Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
M:2/2
K:D
['AF F2', 'A2 a2 fa bged Beed efge f3']  
X:1
L:1/8
M:2/2
K:D
 agfg efdB | AF F2 DFAd | BG G2 EGBG | AF F2 ABdf | agfg efdB | AF F2 DFAd | BG G2 A2 ag |1 
 f2 eg fd d2 :|2 f2 eg fddf |: a2 fd Adfd | efed cA A2 | a2 fa bged | Beed efge | f3 d g3 e | 
 f2 df edBA | F2 AF G2 BG |1 A2 ag fddf :|2 Aag fd d2 ||  