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May 31, 2009

The Mathematics of Music at Chicago

Posted by John Baez

As a card-carrying Pythagorean, I’m fascinated by the mathematics of music… even though I’ve never studied it very deeply. So, my fascination was piqued when I learned a bit of ‘neo-Riemannian theory’ from Tom Fiore, a topology postdoc who works on double categories at the University of Chicago.

Neo-Riemannian theory is not an updated version of Riemannian geometry… it goes back to the work of the musicologist Hugo Riemann. The basic idea is that it’s fun to consider things like the 24-element group generated by transpositions (music jargon for what mathematicians call translations in /12\mathbb{Z}/12) and inversion (music jargon for negation in /12\mathbb{Z}/12). And then it’s fun to study operations on triads that commute with transposition and inversion. These operations are generated by three musically significant ones called P, L, and R. Even better, these operations form a 24-element group in their own right! I explained why in week234 of This Week’s Finds. For more details try this:

Yes, that’s my student Alissa Crans, of Lie 2-algebra fame!

On June 11th, Thomas Noll is giving some interesting talks on music theory at Chicago, which will delve deeper into such issues. They’ll even get into some topos theory!

  • Thomas Noll (Escola Superior de Musica de Catalunya), The Triad as Place and Action: a Transformational Perspective on Stability, Thursday, June 11, 1:30 pm, Department of Mathematics, University of Chicago.

    Abstract: Recent transformational approaches to the study of triads are based on group actions on the set of the major and minor triads. A particular music-theoretical interest in this subject is driven by the possibility of parsimonious voice leadings between certain triads. To each triad X, say X = {C, E, G}, (considered modulo octave) there are three triads P(X)= {C, Eb, G}, L(X)= {E, G, B}, R(X)= {A, C, E}, each sharing two tones with X. What distinguishes triads from arbitrary 3-chords is the small amount by which the third tone has to be displaced: In the case of P(X) it is an augmented prime (E → Eb), in the case of L(X) it is a minor second (C → B) and in the case of R(X) it is a major second (G → A). Richard Cohn therefore speaks of the “over-determined triad”, as - traditionally - the music-theoretical prominence of the triad is explained in terms of consonance.

    My pre-talk is dedicated to yet another property which can be added to the list of over-determining decorations of the triad. This property provides a conceptual link between (the discussion about) Hugo Riemann’s concept of consonance on the one hand and the Neo-Hugo-Riemannian transformations P, L, R as mentioned above, on the other. This approach is based on a transformational investigation of the intervallic constitution of the triads. Each triad is studied as a sub-action of a monoid action of an 8-element monoid on /12\mathbb{Z}/12. Each transformation is a Twelve-Tone-Operation (an affine endomorphism of /12\mathbb{Z}/12) which stabilizes the triad in question and which extrapolates an association of an internal interval of the triad with its fifth. With this approach I hope to make a contribution to an abandoned discourse between Hugo Riemann and Carl Stumpf and in particular to an elaboration of Stumpf’s concept of the triad as a concord of consonances.

    Mathematically the approach is an application of elementary topos theory. The Neo-Riemannian transformations P, L, R can be studied as equivariant maps between monoid actions (i.e. as arrows in an associated topos). The structure of the sub-object classifier and its Lawvere–Tierney topologies allow to draw links between the different qualities of tones in the complement of a triad as such, and the roles of these tones as images of proper triad tones under the transformations P, L, R on the other. In a way, this approach is an attempt to actualize Hugo Riemann’s vision of the theory of harmony as a “musical logic”.

    I will refer to two musical examples and related discussions in music theory: the first movement of Schubert’s sonata in Bb (D 960) as discussed by Richard Cohn and by Balz Trümpy and the last study of Alexander Skriabin Op. 65 No. 3 as discussed by Clifton Callender.

  • Thomas Noll (Escola Superior de Musica de Catalunya), Diatonic and Tetractys Modes as instances of Christoffel Duality, Thursday, June 11, 3:00 pm, Department of Mathematics, University of Chicago.

