COMPLEXITY: Physics of Life

Dmitri Tymoczko on The Shape of Music: Mathematical Order in Western Tonality

Episode Notes

Math and music share their mystery and magic. Three notes, played together, make a chord whose properties could not be predicted from those of the separate notes. In the West, music theory and mathematics have common origins and a rich history of shaping and informing one another’s field of inquiry. And, curiously, Western composition has evolved over several hundred years in much the same way economies and agents in long-running simulations have: becoming measurably more complex; encoding more and more environmental structure. (But then, sometimes collapses happen, and everything gets simpler.) Music theorists, like the alchemists that came before them, are engaged in a centuries-long project of deciphering the invisible geometry of these relationships. What is the hidden grammar that connects The Beatles to Johann Sebastian Bach — and how similar is it to the hidden order disclosed by complex systems science? In other words, what makes for “good” music, and what does it have to do with the coherence of the natural world?

Welcome to COMPLEXITY, the official podcast of the Santa Fe Institute. I’m your host, Michael Garfield, and every other week we’ll bring you with us for far-ranging conversations with our worldwide network of rigorous researchers developing new frameworks to explain the deepest mysteries of the universe.

This week on the show, we speak with mathematician and composer Dmitri Tymozcko at Princeton University, whose work provides a new rigor to the study of the Western canon and illuminates “the shape of music” — a hyperspatial object from which all works of baroque, classical, romantic, modern, jazz, and pop are all low-dimensional projections. In the first conversation for this podcast with MIDI keyboard accompaniment, we follow upon Gottfried Leibniz’s assertion that music is “the unconscious exercise of our mathematical powers.” We explore how melodies and harmonies move through mathematical space in ways quite like the metamorphoses of living systems as they traverse evolutionary fitness landscapes. We examine the application of information theory to chord categorization and functional harmony. And we ask about the nature of randomness, the roles of parsimony and consilience in both art and life.

If you value our research and communication efforts, please subscribe, rate and review us at Apple Podcasts, and consider making a donation — or finding other ways to engage with us — at You can find the complete show notes for every episode, with transcripts and links to cited works, at

Thank you for listening!

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Mentions and additional resources:

All of Tymoczko’s writings mentioned in this conversation can be found on his website

You can explore his interactive music software at

The Geometry of Musical Chords
by Dmitri Tymoczko

An Information Theoretic Approach to Chord Categorization and Functional Harmony
by Nori Jacoby, Naftali Tishby and Dmitri Tymoczko

This Mathematical Song of the Emotions
by Dmitri Tymoczko

The Sound of Philosophy
by Dmitri Tymoczko

Select Tymoczko Video Lectures:
Spacious Spatiality (SEMF) 2022
The Quadruple Hierarchy
The Shape of Music (2014)

On the 2020 SFI Music & Complexity Working Group (with a link to the entire video playlist of public presentations).

On the 2022 SFI Music & Complexity Working Group

Foundations and Applications of Humanities Analytics Institute at SFI

Short explainer animation on SFI Professor Sidney Redner’s work on “Sleeping Beauties of Science”

The evolution of syntactic communication
by Martin Nowak, Joshua Plotkin, Vincent Jansen

The Majesty of Music and Math (PBS special with SFI’s Cris Moore)

The physical limits of communication
by Michael Lachmann, Mark Newman, Cristopher Moore

Supertheories and Consilience from Alchemy to Electromagnetism
SFI Seminar by Simon DeDeo

Will brains or algorithms rule the kingdom of science?
by David Krakauer at Aeon Magazine

Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System
by Michael Bank and Nicola Scafetta

“The reward system for people who do a really wonderful job of extracting knowledge and understanding and wisdom…is skewed in the wrong way. If left to the so-called free market, it’s mainly skewed toward entertainment or something that’s narrowly utilitarian for some business firm or set of business firms.”
Murray Gell-Mann, A Crude Look at The Whole Part 180/200 (1997)

Related Episodes:

Complexity 81 - C. Brandon Ogbunu on Epistasis & The Primacy of Context in Complex Systems
Complexity 72 - Simon DeDeo on Good Explanations & Diseases of Epistemology
Complexity 70 - Lauren F. Klein on Data Feminism: Surfacing Invisible Labor
Complexity 67 - Tyler Marghetis on Breakdowns & Breakthroughs: Critical Transitions in Jazz & Mathematics
Complexity 46 - Helena Miton on Cultural Evolution in Music and Writing Systems
Complexity 29 - On Coronavirus, Crisis, and Creative Opportunity with David Krakauer

Episode Transcription

Dmitri Tymoczko (0s): There really was a centuries long tradition of thinking of music as fundamentally a kind of science. So if you go back to the old quadrivium, music is right up there with mathematics and astronomy. It's really sort of shocking and difficult to accept the fact that it might be more like the grammar of a language and that other languages might have a very different grammar. So there's this perception that some part of the rules of traditional Western music are no longer valid, but at the same time, there's real confusion about which rules those are. 


There's this feeling that we have to get rid of the irrelevant stuff, save only the core, expand our horizons, draw the lessons that we can draw from all these other ways of making music. But because so much musical understanding is implicit, this turns out to be a giant and enormously difficult problem. And I would say we are still struggling with that problem today, 130 years later. 


Michael Garfield (1m 29s): Math and music share their mystery and magic. Three notes played together, make a chord whose properties could not be predicted from those of the separate notes. In the west music theory and mathematics have common origins and a rich history of shaping and informing one another's field of inquiry and curiously Western composition has evolved over several hundred years in much the same way economies and agents in long running simulations have becoming measurably more complex and coding more and more environmental structure, but then sometimes collapses happen and everything gets simpler. 


Music theorists like the alchemists that came before them are engaged in a centuries long project of deciphering the invisible geometry of these relationships. What is the hidden grammar that connects the Beatles to Johann Sebastian Bach and how similar is it to the hidden order disclosed by complex systems science? In other words, what makes for good music and what does it have to do with the coherence of the natural world? Welcome to Complexity, the official podcast of the Santa Fe Institute. 


I'm your host, Michael Garfield, and every other week, we'll bring you with us for far ranging conversations with our worldwide network of rigorous researchers, developing new frameworks, to explain the deepest mysteries of the universe. This week on complexity, we speak with mathematician and composer, Dmitri Tymoczko at Princeton University. His work provides a new rigor to the study of the Western Canon and illuminates the shape of music, a hyper spatial object from which all works of baroque, classical, romantic, modern, jazz, and pop are all low dimensional projections. 


In the first conversation for this podcast with mini keyboard accompaniment, we follow upon Gottfried Leibniz’s assertion that music is the unconscious exercise of our mathematical powers. We explore how melodies and harmonies moved through mathematical space in ways quite like the metamorphosis of living systems as they traverse evolutionary fitness landscapes. We examine the application of information theory to chord categorization and functional harmony. And we ask about the nature of randomness, the roles of parsimony, and consilience in both art and life. 


If you value our research and communication efforts, please subscribe, rate, and review us at applepodcasts and consider making a donation or finding other ways to engage with us at You can find the complete show notes for every episode with transcripts and links to sighted works at Thank you for listening. Dmitri Tymoczko, it's a pleasure to have you on Complexity podcast. 


Dmitri Tymoczko (4m 21s): Thank you very much for inviting me. 


Michael Garfield (4m 23s): This is a diversion from the standard fare on this show, but I think that it will nonetheless provide us with a kind of like rotating out of the plane of our normal conversations and we'll be able to take a perspective that hopefully is illuminated by and illuminates the normal episodes of complexity in an interesting way. 


