← Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas
Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas
332 | Dmitri Tymoczko on the Mathematics Behind Music
01:21:11
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- [00:00]
A
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 Thelonious Monk, the famous jazz composer and pianist who said all musicians are subconsciously mathematicians. I learned about the existence both these quotes from today's guest, Dimitri Tomosko, 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 and has been explored in great details. But 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. Dimitri Tomosko is a musicologist, a composer, performer, a professor at Princeton, and. And he's been thinking very deeply about the relationships between mathematics and 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 User's Manual is another one. Thinking 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.
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It.
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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 Tomaszko, welcome to the Mindscape podcast.
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Thank you very much. Happy to be here.
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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 a correct theory of music like there is in physics? Are we theorizing about what happens, or are we searching for better and better music via theorizing?
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B
Yeah, that's a really great question. I'd say that probably people disagree about that as well as all the substantive issues they confront. My own belief is that human beings 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, 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.
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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.
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B
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 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, 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.
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A
Well, 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?
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B
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. And there's a. There's a long and. And not that glorious history of. Of theorists, 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. This idea of a composer as an explorer, I think is an. Is an interesting idea. I would. 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.
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Right.
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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. And if you. If you press the wrong one, so you have. You have 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.
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Right.
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Like, 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. 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. 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.
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A
I think that makes sense. I do remember the moment in my life when I figured out that improvisers in music weren't just randomly going or even that they had no idea where they were going. They know what key they're in, they know a whole bunch of possibilities, and they're making conscious choices along the way.
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B
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 Mozart. It's basically what every saxophonist did with Charlie Parker. Right. And so the thing that creates a challenge is what do you do if you want to, you know, you want to do something really different from. From a previous model, and they're 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 a Haydn 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. And so that the. The earlier tradition of, like, finding a teacher and copying them stopped seeming attractive.
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A lot of weird things happened in the 20th century, and we're still trying, still seeking our way in some sense. But. Okay, let's go back way earlier in 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 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?
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Right, right. So, yeah, I mean, we start with a continuum which, 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, to some smaller collection. And, and this does not seem to be true of animal music. So we use, 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 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.
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Can I but 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?
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B
Yeah. So, you know, this is not my area of totally unfair, but I have recently 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. There are birds that will maybe chirp back and forth between two notes. You know, 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.
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You know, there's 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.
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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 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 amount of signaling in the animal world, right. And, 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, 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.
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Fair enough. Okay, back to the scale.
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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, 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, that the pairs that sounded good were, 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, they formed the perfect fifth and four to three, five to four. And, 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 abstract things. And then from that we get Platonism and maybe even Christianity, if need is to be believed.
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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 frequency, that's a very obvious connection there. But then there's sort of these vaguer connections. Three to two, five to four, whatever.
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B
Yeah, exactly. And so, so the simple, one simple story about where scales come from is basically you, you start with one note that's a middle C. And then you kind of add to it, say the three over two, right? And then maybe you add to that the three over two of the three over two. Or you could add a four over three 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 P. Piano keys. Right. So one story about scales is they come from you. 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, and when they're too close together, it becomes very hard to. It becomes very hard to sing.
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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.
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Oh yeah, I love that. I think like one of the things that's really cool about, about music is it has this abstract mathematical quality to it. You know, if you listen, if you listen to Bach or 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, 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, that is rewarding in a different way.
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Okay, so tell me about the chromatic scale. I'm dividing up the octave into 12 notes, right?
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Well, we've started with five.
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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 to just start with the five and build out well.
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B
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 can go to every continent. If you told me about a human culture I'd, 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, 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, it's in China, it's in India, it's in the West.
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So, so is that the diatonic scale? Am I getting names right? I'm very bad. Okay, good.
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No, diatonic is excellent. Diatonic is kind of. Some people would call it the major scale. But, but major means those seven notes with a particular note chosen as what we would call the tonic or the, 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.
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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?
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Yeah, I mean, you know, we do have a concept of a non octave 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 scale.
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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.
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It does have a cma. Well, the first three notes are not quite. It has a C major chord within 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, and then you've got, you've got your pentatonic scale. If you do it seven times, you get the diatonic scale. And then to answer your earlier questions, question, if you do it 12 times, you get a chromatic scale.
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Good. That's actually super duper helpful because it answers my actual question, which is why those seven notes in the diatonic scale? I mean, why the major scale. And somehow you just give me a little mathematical formula for making it.
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B
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 1550 or so, there were people who argued for, for keyboards that had 19 tones per octave. And they're like, 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.
