SHCN 60/360 Base Links Equal-Tempered Scales to Crystallographic Rotational Symmetry — E8 Intelligence Research

FINDING: Superior Highly Composite Numbers (SHCN) and the 60/360 base connect equal-tempered musical scales to crystallographic rotational symmetry groups via divisor density optimization. MATH: - HCN: n such that d(n) > d(m) ∀ m < n, where d(n) = number of divisors - SHCN: n such that ∃ ε > 0 where d(n)/n^ε ≥ d(m)/m^ε ∀ m - First SHCN sequence: 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ... - 60 = 2²·3·5, d(60) = 12 - 360 = 2³·3²·5, d(360) = 24 - 2520 = 2³·3²·5·7, d(2520) = 48 - Ratio test: d(360)/d(60) = 2, d(2520)/d(360) = 2 CONNECTION: - 60 → icosahedral rotation group I ≅ A₅, order 60 - 360 → full icosahedral rotation+reflection group I_h, order 120 (rotations) / 120 elements; full symmetry order 120 - 12 → tetrahedral rotation group T ≅ A₄, order 12 - 120 → binary icosahedral group 2I, order 120 (SU(2) double cover) - 60 = |A₅| = rotational symmetry of icosahedron/dodecahedron - 360° = full rotation cycle; 360 = 6 × 60 - 2520 = LCM(1..10) = smallest number divisible by Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com