  Trapped ions are ideally suited for precision spectroscopy, as is evident
from the remarkably low systematic uncertainties of single-ion clocks. The
major weakness of these clocks is the long averaging time, necessitated by the
low signal of a single atom. An increased number of ions can overcome this
limitation and allow for the implementation of novel clock schemes. However,
this presents the challenge to maintain the excellent control over systematic
shifts of a single particle in spatially extended and strongly coupled
many-body systems. We measure and deduce systematic frequency uncertainties
related to spectroscopy with ion chains in a newly developed rf trap array
designed for precision spectroscopy on simultaneously trapped ion ensembles.
For the example of an In${}^+$ clock, sympathetically cooled with Yb${}^+$
ions, we show in our system that the expected systematic frequency
uncertainties related to multi-ion operation can be below $1\times10^{-19}$.
Our results pave the way to advanced spectroscopy schemes such as entangled
clock spectroscopy and cascaded clock operation.
