Some simple principles for good
It is best to have a few simple principles for doubling in chords, rather than remembering a lot of complicated specific rules. In general, you are trying to reinforce the function of chords in the tonality, as well as work with the natural acoustics (overtones) of the notes in the chords, in order to have well blended harmonies. Here are some basic principles:
1. Double the primary tones of the scale, scale degrees 1, 4, and 5. (Those tones reinforce the tonic, pre-dominant, or dominant function of the chord)
2. Don’t double active tones that require resolution, such as leading tones, chord sevenths, and chromatic notes. (This would create parallel octaves when the tones would resolve)
3. ALWAYS double the fifth of a 6/4 chord (this is the one very specific rule to remember.)
4. In major triads, the root is most often the best choice to double, followed by the fifth. Avoid doubling the third of major triads particularly when in first inversion. (The overtones of the third of the chord work against the other notes. The lower the first inversion bass note of the chord is, the worse it sounds.)
5. In minor triads, there are not as many acoustical issues to worry about between the notes, so the third can be a very good note to double, especially in a major key, where the thirds of the minor triads are the primary tones of the scale. Also the root is a good note to double. The fifth is the least used doubling (except in a minor 6/4 triad of course).
6. In diminished triads, the third is always the preferred choice to double, since it is not a member of the dissonant tritone formed by the root and fifth (In a major key, the root of the diminished triad is the leading tone; that is a bad choice to double anyway for reasons described in #2 above. In minor, the diminished chord occurs on scale degree 2, so the third of that triad is a primary scale tone (4), reinforcing the pre-dominant function of the chord as described in #1 above)
7. In augmented triads, it doesn’t really matter which note you double from the acoustic point of view, since by sound, any note could be the root. To help clarify which note is the root, you could double it, or, double a primary tone in the chord.
Good voice leading always takes precedence over doubling. So you may sometimes need to “break” these preferred doubling rules in order to avoid creating parallel fifths or other poor voice leading.