Mode Density Function

The Mode Density Function is the continuous approximation to how many harmonic modes fall near a given frequency when harmonics follow BPT's quadratic spacing. It is the last step of the book's account of spectral density and frequency mapping, which moves from discrete quadratic scaling, through amplitude-modulated spectral distributions, to a smooth distribution of modes. The book says the square-root form reflects the quadratic frequency relation in the continuous limit, consistent with string-theory applications, and it lists it among the Chapter 7.2 predictions: fractal lattices should show this mode density, reflecting Harmonic Origin Pulse structure, detectable through statistical frequency analysis. The formal statement is g(f) ≈ 1/(2ν_0) √(ν_0/f), where g(f) is the Mode Density Function, f the frequency variable and ν_0 the fundamental frequency. Dimensions: g(f) [𝕋], f and ν_0 [𝕋⁻¹]; the book's check is consistent. It connects to Amplitude Scaling, the Spectral Density Function, the Harmonic Origin Pulse, Density-Dependent Planck Time and Harmonic Complexity Scaling.