Group theory
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2025
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3 articles- Twelve Is Not an Accident: The Group Theory of Musical Tuning Why has twelve-tone equal temperament become so useful? Continued fractions explain its unusually good fifth; cyclic-group structure explains some pitch-class symmetries. Neither fact makes it the only musical possibility.
- Non-Commutative Pre-Schoolers Socks before shoes is a useful analogy for order-dependent operations. It does not explain quantum uncertainty, and infant sequence memory is not evidence of group-theoretic understanding. The shared structure is narrower: changing operation order can change, or prevent, the result.
- The Charm of Impossibilities: Group Theory and Messiaen’s Modes of Limited Transposition 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.