Multidimensional Continued Fractions

There was some discussion by John Chalmers and others a while ago about
multidimensional continued fractions (such as ternary continued fractions)
in the context of approximating a set of JI intervals with an ET.
Multidimensional continued fractions could also be useful in approximating
an irrational chord with JI. I was thinking that, since for dyads the Farey
mediants could probably be derived by using an appropriately truncated
continued fraction expansion for every interval in the continuum, perhaps
the triadic harmonic entropy problem could be approached with a similar
approach. Discouragingly, I found the following website:
http://www.mathematik.uni-bielefeld.de/~sillke/NEWS/continued-fractions
which says,
"Alas, it turns out that in higher dimensions there are MANY competing
algorithms, all equally disappointing ;)"
although my particular problem might escape these disappointments. Anyone
know anything about the Jacobi-Perron algorithm? Or if any of the algorithms
discussed by Barbour might be appropriate?