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
['B2 d', 'd2 B', 'D2 D C2 B,']  
X:1
L:1/8
M:6/8
K:G
 G2 d B2 d | c2 d A2 c | B2 d G2 B | ABG FED | G2 g e2 g | f2 a d2 f | egf ed^c | d3 D3 :: 
 d2 B d2 B | c2 A c2 A | gdB cAG | FAG FED | d2 B d2 B | c2 A c2 A | gdB cAF | G3 G,3 :: 
 D2 D C2 B, | D2 D C2 B, | GAB ABc | BAG AFD | D2 D C2 B, | D2 D C2 B, | B,cB AGF | G3 G,3 :|  