No. Even ignoring additional drag from wing interference, there's little or nothing to be gained by going to a triplane configuration. In the case of a vehicle with multiple wings, the effective aspect ratio of the vehicle is different than the aspect ratio of a single wing.
The induced drag of the vehicle strongly depends on span; aspect ratio is just a convenient dimensionless parameter to represent this. , If you keep the same wing area and don't change the span, the induced drag will not change significantly, but you will add lots of interference drag. This is the case with the triplane you describe.
For some intuition on why this is the case, remember that induced drag is an effect of downwash caused by wingtip vorticies, the strength of which is related to the total amount of lift the vehicle produces. This is related mainly to the vehicle wing loading $W/S$. Vehicles with large spans reduce this effect by moving the tip vorticies out further from the center of the vehicle.
In the case where you have the smaller wings, the induced drag of each is reduced by ~1/3 but you need to sum the drag from each wing into the total vehicle drag.
$C_D = C_{D_0} + C_{D_{i,1}}+ C_{D_{i,2}}+ C_{D_{i,3}} + C_{D_\mathrm{interference}}$
You can see this mathematically by considering the dimensional form of the equation for induced drag $D_i$.
$D_i = qS\frac{C_L^2}{\pi e \mathrm{AR}}$
In level flight, $C_L = \frac{W}{qS}$ and recalling $AR = \frac{b^2}{S}$,
$D_i = \frac{1}{\pi e}\frac{1}{q}\frac{W^2}{b^2}$
This shows it's really the span loading, not the aspect ratio, that drives induced drag. Aspect ratio is just convenient shorthand.