Channel charting is a data-driven baseband processing technique consisting in applying self-supervised machine learning techniques to channel state information (CSI), with the objective of reducing the dimension of the data and extracting the fundamental parameters governing its distribution. We introduce a novel channel charting approach based on triplets of samples. The proposed algorithm learns a meaningful similarity metric between CSI samples on the basis of proximity in their respective acquisition times, and simultaneously performs dimensionality reduction. We present an extensive experimental validation of the proposed approach on data obtained from a commercial Massive MIMO system; in particular, we evaluate to which extent the obtained channel chart is similar to the user location information, although it is not supervised by any geographical data. Finally, we propose and evaluate variations in the channel charting process, including the partially supervised case where some labels are available for part of the dataset.