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Times.
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So that 3, 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 has, which is just generated by the 3 over 2 ratio, to this compromised diatonic scale called the just intonation diatonic scale, where you have, you have 3, 3 over two ratios, F, C, G, D. And then you also have some five over four ratios, ce, FA and gb. 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, that don't exactly line up well.
- [26:45]
A
And the chromatic scale would seem to be the worst offender. 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.
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B
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 instruments. So what happened is around. Well, mathematically this happened around the time of Bach, so 1720, 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, 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 tuner 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 close to three over two. It's. The difference is about 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 3 over 2 ratio. It's not a very good approximation to the 5 over 4.
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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.
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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.
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I was going to ask, like, are they good enough to do that? Is like, how. How precise is the distinction?
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They are good enough to get better than the piano keyboard. No question. And some people specialize in this and they're amazing.
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Yeah.
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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 3 over 2 and the 5 over 4 ratios, you can drift 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, 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 Aronov effect, where maybe you go around a solenoid quantum phase. Yeah, yeah, yeah. So. So there can be these that 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.
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We should just mention for the audience that Dimitri has written a book called the Geometry of Music. Is that what it was called?
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It's actually called A Geometry of Music. I didn't want to be hegemonic, but.
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It'S all about, you know, unearthing the geometric structures in a very advanced, highly non trivial way behind these journeys through the various scales and musical constructions.
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And actually I've written a second book called Tonality and Owner's Manual, which kind of maybe is a little more hands on in terms of like how some of these phenomena mass manifest themselves in, in rock music or in Beethoven.
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A
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, like 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.
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B
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, 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 3 over 2 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, it. 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, and essentially so we call those, those simpler vibrations, we call them partials or overtones. And essentially the 3 over 2 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 3 over 2 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. You know, there are lots of things that vibrate inharmonically. And for those instruments. So if you go to, if you go to Indonesia and you listen to a gamelan orchestra, they are using non inharmonic bell like Metallic instruments. And they tune their gamelans in very different ways from the ways Western instruments are tuned. And my friend Bill Sathairs has written a wonderful book about this called Tuning Timbre 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.
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A
And I take it that 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 the frequency, three times, whatever. Whereas a bell is a complicated shape and its higher frequencies are not going to be simple multiples.
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B
Yeah, exactly right. And even if I can. Just like if you wanted to get into the physics of it, when you said a string is the simplest possible shape, you're really talking about an ideal string, right? Yes, 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 2 to 1 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.
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A
So annoying when the messiness of the real world gets in the way of my spherical cows. That made everything easy to understand.
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B
Well, again, that's like. That's both the blessing and the curse of music is that the mathematics breaks down maybe earlier 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, this is a cool idea, but how does it actually sound?
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A
Well, as you. Yeah, with exactly that in mind, I have to mention the fact that there is 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 a different emotional resonance to us?
- [37:04]
B
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. Is play it a lot. Right. I'm just gonna do that as a demonstration. Right. So first I'll play the white notes, and I'm just gonna really, just randomly play white notes. But I'll play. I play a lot of Cs. 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. Phrygian. We call it the E Phrygian. Okay, so here is. Here are the white notes, just with a lot of Cs. And here are now basically the same selection from the piano keyboard, but with a lot of ease. 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. You're getting different collections of notes relative to that origin.
- [38:23]
A
And. And it sounds different. I mean, it's just like the. The physicist in me wants to say it's just an overall scaling. How could it matter that much? But our bodies care.
- [38:33]
B
Yeah. And I mean, one way to think about it. Okay. Is we're. 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.
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Right.
- [39:08]
B
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. People experience that as being sad. And, and so it's connected to low affect. Right. When you're sad, you're not, you know, your, Your. Your vocal contour is maybe a little bit suppressed. And, and, and so it might be something like that.
- [39:50]
A
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 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?
- [40:21]
B
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, and these are all different ways of taking a pattern and, and, 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.
- [42:12]
A
Because the scales are embedded within each other.
- [42:16]
B
Exactly. So you can, you can say, okay, I'll take my ceg and I'm going to take that up one, up one pattern and I'll move it to egc. Okay, that's not, that's not sophisticated. Okay, but then I can take that EGC 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.
- [43:08]
A
Yeah, you're actually writing music.
- [43:12]
B
What's tricky and what's complicated is that the earth 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, that transformation from this much smaller Alphabet.
- [43:42]
A
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.
- [44:03]
B
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. Of the first thing.
