The fundamental frequencies of vibration of 12 first-row closed-shell diatomics have been predicted using both full and composite levels of CCSD(T) ab initio theory. For a given CPU time budget, composite levels of theory were found to predict harmonic frequencies significantly better than full ab initio methods. However, little improvement was obtained in the computation of the anharmonic correction with composite methods, its being already well predicted at the CCSD(T) small basis set level. It was found that for a given CPU time budget the most accurate fundamental frequencies are obtained by performing a calculation of the harmonic frequencies using a composite method where the optimal choice of basis sets involved a larger cc-pVXZ basis set only I greater in the valence designation compared with the smaller basis set. An anharmonic correction computed using a small basis set or a low-level composite method could then be added to these harmonic frequencies. The implication of these findings is that accurate fundamental frequencies can be computed cheaply and efficiently by first computing the harmonic frequencies using an accurate composite method and then correcting these frequencies with an anharmonic correction obtained by solving the nuclear Schrodinger equation by some means on a potential energy surface generated using a CCSD(T) small basis set or a lowlevel composite method.