  X-ray diffraction, thermodynamic measurements, and density-functional
band-structure calculations are used to study the magnetic behavior of
BaCuTe$_{2}$O$_{6}$, a member of the $A$CuTe$_2$O$_6$ structural family that
hosts complex three-dimensional frustrated spin networks with possible
spin-liquid physics. Temperature-dependent magnetic susceptibility and heat
capacity of the Ba compound are well described by the one-dimensional
spin-$\frac12$ Heisenberg chain model reminiscent of the Sr analog
SrCuTe$_2$O$_6$. While the intrachain coupling $J/k_{\rm B}\simeq 37$ K is
reduced compared to 49 K in the Sr compound, the N\'eel temperature increases
from 5.5 K (Sr) to 6.1 K (Ba). Unlike the Sr compound, BaCuTe$_2$O$_6$
undergoes only one magnetic transition as a function of temperature and shows
signatures of weak spin canting. We elucidate the microscopic difference
between the Sr and Ba compounds and suggest that one of the interchain
couplings changes sign as a result of negative pressure caused by the Sr/Ba
substitution. The N\'eel temperature of BaCuTe$_2$O$_6$ is remarkably
insensitive to the magnetic dilution with Zn$^{2+}$ up to the highest reachable
level of about 20%.