    Abstract: A recent development in the mathematical and music-theoretical study of so called well-formed scales is closely related to research directions within the field of algebraic combinatorics on words, namely Sturmian words and their finite analogues, i.e. Christoffel words and their conjugates. I’m going to report on joint work with David Clampitt (Ohio), Karst de Jong (Barcelona) and Manuel Dominguez (Madrid). There is a general music-theoretical desire to understand the principles which guide or constrain the constitution of musical tone relations. Well-formed scales are generated by a fixed interval modulo some period (typically the octave) and thereby embody a concept of tone kinship - given by the generation order of the scale tones. A second concept of tone relation is given by the pitch height order of the scale (step order). The well-formedness condition (as introduced and studied by Carey and Clampitt 1989) requires that the conversion from step order to generation order is a linear automorphism of /n\mathbb{Z}/n (where nn is the number of tones). A refinement of this condition for musical modes (instead of scales) leads to Christoffel words and their conjugates and to a related concept of duality.

    This word-theoretic approach is tightly connected with traditional accounts to tone kinship. This connection is given trough the projection from the free non-commutative group F 2F_2 with two generators to the commutative group 2\mathbb{Z}^2. Traditionally, the mathematical investigation of tone kinship is based on the free commutative group over two (or more) musical intervals as linearly independent generators. In the case of the two generators fifth and octave this group is called the Pythagorean tone lattice PP. Pitch height enters into this picture as a linear form h:Ph: P \to \mathbb{R}. It is insightful to embed PP into the real plane 2\mathbb{R}^2 with respect to a basis, which is constituted by the gradient (pitch height axis) and the kernel (pitch width axis) of this linear form. Word theory enters here through the approximation of the pitch width axis though polygons in PP, i.e. though elements of F 2F_2. For each Christoffel prefix of the infinite “Pythagorean word” (along the pitch width axis) there is a natural candidate for a metric on PP, respective 2\mathbb{R}^2.

    I will summarize actual results about the interdependence of regions (such as the Guidonian hexachord aabaa) and standard modes (such as the authentic Ionian mode aaba|aab). The underlying duality has a clear expression in terms of Standard morphisms. The generalization of the diatonic case does not only include larger and/or musically counter-factual modes, but also simpler ones, such as the modes of the three note tetractys scale. I will conclude my talk with remarks on a “Neo-Riemannian” approach to the analysis of fundamental progressions.

I think anyone who encounters these ideas is likely to wonder: is topos theory being put to good use in the study of music, or is it just a form of ‘showing off’?

I don’t know! I have no opinion yet. I don’t think topos theory is inherently too abstract to shed light on harmony. And, after years of slow study, I finally understand enough topos theory to follow what’s being said above, at least in principle. But I haven’t understood it yet. One way to dig deeper would be to read this book:

  • Guerino Mazzola, The Topos of Music: Geometric Logic of Concepts, Theory, and Performance, Birkhäuser, 2002.

But reading and understanding Tom Fiore’s review of this paper:

would be much quicker… at least for anyone who already understands Lawvere–Tierney topologies!

Does anyone here dare to venture some comments?

Posted at May 31, 2009 8:58 PM UTC

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17 Comments & 1 Trackback

Re: The Mathematics of Music at Chicago

I tried once to read the Mazzola book. I sincerely hope that my impressions of the book are inaccurate, as that would restore some of my lost hopes for the use of topoi in interesting areas.

I found the book to be ghastly - to fluffy for mathematicians, and probably too mathematical for music theorists - and wrote a review for the Jena-based mathematics magazine that said as much. The review ended up not being published, because it was too harsh, but being widely circulated among the magazine staff, because it was amusingly so.

Posted by: Mikael Vejdemo Johansson on June 1, 2009 2:49 PM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

Mikael wrote:

I found the book to be ghastly — too fluffy for mathematicians, and probably too mathematical for music theorists…

That combination of properties does not necessarily imply the book is bad. Almost any book that applies topos theory to music would suffer from this problem. The real question in my mind is whether he’s found nontrivial applications of topos theory to music. That alone would make the book worthwhile to me.

