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
['BGE D2 B,', 'G3 BGE D2 B, DEG AGE DEG', 'DAD BDD']  
X:1
L:1/8
M:6/8
K:G
 BGE D2 B, | DEG AGA | BGE D2 B, | DEF G3 || BGE D2 B, | DEG AGE | DEG ABA | AGF G3 |: DAD BDD | 
 BDD AGE | DAD BDD | AGF G3 | DAD BDD | BDD AGE | DEG ABA | AGF G3 :|  