Great tuning requires understanding of the harmonic series – and how to affect it with your tone.
An in-tune chord has harmonics that line up from top to bottom. For a major chord, for example, all notes in the chord align with the harmonic series of the root. For a D-major chord, the root is D, the fifth is A, the second partial, and the third is F#, the fourth partial.
All roots in that chord can sing with full tonal richness – all the overtones make sense to our ears as part of that chord. But the fifth can only emphasize some harmonics – the A’s first and third partials are A, but the second is E (not a part of a D major chord) and the fourth is C# (definitely not part of the chord). It’s even worse for the third of the chord. F#’s first four partials are F#, C#, F#, and A#. Playing an A# over a D major chord is jarring, so it’s best not to bring out the overtones if you’re singing the third.
Here’s the contrast. If you’re singing a root, make it rich like hot chocolate – full, warm, and complex. If you’re singing the third, make it clear like sparkling water – simple, subtle, and transparent. Fifths can be somewhere in between.
This, of course, requires, an underling comprehension of theory, knowledge of what part of the chord you’re singing at all times, and an ability to change tone chord by chord.
Alternatively, you can take the vocal jazz approach: Almost always choose sparkling water. There might not be so many overtones or richness in your sound, but the tuning will always lock.