We demonstrate an accurate high sensitivity method for cavity spectroscopy. We measure the frequency intervals of transverse electromagnetic modes relative to a fundamental mode in a high finesse optical resonator, and attribute their mode numbers unambiguously. A laser is frequency locked to a fundamental T E M 00 cavity mode by optical feedback, and phase modulation is used to obtain frequency side bands, which may come to resonance with other transverse cavity modes as the radio-frequency of the modulation is tuned. At these resonances, transmission of the side bands is sensitively detected by heterodyning with the carrier. We also analyze the transverse spatial profile of the heterodyne signal for identification of mode numbers. The adjustment of the Gaussian cavity model to the measured frequency intervals yields values of cavity length, mirror radius of curvature, and mirror ellipticity, with high precision to the ppm level.