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
['z F G2 GE A2 BA GFGE', 'edBG A2', 'e2 fe edBG A2 Bc dBAE FEDF']  
X:1
L:1/8
M:4/4
K:D
 z F | G2 GE A2 BA | GFGE AFDF | GFGE A2 ^GA | Beed e2 :| fe | edBG A2 Bc | dBAE FEDd | 
 edBG A2 ^GA | Beed e2 |] fe | edBG A2 Bc | dBAE FEDF | GFGE A2 ^GA | Be ^d2 e2 |]  