The spectrum of resonance frequencies in so called stand-alone resonators, built with coupled ring or disk resonators is analytically and numerically investigated. These composite resonators which constitute the core of numerous photonic circuits used in channel dropping filters, dispersion compensators, laser mirrors, etc., determine the width of their passband and their free spectral range (FSR). The spectral characteristics of the resonances are determined by the dimensions of the resonators and the strength of coupling between them. Novel relationships between these parameters are described that ensure invariance of the splitting ratios and as a consequence maintain the passband characteristics of the associated devices. Waveguide attenuation is found to have no effect on the spectral characteristics of the composite resonators.