After trying different approaches, I figured out that I was facing non-linear data. Machine learning techniques can handle this situation. And their algorithms are nowadays fairly easy to use.
This article describes the logic behind the approach I do recommend. I will present in a familiar way the limitations of using balance ranges (or concentration ranges), or to use linear statistics. Few paragraphs up from here, I intentionally used the word coordinates not only because it is commonly used to describe the space of isometric log-ratios \cite{2015}, but it also serves the example above.
The ionome as a map
We have seen that isometric log-ratios project compositions into a real orthonormal space, or to coordinates, where euclidean distances can be computed. They are accordingly analogous to geographical maps, like this archipelago (Îles-de-la-Madeleine, Québec).