ELM-free H-mode plasmas with stationary plasma density and radiation level are obtained in the ASDEX Upgrade tokamak with large clearance between the last closed flux surface and the wall, and neutral beam injection in a toroidal direction opposite to that of the plasma current. This behaviour is accompanied by a characteristic narrow-band magnetohydrodynamic (MHD) oscillation with clear harmonics up to n = 11 visible. This mode is localized in the plasma edge region. Conditions and properties of the stationary ELM-free phases and the edge MHD oscillation closely resemble that of the `quiescent H-mode' and the `edge harmonic oscillation' found in the DIII-D tokamak (Burrell K H et al 2002 Plasma Phys. Control. Fusion44 A253). In addition, high-frequency MHD oscillations are found with an amplitude correlated with fluctuations of the divertor Dα intensity, suggesting a possible relevance of these modes for particle transport.