Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
Q:1/8=320
M:6/8
K:Ador
['A AcA AFA EFG A2 A F3', 'A3 GED G3']  
X:1
L:1/8
Q:1/8=320
M:6/8
K:Ador
"^Double Jig" A | AcA AFA | EFG A2 A | F3 E3 | GEA G2 c | c3 cAG | Acd e2 e | edc AGE |1 DEG A2 :|2 
 AGE A2 || d | edc A3 | GED G3 | AGA cAc | ged e2 d | edc A3 | GED G3 |1 AGA ABc | BAG A2 :|2 
 ABA ged | cAG A2 |]  