I'm not sure if you want follow the bass line or the back voices or invent a melody:


It depends of the style:

In classic, Jazz and Pop the you may keep on this rules:

bass: you can choose the pitch you like - most comfortable to your voice (priority: the counter-point to the tune).

back-voicing: keep the common tones or go to the next step up or down (preferably counter movement to the melody or bass line.

Gospel songs: parallels (even fifth parallel) are usual creating a typical effect.

In European Folk Music parallels of thirds and sixths are usual, in English and American Folk Songs and Pop Songs (Beatles) parallel fourth with the tune are also usual.

try this out:

https://www.youtube.com/watch?v=ADN1lLEp3H0

https://www.youtube.com/watch?v=pd_5-2kCzfs

https://youtu.be/2O0k7e5ApFU?list=RD2O0k7e5ApFU

https://www.youtube.com/watch?v=V1bFr2SWP1I&list=RD2O0k7e5ApFU&index=2


> Isn't it weird that even though harmony went down in pitch I sing higher in pitch at the same time?

summary: 

in almost all styles it is even better to lead the voice in opposition to the bassline respectively the chords.