Metastaseis and ruled surfaces

Iannis Xenakis was a Greek-French composer who was also an architect and a mathematician. Before he wrote music full-time, he worked in Le Corbusier's studio designing buildings. In 1954 he wrote an orchestral piece called Metastaseis for 61 musicians where no two players play the same part. The piece is about 8 minutes long and it sounds like nothing that had been composed before it.

The basic idea is that a glissando --- a continuous slide from one pitch to another --- can be represented as a straight line on a graph where the x-axis is time and the y-axis is pitch. One cello sliding from C3 to G5 over four seconds is just a line segment in this plane.

Xenakis realized that if you have 46 string players each performing their own glissando simultaneously, at different rates and in different directions, the collection of lines traces out a ruled surface.

A ruled surface is a surface where through every point, there passes at least one straight line that lies entirely on the surface. The cylinder is one --- you can construct it from a family of vertical lines. The cone is another. The one Xenakis cared about was the hyperbolic paraboloid, which is the saddle-shaped surface defined by

$$ z = \frac{x^2}{a^2} - \frac{y^2}{b^2} $$

A hyperbolic paraboloid is doubly ruled --- there are actually two distinct families of straight lines on it. You can see this if you parametrize it: the surface $z = xy$ contains every line of the form $(t, c, ct)$ and every line of the form $(c, t, ct)$ for constant $c$.

In Metastaseis, each straight line is one player's glissando. The opening section has all 46 strings start on the same pitch and fan outward --- some sliding up, some sliding down, at different speeds.

On the score, this literally looks like the lines generating a cone or paraboloid. Bars 309--314 of the piece draw out a parabola where the glissandi function as tangent lines to the curve.

Xenakis sketched the piece as an architectural blueprint with pitch on one axis, time on the other, and straight lines everywhere. He then translated the geometric sketch into actual notation for 61 players.

A few years later, Xenakis designed the Philips Pavilion for the 1958 Brussels World's Fair while still working in Le Corbusier's studio. The building's shell is a set of hyperbolic paraboloids --- the same ruled surfaces from Metastaseis. The walls of the pavilion are the score of the composition, or the score is the walls, depending on how you want to think about it.

Interestingly, the score's middle section uses a twelve-tone row with durations based on the Fibonacci sequences. Xenakis uses this to generate rhythmic proportions that feel organic and has structure without periodicity.

After Metastaseis, Xenakis composed Pithoprakta (1955--56) using actual probability distributions --- Poisson and Gaussian --- to determine when and how 50 instruments would play, which he calls "stochastic music".