The Munich a cappella group Mundwerk sings Brother Jaques not only with voices in canon, but also with overtones. My student and master class graduate Oliver Zunker (now no longer with the group) sings together with his colleague Jens Ickert the canon exclusively with overtones. Oliver himself is an excellent overtone teacher, as you can hear. I think the whole choral arrangement is superb!
All tones of the melody are already contained in a single keynote (harmonic 6 to 13, cf. composing with overtones). So you can sing them on one fundamental note.