Mathematics and music

As a mathematician who is also a musician, I have been asked countless times why mathematics and music go together. My answer for many years has been one that I know people do not want to hear: both require many thousands of hours of relentless and often soul-destroying practice to do well. Anyone who has the personality to deal with that has the potential to do well at both. Of course, what most people want to hear is that it is a special gift, because that lets them off the hook of examining their own incompetence in both fields. While that may be part of it, the real point is that to do well in any human endeavour requires dedication and hard work. And it is the particular nature of the hard work that seems to me to connect mathematics and playing a musical instrument: both are absolutely unforgiving, and the clear distinction between right and wrong means that most of the practice consists in repeatedly getting things wrong, which many people find hard to deal with.

And in both cases, a faultless technique is both essential and irrelevant at the same time. In mathematics, just as much as in music, a faultless technique without imagination is sterile: I have come across many mathematicians in my career whose technique is far better than mine, but who have no imagination, and cannot play the mathematics. Just as I have come across many musicians in my career whose technique is far better than mine, but whose music-making has no soul. The technique is not an end in itself, it is a means to an end. In music, technique is there to make something difficult and delicate sound easy and assured, so the listener can be transported to a place of beauty and imagination. We do not want music to sound difficult – we already know it is. The same is true of mathematics. A real piece of mathematics sounds like a piece of music – you may not understand a word of it, but it transports you out of this world into a place you can hardly imagine. And a true virtuoso makes it all sound so easy, as years and years of practice appear to slip effortlessly off the chalk onto the blackboard. The goal of the real mathematician is for the listener to say at the end of it: that’s easy, why didn’t I think of that? Too many mathematicians these days revel in a formidable technique, and play a kind of atonal post-modernist music that no-one wants to listen to. The mark of a true master is playing something really new in C major.

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14 Responses to “Mathematics and music”

  1. Robert A. Wilson Says:

    Much the same is true in theoretical physics, of course. There are plenty of people practising Paganini caprices trying to get our attention, but the standard model can’t even play a C major scale in tune. First things first, I say. Don’t try to run before you can walk. And what about all these people predicting new physics at the first sign of a discrepancy in the 10th decimal place? If they can’t even quantise the forces we’ve got already, why are they trying to invent new forces that don’t exist?

  2. Robert A. Wilson Says:

    Symphony in C, in 4 movements? arxiv:2009.14613, arxiv:2011.05171, arxiv:2102.02817, arxiv:2104.10165.

    • Robert A. Wilson Says:

      Or Symphony in B minor (Unfinished)?

    • Robert A. Wilson Says:

      My favourite is the third movement (Scherzo). I love the way the leptons play in 3/4 against the 6/8 of the quarks. I love the way the weak force of the conductor mixes with the strong force of the brass, the electromagnetic force of the wind, and the gravity of the strings. I love the way the colours change, from flute to oboe to clarinet, from violin to viola to cello, from horn to trumpet to trombone, from double bass to bassoon to tuba.

      • Robert A. Wilson Says:

        Or have I got that the wrong way round? Is it the conductor that has the gravity, and the standard model of the orchestra assumes that the orchestra plays independently of the conductor? That wouldn’t be too far from the truth, in my experience. But just try playing something really tricky without a conductor, and you might start to understand why gravity should be not be ignored.

      • Robert A. Wilson Says:

        Yes, I think that must be it. If so, then the problem of quantising gravity is equivalent to the problem of quantising the conductor’s beat, which is a well-known problem in orchestral circles, usually dealt with by ignoring it (the beat, I mean, not the problem). It is called the measurement problem, because the problem is how to tell at which precise point in the continuous motion of the conductor’s arms does the wave-function (I mean the music) collapse onto the next beat. And the answer is exactly the same as in quantum mechanics: it has nothing to do with the conductor’s wave-function, it has everything to do with the environment. If everyone around you says this is where the next beat is, then this is where the next beat is. It has nothing to do with the conductor. That is the difference between theory and experiment.

      • Robert A. Wilson Says:

        Or to put it another way, the conductor waves their arms in time to the music, just as much as the other way around. The worst concerts I have ever played in are the ones in which the orchestra has made the mistake of trying to follow the conductor. Often this results in the conductor following the orchestra, and the orchestra following the conductor (“After you” – “No, after you”), as a result of which everything gets slower and slower and the audience leaves before the end in order to get to the pub before it shuts. Conversely, many of the best results are obtained by conductors who give up any pretence of beating, and simply let the orchestra get on with the business of playing the music.

  3. Math Światek Says:

    Hmm, do you really consider mathematics as hard work? To be honest I was never able to feel that way, because I always loved to make deep thoughts about everything and once i learned the framework of mathematics (or rather logic) it just gave me the perfect toolset to navigate the plane of thoughts with all the details of reality feeding an endless flow of inspiration of observations to think about. Thinking was always an escape for me whilst it is the mundane stuff that really feels like work that i need to force my mind to focus on: groceries, making food and the usual household, budgeting, etc. I would basically define hard work as everything repetitive that requires more routine then thinking or creativity – i.e. everything (pure) math is not. Well, calculating stuff is a different issue but then you can think of how to teach a computer to do that for you in a faster and more reliable way where programming is again a creative thinking process, so…

    I have learned that being lazy and finding creative ways to skip hard work e.g. by automating it via a program, is ironically what will make you popular and successful in the business world. But yeah, being lazy is about presenting a problem in a way where it becomes almost trivial so it can be easily automated and enables others understand the topic more easily so they can quality assure the result much better. I guess that’s the Canon in D-Dur for you.

