To play the F#aug chord, place the little finger on the 3rd fret of the top g-string, middle finger on the 2nd fret of the C-string, ring finger on the 2nd fret of the E-string, and index finger on the 1st fret of the bottom A-string.
The F#aug chord (also known as a Gbaug chord) contains the notes F#-A#-C##.
Yes, you read that right – "C##" – pronounced "C double sharp" which is the enharmonic equivalent of D, meaning C## = D.
Here's how it works:
To build an F#aug chord, take the augmented triad formula (1-3-#5) and apply it to the F# major scale (F#, G#, A#, B, C#, D#, E#).
Identify the first, third, and fifth scale degrees in an F# major scale which are F#-A#-C#.
Because the fifth scale degree is a "#5" (said "sharp five") raise the C# note a half step to C## making the chord augmented with the notes F#-A#-C##, also known as F#-A#-D.
Note: The reason we use a double sharp here is because 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 C#. So in this case, to make an augmented chord, we must raise the fifth scale degree of the chord "C#" up a half step to "C##", which is the enharmonic equivalent of "D".
With that said, some prefer to view the key of F# major as the key of Gb major (I know I do!), so to build a Gbaug chord, take the augmented triad formula (1-3-#5) and apply it to the Gb major scale (Gb, Ab, Bb, Cb, Db, Eb, F).
Identify the first, third, and fifth scale degrees in the Gb major scale, which are Gb-Bb-Db.
Because the fifth scale degree is a "#5" (said "sharp five") raise the Db note a half step to D making the chord augmented with the notes Gb-Bb-D.
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