Harmony is just singing notes that are different than the melody, basically.
Singing "good" harmony is more complicated, but, that's music theory and stuff, and generally dependent on personal taste.
If you have no idea what you're doing, try picking out some songs that have harmonies you like and sing along with them, if you can hear the different parts, try to sing them, like switch from a high part to a low part and back and forth.
That'll give you a general ear for how it works.
Maybe make a CD of your stuff and drive around in your car trying to sing along NOT singing the notes of the melody, and see what sounds good.
Also, harmonies can be in "unison" like singing along with the melody, but different notes, or can be antiphonal or contrapuntal or in "blocks" where you're singing completely different stuff, like "doo wah oooh" kinda stuff.
The poor man's harmony is usually just mostly made up from singing the melody an interval up or down from the actual notes be it a third, fourth, fifth, sixth, or seventh.. depending on the music style or artist/composer's tastes.
If you want to do REALLY crazy shit like the beatles you could try singing an occassional note from a different key, spicing up the melody with dissonant touches.
But "harmony" is just different notes from the melody.
Technically a guitar chord is "harmony" if used to support a melodic line, and speaking of chords, any of those notes should work for a supporting vocal part.
I'd just mess around and see what sounds good.
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