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The A note is not part of G major Triad. Triad is a 3 note chord. If you place a finger on the A string at the second fret the note becomes a Bnote. B is one of the tones in the Gmajor triad. This is a common way this chord is played. By muting the A string Tony takes the middle tone out of the chord. Tony is playing a chord with only two tones G and D. It is a cool sound. All the strings are either G or D. It seems floating and non-committal. That middle tone lends direction to the chord anchoring it one way or another (mostly). Without the anchor the listener is left floating. There is also another reason less obvious in this lesson. When he mutes the A string it frees up his index finger. Having that index free is magic for the G chord. Having this in your tool kit will make space for even cooler stuff later.