Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas - 332 | Dmitri Tymoczko on the Mathematics Behind Music

Episode Date: October 20, 2025

Music is math that you can dance to. The fact that certain notes sound good when played together, or in succession, is related to the mathematical properties of the frequencies to which they correspond, an idea that goes back as far as Pythagoras himself. These days we have a much more intricate understanding of these relationships and how to manipulate them. I talk to composer and music theorist Dmitri Tymoczko about how different musical scales are constructed and the math underlying what sounds good. Blog post with transcript: https://www.preposterousuniverse.com/podcast/2025/10/20/332-dmitri-tymoczko-on-the-mathematics-behind-music/ Support Mindscape on Patreon. Dmitri Tymoczko received a Ph.D. in music composition from the University of California, Berkeley. He is currently a professor of music at Princeton University as well as a composer and performer. He has been the recipient of Rhodes and Guggenheim fellowships. As a composer, his works have been performed by multiple groups, and recorded on several albums. Personal web site Princeton web page Mad Musical Science Spiral diagrams: rock music, classical music Google Scholar publications Amazon author page Wikipedia William Sethares's Tuning Timbre Spectrum Scale

In California, staying compliant means watching the state laws and the city rules at the same time. And no wonder it feels overwhelming. Meal breaks, rest breaks, wage rules, constant updates, it's a lot. And that's why Southern California businesses rely on Guardian HR. They're local in L.A. And they understand this community and they help you stay compliant to avoid costly missteps. You get accurate payroll software and a real HR expert who keeps you ahead of issues. Get your Southern California business protected at GuardianHR.

Aging is real. And so are the benefits of adding vital proteins collagen peptides to your daily routine. New vital proteins collagen sparkling water. Your daily glow-up now in three fresh flavors. Strawberry blossom, lemon, lime, and blood orange. Improved skin health in as little as 30 days thanks to collagen peptides?

Cheers to that. Or go with our classic collagen peptides. So you can stay vital, stay you. Visit vital proteins.com to learn more and where to buy. These statements have not been evaluated by the Food and Drug Administration. This product is not intended to diagnose, treat, cure, or prevent any disease. Hello, everyone. Welcome to the Mindscape Podcast. I'm your host, Sean Carroll. Let me start today's episode with two quotes from two very different important thinkers in human intellectual history.

One is from Gottfried Leibniz, who says, Music is the pleasure, the human mind experiences from counting without being aware that it is counting. The other is from Thelonius Monk, the famous jazz composer and pianist, who says, said, all musicians are subconsciously mathematicians. I learned about the existence of both these quotes from today's guest, Dmitri Tomosco, and they express basically exactly the same idea, right, that there is something in music, in the playing of music, maybe even in the experiencing of music, that is subconsciously mathematical. This connection between math and music is very, very well known, has been explored in great details. But there's

There's something else going on. You know, music is not simply abstract mathematics. It's not theorems and proofs. It's embodied somehow. You play the notes or you sing the notes, and then you hear the notes, you experience the notes. So that's physics, as well as mathematics, not to mention psychology and physiology and things like that. There's a lot going on in the world of music that intersects with science and with mindscapey topics in all sorts of ways. At the same time, There's another famous story that may or may not be apocryphal, but I know that I love telling it. I probably told it before of Albert Einstein, who loved playing the violin. And he became a celebrity, of course, Einstein.

So he got to play the violin with real, classically trained musicians, famous musicians. And at one point, one of them is playing with Einstein and just stops out of frustration and says, Professor, don't you know anything about time? Of course, you can know more about time in its intrinsic nature and its physical reality. than any other person alive at the moment without being able to embody that knowledge in keeping time in a piece of music. So the actuality of it, the physicality of it, the reality of it can be different than the knowledge of all this mathematics underlying music. In today's episode, we're going to be doing a little bit of both. Dmitri Tamosco is a musicologist, a composer, a performer, a professor at Princeton, and he's been thinking very deeply about the relationships between mathematics and physics.

music. He has a background that includes studying a little bit of physics and philosophy, like many mindscape guests end up doing, and then really thinking about where all this mathematics fits in. And getting to very advanced topics mathematically, we're not going to quite get there in this podcast episode, but Dimitri has authored books that you can buy. A geometry of music is one, and tonality, a user's manual, is another one. Thinking of about how to put a metric on a space that is defined by what scale or what chords you're playing and thinking about the topology as you travel around, really using advanced ideas like orbifolds for modern algebraic geometry, even inspired by string theory.

Today, though, we're going to stick with what is a scale? You know, where do you get the particular notes that you play? Why do some notes sound good? How do they fit together? How does that play out over the course of the history of Western music, both classical music and pop music? How does it bump into non-Western traditions and get modified by that when you realize, oh, our way of doing it is not the only way in town? There's some music that sounds good, some music that sounds less good, but there's no music that sounds perfect

because different people are going to hear it differently, and that's part of the pleasure. The history of it matters, your particular place in the musical landscape matters, It's a relationship between something going on out there in the world and something going on inside us, which is really fun to think about. So let's go. Dimitri Tomasco, welcome to the Mindscape Podcast. Thank you very much. Happy to be here. You know, you're described on your own webpage, so I presume it's an accurate description as a music theorist, which is a thing one can be.

But it raises the question in my mind, like, is there supposed to be? to be a correct theory of music like there is in physics? You know, are we theorizing about what happens or are we searching for better and better music via theorizing? Yeah, that's a really great question. I'd say that probably people disagree about that as well as, you know, all the substantive issues they confront. My own, my own belief is that, is that human beings have.

have an enormous amount of musical knowledge that is implicit or embodied. So they know it kind of in a practical sense. They know it in terms of how to get around their instruments, but they don't necessarily know how to express that knowledge in, you know, the terms that scholars would recognize in academia. So one way of understanding the business of music theory is that it's a process of translation and going from a kind of physical embodied knowledge to a kind of conceptual descriptive knowledge.

