Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
M:6/8
K:G
['G2 G', 'D3 d2 d ded afd afd g2']  
X:1
L:1/8
M:6/8
K:G
 G2 G GAG | GBG dBG | c2 B A2 G | FGF AFD | G2 G GBG | Bdg g2 b | agf ed^c | dAF D3 :: d2 d ded | 
 afd afd | g2 d c2 B | ABG FED | G2 G GAG | dBG Bdg | dBG cAF | G3 G3 :|  