- [44:36]
A
Right.
- [44:36]
B
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.
- [44:59]
A
And you can sort of trace how we collectively discovered these different kinds of transformations through the history of Western classical music.
- [45:08]
B
Yeah, basically. Basically, that discovery starts happening maybe sort of like something like 1550-1750. 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 Josquin, didn't not compose at the keyboard and were not keyboardists. And then you have a shift. People like Bach, you know, they are all Mozart, Haydn, they're all keyboardists. So they start to notice these possibilities.
- [45:53]
A
And so if I were Bach, JS Bach, how conscious would I have been about what I was doing? Or was it just, you know, precognitive genius?
- [46:05]
B
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's. He's 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. Of what's going on. And, you know, Bach in particular, just. He loved math and turning things upside down.
- [46:51]
A
Yeah.
- [46:51]
B
You know, he was, he. He really was a kind of mathematician of notes. And, and so I, I bet that they were. There's more consciousness than, than we realize. It's still, though, 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 use totally different chords and scales from the ones, you know, he would just, he wouldn't even know how to start because all of his knowledge was his scales, his key.
- [47:29]
A
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, like, by the time the 20th century came along, we had sort of run out of new places to go in the. On the musical scales.
- [47:48]
B
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, to come up with a point in time at which this became really obvious, it's the 1889 World's Fair in Paris where Debussy hears a gamelan orchestra from Indonesia. Right. Gamelan 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, it's becoming easier to. Globalization is starting to get underway. And so, so Debussy hears this, this, this very strange music and he realizes, oh, my God, you know, all the stuff I've learned in school, it's. It's more like it's conventional.
- [49:11]
A
It's like it's not the only way. Yeah, yeah.
- [49:14]
B
It's not the on. So then, so, so the question. It's not so much that, 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 in the early times.
- [49:36]
A
Yeah.
- [49:36]
B
He sees these African masks and realizes, wow, there's a power in other cultures ways of doing art that, that my culture is not tapped into. Neoclassical 19th century French art doesn't have something that these African artworks have. So I would say there is a tradition of talking about exhaustion and the Western ways of doing things, running out of power. But I see it more of a, 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?
- [50:29]
A
And there does seem to be a difference between 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.
- [50:48]
B
Right. Well, I think a lot of that has to do with the fact that Western music is notated. And so notation becomes the primary vehicle for transmitting Western music really early on, you know, around 1100 or something like that. And so the idea, and it's also connected with the, with the fact that Western music early on is bound up with the church and you have these particular hymns and you know, it's part of religious ritual. So 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 it's, 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 is 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 variation and expression within.
- [52:36]
A
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, the scales and the transpositions and the, the toolkit that is used? I know they also have sort of thematic and instrumental differences as well as rhythmic ones.
- [53:12]
B
Yeah, absolutely. And, and this is a big focus of my, of my second book. So the, the first, this, 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. So and you say those are your only two moves.
- [54:07]
A
Okay.
- [54:08]
B
Right. That's all you get. You can, 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 madmusical science.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 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. 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.
- [55:46]
A
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, 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.
- [56:12]
B
I think you've got a different spiral. Oh, 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.
- [56:27]
A
I was right.
- [56:29]
B
And that's really a circle. So the kind of trick here which relates what I do to stuff you think about is 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 started. 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 chords evenly spaced along that threefold 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.
- [58:18]
A
I will definitely include links and I will definitely vouch for the fun of playing on Dimitri'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 his. He did these educational. Maybe it's not an interview. He's doing these educational shows and he's trying to say nice things about the Beatles. He's like, yes, they're 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 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.
- [59:04]
B
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, you know, with, 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? 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.
- [60:12]
A
Yes, yeah.
- [60:13]
B
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. You know, that's a big thing for someone like him. Say, I would say, if you ask me, who invented rock harmony, who invented the, you know, we were talking about these, these simple ways of moving around the spiral and creating these chord progressions. I mean, I, 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, 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 what, electronica, the harmonic language, even movie music, our harmonic language was really pioneered by the Beatles.
- [61:31]
A
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.
- [61:43]
B
Yeah, exactly. And you can look at Einstein, even mathematical geniuses like Grotendieck, 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, what if the Lorentz transformations just replace.
- [62:05]
A
Well, there's no ether at all. Yeah, yeah. And so, okay, I did want to ask though about the. You have the spiral in the rock music. Does that mean that there's just like a separate country music spiral and a separate jazz spiral?