Posted by: John Baez on June 3, 2009 6:58 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

I agree, and this was the reason I also wrote:

I sincerely hope that my impressions of the book are inaccurate, as that would restore some of my lost hopes for the use of topoi in interesting areas.

I desperately want to be proven wrong in my dislike of Mazzola’s book, and failing that, I would really like to read and then understand an exposition of using topos theory in music theory, as the idea of it sounds marvelous to me.

Posted by: Mikael Vejdemo Johansson on June 3, 2009 11:19 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

You may also be interested in the work of Dmitri Tymoczko, in the music department at Princeton. As I recall (mainly from a couple discussions at bars and parties during philosophy conferences) the idea is that there’s a natural representation of harmonies as points in some sort of orbifold, and that looking for paths on this orbifold with some nice properties helps explain the way that tonal harmony arises from the basic ideas of voice leading.

I think this paper is the one with the relevant interest.

Posted by: Kenny Easwaran on June 1, 2009 6:44 PM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

I’ve corresponded with Dmitri Tymoczko, and I talked a bit about his work in week234. Here’s what I wrote:

To study different tuning systems in a unified way, one first step is replace the group Z/12 by a continuous circle. Points on this circle are “frequencies modulo octaves”, since for many - though certainly not all - purposes it’s good to consider two notes “the same” if they differ by an octave. Mathematically this circle is R+/2, namely the multiplicative group of positive real numbers modulo doubling. As a group, it’s isomorphic to the usual circle group, U(1).

This “pitch class circle” plays a major role in the work of Dmitri Tymoczko, a composer and music theorist from Princeton, who emailed me after I left a grumpy comment on the discussion page for this fascinating but (at the time) slightly obscure article:

17) Wikipedia, Musical set theory,

He’s recently been working on voice leading and orbifolds. They’re related topics, because if you have a choir of n indistinguishable angels, each singing a note, the set of possibilities is:


where Tn is the n-torus - the product of n copies of the pitch class circle - and Sn is the permutation group, acting on n-tuples of notes in the obvious way. This quotient is not usually a manifold, because it has singularities at certain points where more than one voice sings the same note. But, it’s an orbifold. This kind of slightly singular quotient space is precisely what orbifolds were invented to deal with.

Tymoczko is coming out with an article about this in Science magazine. For now, you can learn more about the geometry of music by playing with his “ChordGeometries” software:

18) Dmitri Tymoczko, ChordGeometries,

As for “voice leading”, let me just quote his explanation, suitable for mathematicians, of this musical concept:

BTW, if you’re writing on neo-Riemannian theory in music, it might be helpful to keep the following basic distinction in mind. There are chord progressions, which are essentially functions from unordered chords to unordered chords (e.g. the chord progression (function) that takes C major to E minor).

Then there are voice leadings, which are mappings from the notes of one chord to the notes of the other E.g. “take the C in a C major triad and move it down by semitone to the B.” This voice leading can be written:

(C, E, G) |→ (B, E, G).

This distinction is constantly getting blurred by neo-Riemannian music theorists. But to really understand “neo-Riemannian chord progressions” you have to be quite clear about it.

To form a generalized neo-Riemannian chord progression, start with an ordered pair of chords, say (C major, E minor). Then apply all the transpositions and inversions to this pairs, producing (D major, F# minor), (C minor, Ab major), etc. The result is a function that commutes with the isometries of the pitch class circle. As a result, it identifies pairs of chords that can be linked by exactly similar collections of voice leading motions.

For example, I can transform C major to E minor by moving C down by semitone to B.

Similarly, I can transform D major to F# minor by moving D down by semitone to C#.

Similarly, I can transform C minor to Ab major by moving G up to Ab.

This last voice leading,

(C, Eb, G) |→ (C, Eb, Ab)

is just an inversion (reflection) of the voice leading

(C, E, G)| → (B, E, G).

