Chapter 1: What role does mathematical symmetry play in catchy melodies?
Where intellectual honesty gets witness protection from social prosecution. The Last Show with David Cooper. Your favorite musician to write their catchy tunes. Are they just doing algebra? Well, maybe not on purpose, but a new study suggests that behind every earworm, there might be some sneaky math working in the background.
Things like group theory, symmetry, mathematics are engineering the hook that you've been humming all week. I'm here with someone who's done research in this area at the University of Waterloo. She's a data scientist and her name is Olya Ibrahimova.
Chapter 2: How can we understand symmetry in melodies without a math background?
Olya, welcome to the show. Thank you. So, symmetry in melodies. I've long heard that music is underpinned by math, but what does it actually sound like to someone who doesn't understand the math when they hear symmetry in a melody? Symmetry is just the word to describe what happens to musical pieces under certain types of transformation. Mathematics can be weird.
It can define weird things that are abstract in the terms that seem familiar but actually mean something different. In my case, symmetry is basically transforming a music with some sort of like tonal and positional operations on the notes, and then the melody would come to itself. Basically, it doesn't change under the transformation that symmetry means.
Okay, so the music is broken down into like 12 notes of the Western scale. How do you take a melody and sort of transform it and analyze it using math? How can we do that? Great question. So for our application, we basically take the set of all the melodies and we'll take a close look at how we can transform them.
So it turns out that our melodies that we, some of them find, they possess hidden mathematical properties that are not visible to the naked ear. And this is what was interesting about it. For example, if you shift all the notes pitch up, there exist some melodies that would stay invariant or symmetrical under those transformations.
Well, for example, Bach used to reverse the notes on his canons and they would sound exactly the same. Interesting. So you did this to what sorts of songs, for example? We didn't do it to the songs. I think we simplified the problem to begin with because it's much easier to take a look at this way.
So we defined a set of notes that we want to use, we defined the length of the melody, as well as we took four transformations from music theory and then examined the properties. And how were you able to analyze melodies and figure out that melodies had these commonalities? I feel like I'm missing something here. Okay, so imagine you have a melody that's going to be a sequence of notes.
Now imagine you can transform this melody in a certain way. For example, you can shift all the notes higher or lower, or for example, you can reverse the order backwards, you can flip the notes, or you can rotate where exactly the melody starts. Some melodies, when you apply those transformations, stay the same, and this is exactly what we've been studying. Does it make it clearer? I think so.
I'm hearing words that make it sound a little bit like dance moves. Okay, so what surprised you when you actually ran the math on these melodies? Did it confirm what we already knew about music or were there new things that you kind of discovered? There were some surprising results that genuinely surprised me in a way that I didn't expect.
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Chapter 3: What transformations can be applied to melodies using math?
So what was surprising to begin with is there was a clean separation of certain transformation. So basically every melodic symmetry splits uniquely into tonal parts and the positional part. Tonal symmetries act on what notes are, and positional on where the notes are. And you would expect that those transformations would interact in complicated ways, but no, they actually don't.
They commute, and that means that you can analyze what's happening from two different perspectives. But then what's surprising is that while tonal and positional parts are completely independent, the way that they interact when we take a look at the constraints of symmetry is very surprising.
So basically, under certain transformations that are tonal, you will see some transformations that are positional to the notes that would combine the melody into a symmetrical one. How long have musicians been using these symmetries, I guess without realizing it? Is this something that's been going on in music for hundreds of years, or is this a recent thing? Definitely for hundreds of years.
As I mentioned in our research, and I mentioned earlier in the interview, one of the most well-known examples is Bach, who would play his melodies forward and reverse in some of his canons. Does this mean that great music is less about like inspiration or something super unique and rather kind of discovering these mathematical things without realizing it and putting it together in the right way?
I think being symmetrical doesn't necessarily make music great. I just think that certain Easter eggs that you can put inside of the music can make them a little bit more special. I think one cultural parallel that I can think of, the composer Athex Twin decoded his portraits into the spectrogram of the sound piece he made. So that was another funny thing on how you make music even more special.
But then if it possesses some interesting symmetry, you can also put that inside as well. If you've cracked this new kind of mathematical code about what makes music sound good, what does this say about the future of like AI generated music? Could we integrate what you've learned into models and generate better AI music?
And side note, that sounds like a bad thing for, you know, musicians who still want to do things manually. But could this study and what you've uncovered be used to generate really good quality music, you know, programmatically? This is a great question. The framework that we have in our research builds a new way of structuring the melodies.
So now, instead of relying on AI to learn symmetry from the data like it usually does, you can actually enforce it in the code algebraically. So our paper provides the explicit recipe, pick the desired symmetry, and then the rest of the melody is going to be determined.
So instead of iterating through all the different symmetries you might think of, you can actually force a computing algorithm to compute a melody that possesses the internal symmetry that you desire. I imagine these companies that generate songs but just based on a prompt would be particularly interested in this research if any of them reached out.
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Chapter 4: How do melodies exhibit hidden mathematical properties?
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