Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/16
M:2/4
K:E
['e2 G2GA BEE2 BEdE BEEF G2GA BABd', '(3(gag) (3(fgf)']  
X:1
L:1/16
M:2/4
K:E
 e2 | G2GA BEE2 | BEdE BEEF | G2GA BABd | egfd e2 :| (ef) | (3(gag) (3(fgf) edBc | d2ed Bdd2 | 
 (3(gag) (3(fgf) edBd | dgfd e2 ef | (3(gag) (3(fgf) edBc | d2ed Bddg | bgaf gfed | egfd e2 |]  