Rainfall regimes are experiencing variations due to climate change, and these variations are adequately simulated by Earth System Models at a daily scale for most regions. However, there are not enough raw outputs to study extreme and sub-daily precipitation patterns on a local scale. To address this challenge, Monjo developed the n-index by characterizing the intensity and concentration of precipitation based on mono-fractal theory. In this study, we explore the use of a multi-fractal approach to establish a more accurate method of time scaling useful to study extreme precipitation events at a finer temporal resolution. This study was carried out on the reference station of Barcelona (Spain) and its surroundings in order to be representative of the Mediterranean climate. For return periods between 2 and 50 years, two variables were analyzed: the n-index and the reference intensity $I_0$. Moreover, a new parameter, the so-called “breaking point”, was designed here to describe the reference intensity $I_0$, which is predominant for low time ranges. The results showed that both parameters are dependent on the time steps and the return period, and the scores confirmed the validity of our approach. Finally, the n-index was projected under downscaled CMIP6 climate scenarios by 2100, showing a sustained increase of up to +10%.