Much like, I guess, both literary theory and even like, you know, athletic theory. Like we can do these things. We can read, we can run, but we can learn to do them better or even understanding what we're doing absent any normative judgment. Yeah. I mean, I like the idea of athletic theory because that's something that has maybe really taken off in the last couple of decades. I think, you know, 50 years ago, there was a lot more guesswork,

and now we've got all these ways of measuring athletics and then discovering, oh, you know, if you're trying to hit a baseball, the angle is really more important than we thought or something like that. I will say in this context, it's very important to me that I am both a music theorist and also a composer and, you know, an improviser and I am a musician. And for me, that's important because a large part of what I do involves kind of me watching myself or looking over my own shoulder when I'm doing music and asking, you know,

why do I want this? Why does this sound good? Why did that not sound good? So I'm kind of like an anthropologist who studies himself. Well, that's here. I was going to get to this much later in the podcast, but how conscious, are you about music theory when you are playing or

performing or improvising or even composing? I mean, is there something is there a switch in your brain that goes like I'm just in the mood or in the zone versus I'm thinking about it now? Yeah, so this goes maybe to the second half of your question

which is in the 20th century theorists started exploring and searching for new methods of musical organization. And there's a long and not that glorious history of composers sort of becoming theorists and saying,

boy, wouldn't it be great if like, you know, we did this or we did that? And so this idea of a composer as an explorer, I think is an interesting idea. I would kind of reject the question that you asked. I was actually talking with my kids about this at dinner the other night. I kind of conceive it in terms of the philosophical question of positive versus negative liberty, right? And so when you're sitting at a piano, you have infinite freedom in the sense that you can kind of press any note and there's 88 choices and nobody's going to arrest you if you press the wrong one. So you have you've tremendous. liberty in that sense. On the other hand, it's really, really, really painful. And believe me, I know, to compose music by trying to figure out which of those 88 keys is the one you play next, right? Yeah.

You know, 88 to the power of N is a really big number. And so there's a positive liberty aspect to music theory that I think is really important, which is what is a good guess for something I will do that sounds cool. And so, you know, weirdly enough improvisers are some of the biggest consumers of music theory because they have to get up on a stage in front of people and they have to make music. And, you know, no method is foolproof, but it's worth a lot to have a method that's going to work 80% of the time or to have ideas, I've just done this. What do I do next?

Right? And so what I would say is the business of becoming a musician is really deeply involved in kind of internalizing conceptual procedures to the point where they become automatic and to the point where you don't have to explicitly think about them and they become second nature. I think that makes sense. I do remember like the moment of my life where I figured out that improvisers and music weren't. just randomly going or even that they had no idea where they were going. Like, they know what key they're in. You know, they know a whole bunch of possibilities and they're making conscious choices along the way. Yeah. And what I would say is there's a huge spectrum of improvisers in terms of whether this knowledge is explicit and conceptual or whether it's implicit.

There are great improvisers who might not be able to tell you what key they're in, but they have the knowledge. They know. Yeah, exactly. And that's what's what's so cool. I will say this goes back to kind of the essential paradox of implicit knowledge because in some sense, a lot of, you know, the easiest way to make music is just to sort of find some other musician you really like and kind of copy them and make a few changes here and there. And this is basically what Mozart did with Haydn. It's basically what Beethoven did with with Mozart. It's basically what, you know, every saxophonist did with Charlie Parker. Right. And so the thing that creates a challenge is, is what do you do if, if you want to like, you know, you want to do something really different from,

from a previous model. And there, a little bit of explicit knowledge. And well, you know, that's where the, The book learning theory comes in. You don't need book learning if you've got to hide and you want to copy height, right? But I would say something happened in the 20th century where people felt that they lost their immediate models. And so that the earlier tradition of like finding a teacher and copying them stopped seeming attractive. A lot of weird things happen in the 20th century and we're still seeking our way in some sense. But okay, let's go back way further than the 20th century. You mentioned the piano keyboard, 88 keys.

Here's my chance to get straight on various questions I've had about scales and, like, the major scale and the diatonic scale and things like that. And I know that this is deeply involved in your work in the geometry of music. So why don't you give us the lay of the land in terms of what are the notes on the piano keyboard? How are they organized? Right, right. So, yeah, I mean, we start with a. continuum, which is, you can think of it as like a line. And anyone who's ever tried to sing or play the violin or an instrument like that knows that like really between any two notes,

there are an infinite number of possibilities. So one of the really distinctive and cool things about human music is that we limit those choices to some smaller collection. And this does not seem to be true of animal music. So we use what's sometimes called an alphabet of notes, just like we have a limited number of letters or a limited number of phonemes in a language. We have a limited number of notes that we make our music with. And this is quite possibly the result of a biological adaptation to music making. Nobody is really sure exactly where it comes from or what it means, but it seems to be true of really almost every human musical culture.

Can I put in to ask about the little comment you made about other animals? Is that a known thing? I guess I'm not surprised by it, but I'm now fascinated by the question, do any animals stick to a certain finite set of notes? Yeah. So, you know, this is not my area of expertise. But I have recently, you know, asked this question of people who know, people like Ani Patel at Tufts, who's a world expert in animal music making. And the consensus seems to be basically. No, there are no other animals that have musical scales.

All right, good. There are birds that will maybe chirp back and forth between two notes. So in a brief moment within an animal vocalization, you might go back and forth between two fixed points. But I don't think those fixed points come back later in the song or add up to a scale in a recognizable human. From the writers of parenthood and life as we know it comes, It's Not Like That, a new family drama about starting over and second chances. Scott Foley stars as Malcolm, a recently widowed pastor and dad of three. And Aaron Hayes is Lori, newly divorced with two teens.

Their families used to do everything together. Now they're navigating single parenthood and maybe something more. Watch It's Not Like That, All episodes streaming May 15th on Prime Video. It always happens right before the whistle. There's a little voice that says, what if I mess up? What if I'm not ready? I see a whole highlight reel of everything I don't want to happen.

Missed shots, turnovers, letting my team down. And for a second, there's doubt. But then, I realize I've done enough to be where I'm at. The early mornings, the extra reps, the days I wanted to quit and didn't. So, I smile. self-doubt is natural, but my smile is a reminder that I'm resilient. To put more smiles out into the world, Colgate has supported female athletes for over 50 years with the Colgate Women's Games.

The Colgate Women's Games is the nation's longest running indoor track and field series for girls and women. Colgate, your smile is your strength. You know, this decades, centuries, millennia-long quest to figure out what is different between human beings and other animals. and maybe it's the musical scale. That'd be great. Well, there's a lot of differences. But yeah, this is actually a super interesting topic because it is clear that animals make something that is like music. Sometimes people call it proto music. In the old days, there was a real reluctance to attributing emotion or human-like behaviors to animals. But I think we're now moving back to recognizing, you know, and it is true. There's a huge. huge amount of signaling in the animal world, right? And we as part of the animal world need to be

able to understand that signaling. So if you think about a growl, right, like an animal growling at you, you are going to recognize the meaning of that thing or vocalizations like a mew or a, you know, a friendly kind of sound. And so that sound making is clearly connected. to human music making in ways that I think we don't really understand and people are really starting to explore. Fair enough. Okay. Back to the scale.

Yeah, right. So I think you're leading here because I think you know just as well as I do that one of the big ideas about the origin of scales is really the first discovery of mathematical physics. Right. So there's a legend. The legend is probably not true, but it sort of falls into the category of too good to check. There's a legend that the ancient Greek philosopher Pythagoras was walking by a blacksmith's shop, and he heard the blacksmiths hammering on these swords.

