Modern 12-tone equal temperament is a compromise where every non-octave interval is slightly out of tune(as the 12th root of 2 is irrational) in order to facilitate modulation and playing in any key. Integer ratios (or closer to integer ratios) may sound more in tune, but they are mostly impossible in this temperament.

Keyboard instruments in other temperaments (for example some Baroque tunings) may split the black keys (for example) into separate sharp and notes; sharps are used for sharp keys and flats for flat keys.

Choirs and instrumentalists who can dynamically adjust the pitch of individual notes will often do so for better tuning. (Some software instruments can also adjust tuning dynamically as you play.)

Many (if not most) pieces of music (perhaps most famously Bach's Well-Tempered Clavier) were composed with a particular temperament in mind.

https://en.wikipedia.org/wiki/Musical_temperament

I'm guessing that what they meant by "proper harmony" is just intonation: where thirds and fifths are expressed by small, integer ratios of frequencies (e.g., a fifth is 3:2 and a major third is 5:4).

A just intoned major third is about 14 cents flatter than a major third played on a 12 tone equal temperament tuned instrument (e.g., piano).

I'm not sure how much this matters in terms of having or not having perfect pitch though. Some people with perfect pitch can hear the difference between JI and 12TET and correctly their singing accordingly.