Dmitri Tymoczko (4m 46s): I'll try to be as nerdy and statistical as I can 


Michael Garfield (4m 50s): Originally. I wanted David Krakauer to join us, and he's a little too busy for this call, but I know that the two of you have a long acquaintance, so I'll be sure to invoke him in this conversation. I guess, where I would like to start in the mundane before we take off into hyperspace here with a bit of intellectual biography, because you have charted an interesting and very non-linear course in your career and the life of your mind. 


And I think that giving people a bit of a foundation in who you are as a person will be helpful to understanding what animates the questions that we're going to explore in today's show. So yeah, if you care to offer some background, let's start there. 


Dmitri Tymoczko (5m 42s): Okay. I was born in north Hampton, Massachusetts, which is a lovely little college town in Western Massachusetts. My parents were both professors, which is probably the root of a lot of my problems. I grew up as a pretty standard math nerd. And when I went to college, I went to Harvard. When I went to college, I remember I was very influenced by an older friend of mine. His name is David Bogarts, and he was also, I think, kind of a math computer nerd. 


And I remember he had gone to college and majored in music. And I asked him, I said, well aren't you good at math and physics and stuff like that? Why are you majoring in music? And he said I realized I actually kind of like music, but I don't really love math and physics. And that really stuck with me. And it made me think in my ordinary life, I had never really needed to find the slope of a curve or any of that. Whereas I really  loved music, and that was something that I liked to do just for my own sake. 


And I know a lot of people feel that way about math and science and that's great for them and they should become a scientist. So I went off and I became a musician. I studied music there. This was maybe the very tail end of what I think of as American academic, a tonality. So a lot of my professors wrote this very, very dissonant music that was really, really far from the music I'd grown up with. I'd grown up with progressive jazz rock and new wave and Laurie Anderson and basically cerebral popular music. 


I really liked Bach and Debussy. And this was all music written in the wake of Schoenberg. And I encountered a composer there, Milton Babbitt, who became a mentor of mine. He's a very mathematical, very scientific guy, his music, really, to me sounds essentially random. I have a really hard time understanding what it's doing, even though if you look under the hood, it's filled with these mathematical patterns. 


So while I was there, John Cage came and gave these prestigious lectures called the Norton Lectures. I think you get paid a lot of money. I remember a thousand people packed the room for his first lecture and he just read random words for like three hours. Everybody left except for maybe 15, 25 people. You rarely see 980 people walking out of a lecture like that. So I guess what I'm trying to say is this was an era where people really didn't think of music as something you did with the body. 


I didn't learn anything about jazz. Popular music was not really on the table and we're talking late eighties, early nineties. So I would say there were institutions in America that were not like this. The very prestigious institutions sometimes tend to be the last to change. And so maybe the mindset that prevailed, there was one that was more common elsewhere, a decade earlier. So I then bounced off to philosophy. My dad was a philosopher, so I'd always kind of felt comfortable with philosophy. 


There was a Professor at Harvard named Stanley Cavell, who was a former composer, and who wrote a really cool book called Must We Mean What We Say that included a bunch of essays about sort of both sympathetic and critical essays about modern music. And so I gravitated toward him. He was another real mentor figure for me. And so to cut a long story short, by the end of my college career, I felt a little more comfortable expressing myself philosophically in words than in notes. 


I was fortunate enough to get a road scholarship. I went off to study philosophy at Oxford and I got kicked out. And I had a couple of years in the wilderness wandering around. I became kind of a freelance teaching assistant back at Harvard. I lived in Somerville. I was basically a grad student, except I didn't have to go to any classes. After a few years of that, my father died tragically young, basically at the same age I am now. I applied to philosophy grad school. I applied to Yale Law School and I applied to music composition grad school, and I got into music and I got into Yale and I had this dark night of the soul where I tried to decide between becoming a lawyer and becoming a composer. 


And I feel good about that. I can tell you how I made the decision because I'm proud of it. But I went off to Berkeley. I started composing. I started doing music theory. Berkeley was still a kind of conservative place in terms of the styles of music that they accepted. But I was annoying enough to argue with that. People mostly left me alone. And so then I was fortunate again and got a job at Princeton. They basically leave me alone. And I spend my time now about half and half between making music and thinking about how music works. 


So I'm a composer, but I'm also a music theorist. And I'm definitely, I think I'm pretty influenced by the same sorts of ideas that a lot of the more regular guests on your podcast are influenced by. 


Michael Garfield (10m 57s): Oh, no, there's no question. And one of the things I'm hoping to perform over the course of this conversation is just the webbing back to recognizing the through lines between your work and the work of so many other people at SFI. This is where I feel the real meat of this conversation comes into focus because I actually found you before I realized that there was an SFI connection. I saw a talk that you had given on your work on music and geometry back in 2014, talk, titled The Shape of Music and was just totally floored. 


I mean, unfortunately we don't have your keyboard hooked up. We're going to have to interpolate musical samples in this, but you were talking and you were on the keys at the same time. And it was a beautiful instance of an augmented lecture, which I really enjoyed. And the lecture content was about this revelatory glimpse into this deep unifying mathematical structure underneath the musical patterns that people enjoy and continue to invoke in composition and ways of visualizing these things. 


It had everything for me. It was synesthetic. It lived somewhere in this intertidal zone between music and mathematics and science. And so I'd love to just give you an opportunity to talk about that, about that work. And from there, we can kind of, maybe once you've given us a foundation, the sort of history and current state of thinking about music and geometry, then as you sort of provide that exegesis I hope that we can link out to other ideas from complex systems science and I'll sound things on you. 


I'll bounce them off you and see whether they stick. 


Dmitri Tymoczko (12m 45s): I should say that I am at a disadvantage in the podcast format since a lot of what I have to say not only combines the languages of music and math, but also combines the visual system and the auditory system. So really in some sense, it's a question of translating between what you hear and what you see. And of course the podcast is essentially limited in that department. So your listeners are going to have to imagine really amazing visuals that completely clarify everything I'm talking about. 


Michael Garfield (13m 19s): And then they will go to the show notes and they will watch the talk that you gave at the 2020 SFI musicology and complexity working group and the other videos up there. So this will be more an appetizer, I guess, for that stuff. 


Dmitri Tymoczko (13m 34s): An hour long commercial, really. 


Michael Garfield (13m 36s): The goal is to take it somewhere that your talks and writings may not necessarily have already gone. So let's aim for that. 


Dmitri Tymoczko (13m 45s): So I would say that if I had to back up, I would start with the fundamental problem, which is that musical knowledge is very often a kind of implicit knowledge. So I believe that musicians are generally speaking really smart people who know just an awful lot about music. I remember one of my piano teachers when I was a kid just describing Bach and Mozart as these incredible intellectuals who express their intellectual reality by making music rather than by writing down theorems. 


What I would say though, is that a lot of times this knowledge really isn't something that musicians can translate very easily into word. So a great musician understands music very deeply, but does not necessarily have the ability to express that knowledge in anything other than musical form. And that's really kind of a wonderful and cool thing. There's no reason to feel bad about that, but it does sometimes create problems or obstacles or difficulties. 


I would say if you wind the clock back to maybe 1890, 1900, something like that, you're coming off the tail end of 300 years of just glorious Western musical history. The tradition that starts about with Corelli and goes through Brahms, and it's just really an amazing cultural achievement. The people from that culture were quite aware of music as one of the profound contributions that their culture had made. At about 1900 there starts to become first of all, encounters with non-Western music. So Agomallon visits Paris as part of the 1889, world's fair and Debussy sees it and just has his mind blown. He thinks, wow, there's all these other kinds of musics that are out there. It's kind of a similar experience to Picasso encountering these African masks and just realizing that the aesthetic of European painting is just one small part of the world of possible art. 