    Then again i’m within the the autistic spectrum, so i guess that changes my perspective. And having this autistic reflex to just memorize information… because it’s kind of fun… helps to generate these moments of enlightment when sailing the seas of contemplation as the mind will just always find something i saw/memorized before (usually a detail) that kind of reminds me of a part of the problem i face. abstracting the similarities means identifying a pattern and than, it’s about identifying how the properties of one known thing relate more generally to the pattern and therefore apply to all that have it in common. but that’s just fun i do in my free time anyway so it’s difficult for me to count is as work.

    Sadly, I never learned to play an instrument, so i cannot say anything about music.

    • TJ Wence Says:

      “abstracting the similarities means identifying a pattern and than, it’s about identifying how the properties of one known thing relate more generally to the pattern and therefore apply to all that have it in common”.

      I could not say this better myself, it is exactly this that I never found the words to describe: learning as much as you can, even if it’s just simply that a *new* thing exists, makes building connections so much easier, and you have more confidence when encountering more ideas because you have this toolkit where you’ve seen things before and have connections drawn on an already-started map… Just seeing stuff is the first step in processing, so why not look around and around whilst sailing the seas of contemplation? It’s a lot more fun, and there’s no harm in looking up at the stars whilst sailing a boat through the sea that will always be there… I hope my attempt at saying the same thing didn’t come out as too nonsensical, but I do think that schools, which provide a linear path, are not doing the creative process any favors. My earlier teachers would say imaginary numbers did not exist. That’s quite wrong. And I only realized it was wrong from a lecture by John Baez where he described the history of Hamilton, Graves, and how the doing the 2D rotations on the 2D plane with a circle (2D unit sphere) was natural and therefore the imaginary units where fundamental for going around the circle as the multiplication was the rotations on it. The imaginary numbers add the second axis to the number-line because they are units of 2D system, and the reals are only 1D. I am fairly confident that my fellow students did not unlearn what our school forced into us, and while they get higher test scores, they don’t care to see anything past the blinders strapped to their heads. Well maybe that’s a little harsh, but they can be just as hard on me for me NOT inherently trusting an authority that wrote a curriculum for math, while this same authority itself is not in any way proficient in the subject, and are only trying to get us in and out a credit based system. Though this only applies to my older school that I have been out of for some years now… As I move up the system, things are starting to finally look better…

  4. Robert A. Wilson Says:

    No, I never considered mathematics to be hard work. I always considered it to be fun. But to do it well, requires dedication and a lot of time doing it. That is what I meant by “hard work”. And I completely agree with you, that laziness is an essential characteristic for a successful mathematician: the whole point of mathematics is to avoid doing calculations, whereas most people mistakenly think that mathematics is about doing calculations. The only thing is that, on the whole, you have to understand how to do the calculations before you can work out how to avoid them. It took me many years of doing calculations before I was able to transfer my natural laziness into mathematics successfully. But it is impossible to avoid work altogether, and sometimes there is nothing for it but to do the damn calculations.

  5. Robert A. Wilson Says:

    Perhaps the same is true in music: many amateur musicians, like amateur mathematicians, work too hard, and get hung up on technique. We are so busy trying to play the notes, that we forget to play the music. As one conductor memorably told us, playing in an orchestra is not about getting it right, it is about playing with everybody else. It seems to me that many amateur mathematicians in the Orchestra of Physics are so determined that *they* are right, and everybody else is wrong, that the resulting cacophony is unbearable. If only the strings of particle physics would listen to the winds of general relativity, and vice versa, concord(ance) rather than discord might be achievable. Ultimately, though, as one of our conductors frequently reminds us, the percussion are in charge. Never mind the continuous variables of strings, wind and brass, listen to the discrete variables of the drums, cymbals and triangle. You’ll never be able to play the Rite of Spring if you don’t. Never mind the curvaceous geometry of the seductive melodies, however good the first bassoon is, concentrate on the algebra, and count the beats: ONE two THREE four five ONE two THREE four five ONE two ONE two ONE two three. You’ll never be able to unify SU(2) with SU(3) if you don’t.

  6. Math Światek Says:

    That is an interesting reflection. Thinking back on how many things in physics ultimately came together, it was indeed a consequence to people working too hard and having too much of a result-focused approach forcing the math to yield them the predictions they needed. That just created a horrible frankenstrein of a theory that – while it mostly works calculus wise – it comes with a monster of interpretation and becomes extremely inconvenient to use.

    But with all the rush they never looked back and contemplated if there was a better way. This way they never realized it’s far from the only way to model the theory. They think they listen to percussion that nature plays where in fact they have erected a weird architecture of walls that reflect the beat of drums in such a way that it just creates a complicated cacophony. They think this to be nature, but where it is fact a problem of their own making.

    This metaphor brings immediate a quote to my mind “Mr. Gorbachev, tear down this wall”. I suppose a revision of the fundamental postulates of physics would be in order to check which are indeed the bare minimum and which are in fact just technical and of our own choosing (conscious or not) to model the world in a very specific way.

  7. Robert A. Wilson Says:

    Well, I’ll just keep on banging my drum in 7/8 time until somebody starts listening! Half the orchestra is playing in 3/4 and the other half in 4/4.

    • Robert A. Wilson Says:

      Seven is the magic number. It is the number of Weyl spinors you need in the standard model: one for neutrinos, two each for electrons and up and down quarks. Too many people are trying to add in some non-existent right-handed neutrinos so that they can march in 4/4 time, instead of dancing in 7/8. Or ignoring the neutrinos so they can waltz in 3/4. Four quarks, three leptons, 2+2+2+1, dance to the rhythm of the Rite of Spring. Colour confinement means there aren’t as many quarks as you think there are, so dance (2+2+1)+(2+2+1)+(2+2+2+1), or, if you prefer, think of it as (2+3)+(2+3)+(2+2+3) as Stravinsky wrote it.

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