Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
M:2/4
K:G
['B2 Bd', 'BAGA FAdf a4 fagf egfe', 'B2 Bd', 'G4 z2']  
X:1
L:1/8
M:2/4
K:G
|: DC | B,DGB DGBd | B2 B2 B2 Bd | B2 Bd B2 Bd | dcAF D2 (3DED | CDFA DFAc | f2 f2 f2 ec |1 
 f2 ec f2 ec | edBG D2 :|2 d2 fe dcBA | G2 g2 g2 |: z A | (3ABA ^GA BAGA | FAdf a4 | fagf egfe | 
 dfe^c d^cBA | (3ABA ^GA BAGA | FAdf a4 | fagf egfe | d2 f2 d2 :: DC | B,DGB DGBd | B2 B2 B2 Bd | 
 B2 Bd B2 Bd | dcAF D2 (3DED | CDFA DFAc | f2 f2 f2 ec |1 f2 ec f2 ec | edBG D2 :|2 d2 fe dcBA | 
 G2 g2 g2 |: g2 | g4 z ecA | G4 z2 G=F | EGcG EGcG | DGBG =F2 =f2 | =f4 z dBA |1 G4 z2 (3gag | 
 b3 =f b3 =f | agec B2 :|2 G4 z2 A2 | G2 AG ^FGAB | c2 e2 c2 ||  