So around 1900, there starts to get this widespread impression that the traditional rules of Western music, they no longer have the authority that they once had. So some of them it is gradually realized are just these kind of local cultural conventions, like ways of dressing. They're not laws of nature. You have to understand. There really was a centuries long tradition of thinking of music as fundamentally a kind of science. 


So if you go back to the old quadrivium, music is right up there with mathematics and astronomy. So the idea that music is somehow a scientific pursuit, this is a very deeply held belief in Western culture. It's really sort of shocking and difficult to accept the fact that it might be more like the grammar of a language and that other languages might have a very different grammar. So to make a long story short, there's this perception that some part of the rules of traditional Western music are no longer valid, but at the same time, there's real confusion about which rules those are. 


So there's this feeling that we have to like get rid of the irrelevant stuff, save only the core, expand our horizons, draw the lessons that we can draw from all these other ways of making music. But because so much musical understanding is implicit this turns out to be a giant and enormously difficult problem. And I would say essentially, we are still struggling with that problem today 130 years later. 


Michael Garfield (17m 39s): So you're definitely bringing themes that I want to focus on into focus here. And one of them is, again, that question that was in the working group, at least in the first session of the working group, back in 2020 at SFI on what features of the structure of music are indeed universal and which are specific to various cultural contexts. And I see you dig into this extensively in your work. And for me, these questions of syntax and grammar are what connect, what you're doing at least on one level to what got me into complex system science in the first place, which was the work that David Krakauer was doing with Martin Nowak and Joshua Plotkin on the evolution of syntax and the evolution of a sentence based human communication format in the first place and thinking about it in mathematical terms. So a bit of historical background on given that you gave this talk recently at SCMF on visualizing music's harmonic structure and the relationships in mathematical spaces of different dimensions. A little bit of context there historically, and then how your work builds on that, I think would be a good ramp into this. 


Dmitri Tymoczko (18m 60s): So I would say before I get there, I would like to give a little bit of context that I think is important. And one thing that I would say is there a sort of two questions that we can ask? And one question is about universal musical features, what features of music are truly shared by all forms of human music making. For me, I'm not super interested in that question because there's just a lot of different kinds of music out there. And if you include the music of the 20th and 21st century, you have music, John Cage's music of silence. 


And so true search for universals is going to leave you with something that's quite thin. I would say that a lot of my work is a little bit more specific in that the question that I'm most interested in is something more like what are the core features of Western music as we know it for the last 500 years. So music stretching back to the Renaissance and composers like Shoshscan and stretching forward to the present day and including popular music and jazz. 


And I do think there are some features of that music structures that characterize that music that are not universal in the deep sense, but are widespread enough that they sort of constitute our musical world or some substantial portion of our shared heritage. And that's really the question that gets me going is sort of thinking about the very general grammatical structure of this particular branch of the world's music making. 


And it is one that the west has exported over the last a hundred years in the sense that you can now go to all sorts of different countries. You can go to Korea, Japan, India, and you will, especially in the popular realm, you will hear a kind of popular music that is playing with Western ideas. So that's the question that really animates me, sort of what constitutes Western tonality in the most abstract sense. And I would say that there are different ways of answering this question and in a way, some of the sexiest and most interesting answer the question using geometry and by providing these sort of higher level models that allow you to visualize relationships, but at a more prosaic and more useful level a lot of what I do is just thinking in a kind of very basic way about how to make precise, the kinds of intuitions or the thoughts that lots of musicians have. I do think that one of my strengths is not being afraid to look dumb and being willing to sort of say things that on the one hand seem kind of obvious, but on the other hand, maybe because they're obvious or because they're wrongly dismissed as obvious haven't got as much exploration as they might otherwise have. 


So one thing that we could talk about is just breaking down some of the basic features that distinguish tonal music from non-tonal music or some of the basic features that make Western music what it is. 


Michael Garfield (22m 21s): I definitely do want to do that because at some point in this conversation, I want to bring up the essay that you wrote on bebop and free jazz for transitions. In the one conversation that we've had on this show that lingered on jazz, the conversation I had with Tyler Marghetis, that's the anchor point here for me where it turned out that there was a kind of deep mathematical structure, if not in the atonal experimentation of a free jazz ensemble, then at least in their ability to coordinate like  an emergent collective behavior, like the way that they moved through this space together. 


And so again, this sort of like the differences and sameness here in all of that, I find very interesting. 


Dmitri Tymoczko (23m 7s): The place I think we should start is just with some really basic musical concepts. And so in my first book, which is called the Geometry of Music, I laid out what I consider five really basic components that tend to produce a sense of tonality, which for many, but not all listeners corresponds to pleasing this. So it's probably useful to go through those properties and that will sort of set the stage for how geometry enters the picture. 


So the first of the properties has to do with just the difference between consonance and dissonance. So it's just sort of a basic fact about human beings that we tend to find certain sounds pleasing and certain sounds less pleasing. Some maybe pleasing, not pleasing. This is one of the qualities, one of the adjective pairs we use to describe it, but restful and active is another one. And so the most obvious example is the octave, which is typically considered the most restful stable of the possible combinations of notes you can make followed by the perfect fifth and then the major third. And then you can put these things together in various ways, you can add a fifth and a major third to get what we call a major triad, or you can do that same structure upside down to get a minor triad. You can build out a whole harmonic vocabulary, adding together different sounds and different musical styles, sort of demarcate different regions of this space of sounds as their sort of basic vocabulary. As a general rule, Western music has evolved toward increasing complexity. 

So it started with fairly sort of simple sounds. So that's a kind of early medieval-ish sound and then harmony got richer and richer and richer and richer. And this eventually led jazz on the one hand, which uses very complex harmonies all the time, but also a tonality which uses very dissonant. So if you want, you can make music out of sounds like this. 

And a huge amount of 20th century music has been devoted to exploring the potential in those kinds of dissonant sounds. So our first property, that is just a sliding scale between what we might call the restful soothing pleasing sounds and the more energetic dissonant aggressive sounds. I think it's very important not to think of this as a moral continuum. All of these sounds are great. They all have musical uses. Maybe a way to think of it as the difference between sweet food and really spicy food.

So there are people who love to eat super spicy food, even though it causes them pain and makes them sweat and throw themselves on the floor. And then everybody likes a little bit of spiciness, but different people kind of get off the boat at different points in the chili pepper continuum. 

Michael Garfield (26m 28s): And again, I don't want to jump ahead too far ahead in our conversation, but mathematizing these relationships and then the melodic paths that are taken through harmonic spaces in music as you articulate them and others, it has me wonder about things like activation energies and chemical reactions and the distances between points on evolutionary fitness landscapes that certain trajectories are paths that seem theoretically possible in the way that it's theoretically possible for information to escape a black hole. 

It's not practically viable. And so you've demonstrated some caution in your statements already in this call about trying to avoid this sort of mystical pre-modern music as science kind of transcendental thing. But it is curious to me that there may be reasons why, again, there's like in spite of the notable differences between the distinct musical grammars that have evolved around the world, the different aesthetics that there may still be even so a set of deeper underlying unities. 

Dmitri Tymoczko (27m 41s): Well, that's the hope, I think there's some truth to that. I just want to say like, no one should ever say the words practically viable to a musician. Especially someone who writes string quartets in the 21st century. That is definitely not a practically viable enterprise and making art in the world of Trump is a pretty quixotic project. And I'm a big defender of people following their own mysterious and beautiful and strange paths. 

One of my favorite composers is a physician named Conlon Nacarrow, who wrote all this sort of wonderfully crazy music for the player piano in Mexico City in the fifties and sixties. And that music is extreme, it is weird, you know, probably 90% of the people who hear it, it sounds like a player piano that's out of control and rolling down a hill or something like that. And that is really extremely non-practically viable music. 

