Left: case of a “magic” spin, where the strongest resonances could occur at the same radius. For completeness we present the relevant simple combinational frequencies ν_θ − ν_r, ν_θ + ν_r, ν_K − ν_θ, ν_K − ν_r (grey dashed lines). The “magic” spin a_m = 0.983 represents the only case where the combinational and direct orbital frequencies coincide at the shared resonance radius. Right: another example of the characteristic frequency ratio set 3:2:1 that appears at different radii.