Disclaimer: I'm no music teacher, just a hobbyist musician. However, I like to "fill out" a tune with double stops and there is one double stop interval that I like the sound of the best. First off, a double stop is like a mini two-note chord. When you're playing a melody you can create a double stop by playing a harmonizing note on the string lower than the one with the melody note. Playing both notes simultaneously helps fill out the sound.
To explain the double stop that I like the most, I'll give an example in GDAE mandolin/Irish tenor banjo tuning. Here it is: Whenever you're playing an F# note on the 2nd fret of the E string, you can also play an open A on the A string. You can't really go wrong with these two notes together. This combination of notes - F# and A - is most likely going to function as a D-major chord because a D-major chord has the notes D-F#-A in it. But since you're only playing two notes, this combination of F# and A could also ambiguously work as an F#minor type of chord. Worst case scenario is if the person actually laying down the chords to the melody is playing a B-minor chord (B-D-F#) at the moment you play that F# + A double stop. To my ears this hasn't really conflicted with the harmony because what you're suggesting with the notes F# and A resembles a B-minor7 chord which is just a jazzier version of a B-minor chord. It'll work there too.
What I'm saying is you don't really even need to know what the chords are to the tune to play this double stop since it's entirely based on the melody. I've never really been able to hear chord changes per se and I have zero opinion on what the chords to a tune should be. I have never been able to just strum chords to a song and know where to change from one chord to another. I'm just a melody player. But by using this system I am still almost always able to add a harmonizing note to any melody note in the tune without even having to think about it.
That same shape or concept works when you are playing a 3rd fret G note or a 5th fret A note. When you're playing a 3rd fret G note on the E-string, add a 2nd fret B note on the A string. When you're playing a 5th fret A note on the E string, add either a 3rd fret C or a 4th fret C# on the A string. I say "either" because most Irish tunes use the notes D, E, F#, G, and A, but they can use either a C-natural or a C-sharp note or both, depending. So you kind of have to have studied the tune or experimented to decide whether you like harmonizing that A note with a C natural (to give it a minor-ish sound) or a C# (to give it a major-ish sound). If the tune is Sliabh Russell then A + C is going to sound better/correct, but if the tune is High Reel then A + C# is going to sound better.
By now you may have noticed a pattern. All you're really doing is taking the note that would be a 3rd up from the melody note you are playing (in the context of the scale or mode you're in) and then playing a lower version of that note. In other words, the note D is a 3rd up from B, so if your melody note is on B then play a lower open D as your harmonizing note. Following that same logic, F# is a 3rd up from D, so if your melody note is on the 5th fret of the A-string (a D note) then play a harmonizing note on the 4th fret of the D-string (an F# note). Note: there's a slight chance that the F could be an F-natural if you're playing a D-dorian tune such as Sgt. Early's Dream or Maids of Mitchelstown.
Chords are made up of stacked notes in thirds, so this interval is completely within that theory. There's a reason that it works almost every time. Below I will insert a picture of this concept in tablature format.
One thing I haven't mentioned is what about when your melody note is on an open string? In that case, something I hear banjo players such as Theresa O'Grady doing all the time is playing a harmonizing note on fret 2 of the lower string. So if a melody phrase is ending on the open D of the D-string, a good harmony note to add is fret 2 of the G-string which is an A note. And if your melody note is on the open A of the A-string, then a good harmony note that will work more times than not is fret 2 of the D-string which is an E note.
Give this a go!
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