Scale setting for $N_f=3+1$ QCD

We present the scale setting for a new set of gauge configurations generated with $N_\mathrm{f}=3+1$ Wilson quarks with a non-perturbatively determined clover coefficient in a massive O(a) improvement scheme. The three light quarks are degenerate, with the sum of their masses being equal to its value in nature and the charm quark has its physical mass. We use open boundary conditions in time direction to avoid the problem of topological freezing at small lattice spacings and twisted-mass reweighting for improved stability of the simulations. The decoupling of charm at low energy allows us to set the scale by measuring the value of the low-energy quantity $t_0^\star /a^2$, which is the flow scale $t_0$ at our mass point, and comparing it to an $N_\mathrm{f}=2+1$ result in physical units. We present the details of the algorithmic setup and tuning procedure and give the bare parameters of ensembles with two lattice spacings $a=0.054$fm and $a=0.043$fm. We discuss finite volume effects and lattice artifacts and present physical results for the charmonium spectrum. In particular the hyperfine splitting between the $\eta _c$ and $J/\psi $ mesons agrees very well with its physical value.