Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
Q:1/4=120
M:4/4
K:G
['(ef)(ed) .B2']  
X:1
L:1/8
Q:1/4=120
M:4/4
K:G
"^Quick and Lively" D | .G2 (.G.G) [Bdg]2 .g.f | (ef)(ed) .B2 (Bd) | (ef)(ge){f} (ed)(cB) | 
 (AB)(AG) (.E.E .E2) | (GA)(BG) .g2 gf | (ef)(ed) .B2 (Bd) | (ef)(ge) .d2 (cB) | 
{AB} .c2 (BA) (.G.G .G2) | .G2 (AB) .G2 (AB) |{^c} .d2 (de) (dB)(AG) | .G2 (.G.G) [Bdg]2 (gf) | 
 (ef)(ed) .B2 z2 |{c} (BA)Bd{f} edef | (gf)(gd) (BA)(Bd) | (ef)(ge) (dB)(AB) | (GE)(DE) G2 z2 ||  