High-Gain Common Emitter Circuit
For a genuine common emitter configuration (emitter at AC ground), the small-signal open-circuit voltage gain is approximately

$$A_{voc} = -g_m \cdot R_C||r_o \approx-g_mR_C = -\frac{I_CR_C}{V_T} = -\frac{V_{CC} - V_C}{V_T}$$

Thus, the AC gain is fixed by your choice of supply voltage \$V_{CC}\$ and DC collector voltage \$V_C\$

If, like many, you choose (or require that) \$V_C = \frac{V_{CC}}{2}\$, the gain of the (genuine) common emitter amplifier is just

$$A_{voc} \approx -\frac{V_{CC}}{2V_T}$$

Which is to say that you don't have the necessary degree of freedom to choose the AC gain independent of the supply voltage.

Now, by adding a resistor \$R_4\$ in series \$C_4\$, you gain the degree of freedom for the AC gain:

$$A_{voc} \approx -\frac{\alpha R_C}{r_e + R_4||R_7} $$

where 

$$r_e = \frac{V_T}{I_E}$$

Assuming once again that you choose \$I_CR_C = \frac{V_{CC}}{2}\$, the AC gain is:

$$A_{voc} \approx -\frac{V_{CC}}{2(V_T + I_E\cdot R_4||R_7)} $$