We study the spectral gaps of the Schr{\"o}dinger operators
$$H_{1}=-\frac{d^{2}}{dx^{2}}+\sum^{\infty}_{l=-\infty}(
\beta_{1}\delta^{\prime}(x-\kappa-2\pi l)+\beta_{2}\delta^{\prime}(x-2\pi l))\quad
{\rm in}\quad L^{2}({\mathbb R}),$$
$$H_{2}=-\frac{d^{2}}{dx^{2}}+\sum^{\infty}_{l=-\infty}(
\beta_{1}\delta(x-\kappa-2\pi l)+\beta_{2}\delta(x-2\pi l))\quad
{\rm in}\quad L^{2}({\mathbb R}),$$
where $\kappa\in (0,2\pi)$ and $\beta_{1},\beta_{2}\in{\mathbb R}\backslash\{0\}$ are parameters.
Given $j\in{\mathbb N}$, we determine whether the $j$th gap of $H_{k}$ is absent
or not for $k=1,2$.