Question: If your singing a third above the melody and the last note in a phrase is the highest note in the chord, e.g. chord is A and melody note is E, what should you sing as harmony?

Answer: 
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> chord is A and melody note is E, what should you sing as harmony?
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It depends on what kind of harmonic style you want. G# will make the chord an Amaj7, which might be too jazzy for some styles of music. An A will be a fourth, which isn't as pretty as a third. A C# will make it a sixth above, but at least it's pretty and not too jazzy.
Harmony singing is usually a third above the basic note - but - it will depend on what the rest of the bar is doing.

If the bar is V (in key A) the the note above E will be G♯, but if the bar is I, then G♯ won;t usually sound that good. So a 4th may be a better harmony. A and E. It will sound not as harmonious but at least it will fit. Not really a situation where one harmony in thirds will work.