For a four-point centrally damped string system, the algebraic form of the transcendental frequency equation is deduced and simplified to obtain the solution of the algebraic equation, and then the closed solution of the original frequency equation is obtained through a commutative inverse process. The structure of the closed solution of the system is discussed according to the fundamental theorem of algebra. The results show that there are three sets of closed solutions of the frequency equation, two of which are conjugate to each other. The specific conclusions are as follows: 1) The system has two identical motion characteristics, the logarithmic attenuation rate per unit time is the same, and the frequencies are opposite to each other; 2) The logarithmic attenuation rate and frequency per unit time corresponding to the three deexpenditures in the system always cycle periodically with the increase of order; 3) Each single-valued branch under the same debranching also shows a periodic cycle, that is, with the increase of order, its corresponding frequency increases by an integer multiple of 4π, while the logarithmic attenuation rate per unit time remains unchanged.