Context. Some high-resolution observations have revealed that the active region solar corona is filled with a myriad of thin strands even in apparently uniform regions with no resolved loops. This fine structure can host collective oscillations involving a large portion of the corona due to the coupling of the motions of the neighbouring strands. Aims: We study these vibrations and the possible observational effects. Methods: We theoretically investigated the collective oscillations inherent to the fine structure of the corona. We have called them fundamental vibrations because they cannot exist in a uniform medium. We used the T-matrix technique to find the normal modes of random arrangements of parallel strands. We considered an increasing number of tubes to understand the vibrations of a huge number of tubes of a large portion of the corona. We additionally generated synthetic time-distance Doppler and line-broadening diagrams of the vibrations of a coronal region to compare with observations. Results: We have found that the fundamental vibrations are in the form of clusters of tubes where not all the tubes participate in the collective mode. The periods are distributed over a wide band of values. The width of the band increases with the number of strands but rapidly reaches an approximately constant value. We have found an analytic approximate expression for the minimum and maximum periods of the band. The frequency band associated with the fine structure of the corona depends on the minimum separation between strands. We have found that the coupling between the strands is on a large extent and the motion of one strand is influenced by the motions of distant tubes. The synthetic Dopplergrams and line-broadening maps show signatures of collective vibrations, not present in the case of purely random individual kink vibrations. Conclusions: We conclude that the fundamental vibrations of the corona can contribute to the energy budget of the corona and they may have an observational signature. A movie associated to Fig. 10 is available at http:// https://www.aanda.org