Iannis Xenakis used probability distributions, Markov processes, and game structures as compositional tools. Pithoprakta maps a Gaussian law derived from kinetic theory to glissando slopes; the orchestra is an analogy to a gas, not a thermodynamic system with measured temperature.
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, produces patterns that can be compared with timelines documented in several musical traditions. The mathematical similarity does not establish a shared origin, cultural equivalence or historical transmission.
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. The subgroup lattice constrains the possible symmetry types; it does not select Messiaen’s seven scales.