And he noticed that the swords made all sorts of different pitches. And then he would notice that some pairs of swords, kind of resonated and sounded good together. And he rushed into the blacksmith's shop and he sort of isolated which pairs sounded good. And he found that the pairs that sounded good had lengths that were in simple whole number ratios. So when one sword was twice as long as the other,

it formed a musical interval we call the octave. And when it was, when they were in a three to two ratio, formed the perfect fifth and four to three, five to four. And this really was the first discovery of mathematical physics, as far as I know. And it showed that there was a quantitative law to musical consonants. And he took it in a very philosophical direction. He thought that it showed that humans were directly sensitive to number, and numbers were these abscessively. abstract things, and then from that we get Platonism and maybe even Christianity, if it is to be believed. But, yeah, so that's a great starting point. So there is something called a frequency. I mean,

they probably hadn't even realized that the length of the sword, that makes sense. But probably Pythagoras would not have translated that into frequency of sound. That's not how they thought at the time. But certainly, if one note is twice the first, frequency. That's a very obvious connection there. But then there's sort of these vaguer connections, three to two, five to four, whatever. Yeah, exactly. And so the simple, one simple story about where scales come from is basically you start with one note. That's a middle C. And then you kind to add to it, say the three over two. Right. And then maybe you add to that the three over two. And then maybe you add to that the 3 over 2 of the 3 over 2,

or you could add a 4 over 3, and then you can kind of move these things so they're as close together as possible. And if you keep doing that four times, you get a five-note scale we call the pentatonic, which is still embodied in the layout of our keyboard instruments, in the black notes of the piano keys, right? So one story about scales is they come from, you arbitrarily choose some note,

and then you add notes that kind of sound consonant or resonant with that note, and then you basically stop when it gets too hard to sing. If you keep doing this, the notes get closer and closer together, and when they're too close together, it becomes very hard to, it becomes very hard to sing.

I mean, that's a great point to make because the physical limitations of the human, both sound creation systems and sound recognition systems, are going to play a very large role in how we think about music. Oh, yeah, I love that. I think, like, one of the things

that's really cool about music is it has this abstract mathematical quality to it. You know, if you listen to, if you listen to Bach, or if you, if you listen to, you know, John Coltrane or whatever, you hear a kind of mathematical set of relations, but it's also embodied.

And so it's kind of like, it's a kind of mathematics that is constrained by our biology, our sense of pleasure. And so it's, you know, on the one hand, it means that the math of music can never get that sophisticated. and in comparison to string theory or quantum field theory, it's always going to look simple. But on the other hand, it gives us a kind of direct access to these phenomena that is rewarding in a different way. Okay, so tell me about the chromatic scale. I'm dividing up the octave into 12 notes. Right. Well, we've started with five.

Yeah. You're going backwards. I would have liked to start with 12. and then pick out subsets. But maybe you're, because you know this better than I do, you're maybe suggesting that it's more organic. You just start with the five and build out. Well, five is everywhere in the world.

So five note scales are the pentatonic scale I played is the most popular scale in the whole world. You know, you can go to every continent, you know, if you told me about a human culture, I'd never encountered and asked me to guess about their scale, I would start with the five notes. Now, you can add two more notes. Okay, so let's see. F, C, G, D, A, E, B. And that gives you the white note scale. And that's a seven note scale, which is maybe number two, the second most popular scale.

It's in China. It's in India. It's in the West. So, so. Is that the diet? diatonic scale? Am I getting the names right? I'm very bad. Okay, good. No, diatonic is excellent. Diatonic is kind of, some people would call it the major scale, but major means those seven notes with a particular note chosen as what we would call the tonic or the home, the most stable note.

So diatonic scale is a more neutral description that that doesn't specify one note as the sort of most prominent or stable note. And all of these, as soon as you say a scale, do I take it correctly, that what I mean is I take an octave, two notes doubled in pitch, and I subdivide them. Is that what it means? Yeah, I mean, you know, we do have a concept of a non-octive repeating scale where the notes are different in different octaves. But I would say that that is a fairly obscure and esoteric concept. So for the purposes of an introductory podcast like this, let's just limit ourselves to scales that repeat at the octave, which is like 98% of the human skin is. And okay. And so that the pentatonic scale that we start with, is that, you know, if the first three notes of it are basically the C major chord?

It does have a C major. Well, the first three notes are not quite. It has a C major chord with. in it, but actually the way to think of it is it's just a stack of perfect fifths, of three to two frequency ratios. So start with X and multiply by three over two, four times. That gives you five notes and then you've got your pentatonic scale. If you do it seven times, you get the diatonic scale. And then to answer your earlier question, if you do it 12 times, you get a chromatic scale. Good. That's actually super duper helpful because it answers my actual question, which is why those seven notes in the diatonic scale? Why the major scale? And somehow you just give me a little mathematic formula for making it. Yeah. Well, and now, look, with all of these things, we can go down a very nerdy rabbit hole and so on. The chromatic scale, believe it or not, was stabilized probably later than you think. So even in 15. 50 or so, there were people who argued for keyboards that had 19 tones per octave.

They look like our keyboards, but the black notes are split into separate, you know, D-sharp and E-flat are separate. So seven notes came first, and then the 12 came really around the time of the scientific revolution and stabilizing it there. There is another, there's a very subtle, secondary issue, which is that the 5 to 4 ratio is the ratio of the major third, right? So, 5 to 4. In the pentatonic scale, the way we were talking about it, you get to that ratio by multiplying

3 over 2 a bunch of times. So that 3 over 2 to the power of 4 isn't close to, but is not the same as 5 over 4. when you discard the octaves. And so as music developed, people became interested in that 5 over 4 ratio, and they started to actually move from what's called the Pythagorean diatonic scale, which is just generated by the 3 over 2 ratio to this compromised diatonic scale called the just intonation diatonic scale, where you have three 3 over 2 ratios, F, C, G, D, and then you also have some five over four ratios, C.E., F.A. And G.B. And these are actually two different

ways of tuning the scale. And you get into this pretty complicated mathematical realm where you're trying to compromise between all these different intervals that don't exactly line up. Well, and the chromatic scale would seem to be the word. defender, right? I mean, it's 12 equally spaced in log frequency or whatever. So it's the 12th root of two that comes in, and you're never going to get three halves by doing that. Well, okay, so you've jumped ahead because there's a Pythagorean chromatic scale where you multiply three over two 11 times. And that is not the equal-tempered chromatic scale that we know from our keyboard instrument. So what happened is around, well, mathematically, this happened around the time of