As a result it moves one note up by semitone, rather than moving one note down by semitone.

More generally: if you give me any voice leading between C major and E minor, I can give you an exactly analogous voice leading between D major and F# minor, or C minor and Ab major, etc. So “neo-Riemannian” progressions identify a class of harmonic progressions (functions between unordered collections of points on the circle) that are interesting from a voice leading perspective. (They identify pairs of chord progressions that can be linked by the same voice leadings, to within rotation and reflection.)

You can learn more about this here:

19) Dmitri Tymoczko, Scale theory, serial theory, and voice leading, available at

Posted by: John Baez on June 2, 2009 3:22 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

I work at UC and while I am neither a professional mathematician nor musician, I have some training in both fields, probably just enough that this talk won’t go completely over my head. I’m planning on attending and will post a summary here. Thanks for mentioning this, I probably would have missed it otherwise!

Posted by: Charles G Waldman on June 2, 2009 12:17 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

Cool. If you go, let me know what you think! I doubt anyone but the speaker and Tom Fiore will know much about both neo-Riemannian music theory and Lawvere-Tierney topologies. The first talk is what the topologists at Chicago call a ‘pretalk’ — the idea is that people are supposed to ask lots of questions to get ready for the second talk. So, that may be a good time to learn a lot of stuff.

Posted by: John Baez on June 2, 2009 3:29 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

I am a composer working in algorithmic composition.

First, a thanks to John Baez for his prior “This Weeks Find” week 234 on music theory, which was a gold mine for me.

Second, I have put some of the papers referenced by Professor Baez to practical use in algorithmic composition. If you are interested in the details, check out my personal Web site at

I presume other composers are doing similar work, but the only one I personally know of is my friend Drew Krause.

Musically, I have found most useful the work of Tymoczko, but also neo-Riemannian transformations. I have tried to penetrate The Topos of Music, but I suspect that my self-taught mathematical background is not (yet?) adequate; it may also be that the concepts are not as useful for my style of composition as the ones mentioned above.

Mike Gogins

Posted by: Michael Gogins on June 2, 2009 1:27 PM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

I’m delighted that week234 turned you on to some ideas you’ve found useful for actually composing music. You’ve more than repaid that favor by pointing us to your website.

From just a quick glance I’ve already learned about many interesting things: the Canadian Electroacoustic Society, Csound, TuneCore, the idea parametric composition, and more.

I’m glad you’ve made some samples of your music freely available under a Creative Commons copyright! I’ve just done the same for some of my own music, including some I created using Wolframtones. (I worked out a contract with WRI to make this possible.)

I have tried to penetrate The Topos of Music, but I suspect that my self-taught mathematical background is not (yet?) adequate; it may also be that the concepts are not as useful for my style of composition as the ones mentioned above.

This week Guerino Mazzola sent me a paper on ‘gestures’ and ‘metagestures’ which may be related to parametric composition. For Mazzola, a ‘gesture’ seems to be a 1-parameter family of positions of a hand (for example), while a ‘metagesture’ seems to be a 1-parameter family of gestures. A ‘gesture’ is analogous to a piece of music (and indeed this analogy is what an orchestra conductor exploits). So, a ‘metagesture’ is analogous to a 1-parameter family of pieces of music.

Would you like people to learn to appreciate parametrized families of pieces of music, much as they now appreciate pieces of music? It should be possible, with a little practice — and it could open up an interesting new form of art.

Posted by: John Baez on June 3, 2009 6:52 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

(I worked out a contract with WRI to make this possible.)

It's a bit disconcerting that this was necessary. I see this hidden away in their Terms of Use:

Copyrights–All content offered on this site is copyright Wolfram Research, Inc. All rights reserved. By using this Site, you disclaim any authorship rights to content presented at your request on this site.

And hidden even further away (although perhaps it is brought to the user's attention when a piece is created and downloaded) is an ironclad Content License Agreement that contradicts the open, breezy, ‘Sure you can!’ spirit (but not quite the letter!) of the rest of the site.

