Unless someone can cite examples more recent than the mid-18th century, I would also recommend just ignoring 'harmonical,' which as far as I know is obsolete. (If it isn't quite dead yet, it should be, since it's a needless variant.)
'Harmonic' is an objective, neutral term used in science and mathematics: harmonic series, etc. In Western music, it refers (more or less) to using notes from a recognizable scale to form chords (= harmony, different notes sounding together) that we perceive as a certain key or mode. 'Harmonically' would be its adverb, but it's probably seldom used.
'Harmonious' is the subjective, positive term used in music and in figurative language. Harmonious music is notes that make a pleasing sound together. People can also have harmonious relationships. Its adverb is 'harmoniously.'
For questions like this, it's usually a good idea to check a monolingual English dictionary. For more examples, try the text corpus at the Uni Leipzig: http://corpora.informatik.uni-leipzig.de/
. The fact that 'harmonical' doesn't show up at all there (search on 'harmonic*' to see related forms) is a good clue. (-:
OT: I still have my draft list of words ending in -ic and -ical, but I'll polish it up another week when I'm not sick. :-(