Box, so 17, 20, something like that. The mathematicians realized, well, instead of just doing three over twos, we could divide the octave into 12 perfectly equal parts. Yeah. But, but, but, and then that was not actually a practical solution for piano tuners or organ tuners. And, but, but, but, but, but, and then, that was not actually a practical solution for piano tuners or organ tuners until later than you think it was about mid-19th century that the tuners discovered that the tuners discovered how to actually tune their keyboards in an equal-tempered way. Now with computers it's no problem and so on. But it turns out that the 12th root of two, if you iterate it enough times, becomes very very much. It becomes very, very close to three over two. It's the difference is about, uh, one part in 50 or so. So it's not,

it's much, it's not really, it's a very good approximation to the three over two ratio. It's not a very good approximation to the five over four. Okay, very good. But we can, so we can get that perfect fifth, that three over two, pretty approximately on the piano keyboard, but not exactly. Right. Exactly. And when, string quartets play or when singers sing, they, depending on whether they're playing with a pianist or not, they will actually depart from the piano keyboard intervals to try to get closer to, or weirdly enough, sometimes farther away from these mathematically ideal intervals. I was going to ask, like, are they good enough to do that? How precise is the distinction? They are good enough to get better than the piano keyboard.

No question. And some people specialize in this and they're amazing at it. The thing that's tricky though is, okay, if you don't have a keyboard, the danger is that as you make all these adjustments toward the three over two and the five over four ratios, you can drift away from where you started. So you can start in a place. you know, and basically, weirdly enough, this involves making a kind of non-trivial loop in some kind of abstract space where you return to where you started, but it's not the same. I think in your language, it's the Bohm-Arenov effect where maybe you go around a solenoid. Quantum phase, yeah. Yeah, yeah.

So there can be these, we call them comma pumps or comma shifts. And so this is actually one. reason why singers started playing with instruments is because the instruments can return, the instruments will stay fixed. And so there's this kind of endless compromise between what you can think of as the local, which is trying to make all the notes sound as good as possible together at every moment, and the global, which maybe emphasizes more stability, like returning back to the very note you sang a minute ago. We should just mention for the audience that Dimitri has written a book called the Geometry of Music.

Is that what it was called? It's actually called A Geometry of Music. I didn't want to be hegemonic. But it's all about, you know, unearthing the geometric structures in a very advanced, highly non-trivial way behind all these journeys through the various scales and musical constructions. And actually, I've written a second book called Tonality and Owners Manual, which kind of maybe is a little more hands-on.

in terms of how some of these phenomena manifest themselves in rock music or in Beethoven. Very good. Very good. Okay. So we have different ways of constructing these scales. And one thing about the way that you just set it up is all these things exist simultaneously, right? I mean, on the keyboard, you have the chromatic scale with all the keys, but you certainly have the major scale, the diatonic scale, and the pentatonic scale right there to play with as well as lots of variations upon them. Yeah, exactly. And so I would say early on, the diatonic scale in the West was the main scale. And then the chromatic scale was not really a thing. It was the collection of all of the individual major scales. And then over the course of history, people sort of started using the chromatic scale as the main thing. There is one thing we have to talk about right here that I won't, because, Because you're a physicist and because this is just intrinsically cool, I need to put it out there. Which is all of this stuff about three over two and these ratios, right? All of this stuff is fundamentally dependent on the physics of the instruments that we are playing.

And this was not understood early on. I would say for the ancient Greeks, the number three over two was this abstract platonic ideal that was inherently attractive. And in the 19th century, the 18th and 19th century, so, okay, Fourier invented the Fourier transform, right? And he showed that when a string vibrates or a sword vibrates or when anything vibrates, you can decompose that vibration into a sum of simpler vibrations. And essentially, so we call those simpler vibrations, we call them partials or overtones. And essentially, the three over two ratio sounds good because the instruments themselves are vibrating. at a fundamental frequency and then at twice that fundamental frequency

and then three times that fundamental frequency. And so there's actually a physical mechanism that causes the three over two ratio to sound to sound interesting and right and consonant. And the reason this is important is some instruments don't vibrate with that three over two, with that structure. Bells, metallophones, vibraphones.

Yeah. You know, there are lots of things that vibrate inharmonically. And for those instruments, so if you go to Indonesia and you listen to a gamelon orchestra, they are using non-inharmonic bell-like metallic instruments, and they tune their gamelons in very different ways from the ways Western instruments are tuned. And my friend Bill Sathares has really, written a wonderful book about this called Tuning Tamber Spectrum Scale. And in that book, he has these wonderful demonstrations where he will create inharmonic sounds. And then he plays them

with Western scales and they sound terrible. And then he stretches the scales to match the inharmonic sounds. And they sound bizarrely, they sound bizarrely consonant. When I When I've learned Like the value The Family The Importance

of the life and that the 99% of the people of the people that cause a Culebrilla

Although not all the people in risk I'm I'm sorry the erupion Dolorosa

with ampoyas duros times making that even the things even the

things are all a lot not learn you don't know about the doctor or

pharmaceutical And I take it that's just because a string is the simplest possible thing. Like you have a stretch string with some boundary conditions and its overtones. Harmonics are going to be twice the frequency, four times a frequency, three times, whatever. Whereas a bell is a complicated shape and its higher frequencies are not going to be simple multiples. Yeah, exactly right. And even if I can just, like if you wanted to get into the physics of it, Sure. When you said a string is the simplest possible shape, you're really talking about an ideal string, right?

And a piano string, an actual piano string has wire wound around it, and it's under so much tension that piano tuners actually don't tune octaves to two to one ratios because they sound a little bit bad. The octaves get a little bit stretched. And so the non-idealness of the piano string actually makes a difference to the practice of piano tuning, which is why you should find a good piano tuner and not just someone who. So annoying when the messiness of the real world gets in the way of my spherical cows that made everything easy to understand. Well, again, that's like that's both the blessing and the curse of music is that the mathematics breaks down maybe early. than you would expect it. Like, in physics, the standard model is not going to break down until, you know, for centuries, right?

And that's both really cool, but kind of frustrating. And in the musical case, you always have to ask yourself, okay, this is a cool idea, but how does it actually sound? Well, as you, yeah, with exactly that in mind, I have to mention the fact that there's not only the major scale hidden within, the diatonic scale, but also the minor scale, which is another choice of seven notes. And it sounds sad to us in some ways. And that's just, that fact has always blown my mind. It's unmistakable, but how can a different choice of a subset of notes have different emotional resonance to us? Yeah. I mean, and it's even more than that. If you take the white notes of the piano keyboard, you can kind of make any one of those the most stable note.

