Low-energy magnons in a classical Heisenberg chain with random nearest-neighbor interactions are investigated using the negative eigenvalue counting technique. A detailed study is made for a range of concentrations of "wrong sign" bonds (i.e., c less than or equal to 0.1 and c greater than or equal to 0.9). It is found that low-energy magnons have an anomalous power-law (2/3) behavior in the spectrum for random sign-fixed magnitude exchange interactions. Combining these and previous results, a phenomenological fit is established describing the concentration dependence of the spectrum for a wide range of values of c. A discussion based on scaling arguments is presented to account for the anomalous power law for arbitrary concentrations. The general case where the magnitudes and the signs of the exchange interactions and the magnitudes of the spins are all random quantities is further investigated to check whether the anomalous power-law behavior holds. For uniform (flat), nonsingular distributions of the random quantities, the 2/3 power in the spectrum is obtained but for singular distributions and mixed distributions (e.g., the magnitude of the spin has a singular distribution while the magnitude of the exchange interaction has a regular distribution) significant deviations from the 2/3 power law occur.