acid base - What is the formula for theoretical buffer capacity for a diprotic buffer system?

According to Chembuddy, the formula for theoretical buffer capacity for a monoprotic buffer system is as follows:

$$β = 2.303\left(\frac{K_\mathrm{w}}{[\ce{H+}]} + [\ce{H+}] + \frac{C_\mathrm{buf}K_\mathrm{a}[\ce{H+}]}{(K_\mathrm{a} + [\ce{H+}])^2}\right)$$

where $C_\mathrm{buf}$ is the total concentration of buffer and $K_\mathrm{w}$ is the water ionization constant.

However, I have conducted a lab in which I mix $\ce{C4H6O4}$ (tartaric acid; diprotic) with its double salt $\ce{NaKC4H4O4}$ to create a 1.0 M potassium sodium tartrate buffer solution where there are two simultaneous buffer systems. Also, $\mathrm{p}K_\mathrm{a1}$ and $\mathrm{p}K_\mathrm{a2}$ of tartaric acid are quite similar (around 1.6 difference), so I believe that this alternative equation that provides the theoretical buffer capacity for a system with multiple buffers:

$$β = 2.303\left(\frac{K_\mathrm{w}}{[\ce{H+}]} + [\ce{H+}] + \sum\frac{C_\mathrm{buf}K_\mathrm{a}[\ce{H+}]}{(K_\mathrm{a} + [\ce{H+}])^2}\right)$$

will not be correct in my case.

Is this right? If so, is there a proper way to theoretically calculate the buffer capacity of the composite system?