All you really have to do is play it a lot, right? I'm just going to do that as a demonstration, right? So first I'll play the white notes, and I'm just going to really just randomly play white notes, but I'll play a lot of C's, and you'll hear it kind of as a C major thing. And then I'll do it again, but I'll play a lot of E's,

and so you'll hear it as an E-Frigion. We call it the E-Frigion mode. Okay, so here are the, the white notes just with a lot of Cs. And here and now basically the same selection from the piano keyboard, but with a lot of E's. So in the two cases, you're kind of judging the notes relative to a different origin, you know, if you think about the coordinate system, right? And so you're getting different collections of notes relative to that origin.

And it sounds different. I mean, it's just like the physicist of me wants to say, it's just an overall scale. How can it matter that much? But our bodies care. Yeah. And I mean, one way to think about it, okay, is we are, most humans are not that susceptible to absolute frequency. We're always judging frequency relative to another frequency.

And I think what you can say is the intervals of the major scale are higher relative to the note you're judging them from, and the intervals of the minor scale are lower relative to the note you're judging them from the tonic, right? And so if you think more about that relative arrangement, it starts to make a little more sense. And the higher and the lower thing, there are some psychology experiments that suggest if you kind of, if you present people a default set of a scale, you just make a random scale. And, then you kind of lower its notes relative to the tonic. People experience that as being sad. And so it's connected to low affect, right? When you're sad, you're not, you know, your vocal

contour is maybe a little bit suppressed. And so it might be something like that. Yeah. Okay. Very, very good. Very good. So, but let's get into the music theory of it all. We've been doing that a little bit. but you make the wonderful point about transpositions of along these different alphabets or along these different metrics. Like once you have a scale, whether it's the diatonic or the chromatic or the pentatonic, you can then say, let's move up one within that scale. That sounds very simple, but it's the basis of everything. Right, exactly.

So one way of thinking about a scale is it's like a musical ruler, right? So it tells us how to go up or down by one. And it also tells us how to take a pattern and move that around. So, you know, we can have a simple pattern like up one, up one. I started on C. I moved up one to D and I moved up one to E. So I can take that pattern and move that whole pattern, move it up along the white note scale. Right.

But I can also move that pattern along the pentatonic. scale, or I could move that pattern along the chromatic scale, right? And these are all different ways of taking a pattern and shifting it. And actually, so one thing I think is really important, and I'd say this is something I'm actively thinking a lot about, is the things that we commonly think of as a chord, like a C major chord. Those can also be thought of as very little scales. So if we take the pattern up one, up one, let me do that again.

If we take that up one, up one pattern, we can move it along the C-E-G chord as well. And where things start to get really, really complicated is that people start moving patterns along multiple collections simultaneously. because the scales are embedded within each other. Exactly. So you can say, okay, I'll take my C-E-G, and I'm going to take that up-one, up-one pattern, and I'll move it to EG-C. Okay, that's not sophisticated.

Okay, but then I can take that E-G-C, and then I can move it down along the seven-note white-note collection. So first we moved our pattern up along the three-note collection, But now let's try to move it down along the seven-note collection. And now we'll erase the intermediate steps. So instead of going up one along the chord and then down along the scale, we'll just go from the beginning to the end and we get... Right, and that's starting to be almost like a little word of music.

Yeah, you're actually writing music. What's tricky and what's complicated is that the ear doesn't, immediately decompose into its component motions. And I would say even most musicians, they kind of instinctively they instinctively play that, but they don't really think about the fact

that you can generate that transformation from this much smaller alphabet. And it's interesting because I think that the first thing the non-musicians such as myself would be thinking is you're taking some fixed set of intervals and starting it somewhere different. But by the fact that you've moved within either the pentatonic or the diatonic scales, that's not exactly true. They're not equally spaced in interval. Right. If you're counting white notes, that is you go up two and then up two,

whereas this, you go up two and then up three. Right? And so, So I would say most musicians probably intuitively, they think of the second thing as maybe a deformation or a variation or a change to the first thing. And it's actually really important to be able to think of it as a rigid transformation of the first thing, right? You can't tell computers, oh, you know, make a variation because computers don't know what that means. but you can tell a computer move it up one along this collection and move it down three along that collection. And so it's a kind of, it seems like it's not a big deal, but it gives you a much more powerful structural description

of what's going on. And you can sort of trace how we collectively discovered these different kinds of transformations through the history of Western classical music. Yeah, basically that discovery starts happening maybe sort of like something like 1550 to

1750 people gradually start becoming more and more aware of these nested scales these nested alphabets these nested rulers within the space of musical possibilities the development of the keyboard probably

helps because they can you know early composers like Renaissance composers like Palestrina and Joscan didn't not compose at the keyboard and we're not keyboardists. And then you have a shift people like Bach, you know, they're all Mozart Hayden, they're all, they're all keyboardists.

So they start to notice these possibilities. And so if I were Bach, J.S. Bach, how conscious what I have been about what I was doing? Or was it just, you know, pre-cognitive genius? I mean, that's a really great question. I would say we don't really know. No, nobody knows the answer. Yeah. My own view is that Bach was an incredibly smart guy.

So, you know, I put Bach in that Einstein kind of did you just don't really get smarter than him for any. There's no point. He sort of topped out that the. So, you know, you see Bach doing a lot of really intelligent stuff that suggests to me that it wasn't unconscious. that he had some kind of knowledge of what's going on. And, you know, Bach in particular just he loved math and turning things upside down. You know, he really was a kind of mathematician of notes. And so I bet that there's more consciousness than we realize.

It's still kind of tricky because for Bach, all of his knowledge, was constrained to his particular system. So if you asked Bach, compose me a piece, but I used totally different chords and scales from the ones you know, he wouldn't even know how to start because all of his knowledge was his scales, his key. Fair enough, yeah.

But then as time goes on, there's an impetus to do new things and be creative. And I think you give the impression that by the time the 20th century came, long, we had sort of run out of new places to go on the musical scales. Yeah, that's, you know, I probably wouldn't say run out of new places to go. I don't, I don't love that formulation. The way I would put it is that around 1900, it became clear that the way

Western composers had organized their music was profoundly culture-dependent and in some ways arbitrary, right? And, you know, if you really wanted to come up with a point in time at which this became really obvious, it's the 1889 World's Fair in Paris, where WC here is a Gamelan orchestra from Indonesia, right? Gamalán Orchestra has all these metal, giant metal instruments. And so you couldn't

really bring one from Indonesia to Paris, you know, in 1500. By 1889, you know, it's becoming easier to globalization is starting to get underway. And so, so WC hears this, this very strange music. And he realizes, oh my God, you know, all the stuff I've learned in school, it's more like, it's conventional. It's like, you know, drive. Yeah. It's not the only way. So then, so the question, it's not so much that what they were doing had been exhausted or run out of steam. But the question, what are the other ways became really pressing because even you think about Picasso, right?

