This paper presents the results of experiments carried out in JET with the Mark II divertor to study ELMy H modes at high density. In these experiments we varied the effective ion mass of the plasma (pure H, D and T), the input power, plasma current and the edge magnetic shear. The variation of density was achieved by gas fuelling. The paper focuses on two main issues: the variation of the global energy confinement with density and the scaling of the H mode pedestal. We find that the density dependence in the ITER97-p(y) scaling is not confirmed by the experiment when one approaches the density limit. For the high density regime in ELMy H-modes we propose a density independent scaling that includes the plasma triangularity in the regression variables. The core energy content of the plasma is correlated to the edge pedestal parameters, consistently with a mixed Bohm-gyroBohm transport model for H mode plasmas. It is found that with type I ELMs, the edge pedestal pressure increases with the ion mass of the plasma, the input power and the edge magnetic shear. We have derived a scaling for the edge pedestal width starting from the measured pressure at the top of the pedestal and assuming that the critical pressure gradient in the pedestal is determined by ideal ballooning modes. We find that the measured decrease of the edge pedestal pressure with density during Type I ELMs can be described by a model where the reduction of the edge pedestal width is due to the decrease of the ion temperature at the edge. In particular, the best fit to the data is obtained assuming that the pedestal width scales with the poloidal Larmor radius of the ions.