Interpretation of $\frac{\partial }{\partial p_1}Q_1(p_1, p_2)/\frac{\partial}{\partial p_2} Q_1(p_1, p_2)$
One interpretation I can offer. The demand function can be expressed as:

$$Q_1 = Q_1(p_1,p_2)$$

Let us take the total differential:

$$dQ_1 = \frac{\partial Q_1(p_1,p_2)}{\partial p_1}dp_1+\frac{\partial Q_1(p_1,p_2)}{\partial p_2}dp_2$$

Assume that $Q_1$ remains unchanged with respect to a change in prices. This implies that $dQ_1=0$. Solving the equation:

$$\frac{d p_2}{d p_1} = - \frac{\frac{\partial Q_1(p_1,p_2)}{\partial p_1}}{\frac{\partial Q_1(p_1,p_2)}{\partial p_2}}$$

The above expression implies the following (since both partial derivatives are negative):

$$t = \left|\frac{dp_2}{dp_1}\right| = \frac{\frac{\partial Q_1(p_1,p_2)}{\partial p_1}}{\frac{\partial Q_1(p_1,p_2)}{\partial p_2}}$$

$t$ can be interpreted as follows: if good 1 became more expensive, how much would the price of good 2 need to change (decrease) such that the demand for good 1 remains unchanged. 

Mathematically, $t$ represents the magnitude of the slope at ($p_1,p_2$) on the level curve when we fix $Q_1$.