# Structure in jazz and math

Last night I went to a concert by the Branford Marsalis Quartet. One of the things that impressed me about the quartet was how creative they are while also being squarely within a tradition. People who are not familiar with jazz may not realize how structured it is and how much it respects tradition. The spontaneous and creative aspects of jazz are more obvious than the structure. In some ways jazz is more tightly structured than classical music. To use Francis Schaeffer’s phrase, there is form and freedom, freedom within form.

Every field has its own structure, its tropes, its traditions. Someone unfamiliar with the field can be overwhelmed, not having the framework that an insider has to understand things. They may think something is completely original when in fact the original portion is small.

In college I used to browse the journals in the math library and be completely overwhelmed. I didn’t learn until later that usually very little in a journal article is original, and even the original part isn’t that original. There’s a typical structure for a paper in PDEs, for example, just as there are typical structures for romantic comedies, symphonies, or space operas. A paper in partial differential equations might look like this:

1. Motivation / previous work
2. Weak formulation of PDE
3. Craft function spaces and PDE as operator
4. A priori estimates imply operator properties
5. Well posedness results
6. Regularity

An expert knows these structures. They know what’s boilerplate, what’s new, and just how new the new part is. When I wrote something up for my PhD advisor I remember him saying “You know what I find most interesting?” and pointing to one inequality. The part he found interesting, the only part he found interesting, was not that special from my perspective. It was all hard work for me, but only one part of it stood out as slightly original to him. An expert in partial differential equations sees a PDE paper the way a professional musician listens to another or the way a chess master sees a chess board.

While a math journal article may look totally incomprehensible, an expert in that specialization might see 10% of it as somewhat new. An interesting contrast to this is the “abc conjecture.” Three and a half years ago Shinichi Mochizuki proposed a proof of this conjecture. But his approach is so entirely idiosyncratic that nobody has been able to understand it. Even after a recent conference held for the sole purpose of penetrating this proof, nobody but Mochizuki really understands it. So even though most original research is not that original, once in a while something really new comes out.

Related:

## 8 thoughts on “Structure in jazz and math”

1. Jazz actually has a lot in common with baroque music, it has its own sort of counterpoint which facilitates lots of spontaneous embellishments (also vaguely similar to a raga). If nothing else I think it’s an inspiring example of the power of constraints.

2. Indeed. And improvisation was part of baroque music too.

3. David W. Locke

The old-new rhetorical contract stands. You can’t explain the new without reference to the old, a pasts future, but only one. There are other rhetorical contracts, but they are slower. The new-old rhetorical contract, the old as seen from the new gets written long after adoption.

Who thinks in those terms? But, when we read something totally new to us, we don’t have any concepts to link to, so we take more notes and take more time to digest what we are reading. If we do have concepts to link to the volume of your notes goes down as does the time it takes to read that content.

4. John Dyer

As a software engineer, I’ve had similar thoughts about things like conferences. Most of what you hear is the same old same old. But as you listen and interact, you find those ah ha nuggets. Those new bits of information or new ways to combine existing ideas that you hadn’t thought of before. The stuff that makes the entire event worth while. The 10% that’s new to me.

5. David Tate

In some ways jazz is more tightly structured than classical music.

That depends on what you mean by ‘jazz’ and what you mean by ‘classical music’. Nearly all traditional jazz is less structured than a Bach fugue; almost any piece from the Western formal canon is more structured than an Ornette Coleman performance. If you think of the distributions of ‘structure’ in both traditions, jazz has a fat left tail and classical has a fat right tail. I suspect that the cdf for classical lies strictly below the cdf for jazz.

6. I liked the Francis Schaeffer quote. It is a superb description of what goes on in the classical music of southern India . A song performed for 10 minutes by one musician may be performed for more than 20 minutes by another . There is a form demanded by tradition but plenty of freedom within that form