It will be interesting to see if this holds up in court. It's not clear that WRI provides any creative input to the piece of music that it calculates at the user's request. Of course, the user has agreed that WRI has copyright, but aside from the legally shaky nature of clickwrap agreements in general, the terms listed do not make it clear that the user is giving WRI rights that the user actually possesses, at least in part.

To be specific, both the Content License Agreement and the Terms of Use make it sound as if WRI already owns the content that it presents to the user at the user's direction; it seems as if the user is merely agreeing that they have been informed of this fact. While this makes sense for the CLA (which only comes in when WRI licenses the content back to the user), it doesn't really make sense for the ToU (which comes in before the user has instructed WRI to calculate the new piece of music). A court could rule that the user did not know what they were signing away.

On the other hand, WRI clearly seems to be happy to work out deals with people like John! They are probably not trying to be evil.

Posted by: Toby Bartels on June 3, 2009 7:51 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

(Maybe you want to move my comment to Treq Lila, where it's more directly relevant.)

Posted by: Toby Bartels on June 3, 2009 8:00 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

If you’re using WRI’s (or anyone’s) stuff for research/scholarly purposes, for your own enjoyment, or as long as it’s not commercial, you don’t need a license agreement thanks to FAIR USE (Section 107 of the Copyright Act, But if you really want to be sure, permission is always the nicest way).

- Alex

Posted by: Alex Korbonits on August 10, 2009 2:28 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

Research and personal use, yes. But if you want to post it on your website, then you’re violating their copyright (if they really have one), whether it’s commercial or not.

Posted by: Toby Bartels on August 10, 2009 9:42 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

Thanks for the reference to Mazzola’s recent work on gestures and metagestures. I will certainly check it out!

I will ask Mazzola for this, if I can’t find it online somewhere.


Posted by: Michael Gogins on June 3, 2009 1:29 PM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

I’m completely ignorant of even basic music theory, so I can’t tell how much Paul Hudak’s work on Haskore is about “just” expressing nuts-and-bolts musical ideas in a slightly more algorithmic way via functional language transformations, and how much it’s actually about enabling more sophisticated notions in actual compositions. To make matters worse the publications page on his website is “under reconstruction” so you’ve pretty much got to google around his name or Haskore to find stuff about it. However, the tutorial appears to give a rough idea of the kind of thing this music “domain specific language” appears to work with:
citeseer version of the tutorial.

(This is almost certainly much more elementary than the topos based stuff, but might be of interest.)

Posted by: bane on June 3, 2009 9:42 AM | Permalink | Reply to this

Re: The Mathematics of Music at Chicago

I do not know enough to make a qualified mathematical comment, but since you mention Pythagoras and admit to being a Pythagorean of sorts, I take this as an opportunity to point out a wiki-entry that I have written:

a href = “”

(I’m responsible for the subsections “Natural Philosophy” and “Cosmology.”) The section “Natural Philosophy” discusses some Pythagorean music philosophy. The history of Pythagoreanism is one of my long-time hobbies.

Posted by: Johan Alm on June 3, 2009 8:34 AM | Permalink | Reply to this
Read the post Thomas Noll's Talks at Chicago on Mathematical Music Theory
Weblog: The n-Category Café
Excerpt: Thomas Fiore has written a guest post about Thomas Noll's talks on mathematical music theory.
Tracked: July 23, 2009 8:48 AM

Re: The Mathematics of Music at Chicago

I’ve always thought there should be Riemannian geometry associated with a sort of rubato.

There is a sort of rubato where the left hand keeps strict time, and the right is free. I imagine the strict time is like flat space.

There is another sort of rubato where both hands are free - perhaps some call these “agogic distortions”. I imagine these to be like curved space, where each little bit is approximately flat, but large portions. If there is an “Einstein Field Equation for this”, maybe the stress-energy tensor would be some of Mazzola’s “gestures”?

OK, I will try to read the Mazzola book.

Posted by: Andrew on September 5, 2009 5:16 AM | Permalink | Reply to this

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