Context. Some high-resolution observations have revealed that the active-region solar corona is filled with myriads of thin strandseven in apparently uniform regions with no resolved loops. This fine structure can host collective oscillations involving a large portionof the corona due to the coupling of the motions of the neighbouring strands.Aims. We study these vibrations and the possible observational effects.Methods. Here we theoretically investigate the collective oscillations inherent to the fine structure of the corona. We have called themfundamental vibrations because they cannot exist in a uniform medium. We use the T-matrix technique to find the normal modes ofrandom arrangements of parallel strands. We consider an increasing number of tubes to understand the vibrations of a huge numberof tubes of a large portion of the corona. We additionally generate synthetic time-distance Doppler and line broadening diagrams ofthe 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 thecollective mode. The periods are distributed over a wide band of values. The width of the band increases with the number of strands butrapidly reaches an approximately constant value. We have found an analytic approximate expression for the minimum and maximumperiods of the band. The frequency band associated with the fine structure of the corona depends on the minimum separation betweenstrands. We have found that the coupling between the strands is of large extent and the motion of one strand is influenced by themotions of distant tubes. The synthetic Dopplergrams and line-broadening maps show signatures of collective vibrations, not presentin 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 theymay have an observational signature.