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 santafe.edu/engage. You can find the complete show notes for every episode, with transcripts and links to cited works, at complexity.simplecast.com.Thank you for listening!Join our Facebook discussion group to meet like minds and talk about each episode.Podcast theme music by Mitch Mignano.Follow us on social media:Twitter • YouTube • Facebook • Instagram • LinkedInMentions and additional resources:All of Tymoczko’s writings mentioned in this conversation can be found on his Princeton.edu websiteYou can explore his interactive music software at MadMusicalScience.comThe Geometry of Musical Chordsby Dmitri TymoczkoAn Information Theoretic Approach to Chord Categorization and Functional Harmonyby Nori Jacoby, Naftali Tishby and Dmitri TymoczkoThis Mathematical Song of the Emotionsby Dmitri TymoczkoThe Sound of Philosophyby Dmitri TymoczkoSelect Tymoczko Video Lectures:Spacious Spatiality (SEMF) 2022The Quadruple HierarchyThe 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 GroupFoundations and Applications of Humanities Analytics Institute at SFIShort explainer animation on SFI Professor Sidney Redner’s work on “Sleeping Beauties of Science”The evolution of syntactic communicationby Martin Nowak, Joshua Plotkin, Vincent JansenThe Majesty of Music and Math (PBS special with SFI’s Cris Moore)The physical limits of communicationby Michael Lachmann, Mark Newman, Cristopher MooreSupertheories and Consilience from