The n-Category Café

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.

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, http://en.wikipedia.org/wiki/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:

Tⁿ/S_n

where Tⁿ is the n-torus - the product of n copies of the pitch class circle - and S_n 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, http://music.princeton.edu/~dmitri/ChordGeometries.html

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 http://music.princeton.edu/~dmitri/scalesarrays.pdf

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.

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.

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.)