Picasso is doing realist painting. Right, in the early times, yeah. He sees these African, you know, masks, right, and realizes, wow, there's a power in other cultures' ways of doing art that my culture's not tapped into neo-classical, you know, 19th century French art doesn't have something that these African, you know, artworks have. So I would say there is a tradition of talking about exhaustion and, you know, these, the Western ways of doing things running out of power. But I see a more of a more expansive phenomenon where people realize there are these alternatives. But then it raises this question. What are the deep rules of music versus the more conventional and cultural ones? From the writers of parenthood and life as we know it comes,

it's not like that. A new family drama about starting over and second chances. Scott Foley stars as Malcolm, a recently widowed pastor and dad of three. And Aaron Hayes is Lori, newly divorced with two teens. Their families used to do everything together. Now they're navigating single parenthood and maybe something more. Watch It's Not Like That. All episodes streaming May 15th on Prime Video.

It always happens right before the whistle. There's a little voice that says, what if I mess up? What if I'm not ready? I see a whole highlight reel of everything I don't want to happen. Missed shots, turnovers, letting my team down. And for a second, there's doubt. But then I realize I've done enough to be where I'm at.

The early mornings, the extra reps, the days I wanted to quit and didn't. So I smile. Self-doubt is natural. but my smile is a reminder that I'm resilient. To put more smiles out into the world, Colgate has supported female athletes for over 50 years with the Colgate Women's Games. The Colgate Women's Games is the nation's longest running indoor track and field series for girls and women.

Colgate, your smile is your strength. And there does seem to be a difference between, you know, Western traditions and other traditions in this sort of depth of exploration into these different, scales that you're talking about. I'm not sure what way to say is that. Is it melodic exploration? I don't know. Like at an expense of rhythmic exploration, for example.

Right. Well, I think a lot of that has to do with the fact that Western music is notated. Yeah. And so notation becomes the primary vehicle for transmitting Western music really early on, you know, around 1,100 or something like that. And so the idea, and it's also connected with the fact that Western music early on is bound up with the church and you have these particular hymns and you, you know, it's part of religious ritual. So in, I would say the default, the human norm is to make music in a more participatory and in a non-notated fashion, right? and this is actually something I remember when I was a kid, I started playing classical piano.

And my daughter is going through this now. At a certain point in junior high, you face the challenge of kind of taking it to the next level in Western music, which means going from playing 15 minutes a day to playing an hour a day and playing these really advanced pieces and becoming a professional music maker. Or maybe you pick up the guitar and you start singing rock songs

And it's a switch to a more informal, a less professionalized. I mean, there's an incredible amount of expertise. But the key point is that there's a real difference between musical styles in which every note is specified as written down and musical styles in which, you know, writing maybe you write down an outline, but there's a lot of room for individual various. variation and expression within it. Well, I was going to ask about popular music. I mean, I presume it seems to me that in the modern world with technology, et cetera, and like you said, globalization, there are more concurrently vibrant forms of popular music maybe than ever before. We have rock and jazz and folk and blues and country and hip-hop.

And can you think about the differences between these in terms of the scales and the transmissions, in the toolkit that is used? I know they also have sort of thematic and instrumental differences as well as rhythmic ones. Yeah, absolutely. And this is a big focus of my of my second book.

So the first substantive chapter, chapter two of my second book, is all about rock harmony. And there's a kind of, one thing we haven't talked about too much and is difficult to talk about in a podcast context, but there is an inherent geometry to hierarchical transposition.

So if you say, okay, start with a major chord, and you can do two things to it. You can move the notes up along the major scale, major chord, sorry. So that's thing one. And the other thing you can do is you can move your chord down chromatically or up chromatically. And you say, those are your only two. moves, right? That's all you get. You can go up and down along the chord or you can go up and down along the piano keyboard and that's it. You can actually write down a geometry of those two moves and it's a kind of two-dimensional spiral geometry. And you could go to my website,

which is called madmusicalscience.com. You can Google, you can look at my second book. It's got all this stuff. It turns out that there's this spiral shape that that organizes those two kinds of hierarchical transposition. And then if you look at rock idioms and rock harmony, there's a lot of really common progressions that are kind of moving, they're basically moving by minimal distances inside this spiral geometry. And so a lot of rock harmony comes from this geometry, which is being manifested in intuitive considerations like,

okay, I want to sing a melody that moves downward by small distances, and then I want to put chords under it. So my melody is, right, three blind mice. What are my choices? Well, you know, that's one of my choices. There's just really not that many choices in there. They're modeled geometrically in a way. that maybe reveals a new perspective on this intuitive knowledge that humans have.

And that spiral, I want to see this is going to be ambitious. Like you said, it's hard on a podcast. I want to visualize the spiral to our audience members. So basically, you know, the movement within the pentatonic scale looks like we're skipping several notes at a time on the chromatic scale. so I can sort of go around in a circle and come back to where the next note. I think you've got a different spiral. No, no. The wrong spiral.

On this spiral, in my spiral models, each point is a chord. And so this is actually, so you're thinking of maybe a spiral where each point is its own note. I was. Right? And that's really a circle. So the kind of trick here, which relates what I do to stuff you think about, is to exploit the notion of what mathematicians call a configuration space. So instead of thinking of a chord as a bunch of points on the piano keyboard,

we're going to think of the chord as a single point in some higher dimensional space of musical possibilities. And these diagrams actually date back hundreds of years. So you can find 17. 18th century music theorists who are drawing pictures of keys and then eventually chords and that kind of thing. And those pictures are pictures of musical configurations. So if you wanted to draw the spiral diagram of rock music, it would be you sort of start at 12 o'clock on the clock, draw a circle but don't come back to where you

start, go around to the outside once, twice, and then on the third time around, or I guess the fourth time around, you join up back with your original starting point. And then you place your cords evenly spaced along that three-fold spiral.

And then you can kind of move around on that space and generate the patterns of rock harmony. And just to repeat listeners who are curious about this, you can go to my website and I have a little web app that will draw the thing and you can click on it. And I don't know if you have the capacity to include links, but we could. I will definitely include links and I will definitely vouch for the fun of playing on Demetri's website. There's a lot of fun things going on there.

I can't help but mention. I don't know if you've ever seen the interview with Leonard Bernstein where he's talking about the Beatles. And he says, you know, he's doing one of of his, he did these educational, maybe he's not an interview, he's doing these educational shows, and he's trying to say nice things about the Beatles. He's like, you know, yes, they're, there's kind of simple and they're emotional, but his view, his version of saying nice things was they have some interesting chords that I as a classical musician recognize. And I, I kind of had, even though I love Leonard Bernstein and the Beatles separately, I kind of had the feeling he was missing what made them good.