- [62:18]
B
Yeah, I mean, essentially, yes. So for instance, we talked about the, there's the, what I call the 3 and 12 spiral of 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 gonna, we're gonna take that diatonic scale and we'll just think of the major third, C, E. Right. And, 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, 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, it's just a geometry, it's just a path through, 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, there's the hints of, of a deep structure or a deep grammar that's uniting these superficially different musical styles.
- [64:51]
A
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 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.
- [65:15]
B
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, and some of those three note chords are pretty crunchy and some of those three note chords are, are pretty like nice sounding. And so literally the space of chords of a, of a certain size is a fitness landscape. When you, 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. An interesting way of thinking about the paradox of 20th century music is that in the late 19th century, we found ourselves at the peak of a little mountain in fitness space. And then Debussy heard this gamelan, and he sees a peak at some other.
- [66:35]
A
Mountain range way over there.
- [66:37]
B
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. You could make really awesome music. And maybe that didn't work out quite as well as people had hoped.
- [67:14]
A
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?
- [67:29]
B
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 100 years ago. We thought that there were tons of habitable planets and it would be just really easy to get in a 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 generally, 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. And I would say between my first book and my second book, I've started to realize it like, okay, if we think about, okay, moving along a chord and moving along a scale, what if we take some, you know, some just different. Some different chord and try to move along it? I actually think that that that is a pretty fertile territory. Combining 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.
- [69:51]
A
Well, good, because I was going to ask, as the working composer as well as music theorists, Are there times when you write something down, you go like, ah, that's just this again. That's just this move. Debussy did it or Mozart did it. Come on.
- [70:07]
B
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. 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 doing 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. 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. But doing it with new instruments, with new rhythms, with new sounds, with new chords, that doesn't feel frustrating at all.
- [71:55]
A
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 Jimi Hendrix, 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. Better.
- [72:25]
B
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. I've thought a lot about how to. How his sound arises. Right. His. He's an example of a. 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. And you look at. You look at the kinds of relations that exist in his music, and it's clear he developed a kind of musical system. I. 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, then seven, then five syllables, because otherwise there's just no way to always end up with 5, 7, 5. 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 there's basically no benefit in telling anyone your secrets.
- [74:09]
A
I know a little bit about McCoy Tyner's music. The one thing I would say is it does seem to make sense. Everything he does. Right. It makes sense. It fits in. He's not just. Just completely wandering out.
- [74:21]
B
No. And then, you know, I went for. For about while, you know, for 10, 15 years, while I was trying to figure out what was going on, I would ask all these jazz musicians, I said, how do you make that McCoy sound? Right? And they would say various things that were never helpful. I would go. They said, play a lot of pentatonic scales. And you go. And you play pentatonic scales and it doesn't quite work. And even some people would. I would ask someone, they'd say, oh, will 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. 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.
- [75:19]
A
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. Can we just give it a formula and ask it to play some variations?
- [75:42]
B
Yeah, so, I mean, no one knows, right? We. 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 mediocre pop song.
- [76:04]
A
Yes. Right.
- [76:07]
B
So 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 GarageBand or Logic or Pro Tools that allow you to combine notation with improvisation with audio. Hollywood has basically switched from scores to the daw. So. And I think actually my student Brendan Zellickman and I, we have a little prototype program called Harmonia that is kind of 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 could offload some of the musical calculations that would take you a week to learn how to do. At end, the piano or at the guitar, you 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, is the increasing mechanization and AI ification of everything is also driving a hunger for. For people who make music without any machines whatsoever. And the idea of, like, just some people with acoustic instruments singing their song as a reprieve from the iPhone ification 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.
- [78:42]
A
I mean, I absolutely can imagine part of the future being that real human created and performed music as 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.
- [79:12]
B
Yeah, that's absolutely right. And one way in which I particularly hope that. 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 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 a pedagogy where you say, okay, here's this complex motive in this Bach piece. You take it and you transform it and move it around 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. 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 their ear, I do think that, that, that could lead us to maybe a more democratic and participatory musical culture.
- [80:39]
A
I love ending on an optimistic note, and that was a great one right there. Dimitri Tomosko, thanks so much for being on the Mindscape podcast.
- [80:46]
B
Thank you. It's been a lot of fun. Wherever you go, whatever they get into, from chill time to everyday adventures, protect your dog from parasites with Cridelio Quattro. For full safety information, side effects and warnings, visit cordelioquatrolabel.com consult your vet or call 1-888-545-5973. Ask your vet for Cordelia Quattro and visit quattrodog.com.