Music-Theory

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.

Alongside physics and astronomy, two other things have occupied an unreasonable share of my attention since adolescence: mathematics and music. Lateralus by Tool — released 2001, still in rotation — is the piece that most conspicuously occupies the intersection. The song is structurally built around the Fibonacci sequence, from the syllable counts in Maynard Keenan’s vocals to the time signature pattern that concatenates to F(16). This post works through the mathematics in some detail and asks why it works musically.