And yet I find an enormous beauty and also in the thought that someone would dedicate their entire life to this really quixotic quest. So there's sort of two poles. On the one hand, some features of music may be rooted in features of our biology. And so you depart from them at your own risk. On the other hand, I really am a believer in the imagination and in what I guess you could call perversity. There's a lot of what gets people excited is in various ways unnatural. 

And I am a big supporter in not trying to constrain the imagination. So this is something I struggle with. I'm always trying to, on the one hand, be faithful to my like scientific interest, my theoretical interest, my trying to figure out how do these different building blocks of musical structure? How do they fit together? On the other hand, I don't really want to be contributing to a kind of police force mentality or some kind of oppression. And this is very tricky. It's a very tricky balance to draw. 

And I do think about, right now we're in an era of sexual liberation and people are doing all sorts of stuff that was considered unnatural and frowned upon 70 years ago. And we've learned, nobody gets hurt as long as you let people follow their bliss and do what they want. Age of the internet has created these niche for people who like very strange music, very strange food, very strange hearts of all sorts. And so I am a big believer in that. I think the way I solve this dilemma for myself is I'm a believer in consequences. 

So if you want to write totally nasty, strange music with like the weirdest harmonies, I am your biggest backer. But if you then complain that only a small number of people like your music, that's maybe where I start to get off the boat. So you do sometimes have this maybe I think someone like Arnold Schoenberg, he was quite aware that he was a great, great composer, but he would often complain about how few people liked his incredibly dissonant and gnarly music. 

And that I find a little bit that is maybe a place where I can help and say, look, if you're dedicated to writing music with the nastiest sounds you can think of, then you have to know that that's probably going to have some consequences in terms of your record sales. And as long as you're aware of those consequences and prepared to pay the price, then all power to you. 

Michael Garfield (31m 11s): So actually, you know, there's a talk, I'll try to dig up and link to in the show notes, an interview with Murray Gell-Mann about the challenge of funding, fundamental, theoretical science, like the kind that is practiced at SFI and how truly novel work is often commercially invisible. This is something that we talked about with Lauren Klein when we had her on the show to talk about bringing techniques from the digital humanities to bear on surfacing, invisible labor in history, a data feminism approach and how that's, why, this show is itself kind of peculiar in that it is a vehicle for the diffusion of work that in some respects can only be understood in retrospect, in history, in terms of its significance, like Sid Redner’s work on papers that go unrecognized for 50 years, and then suddenly there's a bloom of citations. 

This is like, it's recognized as formative work for an emergent discipline. It was ahead of its time. And so I hear you that one way to think about this might be that there is this extremely intricate landscape of possibility to explore whether in the arts or in the sciences. And there are a lot of nasty, difficult, hard to climb roads on that landscape. And there may be great reward on some of those roads, but it's harder to find resources to support a pilgrimage to a destination that cannot be seen, along a treacherous pathway. 

And in that rate, that gives us a way to kind of fold this back into again, the way that you think about the geometry of harmonic spaces and of the paths that music can take through those spaces. 

Dmitri Tymoczko (32m 57s): I know you're trying to get me back on track but this is the place where I have to say that Murray Gell-Mann was mean to me when I visited SFI first time, somewhere around 2009. He said, isn't all this musical geometry, just group theory that I rediscovered. And I said, no, it really isn't. I feel that for the record, it's important to get that down there. So we're talking about what are these like basic properties that contribute to musical pleasure? One of them is just consonance and dissonance and the intrinsic pleasure we take in some sounds and intrinsic tension we hear in other sounds and essentially balancing those, in an interesting way. 

Another very simple one and here we go back to me, not being afraid to say dumb things as what I call conjunct melodic motion. So a lot of times when we make a melody, we tend to move around by small distances on the key. So that is a melody, whereas something like that's not so a lot. And so a lot of people have written music that hops all around the keyboard all the time, but that they are sort of deliberately thwarting the melodic impulse, another feature. 

And by the way, if listeners are interested, they can type what makes music sound good into Google and maybe add my name and you'll find some cool audio demonstrations that you can play around with to explore these features. Another feature that I talk about as being particularly important is what I call harmonic consistency. So this is, let's just say a chord is a collection of notes that happened at one time in music. And even a lot of musical styles, cords tend to sound similar to one another. 

So all of these cords are in some sense, the same they're transformations of each other we could say. Now other musical styles use very different kinds of harmony, this kind of thing. Maybe you have complex styles like jazz, that kind of thing. Maybe you have very atonal stuff that these, by the way, your listeners need to know him playing with one hand. 

So because of how my keyboard is so little bit, I have one hand tied behind my back literally. What you don't tend to find is music that mixes these different sound worlds kind of indiscriminately. So, something like that almost sounds funnier. Sounds kind of pointless. So harmonic consistency is you use a collection of chords that somehow feel like they belong together. 

And there's a few more of these. If you go to the website, if you want to explore, you can go and figure them out. I don't think I'm going to go through all of them here, but the point is just these dumb little properties they're actually enough to set up a math problem because it turns out that it's not exactly automatic about. On the one hand, you can make little melodies that move around by short distances. And at the same time, these melodies combined to make a sequence of harmonies that sounds similar. That's enough to make a math problem and actually ask the question, well, what are the conditions under which we can have these properties that make music sound pleasing simultaneously? And there, it turns out that the answer to the question involves geometry in a kind of essential and surprising way and hooks into some of these deep themes of 20th century science, which involve, I would say geomatrizing knowledge of various sorts. 

Michael Garfield (37m 9s): In fact, I liked the way that you wrapped up a piece that you wrote called This Mathematical Song of the Emotions, which will link to where you said music can be as livenets said “the unconscious exercise of our mathematical powers and uncanny mixture of logic and emotion.” And again, so that's about, explicated what you said, you know, the implicit knowledge, the embodied knowledge of music. And so here again, like I want to make sure that we linger and unpack this for people that what you've done in your work over the last, at least the last 15 years is show how musical harmonic structure looks for people that are familiar with this podcast alot, like what we were talking about with Brandon Ogbunu a couple episodes ago when he was talking about John Maynard Smith's protein space and the paths through random letter strings and how you can find these paths that are actually maybe practically viable, isn't the right way to talk about it among musicians, but like paths that are functionally related and therefore there are ridges from one part of this abstract space to another part of the abstract space that emerge from this naturally. 

I'd love to hear you talk about the way that you're building on like previous efforts, like the circle of fifths, for instance, in your work to provide a robust approach to visualizing these high dimensional spaces. 

Dmitri Tymoczko (38m 48s): So I think one thing we can say is that one of the deep concepts of 20th century thinking is the concept of a configuration space. And this is the idea that you have some geometry, some space where a point doesn't represent, say a planet flying around in a solar system. It doesn't represent an animal moving through space. Instead a point represents a complex configuration. 

So you use the word fitness landscape. What is a fitness landscape? Well, fitness landscape is a term we use to refer to a configuration of properties that jointly make up our fitness or our ability to reproduce. So the concept of fitness landscape is fundamentally a concept of a configuration space. Configuration spaces play a huge role in 20th century physics. Quantum mechanics is sort of very naturally formulated in terms of these things. 

So in music what this means is that we think of a configuration space as a space whose points represent collections of notes. So instead of thinking of a C major chord as three keys being held on the piano, we think of it as a location in the space of all possible chords. And so then when we're moving from one cord to another, we're taking a path through a configuration space, which is essentially, if you thinking about fitness landscapes, this is kind of like evolving from one state, one organism to another. 