Yeah, you know, he's a funny figure. I think he was maybe a little bit trapped between different musical cultures. He's a great musician. So, okay, in the 20th century, there is this phenomenon of modernism. And modernism is, you know, in music, it manifested itself by the desire to create your own musical language. and it often manifested itself by a valorization of complexity. Okay. And I think one of the truths about music is that simple is often better.

And so you know, with Bernstein, you see this, even when he's talking about Beethoven, there's a wonderful passage where he's talking about Beethoven. He says, what makes Beethoven great? You know, is it the chords? No, they're simple. Is it the melodies? No, they're simple too. Is it the rhythm?

Yeah, simple. And, you know, he doesn't even consider the fact that maybe people like Beethoven because of the simplicity because he paired things down to this just incredibly powerful. Visceral, yes. Yeah. I mean, and I would say, you know, he clearly did think that the Beatles were great musicians. And at various points, he even says, you know, I take more pleasure from the Beatles than from all of notated composition. that's a big thing for someone like him to say. I would say if you asked me who invented rock harmony,

who invented the, you know, we were talking about these simple ways of moving around the spiral and creating these chord progressions. I mean, I basically think the Beatles either invented it or brought it to the world. You know, you can certainly find examples before the Beatles, but the Beatles were, these unbelievably

brilliant musicians who discovered a very natural way to organize music, one that had weirdly been maybe under-exploited by classical composers. And they just, they created

a musical language that I think underneath the differences between, you know, rock and hip-hop or electronica, the harmonic language, even movie music, our harmonic language was really pioneered by the Beatles. Well, it's one way of being a genius is to invent something incredibly complicated and unique.

But another way is to figure something out that in retrospect just seems natural and inevitable, right? But you were there first. Yeah, exactly. And you can look at Einstein, you know, even, you know, mathematical geniuses like Grotendik, right? People are, there are people who just kind of cut to the essence. Everybody else is adding too many. frills and making things too complicated. And, you know, Einstein did what if the Lorentz transformations just replace the galerrean?

Yeah. Yeah. And so, okay, I want to ask, though, about the, you had the spiral in the rock music. Does that mean that there's just like a separate country music spiral and a separate jazz spiral? Yeah. I mean, essentially, yes. So, for instance, we talked about that there's the, what I call the three and 12 spiral of a, major chords in a

chromatic scale and you get progressions like right so that's your kind of if one of the interesting things is if you look at two-note chords inside the seven-note diatonic scale so we're going to we're going to take that diatonic scale and we'll just think of the major

third C-E right and you ask well what is the what's the analog of these motions that produce the Beatles chord progressions or the rock chord progressions, you get things like, which is really a very basic classical music paradigm, right? And so you get the, you know,

and there's a variety of things. There's even, there's a way, a slightly more complicated way of moving around this space that I think Beethoven was particularly fond of and makes a very Beethovenian sound. I'll just play it for you, without explaining it. You can kind of feel the Beethoven juices flowing, you know,

and it's just a geometry. It's just a path through a space. So, yeah, I would say, you know, it's not quite like, okay, this geometry gives you this style, but I would say that there are different styles have different characteristic harmonic moves, and it is sometimes the case, maybe often the case that these characteristic moves live in different geometrical spaces of possibility.

And the really exciting idea here is that you sometimes find similar patterns of motion in these different spaces producing very different musical results, but there's a kind of unity or universality underneath them. And so you're sort of, there's the hints of a deep structure or a deep grammar that's uniting these superficially different musical styles. You know, in both physics and biology, we have the idea of a landscape, like a fitness landscape, right? There's a million things that can happen, but some of them are either lower energy if you're a physicist or higher fitness, if you're a biologist. And I've often wondered if there's a similar thing in music where there's, like, there's like, Like one way of doing it sounds good, this very different way of doing it also sounds good, but halfway in between it would sound bad. Yeah.

I mean, I would say two things. First, the metaphor of the fitness landscape is kind of similar to this idea of the configuration space, right? And so that whole way of thinking where we do really have fitness landscapes in the sense of like all the possible three note chords creates a. kind of fitness landscape. And some of those three-note chords are pretty crunchy, and some of those three-note chords are pretty, like, nice sounding. And so literally, the space of chords of a certain size is a fitness landscape when you think about each point as having a kind of pleasantness quality or something like that. But then, yes, in a more general way, you can think of a fitness landscape that's kind of grouping together many different features of a style.

And I guess I would say is right. What an interesting way of thinking about the paradox of, of 20th century music is that in the late 19th century, we found ourselves at the peak of a little mountain and fitness space. And then Debussy heard this gamel on and he sees a peak at some other. The whole mountain range way over there. But then everybody sort of set off. They left their little peak looking for the other peaks. And yeah, maybe the space turned out to be a little bit more mountainous than they thought.

And it took longer. I think there was a kind of optimism that you could just declare, okay, my musical style is going to work like this. And you sort of, you know, by just willing it to be so, you could make really awesome music. And maybe that didn't work out quite as well as people had hoped. Well, does thinking carefully about the geometry, the topology, the hierarchical sets of transpositions, does it suggest entirely new ways of making music that are not rock or classical or anything like that? I mean, if you had asked me this 15 years ago, I probably would have said no, and I probably would have said that what it shows you is that our fitness peak is kind of the only one in the neighborhood. And so I think in my first book, I gave the metaphor of, you know, 100 years ago, we thought that there were tons of habitable planets.

and it would be just really easy to get into rocket ship and go to Mars and live there. And it might be more like the Earth is the only habitable planet, and so we have to maybe not take it for granted. Okay. So one way of thinking about it is like in its really basic, if you really get very abstract and think very general, there's not an infinite number of easily accessible, really effective musical styles out there. I would say that, so this is what I would have said 15 years ago.

I think in the last 15 years, my thinking has changed a little bit. And I would say between my first book and my second book, I've started to realize, like, okay, if we think about, okay, moving along a chord and moving along a scale, what if we take some just different chord and try to move along it? I actually think that that is a pretty fertile territory combining motions, hierarchical transpositions, motions along these different alphabets,

but choosing alphabets that are totally new, you can find possibilities there that people haven't used that the ear responds to. So I think I'm becoming more enthusiastic about how thinking theoretically might lead to new musical possibilities that are actually enjoyable as opposed to just intellectually interesting. Good, because I was going to ask, Like as the working composer as well as music theorists,

like are there times when you write something down and you go like, eh, that's just this again. That's just this move. Debussy did it or Mozart did it. You know, come on. Yeah. Well, okay, I would say the really, I mean, yes and no, right? They're stealing and nobody wants to steal, right?

