To play the Eaug chord, place the index finger on the 1st fret of the top g-string and ring finger on the 3rd fret of the bottom A-string. Let the other strings ring open.
The Eaug chord contains the notes E-G#-B#.
To build an Eaug chord, take the augmented triad formula (1-3-#5) and apply it to the E major scale (E, F#, G#, A, B, C#, D#).
Identify the first, third, and fifth scale degrees in an E major scale which are E-G#-B.
Because the fifth scale degree is a "#5" (said "sharp five") raise the B note a half step to B# making the chord augmented with the notes E-G#-B#.
Note: Music theory rules dictate that each scale degree be represented by a different letter, which is the reason we identify the fifth scale degree as B#. The enharmonic equivalent of B# is C (that is to say, B# = C).
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