So I would say one thing I did is just take this very common concept of the fitness landscape and just sort of translate it into music. Now, the thing that's interesting in this work, the reason people like to do this, as it turns out that fitness landscapes have complex geometrical features. And when we're talking about fitness, we're saying, well, certain combinations of properties really make you reproduce very well. And so they're like a top of a mountain in the fitness landscape. 

In other situations, the configuration space has sort of interesting non-Euclidean properties. And that's what we find in music. It turns out that when you get serious and you model the geometries that represent common musical objects, like three note chords or two note chords, you end up with these very cool geometrical spaces. The easiest example is that the space of two note cords, the space you can play with two fingers on a keyboard. If you think of them as chords in a very natural musical way, this space is a kind of Mobius strip, furthermore, which has singularities at its boundary and these spaces, they sound very cool. 

They use relatively recent mathematics, but they are very common spaces. They're the geometries that arise when you're modeling unordered objects, that the geometries that arise if you're modeling wallpaper and motion along the wallpaper. So what's interesting here is there's a package of ideas, all sort of surrounding the notion of a configuration space that have allowed us to think about the interesting mathematics of things like wallpaper or basket weaving patterns or tiling patterns that just sort of applies very naturally to music. 

And so, yes, you can think of musicians as moving around in these higher dimensional configuration spaces, which are sort of analogous to fitness landscapes and finding their way around these spaces, musicians, intuit this knowledge, usually in terms of fingers on a keyboard or some other instrument, but we can now given where we are intellectual history, we can translate that knowledge into all sorts of other forms. 

Michael Garfield (42m 42s): So I wanted to double back for just a moment. And one thing I really want to explore with you is, as you mentioned earlier, that there has been this increasing complexity in the harmonic palette of Western composers over the last several hundred years. This is something that several people discussed in the 2020 SFI working group video series that we'll link to. So there's a point, and this is a point that you mentioned, in this transition to 20th century work and the getting to a point where we've got people like Mahler that are really branching out and exploring this in unprecedented ways, and then things flip and then you get, like you said, you get all of this eight tonality, you get Schoenberg, you get free jazz. 

You kind of compare that to stuff like Jackson Pollock in the visual arts. And I've heard musicologists talk about how they saw a relationship between that shift in the focus of artists with respect to, and we brought this up in the Tyler Marghetis episode, it seems kind of synchronous with challenges to the narrative of a kind of ongoing March of progress that was presented by the world wars and the awakening of Western society to nonlinear effects in complex systems and the cybernetics and all of this stuff. 

I'm being very sort of hand-wavy here. It got interesting for me because you have this presentation. But could the Martians understand bar on syntax and epistemology and elsewhere also in talking about your mentor Milton Babbitt, you say that “following the relationships in a Babbitt composition might be compared to attempting to count the cards in three simultaneous games, a bridge I'll play it in less than 30 seconds. Babbit’s music is poetry written in a language that no human can understand. It is invisible architecture. The relationships are out there in the objective world, but we cannot apprehend them.” And so I guess what I'm trying to do here is elicit your thoughts on whether the relationship between tonality and atonality or between signal and noise. Whether that distinction really is a kind of like ontological distinction. Another way to think about this in complex systems in a language is whether randomness really is a thing or whether it is merely contingent on the observer, because we know from work like the physical limits of communication in 1999, that came out of SFI that optimally encoded alien communication would be indistinguishable from black body radiation, for instance. 

And so, you know, is the movement into eight tonality and sort of off of the playing board of is more ordered musical spaces that you're exploring in much of your work, is that really a shift in kind, or are we really doing what we have always been doing, which is pushing the frontier of tonal exploration into areas that for which we lack the appropriate sort of Bayesian priors to the conditioning to recognize a pattern. 

Dmitri Tymoczko (46m 1s): Okay, well, there's a lot in what you just said. We could probably teach an entire graduate seminar at Princeton, with one class on each of your sentences. 

Michael Garfield (46m 12s): Sorry about that. 

Dmitri Tymoczko (46m 13s): That's all right. No, it's great. I'm going to maybe go backwards a little bit and okay, you know, you picked one of my favorite quotes that Babbitt quote that I wrote is one of the favorite sort of couple sentences I've written. First thing I want to say is there's no such thing as a tonality, there are many different things that are a tonality. And the eight tonality of free jazz is really, really, really different from the atonality of Milton Babbitt. And one thing I would like to do is get people to understand the similarity between maybe Schoenberg's early music and free jazz because those seem somewhat similar to me. I would say that Milton Babbitt's version of a tonality is sort of, is very apotheosis of a very deep feature of a certain strand of thinking about music in the west. And that is a fear of the human and a fear of the body and a fear of just how simple music can be. So there's this long trend. I think it probably goes back to Kant maybe historian of philosophy could probably trace it back further, but there is a real trend and it really comes up in music of trying to take all the fun out of it one way or another, and to turn it into a purely cerebral exercise of our minds. 

And the fact is a lot of what makes music really powerful and enjoyable is kind of animalistic and non-cerebral. We really like to hear just like that bass drum beating regularly and to move our bodies in time with the music. And we really just love relatively simple, repetitive musical structures. One of my favorite examples, song I talk about a lot is Helpless by Neil Young, just got a melody with basically three notes and just three chords and it just repeats over and over. 

And it's a really great song. And so what I would say is there is a fundamental philosophical fear of music out there and a tendency to valorize those features of music that look the most like math and to downgrade those features that look like dancing or look that are sweaty. You know what I mean? And so the thing about Beethoven and Bach and Mueller and all those guys is they've got the sweat and they've got the expression and they've got the drama and the screaming and all of this like really earthy human stuff. 

And they also have the math and that's why this music stays with us. And that's also true of Charlie Parker and Art Tatum and John Coltrane. And it's also true of the Beatles. And it's true of a lot of the electronic dance music you can hear now. So I think the great music always has this math quality and this athletics quality. Milton Babbitt's music is maybe the very apogee of a centuries long tradition of asking what happens if we strip away all of the athletics, all the sweat and all the sport out of music. 

And we try to make something that is as much like math as possible. And in fact, the technical procedure of composing the way Milton Babbitt composed is really very much like Sudoku only instead of nine numbers you use 12, but if you want to know what he was doing, he was essentially doing incredibly complicated Sudoku puzzles and then translating them into sound with no regard whatsoever for whether people found those sounds to be intrinsically pleasurable because that's this animalistic and sweaty and for him irrelevant question. 

So you see now why I loath to associate that with free jazz. Free jazz can be very much about the body, very spontaneous, very sweaty, two people get together and let's just play music without any preconceptions. That can be what free jazz is. And so maybe on the surface, they're both dissonant they’re both atonal, but they're coming from very different places. I would say that the stuff I'm interested in is coming up with ways of thinking about the cerebral quality in music that are maybe a little bit demystifying. 

There is this structure and Bach and Beethoven it's there. It's cool. It contributes value to the music, but it's not so amazing and so special that we have to let it sort of eclipse all the other wonderful features about that music. So paradoxically understanding those aspects even better, maybe can help us demystify them a little bit and put them along on the shelf alongside other musical values that are less cerebral, but equally important. 

Michael Garfield (51m 10s): So thank you. I mean, maybe what I'd like to do is just actually take a tiny slice of that question out of that giant manifold of possible questions and just again, to cite talks that I've seen David Krakauer give on his work to find a general theory of intelligence. He's talking about the algorithms that someone might employ to solve a Rubik's cube, for instance, as you know, an actual sort of concrete physical instantiation of all of this conversation about abstract mathematical hyper spaces, that this is about finding a short path from one point to another through these spaces. 

