This is the exercise I'm trying to figure out. I don't really understand how to do this.
A chord is typically defined by intervals of thirds, minor or major. Intervals are the distance between two notes - if a note is three half steps apart, it is a minor third. If it is four, it is a major. Here is a little visual representation of intervals on a piano in the key of C. Not sure what those numbers on the right are...
A typical chord is typically the 1 (root) - 3 - 5 - 7 intervals, the numbers are coming each note's interval from the root. So as we see here, the third is the second note in the chord. This is generally how they are set up.
So for example, A major chord is 1 - Major3 - 5. The third is the second note of the chord, and is a major 3rd away from the root. This is the note we're looking for.
The staff shows B, G#, E#, and C#. This chord is inverted, meaning the root of the chord is not the bass note, so we'll have to arrange it. This takes a little trial and error, testing to see which chords make the most sense with the note combination, but what stands out to me is that B would be the minor 7th of C#, so C# can be the root here.
If C# is the root, then E# is the third. It is a major third away from the root of the note. This makes G# a perfect 5th and B a minor 7th. This gives us a Dominant 7th chord, where the third is E# and B is in the bass. This is written as C#7/B.
You have the notes B, G#, E#, C#. Chords are built in 3rds so you need to arrange the chords until the are in 3rds. Thirds in general are when you skip a letter name. So all the possible thirds are
A - C - E - G - B - D - F - A. Only the letter name matters not the quality i.e. if it is flat, sharp or natural.
In this case you would rearrange them to C#, E#, G#, and B so the 3rd would be E#. They follow the pattern above just there are added accidentals.