The nature of molecular vibrational coordinates is elucidated through the use of canonical coordinate transformations of the Hamiltonian. The importance of studying molecular vibrations with quantum mechanical calculational techniques is discussed and the methods for doing these calculations in rectilinear rotations of the bond coordinates are presented. Chief among these methods are prediagonalization and the transformation method, also known as the discrete variable representation. The results of coordinate rotation calculations on a three-coordinate stretching-only model of acetylene are presented and discussed. The results of calculations on two two-coordinate subsystems of acetylene are also presented. In all of these cases prediagonalization and rotated coordinates result in a substantial improvement in the quality of the zero order basis as measured by the projection of the eigenstate of interest onto the product function basis state with the same quantum numbers. Optimized coordinate rotation calculations for a two-coordinate stretching-only model of hydrogen cyanide are presented. The results are similar to those of the acetylene calculations. The physical interpretation of the hydrogen cyanide calculations is easier because contour plots of the entire system are possible. An unexpected and previously unexplained intensity pattern in the hydrogen cyanide vibrational overtone spectrum is analyzed and clarified. Finally, drawbacks to these calculational techniques are discussed and possible improvements and future uses are suggested.