Math

Iannis Xenakis applied the Maxwell-Boltzmann velocity distribution, Markov chains, and game theory to orchestral composition. In Pithoprakta (1955–56), 46 string parts are molecules of a gas, each following the kinetic theory distribution. In Duel and Stratégie (1959–62), two conductors play a zero-sum game with payoff matrices on stage. This post works through the physics and mathematics, and asks what it means when a composer treats an orchestra as a thermodynamic system.

Euclid’s algorithm for computing greatest common divisors, applied to the problem of distributing k drum beats as evenly as possible among n time slots, generates rhythmic patterns that match traditional timelines from West Africa, Cuba, Brazil, Turkey, and the Balkans. An algorithm devised in Alexandria around 300 BCE encodes the rhythmic structure of musical cultures that had no contact with ancient Greek mathematics.

Messiaen’s seven modes of limited transposition cannot be fully transposed through all twelve keys — not by convention, but because of group theory. The modes are pitch-class sets whose stabiliser subgroups in ℤ₁₂ are non-trivial. The orbit–stabiliser theorem gives the exact count of distinct transpositions for each mode, and the subgroup lattice of ℤ₁₂ maps directly onto the hierarchy of the seven modes.