For the cable-damper system， the algebraic form of the transcendental frequency equation of the system is derived when the damper is located at a unit-fraction-span （1/n span of the cable）. According to the fundamental theorem of algebra， the structure of the eigensolutions of the system is discussed， and the properties of the solution are analyzed with four examples. The results show that： 1） The eigensolutions can be divided into n-1 branches. 2） Within one solution branch， all eigenvalues take an identical value in their real part （as an additive inverse of the logarithmic decrement ratio per unit time）， while their imaginary parts （meaning in physics， the frequency） form an arithmetic sequence. 3） According to the way that the frequencies vary with the damping， the solution branches can be classified as three types： The frequency of type 1 solutions is related on damping； The frequency of type 2 solutions is not affected by damping. The frequency of type 3 solutions may or may not vary with the damping， which means that a type 3 solution may behave like a type 1 or a type 2 solution， depending on the damping is under or over some certain critical value。