So I'm curious if you think that the increasing harmonic complexity of Western music suggests that we are collectively getting smarter about identifying the structures in this musical hyperspace better at pattern recognition, generally doing an expansion of the training data. When people say, oh, it's an acquired taste. The world that we live in now, again, to call on, you know, Stuart Kauffman and his phrase, the adjacent possible, which comes up in the show a lot as I'm sure you can imagine, that the adjacent possible in a world of so many different technologies and languages and designed and undesigned affordances, that possibility space is much greater. 

And so there's a pressure on the individual and collective intellect to have the vocabulary and syntax to explore these spaces. And so it seems again that both in the ways that we've already interrogated, tonal quote unquote and atonal quote, unquote, music, both evolve as an instance of this much more general process of the evolution of intelligence broadly. And I'm curious what your thoughts are on that. 

Dmitri Tymoczko (52m 55s): Yeah, that's a great question. So one of the topics I want to circle around to is how experimentation manifests itself differently across the different arts. So you talked about Pollock versus Cecil Taylor or someone like that. So I want to come back to that, because it turns out that the different arts, I think are cognitively situated in different ways. And maybe as a sort of corollary to that, there's also a technological component to how experimentation manifests itself. 

So if you think about what happened to drama broadly construed, well, movies transformed it bBecause they gave a possibility that wasn't present prior to 1900 or so. Even something as simple as changes in printing technology made the comic book possible somewhere around 1900. And so suddenly we have that as a new art form. In music, the gigantic technological change that really changes everything is the invention of recording technology somewhere around 1920s, basically the same thing as the movies, but for sound. 

So that is really a radical change here in the following sense.Prior to 1900, if you wanted your music to be remembered, there was one way to do it. And that was to write it down. And so what that meant is people would not remember the specific way in which you sang a melodic line. They would not remember all of the things that couldn't be written down. They would remember your notes and rhythms. And so within that technological regime, you can see a kind of increasing complexity over time. 

But it's important to remember that there is a back and forth. I think of it a little bit like the ecological parable of the wolves and the rabbits. So the rabbits get more populous and then that's more food for the wolves so that you get more wolves and that makes less rabbits and then the wolves die off. So the topic of complexity exhibits, something of that dynamic where you've got the listeners and the composers and the composers are always wanting to get more complex and do stuff that's more interesting for them. And the listeners are willing to go with them up to a point, but eventually the composers go too far and someone else, a new style shows up that's simpler and grabs people's attention. 

So you have people like Johann Strauss, who's writing waltzes that are much simpler than the music of Wagner. You have Satie, who's deliberately exploring simplicity. Historically there's a repeated trend of people getting mileage out of new and simple kinds of music. You know, the minimalists, Terry Riley, Steve Reisch, Phillip Glass, the minimalists filled this role within the realm of notated composition in the 20th century. The thing that happens in the 20th century alongside that is recording technology suddenly allows everyone to bypass writing things down. 


You can bypass the conservatory, which was, you went to the conservatory and they told you how to write down music. And along the way they indoctrinated you with all sorts of other aesthetic prejudices. So suddenly anyone who can get time in the recording studio can make their music, can sell music, can make their music remembered. Furthermore, the recording studio is able to capture all those nuances of performance that were lost in notation. So the precise way in which Louis Armstrong sings suddenly becomes part of his musical personality.


So this is a force that really just totally transforms our perception of music and our relation to music. And it also totally is a giant force toward vernacular styles and is a force of simplification. And also suddenly it puts classical composers in competition with vernacular musicians in a way they never were before because suddenly people have a choice about, do you, by yet another recording of Beethoven's Fifth, or do you buy the latest Benny Goodman album or whatever? 

And so I think you can say that Western culture has essentially voted with its feet for non-notated music in the sense that the vast majority of music that is now consumed the most popular styles of music have their roots one way or another in some kind of vernacular tradition, folk music, rock music, jazz, that kind of thing. 

Michael Garfield (57m 26s): May I just interrupt to ask you your thoughts on the work of people? I don't know if you're familiar with like Keith McMillen of McMillen instruments, but his entire career in designing new musical software and hardware has been to try and unify those two domains. If you are a musician and you're listening to the show, you may be familiar with MPE, which is this new format that allows several more dimensions of expressivity into the transcription of a musical performance in Mitie into this digital recording format that doesn't actually capture the music in the sense that you're talking about musical recording, but creates in a digital record, nonetheless. 

And now we have all of these additional dimensions to it. And so even the musically illiterate can perform on a digital keyboard now and leave a record that captures all of that vernacular stuff that can then also be played back on different instruments or by different musicians in a way that preserves all of that stuff. And so it seems like there's a kind of a dialectic going on here and that we're on the cusp of a synthesis. And I don't mean to distract you, but I feel like that's interesting to me, the way that these things might be swirling back together. 

Dmitri Tymoczko (58m 40s): Well, sure. I'm super interested in that. I actually do a lot of programming myself. Some of the projects I'm most proud of involve taking these complex theoretical spaces and making them into something like a video game or a musical instrument, or maybe halfway between those. And actually when I was out at SFI, I did a concert that involved some of this work. I mean the idea that maybe instead of a piece of music, you create an environment, a software environment, and then your audience, instead of passively listening to your piece of music, they navigate paths through your environment like playing a video game, but instead of accumulating points, they make cool sounds. 

This is, I think an incredibly, incredibly interesting opportunity is a very 21st century way to think about music. If we're talking about new musical instruments I think the big one that I would mention is what's called the digital audio workstation or the Daw as we call it in music land. These are software programs like Protools or logic, or there's a million of them. And basically what they are is they are a software generalization of multi-track recording tape. 

They are essentially a form of notation that allows you to represent music visually only instead of like little dots on a musical staff that represent notes you actually have little pictures that represent they can represent notes, but they can also represent actual recordings, found sounds the sound of a cat yowling, whatever you record, you can put in there. And then you can manipulate them visually and create music in that way. And I think it's very, very important for academic musicians, for teachers, such as myself to recognize that the Daw is a kind of notation system and rather than prioritizing, I mean, it's great to learn how to read music. 

I'm a big supporter of reading traditional musical notation, but there's a tendency to sort of valorize that particular form of notation because it happens to be centuries old. Whereas the Daw is a 21st century form of notation that is the native language for a whole generation of musicians. And it itself is sort of creates opportunities that traditional notation. I mean, that's the like 800-pound gorilla in terms of, I mean, I make weird little software instruments that, and people who are out on the front lines, redesigning piano keyboards, that's a niche activity, but like the digital audio workstation, you've got tens or hundreds of thousands of users every day sort of. 

And then that representation really is something very special and something very new. And it's one reason why you hear so much complicated, popular music now is you can create these incredibly elaborate textures and loops and patterns and just are non-repeating patterns just by dragging around little pictures of sounds on your Daw. And it's limited only by the amount of time you're willing to put into it. 

Michael Garfield (1h 1m 47s): So this may be a good place to touch back on your essay on the end of jazz and the way that you argue in this. I liked this, that you argue that bebop is in some ways kindred to classical music, precisely because of its intentional inaccessibility, its emphasis on the learned ear and the learned instrumentalist the effort required to perform and to appreciate this music critically and that there is pretty significant change that's been discussed widely about exactly what you're talking about, about the fact that a child can learn to use one of these digital audio workstations on an iPad. And when I think about this stuff, it reminds me of the conversation I had with, I bring this up all the time. I'm sorry, folks who are regular listeners of the show, but episode 29, when I was talking with David Krakauer about mass extinctions and how, sometimes the systems get so to kind of speak casually about this, so Baroque, an ecosystem is full of all of these very fragile interdependencies and subject to indogenous shocks just through the surprises created by a system like that. 

