Iannis Xenakis' 'Nomos Alpha' was performed by cellist Arne Deforce for Birmingham Contemporary Music Group's Music & Maths Festival, celebrating the composer's centenary.


The piece was performed with live animation created by mathematician Marcus du Sautoy and motion designer Simon Russell on 29 May 2022 at CBSO Centre, Birmingham. Music & Maths was a day-long festival celebrating the centenary of composer Iannis Xenakis, in partnership with Sound and Music, University of Birmingham, and PRiSM (Royal Northern College of Music).

"'Nomos Alpha' is a perfect fusion of my two passions: the mathematics of symmetry and contemporary music. To celebrate Xenakis’s 100th anniversary I’ve been working with artist Simon Russell to create an animation to guide listeners (and players) through the piece. The composition is based on the symmetries of a cube. There are 24 ways to rotate a cube from its starting position. 'Nomos Alpha' is divided into 24 sections. My immediate thought was there is a section for each symmetry. However, as a dug deeper into the piece I discovered something much more interesting is going on. The piece is divided into 6 groups of 4 sections. The first 3 sections in each group correspond to 3 symmetries of the cube.

A label in the animation keeps track of the symmetry being played. The 4th section is more fluid and imagines the cube morphing. Xenakis places 8 musical textures for the cello on the corners of the cube. In our animation, these are called S1,…, S8. In each section, these corners are played in a fixed order corresponding to two tetrahedra that can be embedded inside the cube. At the end of each section, the cube then undergoes a rotation to rearrange the musical textures. The same path is mapped out in the new section but the textures are now played in a new order. A second cube (not shown in our animation) keeps track of the time spent on each corner.

The fascinating thing for me is how Xenakis chooses the symmetries. He starts with two seed symmetries called D and Q12. These control the first two sections. But to get the symmetry for the third section he rotates the cube using symmetry D and then rotates again using symmetry Q12. The combined effect is a new arrangement that you can get in one go by doing symmetry Q4. For Xenakis, each new symmetry is got by combining the two previous symmetries. This is a bit like the way the Fibonacci numbers are defined: 1,2,3,5,8,13…You get the next number by adding the previous two numbers. It takes 18 symmetries before the sequence repeats. This is the longest cycle of symmetries you can realise in the cube. Some of these symmetries are repeated twice. The animation keeps track of the path that this maps out in the 24 symmetries of the cube.

This idea of a Fibonacci sequence of symmetries is not something that has ever been considered in mathematics. It is a new idea that Xenakis contributes. Often mathematics inspires music but here we see music inspiring new mathematical ideas."

Marcus du Sautoy, Professor of Mathematics, University of Oxford.