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We present a comprehensive examination of the X-ray variability of the narrow-line Seyfert 1 (NLS1) galaxy NGC 4051, one of the most variable active galactic nuclei (AGN) in the sky. We combine over 6.5 years of frequent monitoring observations by RXTE with a >100-ks continuous observation by XMM–Newton and so present an overall 2–10 keV power spectral density (PSD) covering an unprecedented frequency range of over 6.5 decades from <10−8 to >10−2 Hz. The combined RXTE and XMM–Newton PSD is a very good match to the PSD of the galactic black-hole binary system (GBH) Cygnus X-1 when in a ‘high’, rather than ‘low’, state, providing the first definite confirmation of an AGN in a ‘high’ state. We also find that a bending power law, rather than a sharply broken power law, besides being more physical, is a much better description of the high-state PSD of Cygnus X-1 and is also a better description of the PSD of NGC 4051. At low frequencies the PSD of NGC 4051 has a slope of −1.1 bending, at a frequency Hz, to a slope of . Although does not depend on photon energy, is steeper at lower energies. If scales with mass, we imply a black-hole mass of in NGC 4051, which is consistent with the recently reported reverberation value of . Hence NGC 4051 is emitting at ∼30 per cent . NGC 4051 follows the same rms–flux relationship as GBHs, consistent with higher Fourier frequencies being associated with smaller radii. From the cross-power spectra and cross-correlation functions between XMM–Newton light curves in different energy bands, we note that the higher-energy photons lag the lower-energy ones. We also note that the lag is greater for variations of longer Fourier period and increases with the energy separation of the bands. Variations in different wavebands are very coherent at long Fourier periods but the coherence decreases at shorter periods and as the energy separation between bands increases. This behaviour is again similar to that of GBHs, and of MCG-6-30-15, and suggests a radial distribution of frequencies and photon energies with higher energies and higher frequencies being associated with smaller radii. Combining our observations with observations from the literature we find it is not possible to fit all AGN to the same linear scaling of break time-scale with black-hole mass. However, broad-line AGN are consistent with a linear scaling of break time-scale with mass from Cygnus X-1 in its low state and NLS1 galaxies scale better with Cygnus X-1 in its high state. We suggest that the relationship between black-hole mass and break time-scale is a function of at least one other underlying parameter which may be accretion rate or black-hole spin or both. [ABSTRACT FROM AUTHOR]