You think about like the nature of technology and how innovation always outpaces regulation. And so, you know, when we talk about the proliferation of when you mentioned these moments like with, I think you said with Strauss, that we kind of reached this crescendo of music of harmonic complexity, and then things pull back and become simple again, that there are these like waves that remind me of the talk that Simon DeDeo I gave at SFI a few years ago on a very similar process going on in science itself where every 150 years or so we see this rise and fall of confidence that there will be a vast unifying consilience theory of everything into which we can sweep all of these seemingly disparate domains of scientific knowledge. 

And then we have a moment where like the scale of the information available to us, outpaces our ability to make sense of it and things collapse back into a conceit of pluralism. I think about the way that the rhetoric around the unification of the worldwide web in the nineties has been tempered by the challenges wrought by connecting everything to everything else. And so I'm curious whether you find these kinds of analogies I'm making to hold any water. And then what it means in terms of understanding the evolution of composition in the west and  the processes that kind of drive that 

Dmitri Tymoczko (1h 4m 31s): I'm going to start by talking about bebop. So I think that article you're quoting is maybe the first or second thing I ever published. So it's been a long time since I've thought about that. So bebop is one of those great examples of the wolves and the rabbits where bebop is the place where jazz goes from being the music, a kind of popular music to being an elite taste. And I love that music. I really very deeply respond to that music. 

I think in a thousand years, people will remember that music the way we remember Bach and Beethoven. So I'm a big believer in it. At the same time, it does take a little bit of work to be able to figure it out. There is a great book that I just wouldn't got off my shelf called Bebop, A Social and Musical History by Scott Deveaux. And he makes the case. He says, bebop was valorized by people like Alan Ginsburg as this deliberately lawless incredibly free sort of music, but that's not really the way to think about it. 

He said, you have to understand that the people who made this music were really consummate professionals, unbelievably talented musicians who practiced all the time. And he sets this music in the context of Jim Crow and the incredible kind of oppression of African-Americans in the 20th century. And before and after, of course, and he says, look, imagine you are like a really smart intellectual African-American from Kansas in born in 1920. 

What are your choices? Well, your choices are kind of doctor, lawyer, school, teacher, and musician, right? And I actually think that is something you could say about classical music more generally suppose you're a math guy and you're born in 1710. Well, you might become a math professor, but there's no computer programmer jobs. There's no theoretical biology jobs. You know, there's not even statistician or data scientist or accountant or all these jobs that are an outlet for the people who love math and patterns and stuff like that. 

They're just gone. And so maybe there's one math professor job, but there's a hell of a lot of musician jobs because people need music. And at that time, just pay someone to write some new music rather than pay for a copy of a Corelli book. It's actually easier just to grab a musician out of the local orphanage, pay them to write some new music than to actually pay for, to copy some earlier music. So I think what I'm getting to is throughout history and all the way up, in the 1940s and fifties, music was playing this vital function as being an outlet for a certain sort of quantitative activity. 

And the thing is, is not clear that it needs to play that role anymore because all of those people who really love math and patterns can just go become data scientists or theoretical biologists or whatever. So I remember a charming and slightly, well, I don't want to say heartbreaking, a heart to heart with one of my students 15 years ago. And he said, you know, I love music and I love computer programming and I can get paid really well to do one of those things. And now he works at Google. And I think the other part of this that I would sort of just put on the table is if you spend all your day doing, let's just call it math but you know, math broadly construed and running algorithms and writing programs and doing all that cerebral pattern kind of stuff when you come home at the end of the day, what kind of music do you want to listen to? Do you want to listen to the very mathy music of Bach or Charlie Parker, or do you want to listen to something that, that has really nothing at all like what you do at work and has four on the floor, you know, the bass drum. So you're sort of asking me to comment on this big picture stuff about how we're all getting smarter. 

I basically agree with that. The Flynn Effect and whatnot, we're all getting more quantitative. I basically agree with that. We all spend all our days on computers. And so then you're asking me, what does this mean for music? And it would be nice if the story was while our music is getting smarter and more complex. But I actually think in some ways the pressure goes in the other direction and suddenly the fact that we all spend our days doing math in front of a computer means that at the end of work, when it's time to relax, we might prefer a kind of music that is maybe more balanced toward the drama, more balanced toward the emotional, more directly accessible. 

And again, I'm okay with that. I'm not going to fall into the trap of wishing that everybody practice their piano for an hour a day and played Bach fugues when they came home from work, because that's really not how I feel. I think there's going to be markets. There's going to be cultures that support people who make cerebral music. But I think there are some good reasons for thinking that maybe cerebral music isn't going to play the culturally central role that it did in Vienna in 1810 or something like that. 

Michael Garfield (1h 9m 53s): Interesting, because this episode has given me the opportunity to be in general, because so much more casual than I would ordinarily be on the show. I want to make sure that we pay the piper and that we wrapped us with an opportunity to discuss a specific paper that you co-authored with Nori Jacoby, Naftali Tishbyon an information theoretic approach to chord categorization and functional harmony. Because at SFI, we recently launched this program in the digital humanities and bringing computational approaches to bear on cultural corpora. 

And here you propose that harmonic theories should be evaluated by at least two criteria accuracy, as you say, in the abstract, how well the theory describes the musical surface and complexity, the efficiency of the theory, according to Occam’s razor, and again, to call on Simon DeDeo, this piece I discussed with him on the show back in episode 72, that he co-authored with Zachary Wojtowiczit's on, they used a bayesianapproach, and they broke down how people think about simple explanations and what constitutes the different heuristics that different people apply to considering whether something is or is not a satisfying explanation. 

And they talk about consilience versus parsimony. Elsewhere David Krakauer has written, he wrote a piece at Aon about the relationship between fundamental theory and large predictive models. You know, the relationship between brains and algorithms in science, you know, prediction and understanding. And so this is something that comes up in your work. And I'd love to just give you one more opportunity to talk about how these big complex systems ideas are actually being applied by you and your co-authors in examining comparing, evaluating, different music, theoretical approaches, and then actually measuring them against one another. 

How you're bringing this all in with like a formal quantitative approach.

Dmitri Tymoczko (1h 11m 55s): This actually connects to some work that I'm currently doing with Mark Newman, a great physicist who is very closely associated with SFI. So I would say one of the things that's really interesting about music is on the one hand, it's got all this math, quantitative stuff. It's like the one art that at stem fetishists really like. On the other hand, it really is a culture unto itself. And I would say that the incorporation of basic scientific ways of thinking and basic scientific methodology, that is something that is still extremely controversial within music theory. 

You don't really need a license to go have a theory about music. You can just sort of hang a shingle out, get some students. Maybe if you have an academic job, you can just say, here's how I think music works. And so there's not really a scientific mindset or a culture of debate or critique. So one of the projects that I've been engaged in over the last 20 years or so is big data is not quite the word, I'd call it little data or something is assembling actual digitized musical scores, analyzing them. 

And then kind of doing babies' statistics on the analysis to try to understand things like, how did the tonal language of Corelli and Mock, how did that evolve? When did that happen? Or if you look at, there are many different ways of explaining how the music of the 17th and 18th century works harmonically. And so actually using information theory to evaluate these different kinds of theories. So the paper you mentioned with Jacoby Naftali Tishby we started with my hand analysis of a bunch of different kinds of music. 

And the language I use is very common language where we have basically like 45 different chord symbols that describe all of these different kinds of chords. And then we tried to squeeze that information into a smaller number of categories using Tishby’s notion of the information bottleneck. And it turned out that when you squeeze it into a small number of categories, you get something that's recognizable, you get the categories, tonic, dominant subdominant, which have governed music theory for hundreds of years since Remo and then Remont after him. 

