But in equal temperament this problem doesn't exist. It just sounds as though it does. In any case electronic tuners, which appear to have replaced careful listening these days, don't have the facility for tuning these additional frets conveniently.

I suppose it's not really part of "chord theory", but as a melody instrument player, I like chords underneath that fit the tune! Usually these contain notes that are found in the tune (probably on accented beats), plus others (say 7ths) that suggest where the tune may be going.

As Jack Campin mentions earlier, the choice of lowest note is important. Ideally these should form a musical sequence. Otherwise known I believe as a bass riff. It's good if other notes should form their own tunes as well. More so if the accompanying instrument is melodic (although the concept of two note "chords" may not be appropriate to this thread -) here the 2nd tune is prominent and needs to be carefully constructed.