We analyze time sequences of spectral lines formed at the photospheric level in order to study the correlation of umbral oscillations with continuum brightness and the influence of the fine structure and evolutionary stage of sunspots. The observations were taken using the Gregory-Coude Telescope (GCT) at the Observatorio del Teide (Tenerife). Power integrated in two frequency bands, 5 minutes (2.5-4.5 mHz) and 3 minutes (5-7 mHz), are used as parameters for this study. Our study yields the following results. A clear correlation exists between the power at the 3 minute band and the umbral brightness in dark cores. The light bridges do not follow this behavior. No correlation is apparent between the 5 minute power and the brightness. The same is obtained after normalization to the power in the adjacent light bridges. There is no correlation between the umbral oscillations of different umbral dark cores. The center-to-limb variation of the power (both bands) seems to support the interpretation of the umbral oscillations as slow body modes. Finally, we discuss the physical context of the observations.