings in music come from some pattern. Sounds, notes, and rhythm all follow math rules and can be shown by functions. So music uses math and functions, but it is more than that. Functions are just language. Music is a feeling and an art.
Music is like a language.
Just like we use words and sentences to talk, music uses notes and sounds to express feelings or ideas. Different notes and rhythms can mean different things, like happy or sad. People all over the world can understand music even if they don’t speak the same language. So music can be seen as a way to communicate, like language. So music can be seen as a way to communicate, like language.
But how?
Language has grammar and rules for making sentences. Music has notes and sounds, but does it have grammar? Some say music is just sounds and not real language because it doesn’t have words or a clear meaning. Also, different people feel music in different ways. So it’s hard to say music speaks one language.
Yes, it is true music doesn’t have words like normal language. But music has some rules too, like scales and rhythm patterns, which are like grammar. Also, music can tell stories or express emotions without words. Like film music, it can show fear or joy without any dialogue. So music is a special kind of language, maybe not like a spoken language, but still a language.
So music is a language but different from talking language. It speaks with sounds and feelings, not words. It can connect people without the need for words.
Mathematics and music share striking similarities: levels of complexity, a sense of space, the creation of order, a reliance on axioms that govern style, and fundamental building blocks such as groups. Yet, since the translation between the two is not exact, and no formal formula maps one onto the other, we find ourselves delightfully orbiting a central premise, occasionally struck by meteoric insights that illuminate both realms in unexpected ways.
To establish connections between mathematics and music, we are treated to a seamless transition through mathematical ideas where mathematics flows like a stream into a river, from numbers to the concept of the irrational, to the infinitesimal, and from there, to spaces defined by their axioms, or styles. A few parallels with music are also hinted at, but ultimately it depends totally on the reader to draw those connections. Additionally, the terminology used is suggestive rather than prescriptive.
When defining the definitional characters of music, it can be argued that music is created for specific audiences and is therefore “modelled.” For example, the kind of music accepted as appropriate in a horror film might be received very differently in a concert hall or a church.
However, the model, the object, and the map are all unclear in music. In what ways might music advance its comprehension of an idea or an experience? We may speak metaphorically about film scores "modelling" emotion, of a piece of music serving as a model, but what does this truly mean? Perhaps what we call “modelling” in music is not a representation in the strict sense, but an evocation, an affective mapping, not logical or spatial, but emotional and temporal.
Neither does music appear to involve the kind of reasoning we find in mathematics, nor does it appear to be searching for something to model from the outside world. We don't witness the act of construction in music. Rather, we are submerged in a realm where, at least initially, knowing seems irrelevant.
And yet, every corner of this "muso-mathematics" is touched: distance, connection, contraction and expansion, equilibrium, the vertical and horizontal, modulations, patterns, symmetry, variations, contradictions, ascending sequences, leaps, disorientations, and pulsings. With this tsunamic force of imagery and association, it may hardly matter what concrete conclusions are drawn; what remains is a resonant field of possible meanings.
The product of mathematics, too, is more of a process than a result. Mathematics itself is a model of mathematical activity.
There is something about beauty that is both private because it involves a silent feeling and public because it makes us feel as if it is revealing something universal. It emphasises both our isolation and our feelings of common sense and sensibility. For the same reason, it can also risk inspiring contemplative withdrawal or impassioned absolutism. We believe that the evaluation of beauty is not subjective but rather grounded in something more fundamental. Beauty feels like an aspect of public knowledge.
We may not assert that everyone will agree with our proclamation of beauty, but beauty inspires a feeling that everyone should agree. The feeling makes a claim not only on us, as we view the beautiful object, but also on our sense of others. Of course, we haven't proved that taste is universal, or that beauty is objective; the only assertion is that the judgment of beauty is treated as universal and felt as objective.
Music allows us to dwell in a second nature; it intoxicates, and in that intoxication, it becomes something more powerful than ourselves.
It is not merely abstract; it reflects how we experience the world.
Music can make us feel strong emotions. It can take us to a different world in our minds and create pictures or feelings inside us. It connects with what we feel and think, even when we can’t say it in words.
Mathematics, too, shares these attributes. The mathematical proof, when it loses touch with reality and becomes baroquely outrageous for its own sake, may drift away from traditional notions of mathematical beauty.
However, keep in mind that not all music is beautiful in the sense of being enjoyable.
It may disturb and overwhelm; it is not so much beautiful, as here the word "mystery" is converted to the romantic word "sublime." In music, the sublime subverts our judgment, stirs our imagination, and often overwhelms us with its force. Similar to this, mathematics has the capacity to inspire awe with its vast, unknown structures and exquisite truths, making it sublime in its depth and immensity.
According to Rothstein's book "The Emblems of Mind", “our greatest risk is that our metaphorical interpretations will be willful, arbitrary, unenlightened, that connections will be made of trivial importance.”
This brings us to the subject of the power of mathematics and music.
The ability of mathematics to interpret the universe, revealing the underlying patterns and truths that shape our reality.
While the power of music is interpreted in its equally versatile ability to bear a variety of meanings, and its role as a gesture that makes a voice in our cultures.
Music, on the other hand, reveals emotional truths. Its power lies in its versatility, its ability to convey different meanings to different people, and its ability to communicate through gestures, tones, and movements that form a kind of cultural voice.
Both fields, in their own language, help us navigate the world, one through reason and form, the other through emotion and resonance.
