The Doubling Constant, d, and the Resonance of Triogenesis and Musical Scales

This paper introduces the \emph{doubling constant} d, a structurally emergent exponent that reconciles a 12-step recursive musical scale—built from alternating $\pm \frac{1}{3}$ frequency ratios—with the octave doubling observed in physical and musical systems. Derived from the Triogenesis framework, d is shown to correct the slight deviation between the recursive scale’s terminal ratio and the canonical 2:1 octave, through the structural growth function g(u). Remarkably, the symbolic formulation of d converges to a precision exceeding most known physical constants, without requiring an infinite tail. This finding challenges the presumed primacy of the number 2 in nature, suggesting it may be a resonance threshold arising from deeper layered recursions.