And so what's interesting about that is those categories are usually thought to be psychological categories. Like they describe chords that sound similar, but the machine doesn't know anything about sounding similar. It just knows about sort of coming up with the most parsimonious information, theoretical compression of a signal. And it's really cool. It shows that these theories are actually, they're hooking into something that's not just psychological, but is also has to do with information theory. So the work I'm doing with Mark Newman in many ways, sort of returns to that theme. 

And what we have done is mark is the scientist. He built a hidden Markov model that basically we can feed it in the notes of the Bach chorales in this case is, so these are 371 little Lutheran hymns that are kind of like the epicenter of music theory pedagogy. There's sort of a distillation of the core principles of tonal theory. And they're very easy to analyze so we can feed it in the notes. And then we just say, okay, come up with some probability distributions that are most likely to produce these notes. 

And it comes up with basically major, minor and diminished guards. And then we say, okay, come up with some more probability distributions that compress your chords into two categories. And it comes up with major and minor keys and essentially it analyzes music pretty much the way a human does. And it's a good example of what's called interpretable machine learning. And it basically shows you that all these sort of theories that musicians have proposed over the years, they're really like very solidly grounded in actual regularities that are present in the music. 

So that's really cool. It also turns out that we can like massage the data a little translated into a more recognizable human form and without telling the computer too much about the underlying structure of the music, we can get it to reproduce some very subtle seemingly arbitrary features of human judgment about how this music works. So it might treat one in the same collection of notes. In some case, it would say, well, that is a harmony. In other cases, it would say, ah, that's not a harmony. 

That's just a collection of melodic passing tones. And it does this essentially in the same way that an expert human being would do this kind of reproducing features that you would never believe that a machine could do. So this is an example of a place where science and machine learning and the kind of stuff that normally happens at SFI can really intersect with the meat and potatoes of music, education, and also music scholarship. 

So that's a whole other sort of aspect of exciting work that's going on that requires bringing the two cultures together, getting them to speak each other's language in an interesting way. 

Michael Garfield (1h 17m 13s): Well, Dmitri you've managed to land the flirtation that we've been engaged in here for like the last hour and a half with this question that I like to end it with this. And, and maybe this is the way to leave people appropriately bewildered and awestruck as I feel contemplating this. You say in your Boston review essay on the sound of philosophy about Babbitt and John Cage “Mathematics for all its beauty is about discovering facts and establishing truths while music, despite its complexity is about moving people through the medium of sound.” 

And yet when you're talking about finding these robust statistical relationships whereby these harmonic properties appear to like spill out from machines that have no ears and have never listened to music, and you're hinting at the possibility of actually being able to generate a competent machine music critic, kind of like the pitchfork people are about to have their Garry Kasparov moment here. It makes me wonder again about, you know, just give you one more opportunity to, to wax on just how deeply unified the stuff really is. 

There's, there's one paper that I don't know if you're familiar with this, but I've, I think about this paper a lot. It's by Michael Bank and Nicola Scafetta on scaling mirror, symmetries and musical consonance is among the distances of the planets of the solar system. And here we are back in that sort of Kepler and, you know, the music of the spheres kind of thinking where they're saying it turns out that the ratios of neighboring planetary pairs correspond to musical consonances is having frequency ratios of the major third, perfect fourth, perfect fifth, and minor sixth, that they do a statistical analysis of this. 

And it P value they get is less than one 1000th. So they're saying “it appears that these musical continents is, are pleasing tones, that harmoniously interrelate, when sounded together suggesting the orbits of the planets of our solar system could form some kind of gravitational optimized and coordinated structure.” So we're talking about a completely separate set of constraints and forces generating these geometric relationships then what appeared to be governing the aesthetic principles by which people determine, experience a pleasurable music. 

And yet it's, it's showing up in these strangely well-tempered orbital periods and orbital distances between the planets. And so, yeah, I'm just, just one more opportunity. I, you know, again, we are careful enough now in the 21st century, not to just slump into a kind of mystical fugue with this. But help me square the circle here, not to borrow too much from the language of alchemy. 

Dmitri Tymoczko (1h 19m 59s): Well, I mean, so there's two ways that we could go. When I was at SFI a few weeks ago, we did a little exercise talking about sounds that were memorable for us. And I talked about when I was in college and I was hiking on the Appalachian trail and it was summer, but the mosquitoes would find you while you're sleeping. And I remember, that terrible sound of the mosquito while you're trying to fall asleep and it's in your ear. And then there were like six mosquitoes and they are making a little chord and sort of hearing their wings as this cord, this incredibly dissonant evil cord and trying to huddle in my sleeping bag, even though it's July. And my friend David Bashwiner, who teaches at the University of New Mexico, he said, well, you know, when they're meeting, they align the male and female mosquitoes align their wings. So that instead of like, instead of sort of weird notes, they get to a consonance like that. Those, these same ratios here, you're hearing in, you're talking about with the planets. So, you know, we could talk about the ways in which these music type relationships show up all over in other non-musical contexts or mosquitoes are pretty simple biological organisms. 

The place that I would want to go with this is that I do believe, both live nets and Thelonious Monk talk about music as a kind of unconscious mathematics. And I do believe that the pleasure we often get from music is recognizably a mathematical pleasure. It's a pleasure in hearing certain kinds of relationships, hearing a theme layered against itself, hearing consonance is hearing the transformation of a musical idea. And the thing is that there is this huge need for any musician. 

I don't care who you are. When you sit down at your instrument, you have to have some kind of guidelines. You have to have some thought about where to put those fingers. Improvisation depends on this, but even pen and paper composition, if you want to get somewhere quickly, you gotta be able to do that. And so for Bach and Beethoven, great improvising musicians, for Bill Evans, for Art Tatum, also great improvising musicians for all of these people theory plays this sort of essential and even illimitable role in helping to direct your imagination toward kind of potentially useful musical ideas. 

And so what I take the most pleasure in is the thought that we can actually distill some of these principles. I guess I'm less interested in the planets and the mosquitoes and Bach then I'm interested in Bach and Art Tatum and the Beatles. I'm interested in that kind of deep grammar that links a lot of Western music, even though some of it uses drums. And some has written in Protools, some was improvised on the harpsichord. 

And what's really exciting is the thought that if we do our jobs right, and pay enough attention to the fact that ultimately we're doing this thing for pleasure, we might be able to hit on some principles that are maybe not fully universal, but are widespread and useful enough to be close enough to universal that we won't complain and actually can contribute in a real way to the imagining of new musical possibilities. 

And so freeing ourselves a little bit, I started by talking about intuitive knowledge. The problem with intuitive knowledge is it's often a matter of repeating formulas and idioms. So if we could get some general tools that were freed from idiom that sort of laid bare the deep mathematics of musical communication, then that would be a way in which the scientific and the artistic could come together and help us imagine something truly new. 

Michael Garfield (1h 24m 10s): Well, that's an inspiring place to end this. I mean, we're going to link to your Princeton website as well as So people can play with your interactive software, again, a number of your talks and so on in the show notes. But is there anything else that you'd like to surface for people? 

Dmitri Tymoczko (1h 24m 26s): Nope. I think that those two websites are great and I really appreciate your doing this. It was a lot of fun.  We could do a whole other one, but I think people will enjoy what we talked about. 

Michael Garfield (1h 24m 37s): Well, certainly it's been a pleasure for me to jam with you on these subjects and I hope that you have a wonderful day. 

Dmitri Tymoczko (1h 24m 44s): Cool. I'm hoping to come out to Santa Fe more regularly, so maybe I'll see around there. 

Michael Garfield (1h 24m 48s): Yeah. Thank you for listening. Complexities produced by the Santa Fe Institute, a nonprofit hub for complex system science located in the high desert of New Mexico. For more information, including transcripts research links and educational resources, or to support our science and communication efforts. Visit