This app visualizes just intervals whose frequency ratios can be written as integer pairs m:n.
Each mark starts from a lattice point (u,v) with u >= 0, v >= 0, and r = sqrt(u^2+v^2) <= 30.
The lattice point is converted to the frequency ratio by m = u + v and n = u + 2v.
To form the circular layout, the app writes (u,v) = (r cosθ, r sinθ) and maps the angle to 4θ, so the displayed position is (x,y) = (r cos 4θ, r sin 4θ).
Click or drag on the plot to hear the base frequency together with the interval tone.
An × mark indicates that the corresponding frequency ratio is not in lowest terms; for a ratio m:n, this means gcd(m,n) != 1.
For gray circles, a lighter color indicates greater harmonic complexity, measured by Benedetti height (Tenny norm).
The colored background is controlled by von Mises densities centered at the colored interval-class angles; larger κ makes each colored region more concentrated.
For N sounding tones with integer frequency ratio x_1:x_2:...:x_N, Generalized Benedetti height is (x_1 x_2 ... x_N)^(1/N) / N.