So it is very frustrating as a composer to feel like, okay, I can imitate Mozart by like writing down Mozart stuff, right? That's not rewarding. On the other hand, if you are thinking abstractly enough so, okay, oh, I see Bach doing this really kind of cool thing or I see Stravinsky do this really cool thing, I'm going to take the general idea, but I'm not. going to take his chords, I'm not going to take his sounds, I'm going to take what he's doing and translate it into a different rhythmic realm, into a different harmonic realm. That's not frustrating, right? So it doesn't feel frustrating, oh, I'm using scales again, you know, like that, right? That's like, oh, I'm speaking English again. Oh, I'm using hierarchical transformations, hierarchical transpositions,

oh, I've got some spiral diagram. Like, that doesn't feel frustrating because it's easier to accept that that's just the character of this giant fitness, you know, mountain range that we're in. So there's a real difference between imitating the sound

and the superficial characteristics of another kind of music versus, you know, working with the same underlying concepts and organizational strategies, but doing it with new instruments, with new rhythms, with new sounds, with new chords. That doesn't feel frustrating at all. You've definitely made the point that a lot of this knowledge, or at least partly, it's intuitive, even in people who are very good musicians, but are not musically trained to read the notes or whatever. I mean,

does someone like John Coltrane or Jimmy Hendricks, when they're improvising, are they implicitly taking advantage of these structures, or is it just feel? I know we've already talked about this a little bit, but now you've gone through some of the background and we can understand the answer even better. Yeah, they're implicitly taking advantage of these structures, is what I would say. I mean, someone, well, and even, so John Coltrane's pianist, McCoy Tyner, I know his playing pretty well. I've transcribed some of his solos.

I've thought a lot about how his sound arises, right? He's an example of a musician who I have loved since I first heard him. And it took me a really long time to figure out how to make his sound where it's coming from. With someone like Tyner, I just, there's no question that he was very, thoughtful and you look at the kinds of relations that exist in his music. And it's clear he developed a kind of musical system. You know, in terms of how much he could have explained it to you in scientific or theoretical language, I honestly, I suspect he could have because I just look at this organization and it's like anyone who's always writing haiku can tell you that a haiku

has five than seven than five syllables because otherwise there's just no way to always end up with five seven five right so whether he would have told it to you is a kind of interesting question partly because there's a little bit of the magician's guild thing and then nobody wants no musician wants to be thought of as a math nerd so like there's basically no benefit in in telling anyone your secrets I know a little bit about you know McCoy Tyner's music The one thing I would say is, you know, it does seem to make sense everything he does, right? It makes sense. It fits in. He's not just completely wandering out. No. And then, you know, I went for about, well, you know, for 10, 15 years while I was trying to figure out what was going on, I would ask all these jazz musicians that says, how do you make that McCoy sound, right?

And they would say various things that were never helpful. I would go. They say, play a lot of pent. Pentatonic skills. And you go and you play pentatonic skills and it doesn't quite work. And even some people would, I would ask someone, they'd say, oh, well, you make it like this. And they'd sit down and they'd play it. And it would sound like McCoy Tyner. You know, so there are people who can do it themselves. But again, you say, all right, well, what, what are you doing? And they maybe aren't as good at answering that question. Yeah, there's an incredible rigor and logic that you can hear.

And you can hear the rigor and logic in the Beatles. You can hear the logic, but you can't necessarily understand what you're hearing. And that's really cool. But you've been sitting not only at the piano keyboard, but at the computer keyboard quite a lot. I mean, there's an obvious question about can we put this on a computer and either, does it help human beings compose? Or does it free human beings from the need to compose? Could we just give it a formula and ask it to play some variations? Yeah.

So, I mean, no one knows, right? We definitely, we're getting, we've got these giant AI models that can produce bad but passable pop songs. So, you know, given a pop song, it's not always clear whether it was made by AI or actually. A mediocre pop song, yes, right. So I think that, I think machines are going to be, let me just back up and say, I think in the 21st century, music making is going to be done cooperatively with machines much more than it ever has before. So I think that composers are migrating away from writing scores with pen and paper to using something that's called a digital. audio workstation. These are programs like garage band or logic

or pro tools that allow you to combine notation with improvisation with audio. Hollywood has basically switched from scores to the DAW. And I think, actually,

my student, Brendan Zellickman and I, we have a little prototype program called Harmonia that is a little bit like a DAW, but it allows you to define musical objects like scales and motives and to work with them using some of the processes that we have described. And I think if we find a way to make that intuitive, you could have machines, you know, you could offload some of the musical calculations that would take

you a week to learn how to do at the piano or at the guitar, could offload those to a machine. and that might actually be useful. There's also the possibility that if you give machines the right structural model for how music works, then it'll be easier for them to find the underlying syntactic structure in existing music. So, yeah, I am broadly optimistic about the use of technology in music making. I also think one thing I've started to notice, and I also totally understand, is the increasing mechanization and AIification of everything is also driving a hunger for people who make music without any machines whatsoever. And the idea of just some people with acoustic instruments, you know, singing their song as a reprie from the iPhoneification of everything. that's also, I think, going to be something we see more and more of.

But I personally, I like using technology, and I think that it can play a positive role in creative music making. I mean, I absolutely can imagine that part of the future being that real human created and performed music is a special thing that we value even more than we ever did before. It would be great if we could use some of these technological advances to also democratize. music making and rather than replacing human creativity, let more people be creative. I don't think that's always the way technology is first deployed, but I can choose to be optimistic that it might go that direction. Yeah, that's absolutely right. And one way in which I particularly hope that it could help is, you know, classical music performance can be awfully rigorous and sometimes even brutal, right? You have people like

practicing for hours, playing exactly the notes that are written, the scope of, the scope for play and interpretation and transformation is relatively limited when you compare it to jazz or more participatory musical

styles. And so the idea of, of you know, a pedagogy where you say, okay, here's this complex motive in this Bach piece. Let's, you take it and you transform it and move it around, you know, chord and scale in your own way. And you do that as part of learning how Bach transforms his motive. Or maybe you take a different motive and transform it

in the way the Bach transforms his motive. And so, or you make apps that allow, you know, kids to explore musical possibilities, you know, maybe even without fully understanding what the different transformations are, but getting these, these operations in the, you know, their ear, I do think that that could lead us to maybe a more democratic and participatory musical culture. I love ending on an optimistic note, and that was a great one right there. Dmitri Tomasco, thanks so much for being on the Weinstein podcast. Thank you.

It's been a lot of fun.