Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
M:4/4
K:D
['', 'F2 FG FEDE FAAB AFED d2 eg', 'D2 d2 AG FDFA dfaf gfeg fedf']  
X:1
L:1/8
M:4/4
K:D
 z4 |: F2 FG FEDE | FAAB AFED | d2 eg fdec | dBAG BEED | F2 FG FEDE | FAAB AFED | d2 eg fdec | 
 dBAG FD D2 :: d2 AG FDFA | dfaf gfeg | fedf eAcA | dfed cA A2 | BEGB ADFA | dfaf gfed | 
 Bdce dBAG | FGEF FD D2 :|  