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:Dmix
['BA GEED cBce', 'D2 Adde fded cBca gcec Ad^cd edcd']  
X:1
L:1/8
M:4/4
K:Dmix
 d^cAB =c2 BA | GEED cBce | d^cAB =cdBd | cAGC EDDe | d2 AB c2 BA | GEED cBce | d2 AB cdBd | 
 cAGC ED D2 :| Adde fded | cBca gcec | Ad^cd edcd | eaag edd^c | Adde =f2 ed | cABd efge | 
 (3(agf) (3(gfe) cded | cAGC ED D2 :|  