The Concept of a Configuration Space in Music

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Music can be seen as the unconscious exercise of mathematical powers, combining logic and emotion. The concept of a configuration space plays a significant role in music, where points in space represent collections of notes. Moving from one chord to another is akin to evolving from one state to another in a fitness landscape. Fitness landscapes in music have complex geometrical features and can have non-Euclidean properties. Modeling musical objects such as chords can result in cool geometrical spaces with singularities at their boundaries. These spaces are analogous to fitness landscapes and can be navigated by musicians. The idea of a configuration space applies not only to music but also to areas such as wallpaper patterns and tiling patterns.

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.

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Podcast theme music by Mitch Mignano.

Mentions and additional resources:

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

You can explore his interactive music software at MadMusicalScience.com

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)

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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